
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 two foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is any point on the ellipse. If the area of $\triangle PF\_1F\_2$ is $1$, $\tan \angle P{F\_1}{F\_2} = \frac{1}{2}$, and $\tan \angle PF\_2F\_1 = -2$, find the value of $a$.

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