
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 $10$ units, two chords $PQ$ and $RS$ intersect at point $T$. The chord $PQ$ has a length of $24$ units, and $RS$ bisects $PQ$ at $T$. Given that $RS$ is the only chord starting from $R$ that bisects $PQ$, and that the segment $PT = QT = 12$ units, find the sine of the central angle subtended by the minor arc $PR$. Express the sine of the angle as a fraction $\frac{m}{n}$ in its lowest terms and find the product $mn$.

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