
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 (a > 0, b > 0)\\), the left and right foci are \\(F_{1}\\) and \\(F_{2}\\), respectively. Let \\(P\\) be a point on the hyperbola \\(C\\). If \\(| \overrightarrow{PF_{1}} + \overrightarrow{PF_{2}}| = \sqrt {|\overrightarrow{PF_{1}}|^{2} + |\overrightarrow{PF_{2}}|^{2}}\\) and \\(| \overrightarrow{PF_{1}}| = 2|\overrightarrow{PF_{2}}|\\), then the eccentricity of the hyperbola \\(C\\) is \_\_\_\_\_\_.

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