
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F_{1}, F_{2}$, respectively, and left vertex $A$. The distance between the foci is $|F_{1}F_{2}| = 2$, and the eccentricity of the ellipse is $e = \frac{1}{2}$.

(1) Find the standard equation of the ellipse.

(2) If $P$ is any point on the ellipse, find the range of values for $\overrightarrow{PF_{1}} \cdot \overrightarrow{PA}$.

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