
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 with its focus on the $x$-axis has the equation $\frac{x^2}{4} + \frac{y^2}{b^2} = 1$ and an eccentricity of $e = \frac{1}{2}$. Let $F$ and $A$ be the left focus and right vertex of the ellipse, respectively, and let $P$ be any point on the ellipse. Determine the maximum value of $\overrightarrow{PF} \cdot \overrightarrow{PA}$.

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