
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 that \( F_{1} \) and \( F_{2} \) are the foci of the ellipse \(\frac{x^2}{4} + y^2 = 1\), and \( A \) and \( B \) are the left vertex and the top vertex of this ellipse respectively. Point \( P \) is on the line segment \( AB \). Find the minimum value of \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}\).

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