A very long stick is about to be passed through a special random chopper at speed $1\:m/s$. Before passing the stick through the chopper, we put a special mark at some point on the stick. The special random chopper chops the stick at time $t$ after the previous chop (or time $t$ after starting for the first chop) with probability $5 \cdot e^{-5t}$.

If the expected value of the length of the stick with the special mark is $\frac{p}{q}$ in meters, where $\gcd(p,q) = 1$, find $(p \cdot q^{-1})\:mod\:1000000007$.