
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 $\triangle ABC$ such that $X,Y,Z$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle XEC, \triangle BYD, \triangle AZF$ (where $E,D,F$ are the midpoints of $BC, AC, AB$, respectively) have centers $O_1, O_2, O_3$. Also, $AB = 15, BC = 20, AC = 13$, and $\angle BAC = \angle BXC, \angle ABC = \angle BYC, \angle ACB = \angle AZB$. Assuming the symmetry in this configuration, find the length $BX$ in the simplest fraction form.

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