
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
Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the cosine of the central angle subtended by the minor arc $PS$ if expressed as a fraction $\frac{m}{n}$ in reduced form, and calculate the product $mn$.

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