A particle on a one-dimensional line moves 1 unit to the right with probability $p = 2/3$ and 1 unit to the left with probability $1-p = 1/3$. Let $P_n$ be the probability that the particle eventually reaches 0 starting from position $n \geq 0$. This satisfies the recurrence $P_n = \frac{2}{3} P_{n+1} + \frac{1}{3} P_{n-1}$ with boundary conditions $P_0 = 1$ and $\lim_{n \to \infty} P_n = 0$. Find the value of $P_2$.