
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 a circle with a radius of 7 units, two chords AB and CD intersect at point P such that AB is bisected at P. Given the segment PB is 4 units long, and the sine of the angle subtended by arc AC at the center of the circle is given by a rational number expressed as a fraction \(\frac{m}{n}\) in lowest terms, find the product \(mn\).

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