
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 $13$, two chords $AD$ and $BC$ intersect such that $BC$ bisects $AD$. If $BC = 10$, and it is given that $AD$ is the only chord starting at $A$ which is bisected by $BC$, determine the sine of the central angle subtended by the minor arc $AB$ as a rational number. Express this sine as $\frac{m}{n}$ in lowest terms and find the product $mn$.

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