
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 $\frac {y^{2}}{a^{2}} - \frac {x^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with foci $F_1$ and $F_2$ located above and below the center, and a line passing through $F_1$ intersecting the hyperbola at points $P$ and $Q$ such that $|QF_1| - |PF_1| = 2a$ and $\vec{PF_1} \cdot \vec{PF_2} = 0$, determine the eccentricity $e$ of the hyperbola.

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