
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
A circle with radius 8 has two intersecting chords, where $EF$ bisects $GH$ at point $J$ such that $EJ = JH$ and $JF = 3$. Assume $GH$ is the only chord starting at $G$ which is bisected by $EF$. Determine the cosine of the central angle of minor arc $EG$ and express it 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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