
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 the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the trisection point of $PQ$, then the eccentricity of $C$ is ____.

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