
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
A hyperbola has its left and right foci at ${F_1(-1,0)}$ and ${F_2(1,0)}$, respectively. One of the intersection points of the parabola ${y^2=4x}$ and the hyperbola is P. If ${\left( {\overrightarrow {{F_2}P} + \overrightarrow {{F_2}{F_1}} } \right) \cdot \left( {\overrightarrow {{F_2}P} - \overrightarrow {{F_2}{F_1}} } \right) = 0}$, then calculate the eccentricity of the hyperbola.

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