
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

Consider the ellipse \(C: \frac{x^{2}}{a^{2}}+y^{2}=1 \) where \( a > 0 \).

1. Given \( a > 0 \), let \( P \) be a point on ellipse \( C \) located in the first quadrant. A tangent line to the ellipse \( C \) at point \( P \) intersects the coordinate axes at points \( A \) and \( B \). Determine the equation of the tangent line when the distance \(|AB|\) is minimized.

2. Denote the tangent line obtained in part (1) as \( l \). The reflection of ellipse \( C \) over line \( l \) yields an ellipse \( C^{\prime} \). If ellipse \( C^{\prime} \) intersects the \( x \)-axis, determine the range of values for \( a \).

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