
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\\) (where \\(a > 0, b > 0\\)), the right vertex of the hyperbola is \\(A\\), and \\(O\\) is the origin. A circle centered at \\(A\\) intersects a certain asymptote of the hyperbola \\(C\\) at two points \\(P\\) and \\(Q\\). If \\(\angle PAQ = 60^\circ\\) and \\(\vec{OQ} = 3\vec{OP}\\), then the eccentricity of the hyperbola \\(C\\) is \_\_\_\_\_\_.

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