
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
A circle is inscribed in triangle \(ABC\). On the longest side \(AC\) of the triangle, points \(E\) and \(F\) are marked such that \(AE = AB\) and \(CF = CB\). The segment \(BE\) intersects the inscribed circle at points \(P\) and \(Q\), with \(BP = 1\) and \(PQ = 8\). What is the length of the segment \(EF\)?

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