Josh is given an array ~a~ of length ~N~, where every element is in the range ~[1, N]~ inclusive. However, some elements are missing and are denoted with the value ~0~.

Josh, being the innovator he is, decides to randomly assign the missing elements so that the resulting array is a permutation of the first ~N~ positive integers. Your task is to find the expected number of inversions after Josh has finished filling in missing values!

An inversion is defined to be a pair of indices ~(i, j)~ such that ~i < j~ and ~a_i > a_j~.

Note: It is guaranteed that there will be at least ~1~ way Josh can fill in the missing elements to result in a permutation of the first ~N~ positive integers.

Constraints

~1 \le N \le 5 \cdot 10^5~

Subtask 1 [30%]

~1 \le N \le 2 \cdot 10^3~

Subtask 2 [70%]

No additional constraints.

Input Specification

The first line will contain ~N~, the length of Josh's array.

The next line will contain ~N~ space-separated integers in the range ~[0, N]~, with a value of ~0~ at position ~i~ meaning that the ~i^\text{th}~ element is missing.

Output Specification

If the answer can be expressed in the form ~\frac{P}{Q}~, where ~P~ and ~Q~ are relatively prime, output ~PQ^{-1} \pmod{10^9+7}~, where ~Q^{-1}~ is the modular inverse of ~Q~. It is guaranteed that the answer is rational.

Sample Input 1

5
4 3 0 2 1

Sample Output 1

8

Explanation for Sample Output 1

There is only ~1~ way Josh can assign values to missing elements.

The resulting array is ~[4, 3, 5, 2, 1]~ and there are ~8~ inversions.

Sample Input 2

2
0 0

Sample Output 2

500000004

Explanation for Sample Output 2

There are ~2~ valid arrays, ~[1, 2]~ and ~[2, 1]~.

The expected number of inversions is ~\frac{1}{2!} + \frac{0}{2!} = \frac{1}{2}~. The answer is ~1 \cdot 2^{-1}~, the modular inverse of ~2~ is ~500\,000\,004~, so the answer is ~1 \cdot 500\,000\,004 = 500\,000\,004~.

Sample Input 3

6
5 0 2 0 1 0

Sample Output 3

500000013

Explanation for Sample Output 3

There are ~6~ valid arrays, ~[5, 3, 2, 4, 1, 6]~, ~[5, 3, 2, 6, 1, 4]~, ~[5, 4, 2, 3, 1, 6]~, ~[5, 4, 2, 6, 1, 3]~, ~[5, 6, 2, 3, 1, 4]~, ~[5, 6, 2, 4, 1, 3]~.

The expected number of inversions is ~\frac{8}{3!} + \frac{9}{3!} + \frac{9}{3!} + \frac{10}{3!} + \frac{10}{3!} + \frac{11}{3!} = \frac{57}{6} = \frac{19}{2}~. The answer is ~19 \cdot 2^{-1}~, the modular inverse of ~2~ is ~500\,000\,004~, so the answer is ~(19 \cdot 500\,000\,004) \bmod (10^9+7) = 500\,000\,013~.