
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]


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Given points \( A = (2, 7) \) and \( B = (8, 5) \) lie on circle \( \omega \) in the plane. Suppose that the tangent lines to \( \omega \) at \( A \) and \( B \) intersect at a point on the x-axis. Compute the area of \( \omega \).

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