
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 the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with its left and right foci being $F_1(-c,0)$ and $F_2(c,0)$ respectively. If there exists a point $P$ on the hyperbola such that $\frac{a}{\sin \angle PF_1F_2} = \frac{c}{\sin \angle PF_2F_1}$, determine the range of the eccentricity of this hyperbola.

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