
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
The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $15$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Calculate the value of $OP$, and if $OP=\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, find $p+q$.

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