
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
$\triangle XYZ$ is inscribed inside $\triangle PQR$ such that $X, Y, Z$ lie on $QR, RP, PQ$ respectively. The circumcircles of $\triangle XRC, \triangle PYD, \triangle QZE$ have centers $O_1, O_2, O_3$ respectively. The side lengths are $PQ = 14, QR = 17, RP = 15$, and $\stackrel{\frown}{PY} = \stackrel{\frown}{XQ}, \stackrel{\frown}{QZ} = \stackrel{\frown}{YP}, \stackrel{\frown}{RZ} = \stackrel{\frown}{XQ}$. Find the length of $PZ$, which can be written in the form $\frac{a}{b}$, where $a$ and $b$ are coprime integers. Compute $a \mathbin{\%} b$.

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