
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 triangle $PQR$, side $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ have lengths 15, 36, and 39, respectively. Let $\Omega$ be the circle circumscribed around $\triangle PQR$ and let $S$ be the intersection of $\Omega$ and the perpendicular bisector of $\overline{PR}$ that is not on the same side of $\overline{PR}$ as $Q$. Find the length of $\overline{PS}$, and express your answer in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers.

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