The Weakest Marksman's Opening Move

Ana, Boris, and Cato settle a rivalry with a three-way elimination paintball match, shooting one at a time in the fixed rotation Ana, Boris, Cato, Ana, ... A player who is hit is out. Ana hits her target $\frac{1}{3}$ of the time, Boris hits $\frac{2}{3}$ of the time, and Cato never misses. On each turn a shooter may aim at either opponent — or deliberately fire into the ground. Everyone plays to maximize their own survival probability. What should Ana do with her first shot, and what is her survival probability?