
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
In triangle \(ABC\), a circle \(\omega\) with center \(O\) passes through \(B\) and \(C\) and intersects segments \(\overline{AB}\) and \(\overline{AC}\) again at \(B'\) and \(C'\), respectively. Suppose that the circles with diameters \(BB'\) and \(CC'\) are externally tangent to each other at \(T\). If \(AB = 18\), \(AC = 36\), and \(AT = 12\), compute \(AO\).

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