
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 7 units, two intersecting chords PR and QS meet at point T, where QS bisects PR. Suppose that QS=8 units, and T is the midpoint of PR. If the chord PR is the only chord starting from point P and bisected by QS, determine the cosine of the central angle subtended by the minor arc PS. Express this as a fraction $\frac{p}{q}$ in its simplest form and find the product $pq$.

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