
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 $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1 (a > 0, b > 0)$ with upper and lower foci $F\_1$ and $F\_2$, a line passing through point $F\_1$ intersects the hyperbola at points $P$ and $Q$. The difference in distances $|QF\_1|-|PF\_1|=2a$, and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}=0$. Determine the eccentricity of this hyperbola.

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