
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 a hyperbola $E$, and point $P$ is on $E$, with $\angle F_1 P F_2 = \frac{\pi}{6}$ and $(\overrightarrow{F_2 F_1} + \overrightarrow{F_2 P}) \cdot \overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyperbola $E$.

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