Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into $n$ pieces. She first chooses a point $P$ in the interior (not boundary) of the triangle pizza, and then performs $n$ cuts, which all start from $P$ and extend straight to the boundary of the pizza so that the $n$ pieces are all triangles and all have the same area.

Let $\psi (n)$ be the number of different ways for Mamma Triangolo to cut the pizza, subject to the constraints.
For example, $\psi (3)=7$.

Also $\psi (6)=34$, and $\psi (10)=90$.

Let $\mathrm{\Psi }(m)={\displaystyle \sum _{n=3}^m\psi (n)}$. You are given $\mathrm{\Psi }(10)=345$ and $\mathrm{\Psi }(1000)=172166601$.

Find $\mathrm{\Psi }({10}^8)$. Give your answer modulo $1000000007$.