Jellyfish always uses OEIS to solve math problems, but now she finds a problem that cannot be solved by OEIS:

Count the number of permutations $p$ of $[1, 2, \dots, n]$ such that for all $(l, r)$ such that $l \leq r \leq m_l$, the subarray $[p_l, p_{l+1}, \dots, p_r]$ is not a permutation of $[l, l+1, \dots, r]$.

Since the answer may be large, you only need to find the answer modulo $10^9+7$.

Input

The first line of the input contains a single integer $n$ ($1 \leq n \leq 200$) — the length of the permutation.

The second line of the input contains $n$ integers $m_1, m_2, \dots, m_n$ ($0 \leq m_i \leq n$).

Output

Output the number of different permutations that satisfy the conditions, modulo $10^9+7$.

Note

In the first example, $[2, 3, 1]$ and $[3, 1, 2]$ satisfies the condition.