
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
Suppose \(\triangle ABC\) has lengths \(AB = 5\), \(BC = 8\), and \(CA = 7\), and let \(\omega\) be the circumcircle of \(\triangle ABC\). Let \(X\) be the second intersection of the external angle bisector of \(\angle B\) with \(\omega\), and let \(Y\) be the foot of the perpendicular from \(X\) to \(BC\). Find the length of \(YC\).

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