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e|g,g=(this[h]&d)<<c;for(h=f-1;h>=0;--h)b[h]=0;b[f]=g;b.t=this.t+f+1;b.s=this.s;b.clamp()};f.prototype.rShiftTo=function(a,b){b.s=this.s;var c=Math.floor(a/this.DB);if(c>=this.t)b.t=0;else{var e=a%this.DB,d=this.DB-e,f=(1<<e)-1;b[0]=this[c]>>e;for(var g=c+1;g<this.t;++g)b[g-c-1]|=(this[g]&f)<<d,b[g-c]=this[g]>>e;e>0&&(b[this.t-c-1]|=(this.s&f)<<d);b.t=this.t-c;b.clamp()}};f.prototype.subTo=function(a,b){for(var c=0,e=0,d=Math.min(a.t,this.t);c<d;)e+=this[c]-a[c],b[c++]=e&this.DM,e>>=this.DB;if(a.t<
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0;c<b.t-1;++c){var e=b.am(c,b[c],a,2*c,0,1);if((a[c+b.t]+=b.am(c+1,2*b[c],a,2*c+1,e,b.t-c-1))>=b.DV)a[c+b.t]-=b.DV,a[c+b.t+1]=1}a.t>0&&(a[a.t-1]+=b.am(c,b[c],a,2*c,0,1));a.s=0;a.clamp()};f.prototype.divRemTo=function(a,b,c){var e=a.abs();if(!(e.t<=0)){var d=this.abs();if(d.t<e.t)b!=null&&b.fromInt(0),c!=null&&this.copyTo(c);else{c==null&&(c=p());var g=p(),h=this.s,a=a.s,i=this.DB-ka(e[e.t-1]);i>0?(e.lShiftTo(i,g),d.lShiftTo(i,c)):(e.copyTo(g),d.copyTo(c));e=g.t;d=g[e-1];if(d!=0){var l=d*(1<<this.F1)+
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d.join(""):"0"};f.prototype.negate=function(){var a=p();f.ZERO.subTo(this,a);return a};f.prototype.abs=function(){return this.s<0?this.negate():this};f.prototype.compareTo=Y;f.prototype.bitLength=function(){return this.t<=0?0:this.DB*(this.t-1)+ka(this[this.t-1]^this.s&this.DM)};f.prototype.mod=ya;f.prototype.modPowInt=function(a,b){var c;c=a<256||b.isEven()?new P(b):new Q(b);return this.exp(a,c)};f.ZERO=N(0);f.ONE=N(1);$.prototype.convert=Ha;$.prototype.revert=Ha;$.prototype.mulTo=function(a,b,c){a.multiplyTo(b,
c)};$.prototype.sqrTo=function(a,b){a.squareTo(b)};V.prototype.convert=function(a){if(a.s<0||a.t>2*this.m.t)return a.mod(this.m);else if(a.compareTo(this.m)<0)return a;else{var b=p();a.copyTo(b);this.reduce(b);return b}};V.prototype.revert=function(a){return a};V.prototype.reduce=function(a){a.drShiftTo(this.m.t-1,this.r2);if(a.t>this.m.t+1)a.t=this.m.t+1,a.clamp();this.mu.multiplyUpperTo(this.r2,this.m.t+1,this.q3);for(this.m.multiplyLowerTo(this.q3,this.m.t+1,this.r2);a.compareTo(this.r2)<0;)a.dAddOffset(1,
this.m.t+1);for(a.subTo(this.r2,a);a.compareTo(this.m)>=0;)a.subTo(this.m,a)};V.prototype.mulTo=function(a,b,c){a.multiplyTo(b,c);this.reduce(c)};V.prototype.sqrTo=function(a,b){a.squareTo(b);this.reduce(b)};var C=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,
409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509],ub=67108864/C[C.length-1];f.prototype.chunkSize=function(a){return Math.floor(Math.LN2*this.DB/Math.log(a))};f.prototype.toRadix=function(a){a==null&&(a=10);if(this.signum()==0||a<2||a>36)return"0";var b=this.chunkSize(a),b=Math.pow(a,b),c=N(b),e=p(),d=p(),f="";for(this.divRemTo(c,e,d);e.signum()>0;)f=(b+d.intValue()).toString(a).substr(1)+f,e.divRemTo(c,e,d);return d.intValue().toString(a)+f};f.prototype.fromRadix=function(a,
b){this.fromInt(0);b==null&&(b=10);for(var c=this.chunkSize(b),e=Math.pow(b,c),d=!1,g=0,h=0,i=0;i<a.length;++i){var l=xa(a,i);l<0?a.charAt(i)=="-"&&this.signum()==0&&(d=!0):(h=b*h+l,++g>=c&&(this.dMultiply(e),this.dAddOffset(h,0),h=g=0))}g>0&&(this.dMultiply(Math.pow(b,g)),this.dAddOffset(h,0));d&&f.ZERO.subTo(this,this)};f.prototype.fromNumber=function(a,b,c){if("number"==typeof b)if(a<2)this.fromInt(1);else{this.fromNumber(a,c);this.testBit(a-1)||this.bitwiseTo(f.ONE.shiftLeft(a-1),ua,this);for(this.isEven()&&
this.dAddOffset(1,0);!this.isProbablePrime(b);)this.dAddOffset(2,0),this.bitLength()>a&&this.subTo(f.ONE.shiftLeft(a-1),this)}else{var c=[],e=a&7;c.length=(a>>3)+1;b.nextBytes(c);e>0?c[0]&=(1<<e)-1:c[0]=0;this.fromString(c,256)}};f.prototype.bitwiseTo=function(a,b,c){var e,d,f=Math.min(a.t,this.t);for(e=0;e<f;++e)c[e]=b(this[e],a[e]);if(a.t<this.t){d=a.s&this.DM;for(e=f;e<this.t;++e)c[e]=b(this[e],d);c.t=this.t}else{d=this.s&this.DM;for(e=f;e<a.t;++e)c[e]=b(d,a[e]);c.t=a.t}c.s=b(this.s,a.s);c.clamp()};
f.prototype.changeBit=function(a,b){var c=f.ONE.shiftLeft(a);this.bitwiseTo(c,b,c);return c};f.prototype.addTo=function(a,b){for(var c=0,e=0,d=Math.min(a.t,this.t);c<d;)e+=this[c]+a[c],b[c++]=e&this.DM,e>>=this.DB;if(a.t<this.t){for(e+=a.s;c<this.t;)e+=this[c],b[c++]=e&this.DM,e>>=this.DB;e+=this.s}else{for(e+=this.s;c<a.t;)e+=a[c],b[c++]=e&this.DM,e>>=this.DB;e+=a.s}b.s=e<0?-1:0;e>0?b[c++]=e:e<-1&&(b[c++]=this.DV+e);b.t=c;b.clamp()};f.prototype.dMultiply=function(a){this[this.t]=this.am(0,a-1,this,
0,0,this.t);++this.t;this.clamp()};f.prototype.dAddOffset=function(a,b){if(a!=0){for(;this.t<=b;)this[this.t++]=0;for(this[b]+=a;this[b]>=this.DV;)this[b]-=this.DV,++b>=this.t&&(this[this.t++]=0),++this[b]}};f.prototype.multiplyLowerTo=function(a,b,c){var e=Math.min(this.t+a.t,b);c.s=0;for(c.t=e;e>0;)c[--e]=0;var d;for(d=c.t-this.t;e<d;++e)c[e+this.t]=this.am(0,a[e],c,e,0,this.t);for(d=Math.min(a.t,b);e<d;++e)this.am(0,a[e],c,e,0,b-e);c.clamp()};f.prototype.multiplyUpperTo=function(a,b,c){--b;var e=
c.t=this.t+a.t-b;for(c.s=0;--e>=0;)c[e]=0;for(e=Math.max(b-this.t,0);e<a.t;++e)c[this.t+e-b]=this.am(b-e,a[e],c,0,0,this.t+e-b);c.clamp();c.drShiftTo(1,c)};f.prototype.modInt=function(a){if(a<=0)return 0;var b=this.DV%a,c=this.s<0?a-1:0;if(this.t>0)if(b==0)c=this[0]%a;else for(var e=this.t-1;e>=0;--e)c=(b*c+this[e])%a;return c};f.prototype.millerRabin=function(a){var b=this.subtract(f.ONE),c=b.getLowestSetBit();if(c<=0)return!1;var e=b.shiftRight(c),a=a+1>>1;if(a>C.length)a=C.length;for(var d=p(),
g=0;g<a;++g){d.fromInt(C[g]);var h=d.modPow(e,this);if(h.compareTo(f.ONE)!=0&&h.compareTo(b)!=0){for(var i=1;i++<c&&h.compareTo(b)!=0;)if(h=h.modPowInt(2,this),h.compareTo(f.ONE)==0)return!1;if(h.compareTo(b)!=0)return!1}}return!0};f.prototype.clone=function(){var a=p();this.copyTo(a);return a};f.prototype.intValue=function(){if(this.s<0)if(this.t==1)return this[0]-this.DV;else{if(this.t==0)return-1}else if(this.t==1)return this[0];else if(this.t==0)return 0;return(this[1]&(1<<32-this.DB)-1)<<this.DB|
this[0]};f.prototype.byteValue=function(){return this.t==0?this.s:this[0]<<24>>24};f.prototype.shortValue=function(){return this.t==0?this.s:this[0]<<16>>16};f.prototype.signum=function(){return this.s<0?-1:this.t<=0||this.t==1&&this[0]<=0?0:1};f.prototype.toByteArray=function(){var a=this.t,b=[];b[0]=this.s;var c=this.DB-a*this.DB%8,e,d=0;if(a-- >0){if(c<this.DB&&(e=this[a]>>c)!=(this.s&this.DM)>>c)b[d++]=e|this.s<<this.DB-c;for(;a>=0;)if(c<8?(e=(this[a]&(1<<c)-1)<<8-c,e|=this[--a]>>(c+=this.DB-
8)):(e=this[a]>>(c-=8)&255,c<=0&&(c+=this.DB,--a)),(e&128)!=0&&(e|=-256),d==0&&(this.s&128)!=(e&128)&&++d,d>0||e!=this.s)b[d++]=e}return b};f.prototype.equals=Z;f.prototype.min=function(a){return this.compareTo(a)<0?this:a};f.prototype.max=function(a){return this.compareTo(a)>0?this:a};f.prototype.and=function(a){var b=p();this.bitwiseTo(a,kb,b);return b};f.prototype.or=function(a){var b=p();this.bitwiseTo(a,ua,b);return b};f.prototype.xor=function(a){var b=p();this.bitwiseTo(a,za,b);return b};f.prototype.andNot=
function(a){var b=p();this.bitwiseTo(a,Aa,b);return b};f.prototype.not=function(){for(var a=p(),b=0;b<this.t;++b)a[b]=this.DM&~this[b];a.t=this.t;a.s=~this.s;return a};f.prototype.shiftLeft=function(a){var b=p();a<0?this.rShiftTo(-a,b):this.lShiftTo(a,b);return b};f.prototype.shiftRight=function(a){var b=p();a<0?this.lShiftTo(-a,b):this.rShiftTo(a,b);return b};f.prototype.getLowestSetBit=function(){for(var a=0;a<this.t;++a)if(this[a]!=0){var b=a*this.DB;a=this[a];if(a==0)a=-1;else{var c=0;(a&65535)==
0&&(a>>=16,c+=16);(a&255)==0&&(a>>=8,c+=8);(a&15)==0&&(a>>=4,c+=4);(a&3)==0&&(a>>=2,c+=2);(a&1)==0&&++c;a=c}return b+a}return this.s<0?this.t*this.DB:-1};f.prototype.bitCount=function(){for(var a=0,b=this.s&this.DM,c=0;c<this.t;++c){for(var e=this[c]^b,d=0;e!=0;)e&=e-1,++d;a+=d}return a};f.prototype.testBit=function(a){var b=Math.floor(a/this.DB);return b>=this.t?this.s!=0:(this[b]&1<<a%this.DB)!=0};f.prototype.setBit=function(a){return this.changeBit(a,ua)};f.prototype.clearBit=function(a){return this.changeBit(a,
Aa)};f.prototype.flipBit=function(a){return this.changeBit(a,za)};f.prototype.add=Ba;f.prototype.subtract=Ca;f.prototype.multiply=Da;f.prototype.divide=Ea;f.prototype.remainder=Fa;f.prototype.divideAndRemainder=Ga;f.prototype.modPow=function(a,b){var c=a.bitLength(),e,d=N(1),f;if(c<=0)return d;else e=c<18?1:c<48?3:c<144?4:c<768?5:6;f=c<8?new P(b):b.isEven()?new V(b):new Q(b);var g=[],h=3,i=e-1,l=(1<<e)-1;g[1]=f.convert(this);if(e>1){c=p();for(f.sqrTo(g[1],c);h<=l;)g[h]=p(),f.mulTo(c,g[h-2],g[h]),
h+=2}for(var k=a.t-1,m,o=!0,n=p(),c=ka(a[k])-1;k>=0;){c>=i?m=a[k]>>c-i&l:(m=(a[k]&(1<<c+1)-1)<<i-c,k>0&&(m|=a[k-1]>>this.DB+c-i));for(h=e;(m&1)==0;)m>>=1,--h;if((c-=h)<0)c+=this.DB,--k;if(o)g[m].copyTo(d),o=!1;else{for(;h>1;)f.sqrTo(d,n),f.sqrTo(n,d),h-=2;h>0?f.sqrTo(d,n):(h=d,d=n,n=h);f.mulTo(n,g[m],d)}for(;k>=0&&(a[k]&1<<c)==0;)f.sqrTo(d,n),h=d,d=n,n=h,--c<0&&(c=this.DB-1,--k)}return f.revert(d)};f.prototype.modInverse=function(a){var b=a.isEven();if(this.isEven()&&b||a.signum()==0)return f.ZERO;
for(var c=a.clone(),e=this.clone(),d=N(1),g=N(0),h=N(0),i=N(1);c.signum()!=0;){for(;c.isEven();){c.rShiftTo(1,c);if(b){if(!d.isEven()||!g.isEven())d.addTo(this,d),g.subTo(a,g);d.rShiftTo(1,d)}else g.isEven()||g.subTo(a,g);g.rShiftTo(1,g)}for(;e.isEven();){e.rShiftTo(1,e);if(b){if(!h.isEven()||!i.isEven())h.addTo(this,h),i.subTo(a,i);h.rShiftTo(1,h)}else i.isEven()||i.subTo(a,i);i.rShiftTo(1,i)}c.compareTo(e)>=0?(c.subTo(e,c),b&&d.subTo(h,d),g.subTo(i,g)):(e.subTo(c,e),b&&h.subTo(d,h),i.subTo(g,i))}if(e.compareTo(f.ONE)!=
0)return f.ZERO;if(i.compareTo(a)>=0)return i.subtract(a);if(i.signum()<0)i.addTo(a,i);else return i;return i.signum()<0?i.add(a):i};f.prototype.pow=Ia;f.prototype.gcd=Ja;f.prototype.isProbablePrime=function(a){var b,c=this.abs();if(c.t==1&&c[0]<=C[C.length-1]){for(b=0;b<C.length;++b)if(c[0]==C[b])return!0;return!1}if(c.isEven())return!1;for(b=1;b<C.length;){for(var e=C[b],d=b+1;d<C.length&&e<ub;)e*=C[d++];for(e=c.modInt(e);b<d;)if(e%C[b++]==0)return!1}return c.millerRabin(a)};f.NEGATIVE_ONE=f.ONE.negate();
var v=function(a){typeof a==="number"&&(a+="");a=Na(a);return new f(a,10)};f.prototype.level=0;f.prototype.liftTo=function(a){if(a.level===1)return new i(this,1);return a.level===2?(a=this.toFixnum(),a===Number.POSITIVE_INFINITY?db:a===Number.NEGATIVE_INFINITY?eb:new d(a)):a.level===3?new g(this,0):l("invalid level for BigInteger lift",this,a)};f.prototype.isFinite=function(){return!0};f.prototype.isInteger=function(){return!0};f.prototype.isRational=function(){return!0};f.prototype.isReal=function(){return!0};
f.prototype.isExact=function(){return!0};f.prototype.isInexact=function(){return!1};f.prototype.toExact=function(){return this};f.prototype.toInexact=function(){return d.makeInstance(this.toFixnum())};f.prototype.toFixnum=function(){var a=0,b=this.toString(),c;if(b[0]==="-"){for(c=1;c<b.length;c++)a=a*10+Number(b[c]);return-a}else{for(c=0;c<b.length;c++)a=a*10+Number(b[c]);return a}};f.prototype.greaterThan=function(a){return this.compareTo(a)>0};f.prototype.greaterThanOrEqual=function(a){return this.compareTo(a)>=
0};f.prototype.lessThan=function(a){return this.compareTo(a)<0};f.prototype.lessThanOrEqual=function(a){return this.compareTo(a)<=0};f.prototype.divide=function(a){var b=Ga.call(this,a);return b[1].compareTo(f.ZERO)===0?b[0]:m(b[0],i.makeInstance(b[1],a))};f.prototype.numerator=function(){return this};f.prototype.denominator=function(){return 1};(function(){var a=function(a,c){for(;!da(X(c),a)||!J(a,X(m(c,1)));)c=z(o(m(c,z(o(a,c))),2));return c};f.prototype.integerSqrt=function(){var b;return ha(this)>=
0?a(this,this):(b=this.negate(),g.makeInstance(0,a(b,b)))}})();(function(){f.prototype.sqrt=function(){var a=this.integerSqrt(),b;if(q(X(a),this))return a;b=I(this);return isFinite(b)?b>=0?d.makeInstance(Math.sqrt(b)):g.makeInstance(0,d.makeInstance(Math.sqrt(-b))):a}})();f.prototype.floor=function(){return this};f.prototype.ceiling=function(){return this};f.prototype.expt=function(a){return Ia.call(this,a)};f.prototype.imaginaryPart=function(){return 0};f.prototype.realPart=function(){return this};
n=function(){return function(a,b,c){var e=512;if(c&&typeof c.limit!=="undefined")e=c.limit;t(a)||l("toRepeatingDecimal: n "+a.toString()+" is not an integer.");t(b)||l("toRepeatingDecimal: d "+b.toString()+" is not an integer.");s(b,0)&&l("toRepeatingDecimal: d equals 0");J(b,0)&&l("toRepeatingDecimal: d < 0");c=J(a,0)?"-":"";a=A(a);c+=qa(a,b);a:{var a=ra(a,b),d=e,e=[],f={};for(f[a]=!0;;){if(d--<=0){b=[e.join(""),"..."];break a}var g=qa(k(a,10),b),a=ra(k(a,10),b);e.push(g.toString());if(f[a])break;
else f[a]=!0}f=a;for(d=[];;)if(g=qa(k(a,10),b),a=ra(k(a,10),b),d.push(g.toString()),s(a,f))break;b=e.join("");for(a=d.join("");b.length>=a.length&&b.substring(b.length-a.length)===a;)b=b.substring(0,b.length-a.length);b=[b,a]}return[c].concat(b)}}();h.fromFixnum=function(a){if(isNaN(a)||!isFinite(a))return d.makeInstance(a);var b=Math.floor(a);return b===a?K(b)?v(Na(a+"")):b:d.makeInstance(a)};h.fromString=ia;h.makeBignum=v;h.makeRational=i.makeInstance;h.makeFloat=d.makeInstance;h.makeComplex=g.makeInstance;
h.makeComplexPolar=function(a,b){return q(b,0)?a:g.makeInstance(k(a,oa(b)),k(a,pa(b)))};h.pi=d.pi;h.e=d.e;h.nan=d.nan;h.negative_inf=d.neginf;h.inf=d.inf;h.negative_one=-1;h.zero=0;h.one=1;h.i=G;h.negative_i=gb;h.negative_zero=L;h.onThrowRuntimeError=function(a){throw Error(a);};h.isSchemeNumber=R;h.isRational=va;h.isReal=E;h.isExact=W;h.isInexact=aa;h.isInteger=t;h.toFixnum=I;h.toExact=S;h.toInexact=ba;h.add=m;h.subtract=w;h.multiply=k;h.divide=o;h.equals=s;h.eqv=q;h.approxEquals=function(a,b,c){return J(A(w(a,
b)),c)};h.greaterThanOrEqual=ma;h.lessThanOrEqual=da;h.greaterThan=U;h.lessThan=J;h.expt=lb;h.exp=F;h.modulo=Pa;h.numerator=ea;h.denominator=Qa;h.integerSqrt=Ya;h.sqrt=D;h.abs=A;h.quotient=qa;h.remainder=ra;h.floor=z;h.ceiling=Ra;h.conjugate=Sa;h.magnitude=function(a){return typeof a==="number"?Math.abs(a):a.magnitude()};h.log=na;h.angle=Ta;h.tan=Ua;h.atan=wa;h.cos=oa;h.sin=pa;h.tan=Ua;h.acos=Va;h.asin=Wa;h.cosh=function(a){return q(a,0)?d.makeInstance(1):o(m(F(a),F(x(a))),2)};h.sinh=function(a){return o(w(F(a),
F(x(a))),2)};h.imaginaryPart=ca;h.realPart=M;h.round=Xa;h.sqr=X;h.gcd=function(a,b){t(a)||l("gcd: the argument "+a.toString()+" is not an integer.",a);for(var c=A(a),e,d,f=0;f<b.length;f++){d=A(b[f]);for(t(d)||l("gcd: the argument "+d.toString()+" is not an integer.",d);!r(d);)e=c,c=d,d=Oa(e,d)}return c};h.lcm=function(a,b){t(a)||l("lcm: the argument "+a.toString()+" is not an integer.",a);var c=A(a);if(r(c))return 0;for(var d=0;d<b.length;d++){t(b[d])||l("lcm: the argument "+b[d].toString()+" is not an integer.",
b[d]);var f=sa(c,b[d]);if(r(f))return 0;c=o(k(c,b[d]),f)}return c};h.toRepeatingDecimal=n;h.BigInteger=f;h.Rational=i;h.FloatPoint=d;h.Complex=g;h.MIN_FIXNUM=-9E15;h.MAX_FIXNUM=9E15})();