
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 \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) (where \(a > 0\) and \(b > 0\)), let \(A_1\) and \(A_2\) be the vertices on the real axis, \(F\) be the right focal point, and \(B(0, b)\) be the endpoint of the imaginary axis. If there exist two distinct points \(P_i\) (for \(i=1,2\)) on the line segment \(BF\) (excluding the endpoints), such that the triangles \(\triangle P_i A_1 A_2\) (for \(i=1,2\)) form right triangles with \(A_1 A_2\) as the hypotenuse, then the range of the eccentricity \(e\) of the hyperbola is \(\qquad\).

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