Generalized Pisano Period 1

Let $F_k(n)$ be a linear recurrence sequence such that $$\begin{array}{rl}{F}_k(n)& =1(1\le n\le k),\\ F_k(n)& =\sum _{i=1}^k{F}_k(n-i)(n>k).\end{array}$$

For example, $F_2(1)=1$, $F_2(2)=1$ and $F_2(3)=2$

Let $P(k,m)$ be the (minimum) period of $F_k(n)$ modulo $m$.

You are given that $P(2,10)=60$ and $P(2,{10}^9)=1500000000$.

Find $P(10,{10}^9)$.