
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 the equation of ellipse \(C_{1}\) as \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)\), and the equation of curve \(C_{2}\) as \(y=\frac{1}{x}\), where \(C_{1}\) and \(C_{2}\) have only one common point \(P\) in the first quadrant:
1. Express the coordinates of point \(P\) in terms of \(a\).
2. Let \(F_{1}\) and \(F_{2}\) be the foci of ellipse \(C_{1}\). As \(a\) varies, determine the range of the area function \(f(a)\) of triangle \(P F_{1} F_{2}\).
3. Let \(g(a)\) be the area of a square with side length equal to the semi-major axis of ellipse \(C_{1}\). Find the expression for the function \(y=\min \{f(a), g(a)\}\).

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