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I. Stairs

time limit per test

10 secondsmemory limit per test

1024 megabytesinput

standard inputoutput

standard outputFor a permutation $$$p$$$ of numbers $$$1$$$ through $$$n$$$, we define a stair array $$$a$$$ as follows: $$$a_i$$$ is length of the longest segment of permutation which contains position $$$i$$$ and is made of consecutive values in sorted order: $$$[x, x+1, \ldots, y-1, y]$$$ or $$$[y, y-1, \ldots, x+1, x]$$$ for some $$$x \leq y$$$. For example, for permutation $$$p = [4, 1, 2, 3, 7, 6, 5]$$$ we have $$$a = [1, 3, 3, 3, 3, 3, 3]$$$.

You are given the stair array $$$a$$$. Your task is to calculate the number of permutations which have stair array equal to $$$a$$$. Since the number can be big, compute it modulo $$$998\,244\,353$$$. Note that this number can be equal to zero.

Input

The first line of input contains integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the length of a stair array $$$a$$$.

The second line of input contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$).

Output

Print the number of permutations which have stair array equal to $$$a$$$. Since the number can be big, compute it modulo $$$998\,244\,353$$$.

Examples

Input

6 3 3 3 1 1 1

Output

6

Input

7 4 4 4 4 3 3 3

Output

6

Input

1 1

Output

1

Input

8 2 2 2 2 2 2 1 1

Output

370

Input

4 3 2 3 1

Output

0

Codeforces (c) Copyright 2010-2021 Mike Mirzayanov

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