Asymmetric Step-Size Gambler's Ruin

A gambler starts with $\$3$. Each round she wins $\$1$ with probability $\tfrac{2}{3}$ or loses $\$2$ with probability $\tfrac{1}{3}$. She stops when her fortune first hits $\$0$ or below (ruin) or reaches $\$5$ or above (victory). Note that if her fortune is $\$1$ and she loses, she goes to $-\$1$, which counts as ruin.

What is the probability that she achieves victory?