
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
Consider the ellipse $\frac{x^2}{16} + \frac{y^2}{12} = 1$ whose left and right intersection points are $F_1$ and $F_2$, respectively. Let point $P$ be on the ellipse and satisfy $\vec{PF_1} \cdot \vec{PF_2} = 9$. Find the value of $|\vec{PF_1}| \cdot |\vec{PF_2}|$.

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