The story

Ares had once gifted Hippolyta, the queen of the amazons, a girdle -- a belt. It is this belt that Hercules is now tasked to steal. But, when word of his task spread to the queen herself, she was very impressed with Hercules’ bravery and offered to give him the belt if he won a game of skill against her.

The game

Let $s_i = {a_j} = {\frac{1}{(2 - \frac{i}{9})n-1}}$ be a series for each $i$, $0 \leq i \leq 7$.

And consider $I_n$ = $\int_{0}^{\infty} \prod_{j=1}^{n} \frac{sin(a_jx)}{a_jx}dx$ for each $s_i$.

As an example, let’s play the game with $i=0$. Here, $I_1 = I_2 = \dots = I_7$, but $I_7 \neq I_8$. Each $I_n$ is of the form $\frac{p}{q}\pi$. We say the value of $s_0$ is $8+p+q$ (8 being the first integral to break the pattern).

Play the game with all the indicated values of $i$. Add together the values of each $s_i$ and submit this sum modulo 1000000007 as the answer.