E2. Chiori and Doll Picking (hard version)time limit per test3 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputThis is the hard version of the problem. The only difference between easy and hard versions is the constraint of m. You can make hacks only if both versions are solved.Chiori loves dolls and now she is going to decorate her bedroom! As a doll collector, Chiori has got n dolls. The i-th doll has a non-negative integer value a_i (a_i < 2^m, m is given). Chiori wants to pick some (maybe zero) dolls for the decoration, so there are 2^n different picking ways.Let x be the bitwise-xor-sum of values of dolls Chiori picks (in case Chiori picks no dolls x = 0). The value of this picking way is equal to the number of 1-bits in the binary representation of x. More formally, it is also equal to the number of indices 0 \leq i < m, such that \left\lfloor \frac{x}{2^i} \right\rfloor is odd.Tell her the number of picking ways with value i for each integer i from 0 to m. Due to the answers can be very huge, print them by modulo 998\,244\,353.InputThe first line contains two integers n and m (1 \le n \le 2 \cdot 10^5, 0 \le m \le 53)  — the number of dolls and the maximum value of the picking way.The second line contains n integers a_1, a_2, \ldots, a_n (0 \le a_i < 2^m)  — the values of dolls.OutputPrint m+1 integers p_0, p_1, \ldots, p_m  — p_i is equal to the number of picking ways with value i by modulo 998\,244\,353.ExamplesInput
4 4
3 5 8 14
Output
2 2 6 6 0 Input
6 7
11 45 14 9 19 81
Output
1 2 11 20 15 10 5 0 
4 4
3 5 8 14

2 2 6 6 0 
['bitmasks', 'brute force', 'combinatorics', 'math', '*3500']