
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 \(F_2\) as the left and right foci of hyperbola \(C\): \( \frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1 \) (\(a > 0, b > 0\)), \(O\) as the origin, point \(P\) is on the right branch of hyperbola \(C\), satisfying \(|F_1 F_2|=2|OP|\), and \(|PF_1| \geqslant 3|PF_2|\), calculate the range of the eccentricity \(e\) of hyperbola \(C\).

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