
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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. Now 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  $1000p$ .

*Proposed by Lewis Chen*

Present the answer in LaTex format: \boxed{Your answer}
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