
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 an ellipse with the equation $\frac{x^2}{4} + y^2 = 1$, let $F_1$ and $F_2$ be its left and right foci, respectively. Find the value of $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}$ when the area of quadrilateral $PF_1QF_2$ is maximum.

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