
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 that \\(F_{1}\\) and \\(F_{2}\\) are the left and right foci of the hyperbola \\(C\\): \\(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)\\), if the symmetric point \\(D\\) of point \\(F_{2}\\) with respect to the line \\(bx - ay = 0\\) exactly falls on the circle centered at \\(F_{1}\\) with radius \\(|OF_{1}|\\), then the eccentricity of the hyperbola \\(C\\) is \_\_\_\_\_\_\_\_.

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