
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 a hyperbola _E_ with eccentricity _e_, its left and right foci are _F_<sub>1</sub> and _F_<sub>2</sub> respectively. A parabola _C_ has _F_<sub>2</sub> as its vertex and _F_<sub>1</sub> as its focus. Point _P_ is a point of intersection between the parabola and the right branch of the hyperbola. If $a|PF_{2}| + c|PF_{1}| = 8a^{2}$, then the value of _e_ is ( ).

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