
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with a focal length of $4$, and the ellipse $C$ passes through point $A(2, \sqrt{2})$.

(1) Find the equation of the ellipse $C$;

(2) Let $P$ and $Q$ be two distinct moving points on the ellipse $C$ that are symmetric about the $y$-axis. Find the range of values for $\overrightarrow{AP} \cdot \overrightarrow{AQ}$.

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