
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
Let \(\overline{AB}\) be a diameter of circle \(\omega\). Extend \(\overline{AB}\) through \(A\) to \(C\). Point \(T\) lies on \(\omega\) so that line \(CT\) is tangent to \(\omega\). Point \(P\) is the foot of the perpendicular from \(A\) to line \(CT\). Suppose \(\overline{AB} = 20\), and let \(n\) denote the maximum possible length of segment \(BP\). Find \(n^{2}\).

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