
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<sub>1</sub> and F<sub>2</sub> are the left and right foci of the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$ respectively, if there exists a point P on the left branch of the hyperbola such that |PF<sub>1</sub>|=|F<sub>1</sub>F<sub>2</sub>|, and the distance from F<sub>1</sub> to the line PF<sub>2</sub> is $$\sqrt {7}a$$, then the eccentricity of the hyperbola e is ___.

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