
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
Let \( p \geq 3 \) be a prime number and consider the right triangle with the longer leg \( p^{2}-1 \) and the shorter leg \( 2p \). A semicircle is inscribed in the triangle such that its diameter is along the longer leg and it is tangent to the hypotenuse and the shorter leg. Find the values of \( p \) for which the radius of the semicircle is an integer.

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