We define two sequences $S=\{S(1),S(2),...,S(n)\}$ and $S_2=\{S_2(1),S_2(2),...,S_2(n)\}$:

$S(k)=(p_k{)}^{k}mod10007$ where $p_k$ is the $k$th prime number.

$S_2(k)=S(k)+S(\lfloor \frac{k}{10000}\rfloor +1)$ where $\lfloor \cdot \rfloor$ denotes the floor function.

Then let $M(i,j)$ be the median of elements $S_2(i)$ through $S_2(j)$, inclusive. For example, $M(1,10)=2021.5$ and $M({10}^2,{10}^3)=4715.0$.

Let $F(n,k)=\sum _{i=1}^{n-k+1}M(i,i+k-1)$. For example, $F(100,10)=463628.5$ and $F({10}^5,{10}^4)=675348207.5$.

Find $F({10}^7,{10}^5)$. If the sum is not an integer, use $.5$ to denote a half. Otherwise, use $.0$ instead.