
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
A triangle $PQR$ is inscribed in circle $\Omega$ with $PQ=8$, $QR=13$, and $PR=5$. The bisector of angle $P$ meets side $\overline{QR}$ at $X$ and circle $\Omega$ again at a point $Y$. Let $\Gamma$ be the circle with diameter $\overline{XY}$. Circles $\Omega$ and $\Gamma$ intersect at $Y$ and another point $Z$. Find the area of triangle $PZQ$.

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