
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 $QR, RP, PQ$, respectively. The circumcircles of $\triangle XPR, \triangle QXZ, \triangle RYX$ have centers $O_1, O_2, O_3$, respectively. Also, $PQ = 27, QR = 29, RP = 28$, and $\stackrel{\frown}{RY} = \stackrel{\frown}{XQ},\ \stackrel{\frown}{PY} = \stackrel{\frown}{XZ},\ \stackrel{\frown}{PZ} = \stackrel{\frown}{RX}$. The length of $RX$ can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.

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