NOTE: This is a more difficult version of Problem 114.

A row measuring $n$ units in length has red blocks with a minimum length of $m$ units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.

Let the fill-count function, $F(m,n)$, represent the number of ways that a row can be filled.

For example, $F(3,29)=673135$ and $F(3,30)=1089155$.

That is, for $m=3$, it can be seen that $n=30$ is the smallest value for which the fill-count function first exceeds one million.

In the same way, for $m=10$, it can be verified that $F(10,56)=880711$ and $F(10,57)=1148904$, so $n=57$ is the least value for which the fill-count function first exceeds one million.

For $m=50$, find the least value of $n$ for which the fill-count function first exceeds one million.