Poisson arrivals spaced over a minute until a woman

Customers arrive at a cinema as a Poisson process with rate one per minute. Independently, each arrival is a man with probability $\tfrac23$ and a woman otherwise. Let $N$ count the arrivals up to and including the first woman. What is the probability that each of the first $N$ interarrival times is longer than one minute? Writing the answer as $\dfrac{a}{be-c}$ with $a,b,c$ coprime integers, give $abc$.

6. Here $N$ is geometric with $P(N=n)=\tfrac{2^{n-1}}{3^n}$, independent of the $\text{Exp}(1)$ gaps, each of which exceeds 1 with probability $e^{-1}$. Summing $\sum_n \tfrac{2^{n-1}}{3^n}e^{-n}$ as a geometric series gives $\tfrac{1}{3e-2}$, so $abc=1\cdot3\cdot2=6$.