
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
In right $\triangle PQR$ with hypotenuse $\overline{PR}$, $PQ = 9$, $QR = 40$, and circle $\omega$ touches $PQ$ internally at mid-point $M$ and passes through vertex $R$. Let $S$ be a point outside $\triangle PQR$ such that $\overline{PS}$ and $\overline{RS}$ are both tangent to circle $\omega$. Find the ratio of the perimeter of $\triangle PRS$ to the length $PR$, and express your answer in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers.

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