F. MEX counting
time limit per test
4 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

For an array $c$ of nonnegative integers, $MEX(c)$ denotes the smallest nonnegative integer that doesn't appear in it. For example, $MEX([0, 1, 3]) = 2$, $MEX([42]) = 0$.

You are given integers $n, k$, and an array $[b_1, b_2, \ldots, b_n]$.

Find the number of arrays $[a_1, a_2, \ldots, a_n]$, for which the following conditions hold:

• $0 \le a_i \le n$ for each $i$ for each $i$ from $1$ to $n$.

• $|MEX([a_1, a_2, \ldots, a_i]) - b_i| \le k$ for each $i$ from $1$ to $n$.

As this number can be very big, output it modulo $998\,244\,353$.

Input

The first line of the input contains two integers $n, k$ ($1 \le n \le 2000$, $0 \le k \le 50$).

The second line of the input contains $n$ integers $b_1, b_2, \ldots, b_n$ ($-k \le b_i \le n+k$) — elements of the array $b$.

Output

Output a single integer — the number of arrays which satisfy the conditions from the statement, modulo $998\,244\,353$.

Examples
Input
4 0
0 0 0 0

Output
256

Input
4 1
0 0 0 0

Output
431

Input
4 1
0 0 1 1

Output
509

Input
5 2
0 0 2 2 0

Output
6546

Input
3 2
-2 0 4

Output
11