
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 line segment \(AB\). Determine the minimum value of \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}\).

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