
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 $C: x^2 - \frac{y^2}{b^2} = 1 (b > 0)$, $O$ is the origin, and point $P$ is on the right branch of the hyperbola $C$ and satisfies $|F_{1}F_{2}| = 2|OP|$, $\tan ∠PF_{2}F_{1} \geqslant 4$. Then, the range of the eccentricity of the hyperbola $C$ is ______.

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