Try all characters from 'a' to 'z', and assume that we want to get a string that consists only of this character. What greedy strategy should we follow?

Suppose that there are C colors in total. Then, each a_i must be either C or C - 1, and it means that the difference between the maximum and the minimum is at most one. Do some case-analysis using this fact.

Suppose that H is not a multiple of h. Put a positive number in a cell if its row-number (0-based) is a multiple of h, otherwise put a negative number. Can you choose the "positive number" and the "negative number" properly?

Create a new cell that keeps the xor of all elements. Each operation can be considered as a swap between this cell and another element in the sequence. Reduce it to a graph problem.

Can you get YES/NO for a given pair of vertices (or more generally, a given set of vertices) in O(M)? Restate it in terms of graphs, then find an O(NM + N³) solution.

The task is about grundy numbers. Can you get O(bell_number(N)) solution by trying all possible grundy-number sequences? Can you reduce it to O(3^N) by DP?