
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 points $P$ and $Q$ are both moving on the ellipse $\Gamma: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{a^{2}-1}=1(a > 1)$, and $F\_{1}$, $F\_{2}$ are the left and right foci of the ellipse $\Gamma$, determine the maximum value of $| \overrightarrow{PF\_{1}}+ \overrightarrow{PF\_{2}}-2 \overrightarrow{PQ}|$.

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