Problem: You are working at a production plant of biological weapons. You are a maintainer of a terrible virus weapon with very high reproductive power. The virus has a tendency to build up regular hexagonal colonies. So as a whole, the virus weapon forms a hexagonal grid, each hexagon being a colony of the virus. The grid itself is in the regular hexagonal form with N colonies on each edge.

The virus self-propagates at a constant speed. Self-propagation is performed simultaneously at all colonies. When it is done, for each colony, the same number of viruses are born at every neighboring colony. Note that, after the self-propagation, if the number of viruses in one colony is more than or equal to the limit density M, then the viruses in the colony start self-attacking, and the number reduces modulo M.

Your task is to calculate the total number of viruses after L periods, given the size N of the hexagonal grid and the initial number of viruses in each of the colonies.

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Input

The input consists of multiple test cases.

Each case begins with a line containing three integers N (1 ≤ N ≤ 6), M (2 ≤ M ≤ 109 ), and L (1 ≤ L ≤ 109 ). The following 2N - 1 lines are the description of the initial state. Each non-negative integer (smaller than M) indicates the initial number of viruses in the colony. The first line contains the number of viruses in the N colonies on the topmost row from left to right, and the second line contains those of N + 1 colonies in the next row, and so on.

The end of the input is indicated by a line “0 0 0”.

Output

For each test case, output the test case number followed by the total number of viruses in all colonies after L periods.

Example

Input

3 3 1
1 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 1
3 3 2
1 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 1
0 0 0


Output

Case 1: 8
Case 2: 18

Solution (in CPP):

Solution:
#include "bits/stdc++.h"
#include<unordered_map>
#include<unordered_set>
#pragma warning(disable:4996)
using namespace std;



 int mod = 1000000007;
struct Mod {
public:
	int num;
	Mod() : Mod(0) { ; }
	Mod(long long int n) : num((n % mod + mod) % mod) {
		}
	Mod(int n) : Mod(static_cast<long long int>(n)) { ; }
	operator int() { return num; }
};

Mod operator+(const Mod a, const Mod b) { return Mod((a.num + b.num) % mod); }
Mod operator+(const long long int a, const Mod b) { return Mod(a) + b; }
Mod operator+(const Mod a, const long long int  b) { return b + a; }
Mod operator++(Mod &a) { return a + Mod(1); }
Mod operator-(const Mod a, const Mod b) { return Mod((mod + a.num - b.num) % mod); }
Mod operator-(const long long int a, const Mod b) { return Mod(a) - b; }
Mod operator--(Mod &a) { return a - Mod(1); }
Mod operator*(const Mod a, const Mod b) { return Mod(((long long)a.num * b.num) % mod); }
Mod operator*(const long long int a, const Mod b) { return Mod(a)*b; }
Mod operator*(const Mod a, const long long int b) { return Mod(b)*a; }
Mod operator*(const Mod a, const int b) { return Mod(b)*a; }
Mod operator+=(Mod &a, const Mod b) { return a = a + b; }
Mod operator+=(long long int &a, const Mod b) { return a = a + b; }
Mod operator-=(Mod &a, const Mod b) { return a = a - b; }
Mod operator-=(long long int &a, const Mod b) { return a = a - b; }
Mod operator*=(Mod &a, const Mod b) { return a = a * b; }
Mod operator*=(long long int &a, const Mod b) { return a = a * b; }
Mod operator*=(Mod& a, const long long int &b) { return a = a * b; }
Mod operator^(const Mod a, const int n) {
	if (n == 0) return Mod(1);
	Mod res = (a * a) ^ (n / 2);
	if (n % 2) res = res * a;
	return res;
}
Mod mod_pow(const Mod a, const long long  int n) {
	if (n == 0) return Mod(1);
	Mod res = mod_pow((a * a), (n / 2));
	if (n % 2) res = res * a;
	return res;
}
Mod inv(const Mod a) { return a ^ (mod - 2); }
Mod operator/(const Mod a, const Mod b) {
	assert(b.num != 0);
	return a * inv(b);
}
Mod operator/(const long long int a, const Mod b) {
	return Mod(a) / b;
}
Mod operator/=(Mod &a, const Mod b) {
	return a = a / b;
}

#define MAX_MOD_N 1024000

Mod fact[MAX_MOD_N], factinv[MAX_MOD_N];
void init(const int amax = MAX_MOD_N) {
	fact[0] = Mod(1); factinv[0] = 1;
	for (int i = 0; i < amax - 1; ++i) {
		fact[i + 1] = fact[i] * Mod(i + 1);
		factinv[i + 1] = factinv[i] / Mod(i + 1);
	}
}
Mod comb(const int a, const int b) {
	return fact[a] * factinv[b] * factinv[a - b];
}

vector<pair<int, int>>get_connect(int N, int x, int y) {
	vector<pair<int,int>>anss;
	{
		int ax=x-1;
		int ay=y;
		anss.emplace_back(ax,ay);
	}
	{
		int ax=x+1;
		int ay=y;
		anss.emplace_back(ax, ay);

	}

	//lu
	{
		int ax,ay;
		if (y < N) {
			ax=x-1;
			ay=y-1;
		}
		else {
			ax=x;
			ay=y-1;
		}
		anss.emplace_back(ax, ay);

	}

	//ru
	{
		int ax,ay;
		if (y < N) {
			ax=x;
			ay=y-1;
		}
		else {
			ax=x+1;
			ay=y-1;
		}
		anss.emplace_back(ax, ay);

	}

	//ld
	{
		int ax,ay;
		if (y < N - 1) {
			ax=x;
			ay=y+1;
		}
		else {
			ax=x-1;
			ay=y+1;
		}
		anss.emplace_back(ax, ay);

	}
	//rd
	{
		int ax,ay;
		if (y < N - 1) {
			ax=x+1;
			ay=y+1;
		}
		else {
			ax=x;
			ay=y+1;
		}
		anss.emplace_back(ax, ay);

	}
	vector<pair<int,int>>real_anss;
	for (auto ans : anss) {
		int ax(ans.first);
		int ay(ans.second);
		bool ok=true;
		if(ay<0||ay>=2*N-1)ok=false;
		if (ay < N) {
			if(ax<0||ax>=N+ay)ok=false;
		}
		else {
			if (ax < 0 || ax>=3*N-ay-2)ok=false;
		}
		if(ok)real_anss.emplace_back(ans);
	}
	return real_anss;
}


template<typename T>
vector<vector<T>> keisann(const vector<vector<T>>l, const vector<vector<T>>r) {
	vector<vector<T>>ans(l.size(), vector<T>(r[0].size()));
	assert(l[0].size() == r.size());
	for (unsigned int h = 0; h < l.size(); ++h) {
		for (unsigned int i = 0; i < r.size(); ++i) {
			for (unsigned int w = 0; w < r[0].size(); ++w) {

				ans[h][w] += l[h][i] * r[i][w];
			}
		}
	}
	return ans;
}

template<typename T>
vector<vector<T>>powgyou(vector<vector<T>>a, const long long int n) {
	assert(a.size() == a[0].size());
	if (!n) {
		vector<vector<T>>e(a.size(), vector<T>(a[0].size()));
		for (unsigned int i = 0; i < a.size(); ++i) {
			e[i][i] = 1;
		}
		return e;
	}
	if (n == 1)return a;
	else {
		vector<vector<T>>ans(a.size(), vector<T>(a[0].size(), 0));
		ans = powgyou(a, n / 2);
		ans = keisann(ans, ans);
		if (n % 2) {
			ans = keisann(ans, a);
		}
		return ans;
	}
}


int main() {
	int case_id=0;
	while (true) {
		case_id++;
		int N, M, L; cin >> N >> M >> L;
		mod=M;
		if(!N)break;
		vector<vector<int>>v(2 * N - 1);
		int num = 0;
		map<pair<int, int>, int>mp;
		for (int i = 0; i < v.size(); ++i) {
			int asize = N + i;
			if (i>=N)asize = N + (v.size() - i - 1);
			if (i <= N)v[i].resize(asize);
			else v[i].resize(asize);
			num += asize;
		}
		vector<vector<Mod>>mat(num,vector<Mod>(num));

		{
			int id = 0;
			for (int i = 0; i < v.size(); ++i) {
				for (int j = 0; j < v[i].size(); ++j) {
					mp[make_pair(i, j)] = id++;
				}
			}
		}

		for (int i = 0; i < v.size(); ++i) {
			for (int j = 0; j < v[i].size(); ++j) {
				int now_id = mp[make_pair(i, j)];
				mat[now_id][now_id]=true;
				auto connect(get_connect(N, j, i));
				for (auto con : connect) {
					int next_id = mp[make_pair(con.second, con.first)];
					mat[now_id][next_id] = true;
				}
			}
		}

		vector<vector<Mod>>start(1, vector<Mod >(num));
		{
			int id = 0;
			for (int i = 0; i < v.size(); ++i) {
				for (int j = 0; j < v[i].size(); ++j) {
					int k; cin >> k;
					start[0][id++] = k;
				}
			}
		}
		vector<vector<Mod>>ans_mat = keisann(start, powgyou(mat, L));
		long long int ans = 0;
		for (int i = 0; i < ans_mat.size(); ++i) {
			for (int j = 0; j < ans_mat[i].size(); ++j) {
				ans += ans_mat[i][j].num;
			}
		}
		cout <<"Case "<<case_id<<": "<< ans << endl;
	}
	
	return 0;
}