
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 points $X, Y, Z$ lie on $RQ, QP, PR$, respectively. The circumcircles of $\triangle PXZ, \triangle QYX, \triangle RZY$ have centers $O_4, O_5, O_6$, respectively. $PQ = 26, QR = 29, RP = 25$, and $\stackrel{\frown}{PY} = \stackrel{\frown}{RZ},\ \stackrel{\frown}{QZ} = \stackrel{\frown}{PX},\ \stackrel{\frown}{QY} = \stackrel{\frown}{RX}$. The length of $RQ$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

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