
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
Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ has the left vertex $A$, points $P$ and $Q$ are both on $C$, and are symmetric about the $y$-axis. If the product of the slopes of lines $AP$ and $AQ$ is $\frac{1}{4}$, calculate the eccentricity of $C$.

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