
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
$\triangle PQR$ is inscribed inside $\triangle XYZ$ such that $P, Q, R$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle PQZ, \triangle QRY, \triangle RPX$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 30, YZ = 40, XZ = 35$, and $\stackrel{\frown}{RY} = \stackrel{\frown}{QZ},\ \stackrel{\frown}{RP} = \stackrel{\frown}{QY},\ \stackrel{\frown}{XP} = \stackrel{\frown}{RZ}$. The length of $RZ$ needs to be found and verified whether it can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers.

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