
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
Let the left and right foci of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{12} = 1$ be $F_1$ and $F_2$, respectively. Point $P$ is on the ellipse and satisfies $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 9$. Calculate the value of $|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|$.

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