Given is the function $f(a,n)=\lfloor (\lceil \sqrt{a}\rceil +\sqrt{a}{)}^n\rfloor$.
$\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.
$f(5,2)=27$ and $f(5,5)=3935$.

$G(n)={\displaystyle \sum _{a=1}^{n}f(a,a^2).}$
$G(1000)mod999999937=163861845.$
Find $G(5000000).$ Give your answer modulo $999999937$.