
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ has the left vertex $A$, points $P$ and $Q$ are both on $C$, and symmetric about the $y$-axis. If the product of the slopes of lines $AP$ and $AQ$ is $\frac{1}{3}$, then the eccentricity of $C$ is ______.

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