Find the distribution of numbers of arrivals of the Poisson process $N(t)$ in time interval $[t, t+\tau)$, $\tau \sim Exp(a)$.

Poisson process has rate $$\lambda$$ and $$\tau \perp \!\!\! \perp N(t)$$. To find distribution i've started with $$P(N(t+\tau)-N(t)=k) = P(N(\tau) = k)$$. I know that $$N(t) \sim Poiss(\lambda t)$$, but i don't know what to do next to find distribution.

Hint: use the formula $$P(N(\tau) = k) = \int_0^\infty P(N(s)=k|\tau = s) ae^{-as}\, ds.$$