Problem:
Score : 900 points

Problem Statement

You are given a sequence of N integers: A_1,A_2,...,A_N.

Find the number of permutations p_1,p_2,...,p_N of 1,2,...,N that can be changed to A_1,A_2,...,A_N by performing the following operation some number of times (possibly zero), modulo 998244353:

For each 1\leq i\leq N, let q_i=min(p_{i-1},p_{i}), where p_0=p_N. Replace the sequence p with the sequence q.

Constraints

1 \leq N \leq 3 × 10^5

1 \leq A_i \leq N

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1
:
A_N

Output

Print the number of the sequences that satisfy the condition, modulo 998244353.

Sample Input 1

3
1
2
1

Sample Output 1

2

(2,3,1) and (3,2,1) satisfy the condition.

Sample Input 2

5
3
1
4
1
5

Sample Output 2

0

Sample Input 3

8
4
4
4
1
1
1
2
2

Sample Output 3

24

Sample Input 4

6
1
1
6
2
2
2

Sample Output 4

0

Solution in C (GCC 5.4.1) (judge verdict: Wrong Answer)

#if 0
cat <<EOF >mistaken-paste
#endif
// thanks for @rsk0315_h4x

#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

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

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;

template <typename T, typename U>
pair<T, U> operator+(pair<T, U> l, pair<T, U> r) {
	return pair<T, U>(l.first + r.first, l.second + r.second);
}

#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;
sll va, vb, vc, vd, ve, vf;
ull ua, ub, uc, ud, ue, uf;
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);
// }

#define BEGDEFCMP(NAME) int32_t NAME (const void *left, const void *right)
#define DEFLR(TYPE) TYPE l=*(TYPE*)left,r=*(TYPE*)right;
#define CMPRET(L, R) if((L)<(R))return-1;if((L)>(R))return+1;

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 prevcomp (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 pdoublecomp (const void *left, const void *right) {
	double l = *(double*)left;
	double r = *(double*)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;
}

int32_t ptripleREVcomp (const void *left, const void *right) {
	hwllc l = *(hwllc*)left;
	hwllc r = *(hwllc*)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;
	}
	if (l.c < r.c) {
		return -1;
	}
	if (l.c > r.c) {
		return +1;
	}
	return 0;
}

int32_t pquadcomp (const void *left, const void *right) {
	linell l = *(linell*)left;
	linell r = *(linell*)right;

	sll ac = phwllABcomp(&(l.a), &(r.a));
	if (ac) return ac;
	sll bc = phwllABcomp(&(l.b), &(r.b));
	if (bc) return bc;

	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);
}

sll a[N_MAX + 5];
// ull a[N_MAX];
// sll a[3001][3001];
sll b[N_MAX + 5];
// sll b[3001][3001];
sll c[N_MAX + 5];
sll d[N_MAX + 5];
// sll e[N_MAX];
char s[N_MAX + 1];
// char s[3010][3010];
char t[N_MAX + 1];
// char t[3010][3010];
hwll xy[N_MAX];
hwllc tup[N_MAX];
sll table[3000][3000];
// here we go

hwll apress[N_MAX];
ull len = 0;

sll pos[N_MAX];
sll rev[N_MAX];
sll candi[N_MAX];
sll width[N_MAX];
sll prob[N_MAX];
sll pwid[N_MAX];

bool setpos (ull v, ull x) {
	if (pos[v] >= 0 && pos[v] == x) return true;
	if (pos[v] >= 0 && pos[v] != x) return false;
	if (rev[x] >= 0 && rev[x] != v) return false;

	pos[v] = x;
	rev[x] = v;
	// printf("%lld is in %lld\n", x, v);
	return true;
}

hwll slick[N_MAX];
sll beg = 0, end = 0;
void spush (sll x, sll c) {
	while (beg < end && slick[end - 1].a <= x) {
		end--;
	}
	slick[end] = (hwll){x, c};
	end++;
}
sll swhat () {
	return slick[beg].a;
}
void spop (sll c) {
	if (slick[beg].b == c) beg++;
}
void slider (sll base, sll d) {
	sll i;

	base = (base + d - 1) % n;

	for (i = d; i >= 1; i--) {
		spush(a[(n + base - i) % n], -i);
	}
	for (i = 0; i < n; i++) {
		spush(a[(base + i) % n], i);
		spop(i - d);
		prob[i] = swhat();
		// printf("<%lld>\n", prob[i]);
	}
}

ull frac[N_MAX];

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;

	frac[0] = 1;
	for (i = 1; i <= n; i++) {
		frac[i] = (frac[i - 1] * i) % FOD;
	}

	ull d;
	ull base;
	for (i = 0; i < n; i++) {
		if (a[i ? (i - 1) : (n - 1)] > 0 && a[i] == 0) {
			base = i;
			break;
		}
	}
	if (i == n) {
		for (i = 0; i < n; i++) {
			if (a[i] > 0) goto fail;
		}

		result = frac[n];
		goto success;
	}

	i = 0;
	while (i < n) {
		j = i;
		while (a[(base + i) % n] == a[(base + j) % n]) i++;

		apress[len] = (hwll){a[(base + j) % n], i - j};
		// printf("%lldx%lld\n", a[(base + j) % n], i - j);
		len++;
	}
	d = apress[0].b;

	for (i = 0; i < n; i++) {
		pos[i] = rev[i] = -1;
	}
	pos[0] = 0;
	rev[0] = 0;
	sum = 0;
	for (i = 0; i < len; i++) {
		if (apress[i].b > d) goto fail;

		sum += apress[i].b;
		if (apress[i].a < apress[(i + 1) % len].a) {
			if (!setpos((n + sum - d) % n, apress[i].a)) goto fail;
		} else {
			if (!setpos(sum % n, apress[(i + 1) % len].a)) goto fail;
		}

		if (apress[i ? (i - 1) : (len - 1)].a < apress[i].a
			&& apress[i].a > apress[(i + 1) % len].a) {
			
			rev[apress[i].a] = BIG;
			// printf("%lld is in [%lld %lld]\n", apress[i].a, (n + sum - d) % n, (sum - apress[i].b) % n);
			j = (n + sum - d) % n;
			while (true) {
				candi[j] = apress[i].a;
				if (j == (sum - apress[i].b) % n) break;
				j = (j + 1) % n;
			}
		}

	}



	slider(base, d);

	for (i = 0; i < n; i++) {
		if (pos[i] < 0 && candi[i] >= 0) {
			width[candi[i]]++;
		}

		pwid[prob[i]]++;
	}
	result = 1;
	for (i = 0; i < n; i++) {
		// printf("pwid[%lld: %lld]\n", i, pwid[i]);
		if (rev[i] != BIG) continue;
		// printf("width[%lld]: %lld\n", i, width[i]);
		result = (result * width[i]) % FOD;
	}

	sum = 0;
	for (i = 0; i < n; i++) {
		// if (rev[i] >= 0 && rev[i] != BIG) continue;
		sum += pwid[i];
		// printf("%lld: %lld...\n", i, sum);
		if (!sum) goto fail;

		if (rev[i] < 0) result = (result * sum) % FOD;
			// printf("! x%llu\n", sum);
		sum--;
	}

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

	return 0;

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

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

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



	// scanf("%llu%llu", &h, &w);
	scanf("%llu", &n, &m);
	// scanf("%llu", &k, &n, &m);
	// scanf("%llu%llu", &h, &w);
	// scanf("%llu", &q);
	// scanf("%s", s);
	// scanf("%lld%lld", &va, &vb, &vc, &vd);
	// scanf("%llu%llu", &ua, &ub, &uc, &ud);
	// scanf("%s", t);
	for (i = 0; i < n; i++) {
		// scanf("%lld%lld", &xy[i].a, &xy[i].b);
		// scanf("%lld%lld%lld", &tup[i].a, &tup[i].b, &tup[i].c);
		scanf("%lld", &a[i]);
		// scanf("%lld", &b[i]);
		// scanf("%lld", &c[i]);
		// scanf("%lld", &d[i]);
		a[i]--;
		// b[i]--;
		// c[i]--;
		// d[i]--;
		// tup[i].a--;
		// tup[i].b--;
	}
	// scanf("%llu", &m, &k);
	// scanf("%llu", &q);
	// scanf("%s", s);
	// for (i = 0; i < n; i++) {
	// 	scanf("%lld", &b[i]);
	// 	// b[i]--;
	// }
	// scanf("%llu", &q);

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

	solve();

	return 0;
}