#include<bitset>
#include<stdio.h>
#include<iostream>
#include<vector>
#include<algorithm>
#include<string>
#include<string.h>

#ifdef LOCAL
#define eprintf(...) fprintf(stderr, __VA_ARGS__)
#else
#define NDEBUG
#define eprintf(...) do {} while (0)
#endif
#include<cassert>

using namespace std;

typedef long long LL;
typedef vector<int> VI;

#define REP(i,n) for(int i=0, i##_len=(n); i<i##_len; ++i)
#define EACH(i,c) for(__typeof((c).begin()) i=(c).begin(),i##_end=(c).end();i!=i##_end;++i)

template<class T> inline void amin(T &x, const T &y) { if (y<x) x=y; }
template<class T> inline void amax(T &x, const T &y) { if (x<y) x=y; }
template<class Iter> void rprintf(const char *fmt, Iter begin, Iter end) {
    for (bool sp=0; begin!=end; ++begin) { if (sp) putchar(' '); else sp = true; printf(fmt, *begin); }
    putchar('\n');
}
template<unsigned MOD_> struct ModInt {
    static const unsigned MOD = MOD_;
    unsigned x;
    void undef() { x = (unsigned)-1; }
    bool isnan() const { return x == (unsigned)-1; }
    inline int geti() const { return (int)x; }
    ModInt() { x = 0; }
    ModInt(const ModInt &y) { x = y.x; }
    ModInt(int y) { if (y<0 || (int)MOD<=y) y %= (int)MOD; if (y<0) y += MOD; x=y; }
    ModInt(unsigned y) { if (MOD<=y) x = y % MOD; else x = y; }
    ModInt(long long y) { if (y<0 || MOD<=y) y %= MOD; if (y<0) y += MOD; x=y; }
    ModInt(unsigned long long y) { if (MOD<=y) x = y % MOD; else x = y; }
    ModInt &operator+=(const ModInt y) { if ((x += y.x) >= MOD) x -= MOD; return *this; }
    ModInt &operator-=(const ModInt y) { if ((x -= y.x) & (1u<<31)) x += MOD; return *this; }
    ModInt &operator*=(const ModInt y) { x = (unsigned long long)x * y.x % MOD; return *this; }
    ModInt &operator/=(const ModInt y) { x = (unsigned long long)x * y.inv().x % MOD; return *this; }
    ModInt operator-() const { return (x ? MOD-x: 0); }

    ModInt inv() const { return pow(MOD-2); }
    ModInt pow(long long y) const {
	ModInt b = *this, r = 1;
	if (y < 0) { b = b.inv(); y = -y; }
	for (; y; y>>=1) {
	    if (y&1) r *= b;
	    b *= b;
	}
	return r;
    }

    ModInt extgcd() const {
	unsigned a = MOD, b = x; int u = 0, v = 1;
	while (b) {
	    int t = a / b;
	    a -= t * b; swap(a, b);
	    u -= t * v; swap(u, v);
	}
	if (u < 0) u += MOD;
	return ModInt(u);
    }

    friend ModInt operator+(ModInt x, const ModInt y) { return x += y; }
    friend ModInt operator-(ModInt x, const ModInt y) { return x -= y; }
    friend ModInt operator*(ModInt x, const ModInt y) { return x *= y; }
    friend ModInt operator/(ModInt x, const ModInt y) { return x *= y.inv(); }
    friend bool operator<(const ModInt x, const ModInt y) { return x.x < y.x; }
    friend bool operator==(const ModInt x, const ModInt y) { return x.x == y.x; }
    friend bool operator!=(const ModInt x, const ModInt y) { return x.x != y.x; }
};

const LL MOD = 1000000007;
//const LL MOD = 998244353;
typedef ModInt<MOD> Mint;

const int SIZE = 10800;
using Bits = bitset<SIZE>;

int H, W;
char S[211][211];

Bits B[SIZE];
void add(Bits b) {
    REP (i, SIZE) if (b.test(i)) {
	if (B[i].test(i)) {
	    b ^= B[i];
	} else {
	    B[i] = b;
	    return;
	}
    }
}

void MAIN() {
    scanf("%d%d", &H, &W);
    REP (i, H) scanf("%s", S[i]);

    Mint ans = 1;
    if (H % 2) {
	string g = S[H/2];
	reverse(g.begin(), g.end());
	if (S[H/2] != g) ans *= 2;
	for (int i=H/2; i<H; i++) REP (j, W) S[i][j] = S[i+1][j];
	H--;
    }
    if (W % 2) {
	string g;
	REP (i, H) g.push_back(S[i][W/2]);
	string h = g;
	reverse(g.begin(), g.end());
	if (g != h) ans *= 2;
	REP (i, H) for (int j=W/2; j<W; j++) S[i][j] = S[i][j+1];
	W--;
    }

    REP (i, H/2) REP (j, W/2) {
	int row[] = {i, H-1-i};
	int col[] = {j, W-1-j};
	string g;
	REP (a, 2) REP (b, 2) g.push_back(S[row[a]][col[b]]);
	sort(g.begin(), g.end());
	if (g[0] != g[1] && g[1] != g[2] && g[2] != g[3]) { // 1 1 1 1;
	    ans *= 12;
	} else {
	    if (g[0] == g[3]) { // 4;
		// nope;
	    } else if (g[0] == g[2] || g[1] == g[3]) { // 1 3;
		ans *= 4;
	    } else if (g[0] == g[1] && g[2] == g[3]) { // 2 2;
		ans *= 6;
	    } else { // 1 1 2;
		ans *= 12;
	    }

	    REP (a, 2) REP (b, 2) S[row[a]][col[b]] = '.';
	}
    }

    REP (i, H/2) {
	Bits b;
	REP (j, W/2) if (S[i][j] != '.') b.set(i*(W/2)+j);
	add(b);
    }
    REP (j, W/2) {
	Bits b;
	REP (i, H/2) if (S[i][j] != '.') b.set(i*(W/2)+j);
	add(b);
    }
    REP (i, SIZE) if (B[i].test(i)) ans += ans;
    printf("%d\n", ans.geti());
}

int main() {
    int TC = 1;
//    scanf("%d", &TC);
    REP (tc, TC) MAIN();
    return 0;
}


### Prompt

Construct a CPP code solution to the problem outlined:

Snuke has a grid with H rows and W columns. The square at the i-th row and j-th column contains a character S_{i,j}.

He can perform the following two kinds of operation on the grid:

* Row-reverse: Reverse the order of the squares in a selected row.
* Column-reverse: Reverse the order of the squares in a selected column.



For example, reversing the 2-nd row followed by reversing the 4-th column will result as follows:

<image>

By performing these operations any number of times in any order, how many placements of characters on the grid can be obtained?

Constraints

* 1≦H,W≦200
* S_{i,j} is a lowercase English letter (`a`-`z`).

Input

The input is given from Standard Input in the following format:


H W
S_{1,1}S_{1,2}...S_{1,W}
S_{2,1}S_{2,2}...S_{2,W}
:
S_{H,1}S_{H,2}...S_{H,W}


Output

Print the number of placements of characters on the grid that can be obtained, modulo 1000000007 (=10^9+7).

Examples

Input

2 2
cf
cf


Output

6


Input

1 12
codefestival


Output

2

### Response

```cpp
#include<bitset>
#include<stdio.h>
#include<iostream>
#include<vector>
#include<algorithm>
#include<string>
#include<string.h>

#ifdef LOCAL
#define eprintf(...) fprintf(stderr, __VA_ARGS__)
#else
#define NDEBUG
#define eprintf(...) do {} while (0)
#endif
#include<cassert>

using namespace std;

typedef long long LL;
typedef vector<int> VI;

#define REP(i,n) for(int i=0, i##_len=(n); i<i##_len; ++i)
#define EACH(i,c) for(__typeof((c).begin()) i=(c).begin(),i##_end=(c).end();i!=i##_end;++i)

template<class T> inline void amin(T &x, const T &y) { if (y<x) x=y; }
template<class T> inline void amax(T &x, const T &y) { if (x<y) x=y; }
template<class Iter> void rprintf(const char *fmt, Iter begin, Iter end) {
    for (bool sp=0; begin!=end; ++begin) { if (sp) putchar(' '); else sp = true; printf(fmt, *begin); }
    putchar('\n');
}
template<unsigned MOD_> struct ModInt {
    static const unsigned MOD = MOD_;
    unsigned x;
    void undef() { x = (unsigned)-1; }
    bool isnan() const { return x == (unsigned)-1; }
    inline int geti() const { return (int)x; }
    ModInt() { x = 0; }
    ModInt(const ModInt &y) { x = y.x; }
    ModInt(int y) { if (y<0 || (int)MOD<=y) y %= (int)MOD; if (y<0) y += MOD; x=y; }
    ModInt(unsigned y) { if (MOD<=y) x = y % MOD; else x = y; }
    ModInt(long long y) { if (y<0 || MOD<=y) y %= MOD; if (y<0) y += MOD; x=y; }
    ModInt(unsigned long long y) { if (MOD<=y) x = y % MOD; else x = y; }
    ModInt &operator+=(const ModInt y) { if ((x += y.x) >= MOD) x -= MOD; return *this; }
    ModInt &operator-=(const ModInt y) { if ((x -= y.x) & (1u<<31)) x += MOD; return *this; }
    ModInt &operator*=(const ModInt y) { x = (unsigned long long)x * y.x % MOD; return *this; }
    ModInt &operator/=(const ModInt y) { x = (unsigned long long)x * y.inv().x % MOD; return *this; }
    ModInt operator-() const { return (x ? MOD-x: 0); }

    ModInt inv() const { return pow(MOD-2); }
    ModInt pow(long long y) const {
	ModInt b = *this, r = 1;
	if (y < 0) { b = b.inv(); y = -y; }
	for (; y; y>>=1) {
	    if (y&1) r *= b;
	    b *= b;
	}
	return r;
    }

    ModInt extgcd() const {
	unsigned a = MOD, b = x; int u = 0, v = 1;
	while (b) {
	    int t = a / b;
	    a -= t * b; swap(a, b);
	    u -= t * v; swap(u, v);
	}
	if (u < 0) u += MOD;
	return ModInt(u);
    }

    friend ModInt operator+(ModInt x, const ModInt y) { return x += y; }
    friend ModInt operator-(ModInt x, const ModInt y) { return x -= y; }
    friend ModInt operator*(ModInt x, const ModInt y) { return x *= y; }
    friend ModInt operator/(ModInt x, const ModInt y) { return x *= y.inv(); }
    friend bool operator<(const ModInt x, const ModInt y) { return x.x < y.x; }
    friend bool operator==(const ModInt x, const ModInt y) { return x.x == y.x; }
    friend bool operator!=(const ModInt x, const ModInt y) { return x.x != y.x; }
};

const LL MOD = 1000000007;
//const LL MOD = 998244353;
typedef ModInt<MOD> Mint;

const int SIZE = 10800;
using Bits = bitset<SIZE>;

int H, W;
char S[211][211];

Bits B[SIZE];
void add(Bits b) {
    REP (i, SIZE) if (b.test(i)) {
	if (B[i].test(i)) {
	    b ^= B[i];
	} else {
	    B[i] = b;
	    return;
	}
    }
}

void MAIN() {
    scanf("%d%d", &H, &W);
    REP (i, H) scanf("%s", S[i]);

    Mint ans = 1;
    if (H % 2) {
	string g = S[H/2];
	reverse(g.begin(), g.end());
	if (S[H/2] != g) ans *= 2;
	for (int i=H/2; i<H; i++) REP (j, W) S[i][j] = S[i+1][j];
	H--;
    }
    if (W % 2) {
	string g;
	REP (i, H) g.push_back(S[i][W/2]);
	string h = g;
	reverse(g.begin(), g.end());
	if (g != h) ans *= 2;
	REP (i, H) for (int j=W/2; j<W; j++) S[i][j] = S[i][j+1];
	W--;
    }

    REP (i, H/2) REP (j, W/2) {
	int row[] = {i, H-1-i};
	int col[] = {j, W-1-j};
	string g;
	REP (a, 2) REP (b, 2) g.push_back(S[row[a]][col[b]]);
	sort(g.begin(), g.end());
	if (g[0] != g[1] && g[1] != g[2] && g[2] != g[3]) { // 1 1 1 1;
	    ans *= 12;
	} else {
	    if (g[0] == g[3]) { // 4;
		// nope;
	    } else if (g[0] == g[2] || g[1] == g[3]) { // 1 3;
		ans *= 4;
	    } else if (g[0] == g[1] && g[2] == g[3]) { // 2 2;
		ans *= 6;
	    } else { // 1 1 2;
		ans *= 12;
	    }

	    REP (a, 2) REP (b, 2) S[row[a]][col[b]] = '.';
	}
    }

    REP (i, H/2) {
	Bits b;
	REP (j, W/2) if (S[i][j] != '.') b.set(i*(W/2)+j);
	add(b);
    }
    REP (j, W/2) {
	Bits b;
	REP (i, H/2) if (S[i][j] != '.') b.set(i*(W/2)+j);
	add(b);
    }
    REP (i, SIZE) if (B[i].test(i)) ans += ans;
    printf("%d\n", ans.geti());
}

int main() {
    int TC = 1;
//    scanf("%d", &TC);
    REP (tc, TC) MAIN();
    return 0;
}


```