
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 that point $P$ is the intersection of the line $y=\frac{b}{3a}x$ and the left branch of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0$, $b>0$), and $F\_1$ is the left focus, with $PF\_1$ perpendicular to the $x$-axis. Find the eccentricity of the hyperbola, $e=$ \_\_\_\_\_\_.

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