CF1147D Palindrome XOR

You are given a string $ s $ consisting of characters "1", "0", and "?". The first character of $ s $ is guaranteed to be "1". Let $ m $ be the number of characters in $ s $ . Count the number of ways we can choose a pair of integers $ a, b $ that satisfies the following: - $ 1 \leq a < b < 2^m $ - When written without leading zeros, the base-2 representations of $ a $ and $ b $ are both palindromes. - The base-2 representation of bitwise XOR of $ a $ and $ b $ matches the pattern $ s $ . We say that $ t $ matches $ s $ if the lengths of $ t $ and $ s $ are the same and for every $ i $ , the $ i $ -th character of $ t $ is equal to the $ i $ -th character of $ s $ , or the $ i $ -th character of $ s $ is "?". Compute this count modulo $ 998244353 $ .

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For the first example, the pairs in base-2 are $ (111, 10001), (11, 10101), (1001, 11111) $ .