CF1641E 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.

Find the expected value of this sum.

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 < 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 $ ).