Asymmetric Jump Ruin with Upward Leap

A gambler on $\{0, 1, \ldots, 6\}$ starts at position $3$. Each round, she jumps $+2$ with probability $1/3$ and $-1$ with probability $2/3$. Positions $0$ and $6$ are absorbing (but a jump from $5$ that would land at $7$ is truncated to $6$, i.e., from state $5$ the gambler moves to $6$ with probability $1/3$ and to $4$ with probability $2/3$). What is the probability that the gambler is absorbed at $0$?