Problem C: Nice Prefixes

Consider strings formed from characters from an alphabet of size K. For example, if K = 4, our alphabet might be {a,b,c,d}, and an example string is bbcac.

For a string S, define count(S, k) to be the number of occurrences of the symbol k in S. For example, count(bbcac, b) = 2 and count(bbcac, a) = 1.

A prefix of a string S is any string obtained from S by deleting some (possibly none) of the trailing characters of S. For example, the prefixes of acb are the empty string, a, ac, and acb.

A string S has "nice prefixes" if for every prefix P of S and for every two characters k1 and k2 in the alphabet, |count(P, k1) - count(P, k2)| <= 2. For example, bbcac has nice prefixes, but abbbc does not because count(abbb, b) = 3 and count(abbb, c) = 0.

Count the number of strings of length L on an alphabet of size K that have nice prefixes. This number can be large, so print its remainder when divided by 1000000007.

Input Specification

The input is a single line containing the two integers L and K, separated by spaces, with 1 <= L <= 10¹⁸ and 1 <= K <= 50.

Sample Input

4 2

Output Specification

Output a single line containing the number of strings of length L on an alphabet of size K that have nice prefixes, modulo 1000000007.

Output for Sample Input

Malcolm Sharpe, Ondřej Lhoták

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