
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 $F$ is the left focus of the hyperbola $C: \frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1 (a > 0, b > 0)$, point $P(0, \sqrt{6}a)$, and the line $PF$ intersects the hyperbola $C$ at exactly one point, then the eccentricity of the hyperbola $C$ is ______.

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