
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
An ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ and a hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$ share common foci $F_{1}$, $F_{2}$. If $P$ is an intersection point of the two curves, find the value of $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}$.

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