201464: [AtCoder]ARC146 E - Simple Speed

Problem Statement

You are given a sequence of $N$ positive integers: $A=(A_1,A_2,\dots,A_N)$.

How many integer sequences $B$ consisting of integers between $1$ and $N$ (inclusive) satisfy all of the following conditions? Print the count modulo $998244353$.

• For each integer $i$ such that $1 \le i \le N$, there are exactly $A_i$ occurrences of $i$ in $B$.
• For each integer $i$ such that $1 \le i \le |B|-1$, it holds that $|B_i - B_{i+1}|=1$.

Constraints

• $1 \le N \le 2 \times 10^5$
• $1 \le A_i \le 2 \times 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$
$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer.

Sample Input 1

3
2 3 1

Sample Output 1

6

$B$ can be the following six sequences.

• $(1,2,1,2,3,2)$
• $(1,2,3,2,1,2)$
• $(2,1,2,1,2,3)$
• $(2,1,2,3,2,1)$
• $(2,3,2,1,2,1)$
• $(3,2,1,2,1,2)$

Thus, the answer is $6$.

Sample Input 2

1
200000

Sample Output 2

0

There may be no sequence that satisfies the conditions.

Sample Input 3

6
12100 31602 41387 41498 31863 12250

Sample Output 3

750337372