
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
$\triangle MNP$ is inscribed inside $\triangle XYZ$ such that $M, N, P$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle MZP, \triangle YNM, \triangle XNP$ have centers $O_4, O_5, O_6$, respectively. Also, $XY = 21, YZ = 27, XZ=26$, and $\stackrel{\frown}{NP} = \stackrel{\frown}{MZ},\ \stackrel{\frown}{MP} = \stackrel{\frown}{NZ},\ \stackrel{\frown}{MN} = \stackrel{\frown}{PZ}$. The length of $PZ$ can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

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