
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
In $\triangle XYZ$, a triangle $\triangle MNO$ is inscribed such that vertices $M, N, O$ lie on sides $YZ, XZ, XY$, respectively. The circumcircles of $\triangle XMO$, $\triangle YNM$, and $\triangle ZNO$ have centers $P_1, P_2, P_3$, respectively. Given that $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{MO} = \stackrel{\frown}{YN}, \stackrel{\frown}{NO} = \stackrel{\frown}{XM}, \stackrel{\frown}{NM} = \stackrel{\frown}{ZO}$. The length of $ZO$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.

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