
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 points $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$), and $P$ is any point on the left branch of the hyperbola. If the minimum value of $\frac{|PF_{2}|^{2}}{|PF_{1}|}$ is $9a$, then calculate the eccentricity of the hyperbola.

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