<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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 $\overrightarrow{PF} \cdot \overrightarrow{PA}$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
4
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 $\overrightarrow{PF} \cdot \overrightarrow{PA}$.
test