
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
Jude repeatedly flips a coin. If he has already flipped \( n \) heads, the coin lands heads with probability \(\frac{1}{n+2}\) and tails with probability \(\frac{n+1}{n+2}\). If Jude continues flipping forever, let \( p \) be the probability that he flips 3 heads in a row at some point. Compute \(\lfloor 180 p \rfloor \).

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