
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 the 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 \(a = \frac{\sqrt{15}}{2}\).

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