
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
\(F_{1}\) and \(F_{2}\) are the foci of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) (\(a > b > 0\)). \(P\) is a point on the ellipse. If the area of \(\triangle P F_{1} F_{2}\) is 1, \(\tan \angle P F_{1} F_{2} = \frac{1}{2}\), and \(\tan \angle P F_{2} F_{1} = -2\), then \(a =\) ___.

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