CF1103E Radix sum

Let's define radix sum of number $ a $ consisting of digits $ a_1, \ldots ,a_k $ and number $ b $ consisting of digits $ b_1, \ldots ,b_k $ (we add leading zeroes to the shorter number to match longer length) as number $ s(a,b) $ consisting of digits $ (a_1+b_1)\mod 10, \ldots ,(a_k+b_k)\mod 10 $ . The radix sum of several integers is defined as follows: $ s(t_1, \ldots ,t_n)=s(t_1,s(t_2, \ldots ,t_n)) $ You are given an array $ x_1, \ldots ,x_n $ . The task is to compute for each integer $ i (0 \le i < n) $ number of ways to consequently choose one of the integers from the array $ n $ times, so that the radix sum of these integers is equal to $ i $ . Calculate these values modulo $ 2^{58} $ .

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In the first example there exist sequences: sequence $ (5,5) $ with radix sum $ 0 $ , sequence $ (5,6) $ with radix sum $ 1 $ , sequence $ (6,5) $ with radix sum $ 1 $ , sequence $ (6,6) $ with radix sum $ 2 $ .