There are $n$ balloons in a cyclic room numbered from $1$ to $n$ (Balloon $2$ comes after $1$, Balloon $3$ comes after $2$, $\dots$, Balloon $1$ comes after Balloon $n$). You start at $1$. Every turn you skip one balloon and pop the next balloon until there is only one balloon left.

Let $F(n)$ denote the number on the last remaining balloon. Find the sum of $F(1) + F(2) + F(3) + \dots + F(182721378)$ and submit the answer modulo $10^9 + 7$.