
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 in $\triangle ABC$ such that $X, Y, Z$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle XAC, \triangle YBZ, \triangle ZCY$ have centers $O_1, O_2, O_3$, respectively. Given $AB = 21, BC = 27, AC = 26$, and $\stackrel{\frown}{BZ} = \stackrel{\frown}{XC},\ \stackrel{\frown}{AZ} = \stackrel{\frown}{CY},\ \stackrel{\frown}{AY} = \stackrel{\frown}{BX}$. The length of $BX$ 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}
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