There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$.

Colors are represented by integers between $1$ and $N$, inclusive.

Consider repeating the following operation until all the colors of the balls become the same.

-   There are $2^N$ subsets (including the empty set) of the set consisting of the $N$ balls; choose one of the subsets uniformly at random. Let $X_1,X_2,...,X_K$ be the indices of the chosen balls. Next, choose a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ uniformly at random. Let $P=(P_1,P_2,\dots,P_K)$ be the chosen permutation. For each integer $i$ such that $1 \le i \le K$, change the color of Ball $X_i$ to $P_i$.

Find the expected value of number of operations, modulo $998244353$.

Here a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ is a sequence of $K$ integers between $1$ and $N$, inclusive, whose elements are pairwise distinct.

Input is given from Standard Input in the following format:

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

Print the answer.

-   $2 \le N \le 2000$
-   $1 \le A_i \le N$
-   All values in input are integers.

4

The operation is repeated until a subset of size 1 is chosen and the color of the ball is changed to the color of the ball not contained in the subset. The probability is $\frac{2}{4}×\frac{1}{2}=\frac{1}{4}$, so the expected value is 4.

0