Problem:
Score : 1100 points

Problem Statement

You are given integers N, K, and an integer sequence A of length M.

An integer sequence where each element is between 1 and K (inclusive) is said to be colorful when there exists a contiguous subsequence of length K of the sequence that contains one occurrence of each integer between 1 and K (inclusive).

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then find the sum of all the counts.
Here, the answer can be extremely large, so find the sum modulo 10^9+7.

Constraints

1 \leq N \leq 25000

1 \leq K \leq 400

1 \leq M \leq N

1 \leq A_i \leq K

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K M
A_1 A_2 ... A_M

Output

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then print the sum of all the counts modulo 10^9+7.

Sample Input 1

3 2 1
1

Sample Output 1

9

There are six colorful sequences of length 3: (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2) and (2,2,1).
The numbers of the contiguous subsequences of these sequences that coincide with A=(1) are 2, 2, 1, 2, 1 and 1, respectively.
Thus, the answer is their sum, 9.

Sample Input 2

4 2 2
1 2

Sample Output 2

12

Sample Input 3

7 4 5
1 2 3 1 2

Sample Output 3

17

Sample Input 4

5 4 3
1 1 1

Sample Output 4

0

Sample Input 5

10 3 5
1 1 2 3 3

Sample Output 5

1458

Sample Input 6

25000 400 4
3 7 31 127

Sample Output 6

923966268

Sample Input 7

9954 310 12
267 193 278 294 6 63 86 166 157 193 168 43

Sample Output 7

979180369

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

/*
cat <<EOF >mistaken-paste
*/

#pragma GCC diagnostic ignored "-Wincompatible-pointer-types"
#define _USE_MATH_DEFINES

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <math.h>
#include <time.h>

#define BIG 2000000007
#define VERYBIG 2000000000000007LL

#define MOD 1000000007
#define FOD  998244353
typedef uint64_t ull;
typedef  int64_t sll;

#define N_MAX 1000000
#define M_MAX 200000

#ifdef __cplusplus
#include <queue>
#include <stack>
#include <tuple>
#include <set>
#include <map>
#include <string>

// using namespace std; // HELL
using std::queue;
using std::priority_queue;
using std::stack;
using std::tuple;
using std::set;
using std::map;
using std::vector;
using std::greater;
using std::pair;
using std::string;

#endif

typedef struct {
	int32_t a;
	int32_t b;
} hw;

typedef struct {
	sll a;
	sll b;
} hwll;

typedef struct {
	sll a;
	sll b;
	sll c;
} hwllc;

typedef struct {
	hwll a;
	hwll b;
} linell;

ull n, m;
ull h, w;
ull k;
ull q;
ull vua, vub, vuc, vud, vue, vuf;
sll vsa, vsb, vsc, vsd, vse, vsf;
long double vra, vrb, vrc;
double vda, vdb, vdc;
char ch, dh;

ull umin (ull x, ull y) {
	return (x < y) ? x : y;
}

ull umax (ull x, ull y) {
	return (x > y) ? x : y;
}

sll smin (sll x, sll y) {
	return (x < y) ? x : y;
}

sll smax (sll x, sll y) {
	return (x > y) ? x : y;
}

ull gcd (ull x, ull y) {
	if (x < y) {
		return gcd(y, x);
	} else if (y == 0) {
		return x;
	} else {
		return gcd(y, x % y);
	}
}

ull bitpow (ull a, ull x, ull modulo) {
	ull result = 1;
	while (x) {
		if (x & 1) {
			result *= a;
			result %= modulo;
		}
		x /= 2;
		a = (a * a) % modulo;
	}
	return result;
}

ull divide (ull a, ull b, ull modulo) {
	return (a * bitpow(b, modulo - 2, modulo)) % modulo;
}

ull udiff (ull a, ull b) {
	if (a >= b) {
		return a - b;
	} else {
		return b - a;
	}
}

sll sdiff (sll a, sll b) {
	if (a >= b) {
		return a - b;
	} else {
		return b - a;
	}
}

int bitcount (ull n) {
	int result = 0;
	while (n) {
		if (n & 1) result++;
		n /= 2;
	}
	return result;
}

// double distance (sll x1, sll y1, sll x2, sll y2) {
// 	double xdist2, ydist2, origindist, dist;

// 	xdist2 = (x1 - x2) * (x1 - x2);
// 	ydist2 = (y1 - y2) * (y1 - y2);
// 	return sqrt(xdist2 + ydist2);
// }

int32_t pullcomp (const void *left, const void *right) {
	ull l = *(ull*)left;
	ull r = *(ull*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t psllcomp (const void *left, const void *right) {
	sll l = *(sll*)left;
	sll r = *(sll*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t pcharcomp (const void *left, const void *right) {
	char l = *(char*)left;
	char r = *(char*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t pstrcomp (const void *left, const void *right) {
	char* l = *(char**)left;
	char* r = *(char**)right;

	return strcmp(l, r);
}

int32_t phwllABcomp (const void *left, const void *right) {
	hwll l = *(hwll*)left;
	hwll r = *(hwll*)right;
	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	return 0;
}

int32_t phwllREVcomp (const void *left, const void *right) {
	hwll l = *(hwll*)left;
	hwll r = *(hwll*)right;
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	return 0;
}

int32_t ptriplecomp (const void *left, const void *right) {
	hwllc l = *(hwllc*)left;
	hwllc r = *(hwllc*)right;

	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	if (l.c < r.c) {
		return -1;
	}
	if (l.c > r.c) {
		return +1;
	}
	return 0;
}

bool isinrange (sll left, sll x, sll right) {
	return (left <= x && x <= right);
}

bool isinrange_soft (sll left, sll x, sll right) {
	return (left <= x && x <= right) || (left >= x && x >= right);
}

ull parent[N_MAX];
ull rank[N_MAX];
void uf_init (ull n) {
	for (sll i = 0; i < n; i++) {
		parent[i] = i;
		rank[i] = 0;
	}
}
ull uf_find (ull x) {
	if (parent[x] == x) return x;
	return parent[x] = uf_find(parent[x]);
}
sll uf_union (ull a, ull b) {
	a = uf_find(a);
	b = uf_find(b);
	if (a == b) return -1;

	if (rank[a] > rank[b]) {
		a ^= (b ^= (a ^= b));
	}

	parent[a] = b;
	if (rank[a] == rank[b]) {
		rank[b]++;
	}

	return b;
}


sll a[N_MAX];
// sll a[3001][3001];
sll b[N_MAX];
// sll b[3001][3001];
sll c[N_MAX];
char s[N_MAX + 1];
// char s[3010][3010];
// char t[N_MAX + 1];
// char t[3010][3010];
// hwll xy[N_MAX];

ull dp1[30000][500], dp2[30000][500];
ull ac1[30000][500], ac2[30000][500];

ull checker[30000];

ull colorful () {
	sll i, j;

	dp1[0][0] = 1;
	ac1[0][0] = 1;

	for (i = 0; i <= n; i++) {
		for (j = 1; j < k; j++) {
			if (i) {
				dp1[i][j] = MOD + ac1[i - 1][k - 1] - ac1[i - 1][j - 1] + (dp1[i - 1][j - 1] * (k - j + 1) % MOD);
				dp1[i][j] %= MOD;
				dp2[i][j] = MOD + ac2[i - 1][k - 1] - ac2[i - 1][j - 1] + (dp2[i - 1][j - 1] * (k - j + 1) % MOD);
				dp2[i][j] %= MOD;

				if (j >= m) {
					dp2[i][j] = (dp1[i][j] + dp2[i][j]) % MOD;
				}
			}

			ac1[i][j] = (ac1[i][j - 1] + dp1[i][j]) % MOD;
			ac2[i][j] = (ac2[i][j - 1] + dp2[i][j]) % MOD;
		}
	}

	ull result = ac2[n][k - 1];
	for (i = 0; i < m; i++) {
		result = divide(result, k - i, MOD);
	}

	return result;
}

ull blocked (ull bef, ull aft) {
	sll i, j;

	dp1[0][bef] = 1;
	dp2[0][aft] = 1;

	for (i = 0; i <= n; i++) {
		for (j = 1; j < k; j++) {
			if (i) {
				dp1[i][j] = MOD + ac1[i - 1][k - 1] - ac1[i - 1][j - 1] + (dp1[i - 1][j - 1] * (k - j + 1) % MOD);
				dp1[i][j] %= MOD;
				dp2[i][j] = MOD + ac2[i - 1][k - 1] - ac2[i - 1][j - 1] + (dp2[i - 1][j - 1] * (k - j + 1) % MOD);
				dp2[i][j] %= MOD;
			}

			ac1[i][j] = (ac1[i][j - 1] + dp1[i][j]) % MOD;
			ac2[i][j] = (ac2[i][j - 1] + dp2[i][j]) % MOD;
		}
	}

	ull result = 0;
	for (i = 0; i + m <= n; i++) {
		result += (ac1[i][k - 1] * ac2[n - m - i][k - 1] % MOD);
		result %= MOD;
	}
	return result;
}

bool acol () {
	sll i;
	for (i = 0; i < k; i++) checker[i] = ~0;
	ull nearest = 0;
	for (i = 0; i < m; i++) {
		if (~checker[a[i]] && nearest <= checker[a[i]]) {
			nearest = checker[a[i]] + 1;
		}
		checker[a[i]] = i;

		if (i - nearest + 1 >= k) return true;
	}

	return false;
}

ull solve () {
	sll i, j, ki, li;
	ull result = 0;
	// sll result = 0;
	double dresult = 0;
	// ull maybe = 0;
	sll maybe = 0;
	// ull sum = 0;
	sll sum = 0;
	sll item;
	ull *dpcell;

	ull bef, aft;

	for (i = 0; i < m; i++) {
		if (checker[a[i]] == 1) break;
		checker[a[i]] = 1;
	}
	bef = i;
	for (i = 0; i < m; i++) {
		if (checker[a[m - 1 - i]] == 2) break;
		checker[a[m - 1 - i]] = 2;
	}
	aft = i;

	if (acol()) {
		result = 0;
	} else if (bef == m) {
		result = colorful();
	} else {
		result = blocked(bef, aft);
	}

	result = (MOD + ((n - m + 1) * bitpow(k, n - m, MOD) % MOD) - result) % MOD;

	printf("%llu\n", result);
	// printf("%.15lf\n", dresult);
	// puts(s);

	return 0;

	success:
	// puts("YES");
	// puts("Yes");
	// printf("%llu\n", result);
	// puts("0");
	// puts("Takahashi");
	return 0;

	fail:
	// puts("NO");
	// puts("No");
	puts("0");
	// puts("-1");
	// puts("-1 -1 -1");
	// puts("Aoki");
	return 1;
}

int32_t main (void) {
	int32_t i, j;
	int32_t x, y;

	// scanf("%lf%lf", &vda, &vdb, &vdc);
	// scanf("%lld%lld", &vsa, &vsb, &vsc, &vsd);
	// scanf("%llu%llu", &vua, &vub, &vuc, &vud);
	// scanf("%*llu%*llu");
	// scanf("%llu%llu", &h, &w);
	scanf("%llu", &n, &m);
	scanf("%llu", &k, &n, &m);
	scanf("%llu", &m);
	// scanf("%llu%llu", &vua, &vub, &vuc, &vud, &vue, &vuf);
	// vua--;
	// vub--;
	// vuc--;
	// vud--;
	// scanf("%lld%lld", &vsa, &vsb, &vsc);
	// scanf("%s", s);
	// scanf("%s", t);
	// scanf("%llu", &k);
	// scanf("%llu", &q);
	for (i = 0; i < m; i++) {
		// scanf("%lld%lld", &xy[i].a, &xy[i].b);
		scanf("%lld", &a[i]);
		// scanf("%lld", &b[i]);
		// scanf("%lld", &c[i]);
		// scanf("%lld", &t[i]);
		a[i]--;
		// b[i]--;
		// c[i]--;
	}
	// scanf("%llu%llu", &m, &k);
	// for (i = 0; i < n; i++) {
	// 	scanf("%llu", &b[i]);
	// 	// b[i]--;
	// }
	// scanf("%llu", &q);

	// for (i = 0; i < h; i++) {
	// 	for (j = 0; j < w; j++) {
	// 		scanf("%llu", &a[i][j]);
	// 	}
	// }
	// for (i = 0; i < h; i++) {
	// 	scanf("%s", &s[i]);
	// }

	// scanf("%llu", &k);
	// scanf("%llu", &q);

	solve();

	return 0;
}