
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 radius $6$, two chords $PQ$ and $RS$ intersect at a point $T$. Chord $RS$ bisects chord $PQ$ such that $PT = TQ = 8$ and $PQ$ is the only chord from $P$ that $RS$ bisects. It is known that the sine of the central angle subtended by minor arc $PR$ is a rational number. If expressed as a fraction $\frac{p}{q}$ in lowest terms, what is the product $pq$?

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