P12028 [USACO25OPEN] Moo Decomposition G

You have a long string $S$ of Ms and Os and an integer $K \geq 1$. Count the number of ways of ways to decompose $S$ into subsequences such that each subsequence is MOOOO....O with exactly $K$ Os, modulo $10^9+7$. Since the string is very long, you are not given it explicitly. Instead, you are given an integer $L$ ($1 \leq L \leq 10^{18}$), and a string $T$ of length $N$ ($1 \leq N \leq 10^6$). The string $S$ is the concatenation of $L$ copies of the string $T$.

The first line contains $K$, $N$, and $L$. The second line contains the string $T$ of length $N$. Every character is either an M or an O. It is guaranteed that the number of decompositions of $S$ is nonzero.

Output the number of decompositions of string $S$, modulo $10^9+7$.

##### For Sample 1: The only way to decompose $S$ into MOOs is to let the first three characters form a MOO and the last three characters form another MOO. ##### For Sample 2: There are six distinct ways to decompose the string into subsequences (uppercase letters form one MOO, lowercase letters form another): - MmOOoo - MmOoOo - MmOooO - MmoOOo - MmoOoO - MmooOO ##### For Sample 4: Make sure to take the answer modulo $10^9+7$. #### SCORING: - Inputs 5-7: $K=1$, $L = 1$ - Inputs 8-10: $K=2$, $N\leq 1000$, $L = 1$ - Inputs 11-13: $K=1$ - Inputs 14-19: $L = 1$ - Inputs 20-25: No additional constraints.