
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 the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{12}=1$ with left and right focal points $F\_{1}$ and $F\_{2}$, and a point $P$ on the ellipse satisfying $\overrightarrow{P{{F}\_{1}}} \cdot \overrightarrow{P{{F}\_{2}}}=9$, find the value of $|\overrightarrow{P{{F}\_{1}}}| \cdot |\overrightarrow{P{{F}\_{2}}}|$.

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