
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
Triangle $PQR$ has sides $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ of length 39, 15, and 50, respectively. Let $\gamma$ be the circle circumscribed around $\triangle PQR$ and let $S$ be the intersection of $\gamma$ and the perpendicular bisector of $\overline{PR}$ that is not on the same side of $\overline{PR}$ as $Q$. The length of $\overline{PS}$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find the greatest integer less than or equal to $a + \sqrt{b}$.

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