var bigInt=function(undefined){"use strict";var BASE=1e7,LOG_BASE=7,MAX_INT=9007199254740992,MAX_INT_ARR=smallToArray(MAX_INT),DEFAULT_ALPHABET="0123456789abcdefghijklmnopqrstuvwxyz";var supportsNativeBigInt=typeof BigInt==="function";function Integer(v,radix,alphabet,caseSensitive){if(typeof v==="undefined")return Integer[0];if(typeof radix!=="undefined")return+radix===10&&!alphabet?parseValue(v):parseBase(v,radix,alphabet,caseSensitive);return parseValue(v)}function BigInteger(value,sign){this.value=value;this.sign=sign;this.isSmall=false}BigInteger.prototype=Object.create(Integer.prototype);function SmallInteger(value){this.value=value;this.sign=value<0;this.isSmall=true}SmallInteger.prototype=Object.create(Integer.prototype);function NativeBigInt(value){this.value=value}NativeBigInt.prototype=Object.create(Integer.prototype);function isPrecise(n){return-MAX_INT<n&&n<MAX_INT}function smallToArray(n){if(n<1e7)return[n];if(n<1e14)return[n%1e7,Math.floor(n/1e7)];return[n%1e7,Math.floor(n/1e7)%1e7,Math.floor(n/1e14)]}function arrayToSmall(arr){trim(arr);var length=arr.length;if(length<4&&compareAbs(arr,MAX_INT_ARR)<0){switch(length){case 0:return 0;case 1:return arr[0];case 2:return arr[0]+arr[1]*BASE;default:return arr[0]+(arr[1]+arr[2]*BASE)*BASE}}return arr}function trim(v){var i=v.length;while(v[--i]===0);v.length=i+1}function createArray(length){var x=new Array(length);var i=-1;while(++i<length){x[i]=0}return x}function truncate(n){if(n>0)return Math.floor(n);return Math.ceil(n)}function add(a,b){var l_a=a.length,l_b=b.length,r=new Array(l_a),carry=0,base=BASE,sum,i;for(i=0;i<l_b;i++){sum=a[i]+b[i]+carry;carry=sum>=base?1:0;r[i]=sum-carry*base}while(i<l_a){sum=a[i]+carry;carry=sum===base?1:0;r[i++]=sum-carry*base}if(carry>0)r.push(carry);return r}function addAny(a,b){if(a.length>=b.length)return add(a,b);return add(b,a)}function addSmall(a,carry){var l=a.length,r=new Array(l),base=BASE,sum,i;for(i=0;i<l;i++){sum=a[i]-base+carry;carry=Math.floor(sum/base);r[i]=sum-carry*base;carry+=1}while(carry>0){r[i++]=carry%base;carry=Math.floor(carry/base)}return r}BigInteger.prototype.add=function(v){var n=parseValue(v);if(this.sign!==n.sign){return this.subtract(n.negate())}var a=this.value,b=n.value;if(n.isSmall){return new BigInteger(addSmall(a,Math.abs(b)),this.sign)}return new BigInteger(addAny(a,b),this.sign)};BigInteger.prototype.plus=BigInteger.prototype.add;SmallInteger.prototype.add=function(v){var n=parseValue(v);var a=this.value;if(a<0!==n.sign){return this.subtract(n.negate())}var b=n.value;if(n.isSmall){if(isPrecise(a+b))return new SmallInteger(a+b);b=smallToArray(Math.abs(b))}return new BigInteger(addSmall(b,Math.abs(a)),a<0)};SmallInteger.prototype.plus=SmallInteger.prototype.add;NativeBigInt.prototype.add=function(v){return new NativeBigInt(this.value+parseValue(v).value)};NativeBigInt.prototype.plus=NativeBigInt.prototype.add;function subtract(a,b){var a_l=a.length,b_l=b.length,r=new Array(a_l),borrow=0,base=BASE,i,difference;for(i=0;i<b_l;i++){difference=a[i]-borrow-b[i];if(difference<0){difference+=base;borrow=1}else borrow=0;r[i]=difference}for(i=b_l;i<a_l;i++){difference=a[i]-borrow;if(difference<0)difference+=base;else{r[i++]=difference;break}r[i]=difference}for(;i<a_l;i++){r[i]=a[i]}trim(r);return r}function subtractAny(a,b,sign){var value;if(compareAbs(a,b)>=0){value=subtract(a,b)}else{value=subtract(b,a);sign=!sign}value=arrayToSmall(value);if(typeof value==="number"){if(sign)value=-value;return new SmallInteger(value)}return new BigInteger(value,sign)}function subtractSmall(a,b,sign){var l=a.length,r=new Array(l),carry=-b,base=BASE,i,difference;for(i=0;i<l;i++){difference=a[i]+carry;carry=Math.floor(difference/base);difference%=base;r[i]=difference<0?difference+base:difference}r=arrayToSmall(r);if(typeof r==="number"){if(sign)r=-r;return new SmallInteger(r)}return new BigInteger(r,sign)}BigInteger.prototype.subtract=function(v){var n=parseValue(v);if(this.sign!==n.sign){return this.add(n.negate())}var a=this.value,b=n.value;if(n.isSmall)return subtractSmall(a,Math.abs(b),this.sign);return subtractAny(a,b,this.sign)};BigInteger.prototype.minus=BigInteger.prototype.subtract;SmallInteger.prototype.subtract=function(v){var n=parseValue(v);var a=this.value;if(a<0!==n.sign){return this.add(n.negate())}var b=n.value;if(n.isSmall){return new SmallInteger(a-b)}return subtractSmall(b,Math.abs(a),a>=0)};SmallInteger.prototype.minus=SmallInteger.prototype.subtract;NativeBigInt.prototype.subtract=function(v){return new NativeBigInt(this.value-parseValue(v).value)};NativeBigInt.prototype.minus=NativeBigInt.prototype.subtract;BigInteger.prototype.negate=function(){return new BigInteger(this.value,!this.sign)};SmallInteger.prototype.negate=function(){var sign=this.sign;var small=new SmallInteger(-this.value);small.sign=!sign;return small};NativeBigInt.prototype.negate=function(){return new NativeBigInt(-this.value)};BigInteger.prototype.abs=function(){return new BigInteger(this.value,false)};SmallInteger.prototype.abs=function(){return new SmallInteger(Math.abs(this.value))};NativeBigInt.prototype.abs=function(){return new NativeBigInt(this.value>=0?this.value:-this.value)};function multiplyLong(a,b){var a_l=a.length,b_l=b.length,l=a_l+b_l,r=createArray(l),base=BASE,product,carry,i,a_i,b_j;for(i=0;i<a_l;++i){a_i=a[i];for(var j=0;j<b_l;++j){b_j=b[j];product=a_i*b_j+r[i+j];carry=Math.floor(product/base);r[i+j]=product-carry*base;r[i+j+1]+=carry}}trim(r);return r}function multiplySmall(a,b){var l=a.length,r=new Array(l),base=BASE,carry=0,product,i;for(i=0;i<l;i++){product=a[i]*b+carry;carry=Math.floor(product/base);r[i]=product-carry*base}while(carry>0){r[i++]=carry%base;carry=Math.floor(carry/base)}return r}function shiftLeft(x,n){var r=[];while(n-- >0)r.push(0);return r.concat(x)}function multiplyKaratsuba(x,y){var n=Math.max(x.length,y.length);if(n<=30)return multiplyLong(x,y);n=Math.ceil(n/2);var b=x.slice(n),a=x.slice(0,n),d=y.slice(n),c=y.slice(0,n);var ac=multiplyKaratsuba(a,c),bd=multiplyKaratsuba(b,d),abcd=multiplyKaratsuba(addAny(a,b),addAny(c,d));var product=addAny(addAny(ac,shiftLeft(subtract(subtract(abcd,ac),bd),n)),shiftLeft(bd,2*n));trim(product);return product}function useKaratsuba(l1,l2){return-.012*l1-.012*l2+15e-6*l1*l2>0}BigInteger.prototype.multiply=function(v){var n=parseValue(v),a=this.value,b=n.value,sign=this.sign!==n.sign,abs;if(n.isSmall){if(b===0)return Integer[0];if(b===1)return this;if(b===-1)return this.negate();abs=Math.abs(b);if(abs<BASE){return new BigInteger(multiplySmall(a,abs),sign)}b=smallToArray(abs)}if(useKaratsuba(a.length,b.length))return new BigInteger(multiplyKaratsuba(a,b),sign);return new BigInteger(multiplyLong(a,b),sign)};BigInteger.prototype.times=BigInteger.prototype.multiply;function multiplySmallAndArray(a,b,sign){if(a<BASE){return new BigInteger(multiplySmall(b,a),sign)}return new BigInteger(multiplyLong(b,smallToArray(a)),sign)}SmallInteger.prototype._multiplyBySmall=function(a){if(isPrecise(a.value*this.value)){return new SmallInteger(a.value*this.value)}return multiplySmallAndArray(Math.abs(a.value),smallToArray(Math.abs(this.value)),this.sign!==a.sign)};BigInteger.prototype._multiplyBySmall=function(a){if(a.value===0)return Integer[0];if(a.value===1)return this;if(a.value===-1)return this.negate();return multiplySmallAndArray(Math.abs(a.value),this.value,this.sign!==a.sign)};SmallInteger.prototype.multiply=function(v){return parseValue(v)._multiplyBySmall(this)};SmallInteger.prototype.times=SmallInteger.prototype.multiply;NativeBigInt.prototype.multiply=function(v){return new NativeBigInt(this.value*parseValue(v).value)};NativeBigInt.prototype.times=NativeBigInt.prototype.multiply;function square(a){var l=a.length,r=createArray(l+l),base=BASE,product,carry,i,a_i,a_j;for(i=0;i<l;i++){a_i=a[i];carry=0-a_i*a_i;for(var j=i;j<l;j++){a_j=a[j];product=2*(a_i*a_j)+r[i+j]+carry;carry=Math.floor(product/base);r[i+j]=product-carry*base}r[i+l]=carry}trim(r);return r}BigInteger.prototype.square=function(){return new BigInteger(square(this.value),false)};SmallInteger.prototype.square=function(){var value=this.value*this.value;if(isPrecise(value))return new SmallInteger(value);return new BigInteger(square(smallToArray(Math.abs(this.value))),false)};NativeBigInt.prototype.square=function(v){return new NativeBigInt(this.value*this.value)};function divMod1(a,b){var a_l=a.length,b_l=b.length,base=BASE,result=createArray(b.length),divisorMostSignificantDigit=b[b_l-1],lambda=Math.ceil(base/(2*divisorMostSignificantDigit)),remainder=multiplySmall(a,lambda),divisor=multiplySmall(b,lambda),quotientDigit,shift,carry,borrow,i,l,q;if(remainder.length<=a_l)remainder.push(0);divisor.push(0);divisorMostSignificantDigit=divisor[b_l-1];for(shift=a_l-b_l;shift>=0;shift--){quotientDigit=base-1;if(remainder[shift+b_l]!==divisorMostSignificantDigit){quotientDigit=Math.floor((remainder[shift+b_l]*base+remainder[shift+b_l-1])/divisorMostSignificantDigit)}carry=0;borrow=0;l=divisor.length;for(i=0;i<l;i++){carry+=quotientDigit*divisor[i];q=Math.floor(carry/base);borrow+=remainder[shift+i]-(carry-q*base);carry=q;if(borrow<0){remainder[shift+i]=borrow+base;borrow=-1}else{remainder[shift+i]=borrow;borrow=0}}while(borrow!==0){quotientDigit-=1;carry=0;for(i=0;i<l;i++){carry+=remainder[shift+i]-base+divisor[i];if(carry<0){remainder[shift+i]=carry+base;carry=0}else{remainder[shift+i]=carry;carry=1}}borrow+=carry}result[shift]=quotientDigit}remainder=divModSmall(remainder,lambda)[0];return[arrayToSmall(result),arrayToSmall(remainder)]}function divMod2(a,b){var a_l=a.length,b_l=b.length,result=[],part=[],base=BASE,guess,xlen,highx,highy,check;while(a_l){part.unshift(a[--a_l]);trim(part);if(compareAbs(part,b)<0){result.push(0);continue}xlen=part.length;highx=part[xlen-1]*base+part[xlen-2];highy=b[b_l-1]*base+b[b_l-2];if(xlen>b_l){highx=(highx+1)*base}guess=Math.ceil(highx/highy);do{check=multiplySmall(b,guess);if(compareAbs(check,part)<=0)break;guess--}while(guess);result.push(guess);part=subtract(part,check)}result.reverse();return[arrayToSmall(result),arrayToSmall(part)]}function divModSmall(value,lambda){var length=value.length,quotient=createArray(length),base=BASE,i,q,remainder,divisor;remainder=0;for(i=length-1;i>=0;--i){divisor=remainder*base+value[i];q=truncate(divisor/lambda);remainder=divisor-q*lambda;quotient[i]=q|0}return[quotient,remainder|0]}function divModAny(self,v){var value,n=parseValue(v);if(supportsNativeBigInt){return[new NativeBigInt(self.value/n.value),new NativeBigInt(self.value%n.value)]}var a=self.value,b=n.value;var quotient;if(b===0)throw new Error("Cannot divide by zero");if(self.isSmall){if(n.isSmall){return[new SmallInteger(truncate(a/b)),new SmallInteger(a%b)]}return[Integer[0],self]}if(n.isSmall){if(b===1)return[self,Integer[0]];if(b==-1)return[self.negate(),Integer[0]];var abs=Math.abs(b);if(abs<BASE){value=divModSmall(a,abs);quotient=arrayToSmall(value[0]);var remainder=value[1];if(self.sign)remainder=-remainder;if(typeof quotient==="number"){if(self.sign!==n.sign)quotient=-quotient;return[new SmallInteger(quotient),new SmallInteger(remainder)]}return[new BigInteger(quotient,self.sign!==n.sign),new SmallInteger(remainder)]}b=smallToArray(abs)}var comparison=compareAbs(a,b);if(comparison===-1)return[Integer[0],self];if(comparison===0)return[Integer[self.sign===n.sign?1:-1],Integer[0]];if(a.length+b.length<=200)value=divMod1(a,b);else value=divMod2(a,b);quotient=value[0];var qSign=self.sign!==n.sign,mod=value[1],mSign=self.sign;if(typeof quotient==="number"){if(qSign)quotient=-quotient;quotient=new SmallInteger(quotient)}else quotient=new BigInteger(quotient,qSign);if(typeof mod==="number"){if(mSign)mod=-mod;mod=new SmallInteger(mod)}else mod=new BigInteger(mod,mSign);return[quotient,mod]}BigInteger.prototype.divmod=function(v){var result=divModAny(this,v);return{quotient:result[0],remainder:result[1]}};NativeBigInt.prototype.divmod=SmallInteger.prototype.divmod=BigInteger.prototype.divmod;BigInteger.prototype.divide=function(v){return divModAny(this,v)[0]};NativeBigInt.prototype.over=NativeBigInt.prototype.divide=SmallInteger.prototype.over=SmallInteger.prototype.divide=BigInteger.prototype.over=BigInteger.prototype.divide;BigInteger.prototype.mod=function(v){return divModAny(this,v)[1]};NativeBigInt.prototype.mod=NativeBigInt.prototype.remainder=SmallInteger.prototype.remainder=SmallInteger.prototype.mod=BigInteger.prototype.remainder=BigInteger.prototype.mod;BigInteger.prototype.pow=function(v){var n=parseValue(v),a=this.value,b=n.value,value,x,y;if(b===0)return Integer[1];if(a===0)return Integer[0];if(a===1)return Integer[1];if(a===-1)return n.isEven()?Integer[1]:Integer[-1];if(n.sign){return Integer[0]}if(!n.isSmall)throw new Error("The exponent "+n.toString()+" is too large.");if(this.isSmall){if(isPrecise(value=Math.pow(a,b)))return new SmallInteger(truncate(value))}x=this;y=Integer[1];while(true){if(b&1===1){y=y.times(x);--b}if(b===0)break;b/=2;x=x.square()}return y};SmallInteger.prototype.pow=BigInteger.prototype.pow;var pow;if(supportsNativeBigInt){pow=eval("(a,b)=>a**b")}NativeBigInt.prototype.pow=function(v){var n=parseValue(v);var a=this.value,b=n.value;if(b===BigInt(0))return Integer[1];if(a===BigInt(0))return Integer[0];if(a===BigInt(1))return Integer[1];if(a===BigInt(-1))return n.isEven()?Integer[1]:Integer[-1];if(n.isNegative())return new NativeBigInt(BigInt(0));return new NativeBigInt(pow(a,b))};BigInteger.prototype.modPow=function(exp,mod){exp=parseValue(exp);mod=parseValue(mod);if(mod.isZero())throw new Error("Cannot take modPow with modulus 0");var r=Integer[1],base=this.mod(mod);while(exp.isPositive()){if(base.isZero())return Integer[0];if(exp.isOdd())r=r.multiply(base).mod(mod);exp=exp.divide(2);base=base.square().mod(mod)}return r};NativeBigInt.prototype.modPow=SmallInteger.prototype.modPow=BigInteger.prototype.modPow;function compareAbs(a,b){if(a.length!==b.length){return a.length>b.length?1:-1}for(var i=a.length-1;i>=0;i--){if(a[i]!==b[i])return a[i]>b[i]?1:-1}return 0}BigInteger.prototype.compareAbs=function(v){var n=parseValue(v),a=this.value,b=n.value;if(n.isSmall)return 1;return compareAbs(a,b)};SmallInteger.prototype.compareAbs=function(v){var n=parseValue(v),a=Math.abs(this.value),b=n.value;if(n.isSmall){b=Math.abs(b);return a===b?0:a>b?1:-1}return-1};NativeBigInt.prototype.compareAbs=function(v){var a=this.value;var b=parseValue(v).value;a=a>=0?a:-a;b=b>=0?b:-b;return a===b?0:a>b?1:-1};BigInteger.prototype.compare=function(v){if(v===Infinity){return-1}if(v===-Infinity){return 1}var n=parseValue(v),a=this.value,b=n.value;if(this.sign!==n.sign){return n.sign?1:-1}if(n.isSmall){return this.sign?-1:1}return compareAbs(a,b)*(this.sign?-1:1)};BigInteger.prototype.compareTo=BigInteger.prototype.compare;SmallInteger.prototype.compare=function(v){if(v===Infinity){return-1}if(v===-Infinity){return 1}var n=parseValue(v),a=this.value,b=n.value;if(n.isSmall){return a==b?0:a>b?1:-1}if(a<0!==n.sign){return a<0?-1:1}return a<0?1:-1};SmallInteger.prototype.compareTo=SmallInteger.prototype.compare;NativeBigInt.prototype.compare=function(v){if(v===Infinity){return-1}if(v===-Infinity){return 1}var a=this.value;var b=parseValue(v).value;return a===b?0:a>b?1:-1};NativeBigInt.prototype.compareTo=NativeBigInt.prototype.compare;BigInteger.prototype.equals=function(v){return this.compare(v)===0};NativeBigInt.prototype.eq=NativeBigInt.prototype.equals=SmallInteger.prototype.eq=SmallInteger.prototype.equals=BigInteger.prototype.eq=BigInteger.prototype.equals;BigInteger.prototype.notEquals=function(v){return this.compare(v)!==0};NativeBigInt.prototype.neq=NativeBigInt.prototype.notEquals=SmallInteger.prototype.neq=SmallInteger.prototype.notEquals=BigInteger.prototype.neq=BigInteger.prototype.notEquals;BigInteger.prototype.greater=function(v){return this.compare(v)>0};NativeBigInt.prototype.gt=NativeBigInt.prototype.greater=SmallInteger.prototype.gt=SmallInteger.prototype.greater=BigInteger.prototype.gt=BigInteger.prototype.greater;BigInteger.prototype.lesser=function(v){return this.compare(v)<0};NativeBigInt.prototype.lt=NativeBigInt.prototype.lesser=SmallInteger.prototype.lt=SmallInteger.prototype.lesser=BigInteger.prototype.lt=BigInteger.prototype.lesser;BigInteger.prototype.greaterOrEquals=function(v){return this.compare(v)>=0};NativeBigInt.prototype.geq=NativeBigInt.prototype.greaterOrEquals=SmallInteger.prototype.geq=SmallInteger.prototype.greaterOrEquals=BigInteger.prototype.geq=BigInteger.prototype.greaterOrEquals;BigInteger.prototype.lesserOrEquals=function(v){return this.compare(v)<=0};NativeBigInt.prototype.leq=NativeBigInt.prototype.lesserOrEquals=SmallInteger.prototype.leq=SmallInteger.prototype.lesserOrEquals=BigInteger.prototype.leq=BigInteger.prototype.lesserOrEquals;BigInteger.prototype.isEven=function(){return(this.value[0]&1)===0};SmallInteger.prototype.isEven=function(){return(this.value&1)===0};NativeBigInt.prototype.isEven=function(){return(this.value&BigInt(1))===BigInt(0)};BigInteger.prototype.isOdd=function(){return(this.value[0]&1)===1};SmallInteger.prototype.isOdd=function(){return(this.value&1)===1};NativeBigInt.prototype.isOdd=function(){return(this.value&BigInt(1))===BigInt(1)};BigInteger.prototype.isPositive=function(){return!this.sign};SmallInteger.prototype.isPositive=function(){return this.value>0};NativeBigInt.prototype.isPositive=SmallInteger.prototype.isPositive;BigInteger.prototype.isNegative=function(){return this.sign};SmallInteger.prototype.isNegative=function(){return this.value<0};NativeBigInt.prototype.isNegative=SmallInteger.prototype.isNegative;BigInteger.prototype.isUnit=function(){return false};SmallInteger.prototype.isUnit=function(){return Math.abs(this.value)===1};NativeBigInt.prototype.isUnit=function(){return this.abs().value===BigInt(1)};BigInteger.prototype.isZero=function(){return false};SmallInteger.prototype.isZero=function(){return this.value===0};NativeBigInt.prototype.isZero=function(){return this.value===BigInt(0)};BigInteger.prototype.isDivisibleBy=function(v){var n=parseValue(v);if(n.isZero())return false;if(n.isUnit())return true;if(n.compareAbs(2)===0)return this.isEven();return this.mod(n).isZero()};NativeBigInt.prototype.isDivisibleBy=SmallInteger.prototype.isDivisibleBy=BigInteger.prototype.isDivisibleBy;function isBasicPrime(v){var n=v.abs();if(n.isUnit())return false;if(n.equals(2)||n.equals(3)||n.equals(5))return true;if(n.isEven()||n.isDivisibleBy(3)||n.isDivisibleBy(5))return false;if(n.lesser(49))return true}function millerRabinTest(n,a){var nPrev=n.prev(),b=nPrev,r=0,d,t,i,x;while(b.isEven())b=b.divide(2),r++;next:for(i=0;i<a.length;i++){if(n.lesser(a[i]))continue;x=bigInt(a[i]).modPow(b,n);if(x.isUnit()||x.equals(nPrev))continue;for(d=r-1;d!=0;d--){x=x.square().mod(n);if(x.isUnit())return false;if(x.equals(nPrev))continue next}return false}return true}BigInteger.prototype.isPrime=function(strict){var isPrime=isBasicPrime(this);if(isPrime!==undefined)return isPrime;var n=this.abs();var bits=n.bitLength();if(bits<=64)return millerRabinTest(n,[2,325,9375,28178,450775,9780504,1795265022]);var logN=Math.log(2)*bits.toJSNumber();var t=Math.ceil(strict===true?2*Math.pow(logN,2):logN);for(var a=[],i=0;i<t;i++){a.push(bigInt(i+2))}return millerRabinTest(n,a)};NativeBigInt.prototype.isPrime=SmallInteger.prototype.isPrime=BigInteger.prototype.isPrime;BigInteger.prototype.isProbablePrime=function(iterations){var isPrime=isBasicPrime(this);if(isPrime!==undefined)return isPrime;var n=this.abs();var t=iterations===undefined?5:iterations;for(var a=[],i=0;i<t;i++){a.push(bigInt.randBetween(2,n.minus(2)))}return millerRabinTest(n,a)};NativeBigInt.prototype.isProbablePrime=SmallInteger.prototype.isProbablePrime=BigInteger.prototype.isProbablePrime;BigInteger.prototype.modInv=function(n){var t=bigInt.zero,newT=bigInt.one,r=parseValue(n),newR=this.abs(),q,lastT,lastR;while(!newR.isZero()){q=r.divide(newR);lastT=t;lastR=r;t=newT;r=newR;newT=lastT.subtract(q.multiply(newT));newR=lastR.subtract(q.multiply(newR))}if(!r.isUnit())throw new Error(this.toString()+" and "+n.toString()+" are not co-prime");if(t.compare(0)===-1){t=t.add(n)}if(this.isNegative()){return t.negate()}return t};NativeBigInt.prototype.modInv=SmallInteger.prototype.modInv=BigInteger.prototype.modInv;BigInteger.prototype.next=function(){var value=this.value;if(this.sign){return subtractSmall(value,1,this.sign)}return new BigInteger(addSmall(value,1),this.sign)};SmallInteger.prototype.next=function(){var value=this.value;if(value+1<MAX_INT)return new SmallInteger(value+1);return new BigInteger(MAX_INT_ARR,false)};NativeBigInt.prototype.next=function(){return new NativeBigInt(this.value+BigInt(1))};BigInteger.prototype.prev=function(){var value=this.value;if(this.sign){return new BigInteger(addSmall(value,1),true)}return subtractSmall(value,1,this.sign)};SmallInteger.prototype.prev=function(){var value=this.value;if(value-1>-MAX_INT)return new SmallInteger(value-1);return new BigInteger(MAX_INT_ARR,true)};NativeBigInt.prototype.prev=function(){return new NativeBigInt(this.value-BigInt(1))};var powersOfTwo=[1];while(2*powersOfTwo[powersOfTwo.length-1]<=BASE)powersOfTwo.push(2*powersOfTwo[powersOfTwo.length-1]);var powers2Length=powersOfTwo.length,highestPower2=powersOfTwo[powers2Length-1];function shift_isSmall(n){return Math.abs(n)<=BASE}BigInteger.prototype.shiftLeft=function(v){var n=parseValue(v).toJSNumber();if(!shift_isSmall(n)){throw new Error(String(n)+" is too large for shifting.")}if(n<0)return this.shiftRight(-n);var result=this;if(result.isZero())return result;while(n>=powers2Length){result=result.multiply(highestPower2);n-=powers2Length-1}return result.multiply(powersOfTwo[n])};NativeBigInt.prototype.shiftLeft=SmallInteger.prototype.shiftLeft=BigInteger.prototype.shiftLeft;BigInteger.prototype.shiftRight=function(v){var remQuo;var n=parseValue(v).toJSNumber();if(!shift_isSmall(n)){throw new Error(String(n)+" is too large for shifting.")}if(n<0)return this.shiftLeft(-n);var result=this;while(n>=powers2Length){if(result.isZero()||result.isNegative()&&result.isUnit())return result;remQuo=divModAny(result,highestPower2);result=remQuo[1].isNegative()?remQuo[0].prev():remQuo[0];n-=powers2Length-1}remQuo=divModAny(result,powersOfTwo[n]);return remQuo[1].isNegative()?remQuo[0].prev():remQuo[0]};NativeBigInt.prototype.shiftRight=SmallInteger.prototype.shiftRight=BigInteger.prototype.shiftRight;function bitwise(x,y,fn){y=parseValue(y);var xSign=x.isNegative(),ySign=y.isNegative();var xRem=xSign?x.not():x,yRem=ySign?y.not():y;var xDigit=0,yDigit=0;var xDivMod=null,yDivMod=null;var result=[];while(!xRem.isZero()||!yRem.isZero()){xDivMod=divModAny(xRem,highestPower2);xDigit=xDivMod[1].toJSNumber();if(xSign){xDigit=highestPower2-1-xDigit}yDivMod=divModAny(yRem,highestPower2);yDigit=yDivMod[1].toJSNumber();if(ySign){yDigit=highestPower2-1-yDigit}xRem=xDivMod[0];yRem=yDivMod[0];result.push(fn(xDigit,yDigit))}var sum=fn(xSign?1:0,ySign?1:0)!==0?bigInt(-1):bigInt(0);for(var i=result.length-1;i>=0;i-=1){sum=sum.multiply(highestPower2).add(bigInt(result[i]))}return sum}BigInteger.prototype.not=function(){return this.negate().prev()};NativeBigInt.prototype.not=SmallInteger.prototype.not=BigInteger.prototype.not;BigInteger.prototype.and=function(n){return bitwise(this,n,function(a,b){return a&b})};NativeBigInt.prototype.and=SmallInteger.prototype.and=BigInteger.prototype.and;BigInteger.prototype.or=function(n){return bitwise(this,n,function(a,b){return a|b})};NativeBigInt.prototype.or=SmallInteger.prototype.or=BigInteger.prototype.or;BigInteger.prototype.xor=function(n){return bitwise(this,n,function(a,b){return a^b})};NativeBigInt.prototype.xor=SmallInteger.prototype.xor=BigInteger.prototype.xor;var LOBMASK_I=1<<30,LOBMASK_BI=(BASE&-BASE)*(BASE&-BASE)|LOBMASK_I;function roughLOB(n){var v=n.value,x=typeof v==="number"?v|LOBMASK_I:typeof v==="bigint"?v|BigInt(LOBMASK_I):v[0]+v[1]*BASE|LOBMASK_BI;return x&-x}function integerLogarithm(value,base){if(base.compareTo(value)<=0){var tmp=integerLogarithm(value,base.square(base));var p=tmp.p;var e=tmp.e;var t=p.multiply(base);return t.compareTo(value)<=0?{p:t,e:e*2+1}:{p:p,e:e*2}}return{p:bigInt(1),e:0}}BigInteger.prototype.bitLength=function(){var n=this;if(n.compareTo(bigInt(0))<0){n=n.negate().subtract(bigInt(1))}if(n.compareTo(bigInt(0))===0){return bigInt(0)}return bigInt(integerLogarithm(n,bigInt(2)).e).add(bigInt(1))};NativeBigInt.prototype.bitLength=SmallInteger.prototype.bitLength=BigInteger.prototype.bitLength;function max(a,b){a=parseValue(a);b=parseValue(b);return a.greater(b)?a:b}function min(a,b){a=parseValue(a);b=parseValue(b);return a.lesser(b)?a:b}function gcd(a,b){a=parseValue(a).abs();b=parseValue(b).abs();if(a.equals(b))return a;if(a.isZero())return b;if(b.isZero())return a;var c=Integer[1],d,t;while(a.isEven()&&b.isEven()){d=min(roughLOB(a),roughLOB(b));a=a.divide(d);b=b.divide(d);c=c.multiply(d)}while(a.isEven()){a=a.divide(roughLOB(a))}do{while(b.isEven()){b=b.divide(roughLOB(b))}if(a.greater(b)){t=b;b=a;a=t}b=b.subtract(a)}while(!b.isZero());return c.isUnit()?a:a.multiply(c)}function lcm(a,b){a=parseValue(a).abs();b=parseValue(b).abs();return a.divide(gcd(a,b)).multiply(b)}function randBetween(a,b){a=parseValue(a);b=parseValue(b);var low=min(a,b),high=max(a,b);var range=high.subtract(low).add(1);if(range.isSmall)return low.add(Math.floor(Math.random()*range));var digits=toBase(range,BASE).value;var result=[],restricted=true;for(var i=0;i<digits.length;i++){var top=restricted?digits[i]:BASE;var digit=truncate(Math.random()*top);result.push(digit);if(digit<top)restricted=false}return low.add(Integer.fromArray(result,BASE,false))}var parseBase=function(text,base,alphabet,caseSensitive){alphabet=alphabet||DEFAULT_ALPHABET;text=String(text);if(!caseSensitive){text=text.toLowerCase();alphabet=alphabet.toLowerCase()}var length=text.length;var i;var absBase=Math.abs(base);var alphabetValues={};for(i=0;i<alphabet.length;i++){alphabetValues[alphabet[i]]=i}for(i=0;i<length;i++){var c=text[i];if(c==="-")continue;if(c in alphabetValues){if(alphabetValues[c]>=absBase){if(c==="1"&&absBase===1)continue;throw new Error(c+" is not a valid digit in base "+base+".")}}}base=parseValue(base);var digits=[];var isNegative=text[0]==="-";for(i=isNegative?1:0;i<text.length;i++){var c=text[i];if(c in alphabetValues)digits.push(parseValue(alphabetValues[c]));else if(c==="<"){var start=i;do{i++}while(text[i]!==">"&&i<text.length);digits.push(parseValue(text.slice(start+1,i)))}else throw new Error(c+" is not a valid character")}return parseBaseFromArray(digits,base,isNegative)};function parseBaseFromArray(digits,base,isNegative){var val=Integer[0],pow=Integer[1],i;for(i=digits.length-1;i>=0;i--){val=val.add(digits[i].times(pow));pow=pow.times(base)}return isNegative?val.negate():val}function stringify(digit,alphabet){alphabet=alphabet||DEFAULT_ALPHABET;if(digit<alphabet.length){return alphabet[digit]}return"<"+digit+">"}function toBase(n,base){base=bigInt(base);if(base.isZero()){if(n.isZero())return{value:[0],isNegative:false};throw new Error("Cannot convert nonzero numbers to base 0.")}if(base.equals(-1)){if(n.isZero())return{value:[0],isNegative:false};if(n.isNegative())return{value:[].concat.apply([],Array.apply(null,Array(-n.toJSNumber())).map(Array.prototype.valueOf,[1,0])),isNegative:false};var arr=Array.apply(null,Array(n.toJSNumber()-1)).map(Array.prototype.valueOf,[0,1]);arr.unshift([1]);return{value:[].concat.apply([],arr),isNegative:false}}var neg=false;if(n.isNegative()&&base.isPositive()){neg=true;n=n.abs()}if(base.isUnit()){if(n.isZero())return{value:[0],isNegative:false};return{value:Array.apply(null,Array(n.toJSNumber())).map(Number.prototype.valueOf,1),isNegative:neg}}var out=[];var left=n,divmod;while(left.isNegative()||left.compareAbs(base)>=0){divmod=left.divmod(base);left=divmod.quotient;var digit=divmod.remainder;if(digit.isNegative()){digit=base.minus(digit).abs();left=left.next()}out.push(digit.toJSNumber())}out.push(left.toJSNumber());return{value:out.reverse(),isNegative:neg}}function toBaseString(n,base,alphabet){var arr=toBase(n,base);return(arr.isNegative?"-":"")+arr.value.map(function(x){return stringify(x,alphabet)}).join("")}BigInteger.prototype.toArray=function(radix){return toBase(this,radix)};SmallInteger.prototype.toArray=function(radix){return toBase(this,radix)};NativeBigInt.prototype.toArray=function(radix){return toBase(this,radix)};BigInteger.prototype.toString=function(radix,alphabet){if(radix===undefined)radix=10;if(radix!==10)return toBaseString(this,radix,alphabet);var v=this.value,l=v.length,str=String(v[--l]),zeros="0000000",digit;while(--l>=0){digit=String(v[l]);str+=zeros.slice(digit.length)+digit}var sign=this.sign?"-":"";return sign+str};SmallInteger.prototype.toString=function(radix,alphabet){if(radix===undefined)radix=10;if(radix!=10)return toBaseString(this,radix,alphabet);return String(this.value)};NativeBigInt.prototype.toString=SmallInteger.prototype.toString;NativeBigInt.prototype.toJSON=BigInteger.prototype.toJSON=SmallInteger.prototype.toJSON=function(){return this.toString()};BigInteger.prototype.valueOf=function(){return parseInt(this.toString(),10)};BigInteger.prototype.toJSNumber=BigInteger.prototype.valueOf;SmallInteger.prototype.valueOf=function(){return this.value};SmallInteger.prototype.toJSNumber=SmallInteger.prototype.valueOf;NativeBigInt.prototype.valueOf=NativeBigInt.prototype.toJSNumber=function(){return parseInt(this.toString(),10)};function parseStringValue(v){if(isPrecise(+v)){var x=+v;if(x===truncate(x))return supportsNativeBigInt?new NativeBigInt(BigInt(x)):new SmallInteger(x);throw new Error("Invalid integer: "+v)}var sign=v[0]==="-";if(sign)v=v.slice(1);var split=v.split(/e/i);if(split.length>2)throw new Error("Invalid integer: "+split.join("e"));if(split.length===2){var exp=split[1];if(exp[0]==="+")exp=exp.slice(1);exp=+exp;if(exp!==truncate(exp)||!isPrecise(exp))throw new Error("Invalid integer: "+exp+" is not a valid exponent.");var text=split[0];var decimalPlace=text.indexOf(".");if(decimalPlace>=0){exp-=text.length-decimalPlace-1;text=text.slice(0,decimalPlace)+text.slice(decimalPlace+1)}if(exp<0)throw new Error("Cannot include negative exponent part for integers");text+=new Array(exp+1).join("0");v=text}var isValid=/^([0-9][0-9]*)$/.test(v);if(!isValid)throw new Error("Invalid integer: "+v);if(supportsNativeBigInt){return new NativeBigInt(BigInt(sign?"-"+v:v))}var r=[],max=v.length,l=LOG_BASE,min=max-l;while(max>0){r.push(+v.slice(min,max));min-=l;if(min<0)min=0;max-=l}trim(r);return new BigInteger(r,sign)}function parseNumberValue(v){if(supportsNativeBigInt){return new NativeBigInt(BigInt(v))}if(isPrecise(v)){if(v!==truncate(v))throw new Error(v+" is not an integer.");return new SmallInteger(v)}return parseStringValue(v.toString())}function parseValue(v){if(typeof v==="number"){return parseNumberValue(v)}if(typeof v==="string"){return parseStringValue(v)}if(typeof v==="bigint"){return new NativeBigInt(v)}return v}for(var i=0;i<1e3;i++){Integer[i]=parseValue(i);if(i>0)Integer[-i]=parseValue(-i)}Integer.one=Integer[1];Integer.zero=Integer[0];Integer.minusOne=Integer[-1];Integer.max=max;Integer.min=min;Integer.gcd=gcd;Integer.lcm=lcm;Integer.isInstance=function(x){return x instanceof BigInteger||x instanceof SmallInteger||x instanceof NativeBigInt};Integer.randBetween=randBetween;Integer.fromArray=function(digits,base,isNegative){return parseBaseFromArray(digits.map(parseValue),parseValue(base||10),isNegative)};return Integer}();if(typeof module!=="undefined"&&module.hasOwnProperty("exports")){module.exports=bigInt}if(typeof define==="function"&&define.amd){define("big-integer",[],function(){return bigInt})}var bigRat=function(bigInt){"use strict";function BigRational(num,denom){if(denom.isZero())throw"Denominator cannot be 0.";this.numerator=this.num=num;this.denominator=this.denom=denom}var gcd=bigInt.gcd,lcm=bigInt.lcm;function reduce(n,d){var divisor=gcd(n,d),num=n.over(divisor),denom=d.over(divisor);if(denom.isNegative()){return new BigRational(num.negate(),denom.negate())}return new BigRational(num,denom)}BigRational.prototype.add=function(n,d){var v=interpret(n,d),multiple=lcm(this.denom,v.denom),a=multiple.divide(this.denom),b=multiple.divide(v.denom);a=this.num.times(a);b=v.num.times(b);return reduce(a.add(b),multiple)};BigRational.prototype.plus=BigRational.prototype.add;BigRational.prototype.subtract=function(n,d){var v=interpret(n,d);return this.add(v.negate())};BigRational.prototype.minus=BigRational.prototype.subtract;BigRational.prototype.multiply=function(n,d){var v=interpret(n,d);return reduce(this.num.times(v.num),this.denom.times(v.denom))};BigRational.prototype.times=BigRational.prototype.multiply;BigRational.prototype.divide=function(n,d){var v=interpret(n,d);return reduce(this.num.times(v.denom),this.denom.times(v.num))};BigRational.prototype.over=BigRational.prototype.divide;BigRational.prototype.reciprocate=function(){return new BigRational(this.denom,this.num)};BigRational.prototype.mod=function(n,d){var v=interpret(n,d);return this.minus(v.times(this.over(v).floor()))};BigRational.prototype.pow=function(n){var v=bigInt(n);var num=this.num.pow(v),denom=this.denom.pow(v);return reduce(num,denom)};BigRational.prototype.floor=function(toBigInt){var divmod=this.num.divmod(this.denom),floor;if(divmod.remainder.isZero()||!divmod.quotient.isNegative()){floor=divmod.quotient}else floor=divmod.quotient.prev();if(toBigInt)return floor;return new BigRational(floor,bigInt[1])};BigRational.prototype.ceil=function(toBigInt){var divmod=this.num.divmod(this.denom),ceil;if(divmod.remainder.isZero()||divmod.quotient.isNegative()){ceil=divmod.quotient}else ceil=divmod.quotient.next();if(toBigInt)return ceil;return new BigRational(ceil,bigInt[1])};BigRational.prototype.round=function(toBigInt){return this.add(1,2).floor(toBigInt)};BigRational.prototype.compareAbs=function(n,d){var v=interpret(n,d);if(this.denom.equals(v.denom)){return this.num.compareAbs(v.num)}return this.num.times(v.denom).compareAbs(v.num.times(this.denom))};BigRational.prototype.compare=function(n,d){var v=interpret(n,d);if(this.denom.equals(v.denom)){return this.num.compare(v.num)}var comparison=this.denom.isNegative()===v.denom.isNegative()?1:-1;return comparison*this.num.times(v.denom).compare(v.num.times(this.denom))};BigRational.prototype.compareTo=BigRational.prototype.compare;BigRational.prototype.equals=function(n,d){return this.compare(n,d)===0};BigRational.prototype.eq=BigRational.prototype.equals;BigRational.prototype.notEquals=function(n,d){return this.compare(n,d)!==0};BigRational.prototype.neq=BigRational.prototype.notEquals;BigRational.prototype.lesser=function(n,d){return this.compare(n,d)<0};BigRational.prototype.lt=BigRational.prototype.lesser;BigRational.prototype.lesserOrEquals=function(n,d){return this.compare(n,d)<=0};BigRational.prototype.leq=BigRational.prototype.lesserOrEquals;BigRational.prototype.greater=function(n,d){return this.compare(n,d)>0};BigRational.prototype.gt=BigRational.prototype.greater;BigRational.prototype.greaterOrEquals=function(n,d){return this.compare(n,d)>=0};BigRational.prototype.geq=BigRational.prototype.greaterOrEquals;BigRational.prototype.abs=function(){if(this.isPositive())return this;return this.negate()};BigRational.prototype.negate=function(){if(this.denom.isNegative()){return new BigRational(this.num,this.denom.negate())}return new BigRational(this.num.negate(),this.denom)};BigRational.prototype.isNegative=function(){return this.num.isNegative()!==this.denom.isNegative()&&!this.num.isZero()};BigRational.prototype.isPositive=function(){return this.num.isNegative()===this.denom.isNegative()&&!this.num.isZero()};BigRational.prototype.isZero=function(){return this.num.isZero()};BigRational.prototype.toDecimal=function(digits){digits=typeof digits==="number"?digits:10;var n=this.num.divmod(this.denom);var intPart=n.quotient.abs().toString();var remainder=parse(n.remainder.abs(),this.denom);var shiftedRemainder=remainder.times(bigInt("1e"+digits));var decPart=shiftedRemainder.num.over(shiftedRemainder.denom).toString();if(decPart.length<digits){decPart=new Array(digits-decPart.length+1).join("0")+decPart}if(shiftedRemainder.num.mod(shiftedRemainder.denom).isZero()){while(decPart.slice(-1)==="0"){decPart=decPart.slice(0,-1)}}if(digits<1)decPart="";if(this.isNegative()){intPart="-"+intPart}if(decPart===""){return intPart}return intPart+"."+decPart};BigRational.prototype.toString=function(){return String(this.num)+"/"+String(this.denom)};BigRational.prototype.valueOf=function(){if(!isFinite(+this.num)||!isFinite(+this.denom)){return+this.toDecimal(64)}return this.num/this.denom};function interpret(n,d){return parse(n,d)}function parseDecimal(n){var parts=n.split(/e/i);if(parts.length>2){throw new Error("Invalid input: too many 'e' tokens")}if(parts.length>1){var isPositive=true;if(parts[1][0]==="-"){parts[1]=parts[1].slice(1);isPositive=false}if(parts[1][0]==="+"){parts[1]=parts[1].slice(1)}var significand=parseDecimal(parts[0]);var exponent=new BigRational(bigInt(10).pow(parts[1]),bigInt[1]);if(isPositive){return significand.times(exponent)}else{return significand.over(exponent)}}parts=n.trim().split(".");if(parts.length>2){throw new Error("Invalid input: too many '.' tokens")}if(parts.length>1){var isNegative=parts[0][0]==="-";if(isNegative)parts[0]=parts[0].slice(1);var intPart=new BigRational(bigInt(parts[0]),bigInt[1]);var length=parts[1].length;while(parts[1][0]==="0"){parts[1]=parts[1].slice(1)}var exp="1"+Array(length+1).join("0");var decPart=reduce(bigInt(parts[1]),bigInt(exp));intPart=intPart.add(decPart);if(isNegative)intPart=intPart.negate();return intPart}return new BigRational(bigInt(n),bigInt[1])}function parse(a,b){if(!a){return new BigRational(bigInt(0),bigInt[1])}if(b){return reduce(bigInt(a),bigInt(b))}if(bigInt.isInstance(a)){return new BigRational(a,bigInt[1])}if(a instanceof BigRational)return a;var num;var denom;var text=String(a);var texts=text.split("/");if(texts.length>2){throw new Error("Invalid input: too many '/' tokens")}if(texts.length>1){var parts=texts[0].split("_");if(parts.length>2){throw new Error("Invalid input: too many '_' tokens")}if(parts.length>1){var isPositive=parts[0][0]!=="-";num=bigInt(parts[0]).times(texts[1]);if(isPositive){num=num.add(parts[1])}else{num=num.subtract(parts[1])}denom=bigInt(texts[1]);return reduce(num,denom)}return reduce(bigInt(texts[0]),bigInt(texts[1]))}return parseDecimal(text)}parse.zero=parse(0);parse.one=parse(1);parse.minusOne=parse(-1);return parse}(typeof bigInt!=="undefined"?bigInt:require("big-integer"));if(typeof module!=="undefined"){if(module.hasOwnProperty("exports")){module.exports=bigRat}}