Problem: For each of the K^{NM} ways to write an integer between 1 and K (inclusive) in every square in a square grid with N rows and M columns, find the value defined below, then compute the sum of all those K^{NM} values, modulo D.

* For each of the NM squares, find the minimum among the N+M-1 integers written in the square's row or the square's column. The value defined for the grid is the product of all these NM values.

Constraints

* 1 \leq N,M,K \leq 100
* 10^8 \leq D \leq 10^9
* N,M,K, and D are integers.
* D is prime.

Input

Input is given from Standard Input in the following format:


N M K D


Output

Print the sum of the K^{NM} values, modulo D.

Examples

Input

2 2 2 998244353


Output

35


Input

2 3 4 998244353


Output

127090


Input

31 41 59 998244353


Output

827794103

Solution (in CPP):

Solution:
/*
                                                                                                     
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                                               ``````                                               
*/

#ifdef aimbot
#pragma comment(linker, "/stack:200000000")
#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
#pragma GCC optimize("unroll-loops")
#endif

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <map>
#include <set>
#include <queue>
#include <ostream>
#include <istream>
#include <typeinfo>
#include <iomanip>
#include <cstdio>
#include <cstdlib>
#include <cassert>
#include <limits>
#include <fstream>
#include <array>
#include <list>
#include <bitset>
#include <functional>
#include <random>
#include <cstring>
#include <chrono>

#define random escape__from__random__aetuhoetnuhshe
#define mt make_tuple
#define x first
#define y second
#define pb push_back
#define ppb pop_back
#define mp make_pair
#define umap unordered_map
#define uset unordered_set
#define elif else if
#define len(v) ((int)v.size())
#define f(i, n) for (int i = 0; i < (n); i++)
#define rof(i, n) for (int i = ((n) - 1); i >= 0; i--)
#define apply(v, act) for (auto &x : v) { act; }
#define log(args...) {string s = #args;deque<string> deq;\
string buf = "";int bal = 0;for (char c : s) {\
if (c == '(' || c == '[' || c == '{') {bal++;\
} else if (c == ')' || c == ']' || c == '}') {\
bal--;} else {if (bal == 0) {if (c == ',') {\
deq.pb(buf);buf = "";} else {if (c != ' ') {\
buf += c;}}}}}if (!buf.empty()) {deq.pb(buf);}\
smart_io::precall_print();smart_io::_print(deq, args);}

inline int min(const int &x, const int &y) { return (((y-x)>>(32-1))&(x^y))^x; }
inline int max(const int &x, const int &y) { return (((y-x)>>(32-1))&(x^y))^y; }
inline long long min(const long long &x, const long long &y) { return (((y-x)>>(64-1))&(x^y))^x; }
inline long long max(const long long &x, const long long &y) { return (((y-x)>>(64-1))&(x^y))^y; }

#define print    \
smart_io::precall_print(); \
cout,

#define scan cin,

#ifdef fast_allocator
const int MAXMEM = 200 * 1000 * 1024;
char _memory[MAXMEM];
size_t _ptr = 0;
void* operator new(size_t _x) { _ptr += _x; assert(_ptr < MAXMEM); return _memory + _ptr - _x; }
void operator delete (void*) noexcept {}
#endif

using namespace std;

char string_in_buffer[(int)260];


void fast_scan(int &x) { scanf("%d", &x); }
void fast_scan(long long &x) { scanf("%lld", &x); }
void fast_scan(unsigned long long &x) { scanf("%llu", &x); }
void fast_scan(double &x) { scanf("%lf", &x); }
void fast_scan(long double &x) { scanf("%Lf", &x); }
void fast_scan(char &x) {
	scanf("%c", &x);
	if (x == '\n') {
		fast_scan(x);
	}
}
void fast_scan(string &x) {
	scanf("%s", string_in_buffer);
	x = string(string_in_buffer);
}

template<class TFirst, class TSecond>
void fast_scan(pair<TFirst, TSecond> &p) {
	fast_scan(p.first);
	fast_scan(p.second);
}

template <class T>
void fast_scan(vector<T> &v) {
	for (auto &x : v) fast_scan(x);
}

void fast_print(const int &x) { printf("%d", x); }
void fast_print(const unsigned int &x) { printf("%u", x); }
void fast_print(const long long &x) { printf("%lld", x); }
void fast_print(const unsigned long long &x) { printf("%llu", x); }
void fast_print(const char &x) { printf("%c", x); };
// void fast_print(__int128 x) {
// 	if (x == 0) { fast_print('0'); return; }
// 	if (x < 0) {
// 		fast_print('-');
// 		x = -x;
// 	}
// 	__int128 p = 1;
// 	while (x / (p * 10)) p *= 10;
// 	while (p) {
// 		__int128 symb = x / p;
// 		fast_print((int)symb);
// 		x -= p * symb;
// 		p /= 10;
// 	}
// };
void fast_print(const double &x) { printf("%.15lf", x); }
void fast_print(const long double &x) { printf("%.15Lf", x); }
void fast_print(const string &x) { printf("%s", x.c_str());}
void fast_print(const char v[]) { fast_print((string)v); }

template<class TFirst, class TSecond>
void fast_print(const pair<TFirst, TSecond> &p) {
	fast_print(p.first);
	fast_print(' ');
	fast_print(p.second);
}

template <class T>
void fast_print(const vector<T> &v) {
	if (v.empty()) return;
	fast_print(v[0]);
	for (int i = 1; i < v.size(); i++) {
		fast_print(' ');
		fast_print(v[i]);
	}
}

template <class T>
void fast_print(const vector<vector<T>> &v) {
	if (v.empty()) return;
	fast_print(v[0]);
	for (int i = 1; i < v.size(); i++) {
		fast_print('\n');
		fast_print(v[i]);
	}
}

template <class T>
void fast_print(const T &v) {
	for (const auto &x : v) {
		fast_print(x);
		fast_print(' ');
	}
}


using namespace std;


namespace smart_io {
	string print_start = "";
	string sep = " ";
	bool first_print = false;
	
	void precall_print() {
		fast_print(print_start);
		print_start = "\n";
		first_print = true;
	}
	
	void _print(deque<string>) {}
	template<class T, class... Args>
	void _print(deque<string> names, T elem, Args... args) {
		if (!first_print) {
			fast_print("\n");
		} else {
			first_print = false;
		}
		fast_print(names.front());
		fast_print(" = ");
		fast_print(elem);
		names.pop_front();
		_print(names, args...);
	}
} //namespace smart_io


template <class T>
ostream &operator,(ostream &os, const T &object) {
	if (!smart_io::first_print) {
		fast_print(smart_io::sep);
	} else {
		smart_io::first_print = false;
	}
	fast_print(object);
	return os;
}

template <class T>
istream &operator,(istream &is, T &object) {
	fast_scan(object);
	return is;
}

namespace random {
	using namespace std::chrono;
	mt19937 rng(duration_cast< milliseconds >(
											  system_clock::now().time_since_epoch()
											  ).count());
	uniform_real_distribution<> prob_dist(0.0, 1.0);
};
#define int long long
namespace typedefs {
	typedef long long ll;
	typedef unsigned long long ull;
	typedef pair<int, int> pii;
	typedef long double ld;
}

namespace numbers_operation {
	template<class T>
	inline T floor_mod(T a, const T &b) {
		a %= b;	
		if (a < 0) a += b;
		return a;
	}
}

using namespace numbers_operation;
using namespace typedefs;
using namespace random;

ll MOD;

int n, m, k;

void addS(ll &a, ll b) {
	a += b;
	if (a >= MOD) a -= MOD;
}

ll add(ll a, ll b) {
	addS(a, b);
	return a;
}

void subS(ll &a, ll b) {
	a -= b;
	if (a < 0) a += MOD;
}

ll sub(ll a, ll b) {
	subS(a, b);
	return a;
}

ll mul(ll a, ll b) {
	return (a * b) % MOD;
}

ll fast_pow(ll x, ll p) {
	ll r = 1;
	while (p) {
		if (p & 1)
			r = mul(r, x);
		x = mul(x, x);
		p /= 2;
	}
	return r;
}

ll mem_binom[250][250];
ll binom(int n, int k) {
	if (k > n) return 0LL;
	if (k == 0) return 1;
	if (mem_binom[n][k] != -1) return mem_binom[n][k];
	return mem_binom[n][k] = add(binom(n - 1, k - 1), binom(n - 1, k));
}

ll mem_func[102][102][102];
ll func(int col, int x, int t) {
	if (mem_func[col][x][t] != -1) return mem_func[col][x][t];
	return mem_func[col][x][t] = fast_pow(mul(
					sub(fast_pow(x, m - col), fast_pow(x - 1, m - col)),
						fast_pow(k - x + 1, col)
					)
				, t);
}

ll mem_funcA[102][102][102];
ll funcA(int row, int x, int t) {
	if (mem_funcA[row][x][t] != -1) return mem_funcA[row][x][t];
	return mem_funcA[row][x][t] = fast_pow(
					mul(sub(fast_pow(k - x + 1, row), fast_pow(k - x, row)),
						fast_pow(x, n - row)
					)
				, t);
}

ll mem[102][102][102][2];
ll dp(int row, int col, int x, bool phase) {
	if (x == k + 1) {
		return row == n && col == m;
	}
	if (mem[row][col][x][phase] != -1) return mem[row][col][x][phase];
	ll rez = 0;
	if (phase == 0) {
		for (int t = 0; t <= n - row; t++) 
			addS(rez, mul(dp(row + t, col, x, 1), mul(binom(n - row, t),
				func(col, x, t))));
	} else {
		for (int t = 0; t <= m - col; t++)
			addS(rez, mul(dp(row, col + t, x + 1, 0), mul(binom(m - col, t),
				funcA(row, x, t))));
	}
	return mem[row][col][x][phase] = rez;
}

signed main(signed argc, char *argv[]) {
	memset(mem, -1, sizeof mem);
	memset(mem_binom, -1, sizeof mem_binom);
	memset(mem_func, -1, sizeof mem_func);
	memset(mem_funcA, -1, sizeof mem_funcA);
	scan n, m, k, MOD;
	print dp(0, 0, 1, 0);
}