
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 $ABC$ has $AB=4$, $BC=5$, and $CA=6$. Points $A'$, $B'$, and $C'$ are such that $B'C'$ is tangent to the circumcircle of $\triangle ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.

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