A14948. Special Positions

You are given an array $a$ of length $n$ . Also you are given $m$ distinct positions $p_1, p_2, \ldots, p_m$ ( $1 \leq p_i \leq n$ ).

A non-empty subset of these positions $T$ is randomly selected with equal probability and the following value is calculated: $$$$\sum_{i=1}^{n} (a_i \cdot \min_{j \in T} \left|i - j\right|). $$ In other word, for each index of the array, $a\_i$ and the distance to the closest chosen position are multiplied, and then these values are summed up.</p><p>Find the expected value of this sum.</p><p>This value must be found modulo $998\\,244\\,353$ . More formally, let $M = 998\\,244\\,353$ . It can be shown that the answer can be represented as an irreducible fraction $\\frac{p}{q}$ , where $p$ and $q$ are integers and $q \\neq 0$ (mod $M$ ). Output the integer equal to $p \\cdot q^{-1}$ (mod $M$ ). In other words, output such integer $x$ that $0 \\leq x &lt; M$ and $x \\cdot q = p$ (mod $M$$$).

The first line contains two integers $n$ and $m$ ( $1 \leq m \leq n \leq 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0 \leq a_i < 998\,244\,353$ ).

The third line contains $m$ distinct integers $p_1, p_2, \ldots, p_m$ ( $1 \leq p_i \le n$ ).

For every $1 \leq i < m$ it is guaranteed that $p_i < p_{i+1}$ .

Print a single integer — the answer to the problem.

In the first test:

- If only $1$ is choosen, than the value equals to $1 \cdot 0 + 2 \cdot 1 + 3 \cdot 2 + 4 \cdot 3 = 20$ .
- If only $4$ is choosen, than the value equals to $1 \cdot 3 + 2 \cdot 2 + 3 \cdot 1 + 4 \cdot 0 = 10$ .
- If both positions are chosen, than the value equals to $1 \cdot 0 + 2 \cdot 1 + 3 \cdot 1 + 4 \cdot 0 = 5$ .

The answer to the problem is $\frac{20 + 10 + 5}{3} = \frac{35}{3} = 665\,496\,247$ (modulo $998\,244\,353$ ).