
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 $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($b > a > 0$), its right focus is $F$, and $O$ is the origin. If there exists a line $l$ passing through point $F$ and intersecting the right branch of the hyperbola $C$ at points $A$ and $B$, such that $\vec{OA} \cdot \vec{OB} = 0$, then the range of the eccentricity of the hyperbola is ______.

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