
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
Let P be the intersection point of the line $y= \frac {b}{3a}x$ and the left branch of the hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$). If $F_1$ is the left focus and $PF_1$ is perpendicular to the x-axis, then the eccentricity $e$ of the hyperbola is \_\_\_\_\_\_.

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