Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.

Let's think step by step and output the final answer within \boxed{}.
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