//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////
/*

in the following javascript code I use these conventions:

xgc_ or XGC_ are prefixes for globals
xgc_something is a variable (begins lowercase)
xgc_Something is a function (begins uppercase)
XGC_Something is an object constructor

*/
//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////


/* to simplify maths we don't compute primes */
// you can change this string to meet your needs
// here are the first 1000 primes
var xgc_primesStr = "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, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919";

/* the following three variables absorb the primes string above */
var xgc_primes = xgc_primesStr.split( ", " );
var xgc_maxPrime = xgc_primes[xgc_primes.length - 1];
var xgc_maxFactorizable = xgc_maxPrime * xgc_maxPrime;

/* int xgc_GCD( int a, int b ) */
// gets the Greatest Common Divisor of a and b
function xgc_GCD( a, b ) {
	var remainder = 0;
	var max = Math.max( a, b );
	var min = Math.min( a, b );
	while (min > 0) {
		remainder = max % min;
		max = min;
		min = remainder;
	}
	return max;
}

/* boolean xgc_IsPrime( int p ) */
// true means that p is prime
function xgc_IsPrime( p ) {
	var factorList = xgc_Factorize( p );
	if( factorList.pop() == p )
		return true;
	else return false;
}

/* boolean xgc_IsPrimeTo( int a, int b ) */
// true means that the intersection between xgc_Factorize( a ) 
// and xgc_Factorize( b ) is empty
function xgc_IsPrimeTo( a, b ) {
	return xgc_GCD( a, b ) == 1;
}

/* boolean xgc_Divides( int divisor, int dividend ) */
// true means that divisor divides dividend
function xgc_Divides( divisor, dividend ) {
	return ( dividend % divisor ) == 0;
}

/* Array xgc_Factorize( int n ) */
// undefined means n > xgc_maxFactorizable
// gets the factors of n (without their multiplicities)
function xgc_Factorize( n ) {
	var s = new Array();
	if( n > xgc_maxFactorizable ) // return empty Stack
		return undefined;
	
	var limit = Math.floor( Math.sqrt( n ) );
	for( var i = 0; xgc_primes[i] <= limit; i++ ) {
		if( xgc_Divides( xgc_primes[i], n ) ) {
			s.push( xgc_primes[i] );
			do {
				n /= xgc_primes[i];
			}
			while( xgc_Divides( xgc_primes[i], n ) );
		}
	}
	if( n > limit ) // n is prime
		s.push( n );
	return s;
}


/* Array makeArray( int len, int value ) */
function makeArray( len, value ) {
	if( len <= 0 ) len = 1;  //minimum length
	var array = new Array( len );
	for( var i = 0; i < array.length; i++ ) array[i] = value;
	return array;
}

/* Array copyArray( Array source ) */
function copyArray( source ) {
	if( ! ( source instanceof Array ) ) return false;
	return source.concat();
}


/* constructor XGC_Array() */
function XGC_Array() {
	//////////////////////////////////////////////////////////////////////////
	// object for an array of numbers that can trasparently grow and shrink
	//
	// possible uses:
	// 0: new XGC_Array();
	//    allocate the array [0]
	// 1: new XGC_Array( number n );
	//    allocate an array with n elements, filled with 0s
	// 2: new XGC_Array( Array a );
	//    allocate a copy of a


	//////////////////////////////////////////////////////////////////////////
	// exported

	/* properties */
	this.values = makeArray( 1, 0 );  // the array

	/* methods */
	this.setLength = setLength;
	this.getAt = getAt;
	this.setAt = setAt;
	this.addHead = addHead;
	this.addTail = addTail;
	this.getChoice = getChoice;
	this.sumChoice = sumChoice;
	this.binSearch = binSearch;
	this.toString = toString;

	//////////////////////////////////////////////////////////////////////////
	// initialization

	switch( arguments.length ) {
	case 1:
		if( typeof( arguments[0] ) == "number" ) {
			len = arguments[0];
			this.values = makeArray( len, 0 );
		} else if( arguments[0] instanceof Array ) {
			array = arguments[0];
			this.values = copyArray( array );
		}
		break;
	}


	//////////////////////////////////////////////////////////////////////////
	// private

	/* number getAt( int index ) */
	// gets the value at index position
	// extends the array as needed (with 0s as initial values)
	function getAt( index ) {
		if (index > this.values.length) this.setLength( index );
		return this.values[index-1];
	}

	/* void setAt( int index, int value ) */
	// sets the value at index position
	// extends the array as needed (with 0s as initial values)
	function setAt( index, value ) {
		if (index > this.values.length) this.setLength( index );
		this.values[index-1] = value;
	}

	/* XGC_Array getChoice( XGC_Array selection ) */
	// gets these elements at selection positions as a new XGC_Array
	function getChoice( selection ) {
		var choice = new XGC_Array( selection.values.length );
		for( var i = 1; i <= selection.values.length; i++ ) {
			var value = this.getAt( selection.getAt( i ) );
			choice.setAt( i, value );
		}
		return choice;
	}

	/* int sumChoice( XGC_Array selection ) */
	// gets the sum of these elements at selection positions
	function sumChoice( selection ) {
		var sum = 0;
		for( var i = 1; i <= selection.values.length; i++ ) {
			sum += this.getAt( selection.getAt( i ) );
		}
		return sum;
	}

	/* void addHead( int value ) */
	// prepends value
	function addHead( value ) {
		this.values.unshift( value );
	}

	/* void addTail( int value ) */
	// appends value
	function addTail( value ) {
		this.values.push( value );
	}

	/* string toString() */
	// converts to string
	function toString() {
		return this.values instanceof Array ? "[" + this.values.join( ", " ) + "]" : this.values;
	}

	/* void setLength( int newLength ) */
	// grows or shrinks such that there are newLength elements
	// (with 0s as initial values when growing)
	function setLength( newLength ) {
		if( this.values.length < newLength ) {
			// grow
			this.values = this.values.concat( makeArray( newLength - this.values.length, 0 ) );
		} else {
			// shrink
			this.values = this.values.slice( 0, newLength );
		}
	}

	/* TaggedValue binSearch( int k ) */
	// { index, true } means that k is at index in values
	// { index, false } means that k is not in values, but
	//                  value at index is less than k, and
	//                  value at index + 1 is greater than k
	function binSearch( k ) {
		var low = 1;
		var high = this.values.length;
		var i = 0;
		while( high >= low ) {
			var mid = Math.floor( ( low + high ) / 2 );
			var n = this.getAt( mid );
			if( n == k ) {
				return {value: mid, tag: true};
			}
			else if( n > k ) {
				i = mid - 1;
				high = mid - 1;
			}
			else {
				i = mid;
				low = mid + 1;
			}
		}
		// to gain speed, we use an instant object like TaggedValue
		return {value: i, tag: false};
	}
}

/* constructor XGC_EuclidSet( int c, int m, int tMax ) */
function XGC_EuclidSet( c, m, tMax ) {
	//////////////////////////////////////////////////////////////////////////
	// object for the Euclid Set of numbers
	// see THE_WEBSITE_URL for more information

	//////////////////////////////////////////////////////////////////////////
	// exported

	/* properties */
	this.residue = c;
	this.modulus = m;
	this.terms = tMax;
	this.values = undefined;

	//////////////////////////////////////////////////////////////////////////
	// initialization

	if( ( 0 < c ) && ( c < m ) && xgc_IsPrimeTo( c, m ) && ( tMax > 0 ) ) {
		var temp = new Array();
		var aeTrue = 0;
		var aeFalse = 1;
		var coprime = new XGC_Array( tMax );
		for( var t = 1; t <= tMax; t++ ) {
			if( coprime.getAt( t ) == aeTrue ) {
				var n = c + m * t;
				temp.push( n );
				var factorList = xgc_Factorize( n );
				while( factorList.length > 0 ) {
					var nextFactor = parseInt( factorList.pop(), 10 );
					for( var i = nextFactor + t; i <= tMax; i += nextFactor ) {
						coprime.setAt( i, aeFalse );
					}
				}
			}
		}
		this.values = new XGC_Array( temp );
	}
}

/* constructor TaggedValue() */
function TaggedValue() {
	//////////////////////////////////////////////////////////////////////////
	// object for a value paired to a boolean
	//
	// possible uses:
	// 0: new TaggedValue();                      { null, false }
	// 1: new TaggedValue( boolean );             { null, boolean }
	// 2: new TaggedValue( notBoolean );          { notBoolean, false } *
	// 3: new TaggedValue( whatever, boolean );   { whatever, boolean } *
	//
	// *: if notBoolean (or whatever) is an Array or an XGC_Array
	//    then a new copy of it is put in TaggedValue.value
	//    else javascript rules apply

	
	//////////////////////////////////////////////////////////////////////////
	// exported

	/* properties */
	this.value = null;
	this.tag = false;

	//////////////////////////////////////////////////////////////////////////
	// initialization

	switch( arguments.length ) {
	case 1:
		if( typeof( arguments[0] ) == "boolean" ) {
			this.tag = arguments[0];
		} else {
			this.value = properValue( arguments[0] );
		}
		break;
	case 2:
		this.value = properValue( arguments[0] );
		this.tag = arguments[1];
		break;
	}

	//////////////////////////////////////////////////////////////////////////
	// private

	/* any properValue( any what ) */
	function properValue( what ) {
		if( what instanceof XGC_Array ) {
			return new XGC_Array( what.values );
		} else if( what instanceof Array ) {
			return copyArray( what );
		} else {
			return what;
		}
	}
}

/* constructor XGC_Partition( XGC_EuclidSet euclidSet ) */
function XGC_Partition( euclidSet ) {
	//////////////////////////////////////////////////////////////////////////
	// object for the partition generator for the euclidSet
	// see THE_WEBSITE_URL for more information

	//////////////////////////////////////////////////////////////////////////
	// exported

	/* properties */
	this.euclidSet = euclidSet;     //reference to object XGC_EuclidSet
	this.pXGC = new XGC_Array();
	
	/* methods */
	this.get = get;
	this.fastPart = fastPart;
	this.slowPart = slowPart;

	//////////////////////////////////////////////////////////////////////////
	// private

	/* type get( int n ) */
	// type = false means n is not valid
	// type = undefined means a partition for n doesn't exist in euclidSet
	// type = XGC_Array means n = sum( XGC_Array )
	function get( n ) {
		var source = this.euclidSet.values;
		var sourceLen = source.values.length;
		var sourceMin = source.getAt( 1 );
		var sourceMax = source.getAt( sourceLen );

		if( xgc_Divides( this.euclidSet.modulus, n ) 
		&& ( n >= this.euclidSet.modulus * sourceMin ) 
		&& ( n <= this.euclidSet.modulus * sourceMax ) )
			if( this.fastPart( n ) || this.slowPart( n ) )
				return source.getChoice( this.pXGC );
			else return undefined;
		else return false;
	}

	/* boolean fastPart( int n ) */
	function fastPart( n ) {
		var source = this.euclidSet.values;
		var sourceLen = source.values.length;
		var sourceMin = source.getAt( 1 );
		var sourceMax = source.getAt( sourceLen );

		var lastAddendum = n;
		var found;
		var oneLess = this.euclidSet.modulus - 1;
		this.pXGC = new XGC_Array( oneLess );
		for( var i = oneLess; i > 0; i-- ) {
			  found = source.binSearch( lastAddendum - i * sourceMin );
			  this.pXGC.setAt( i, found.value );
			  lastAddendum -= source.getAt( found.value );
		}
		found = source.binSearch( lastAddendum );
		if( found.tag ) this.pXGC.addHead( found.value );
		return found.tag;
	}

	/* TaggedValue prevV( XGC_Array p ) */
	function prevV( p ) {
		var pp = new TaggedValue( p );
		if( pp.value.getAt( 1 ) > 1 ) {
			pp.value.setAt( 1, pp.value.getAt( 1 ) - 1 );
			pp.tag = true;
			return pp;
		}
		else return prevH( p );
	}

	/* TaggedValue prevH( XGC_Array p ) */
	function prevH( p ) {
		var pp = new TaggedValue( p );
		var i;
		var length = p.values.length;
		if( length > 1 ) {
			for( i = 2; i <= length && p.getAt( i ) == 1; i++ );
			if( i <= length ) {
				pp.value.setAt( i, pp.value.getAt( i ) - 1 );
				for( var j = 1; j < i; j++ )
					pp.value.setAt( j, pp.value.getAt( i ) );
				pp.tag = true;
				return pp;
			}
			else return pp;
		}
		else return pp;
	  }

	/* TaggedValue nextV( XGC_Array p, int max ) */
	function nextV( p, max ) {
		var pp = new TaggedValue( p );
		var length = p.values.length;
		if( length > 1 && p.getAt( 1 ) < p.getAt( 2 ) 
		|| length == 1 && p.getAt( 1 ) < max ) {
			pp.value.setAt( 1, pp.value.getAt( 1 ) + 1 );
			pp.tag = true;
			return pp;
		}
		else return nextH( p, max );
	}

	/* TaggedValue nextH( XGC_Array p, int max ) */
	function nextH( p, max ) {
		var pp = new TaggedValue( p );
		var i;
		var length = p.values.length;
		if( length > 1 ) {
			for( i = 2; i < length && p.getAt( i + 1 ) == p.getAt( i ); i++ );
			if( p.getAt( i ) < max ) {
				pp.value.setAt( i, pp.value.getAt( i ) + 1 );
				for( var j = 1; j < i; j++ )
					pp.value.setAt( j, 1 );
				pp.tag = true;
				return pp;
			}
			else return pp;
		}
		else return pp;
	}

	/* boolean slowPart( int n ) */
	function slowPart( n ) {
		var source = this.euclidSet.values;
		var sourceLen = source.values.length;
		var sourceMin = source.getAt( 1 );
		var sourceMax = source.getAt( sourceLen );

		var lastAddendum = 0;
		var found;

		var pDownward = prevV( this.pXGC );
		var okDownward = pDownward.tag;

		var pUpward = nextV( this.pXGC, sourceLen );
		var okUpward = pUpward.tag;

		while( okDownward || okUpward ) {

			if( okDownward ) {
				lastAddendum = n - source.sumChoice( pDownward.value );
				found = source.binSearch( lastAddendum );
				if( found.tag ) {
					this.pXGC = pDownward.value;
					this.pXGC.addHead( found.value );
					return true;
				}
				if( lastAddendum > sourceMax ) {
					pDownward = prevH( pDownward.value );
				} else {
					pDownward = prevV( pDownward.value );
				}
				okDownward = pDownward.tag;
			}

			if( okUpward ) {
				lastAddendum = n - source.sumChoice( pUpward.value );
				found = source.binSearch( lastAddendum );
				if( found.tag ) {
					this.pXGC = pUpward.value;
					this.pXGC.addHead( found.value );
					return true;
				}
				if( lastAddendum < sourceMin ) {
					pUpward = nextH( pUpward.value, sourceLen );
				} else {
					pUpward = nextV( pUpward.value, sourceLen );
				}
				okUpward = pUpward.tag;
			}

		}
		return false;
	}

}