
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 \( a \) with \( \sqrt{2} < a < 2 \). A convex quadrilateral \( ABCD \) is inscribed in a unit circle \( \Gamma \) and satisfies the following conditions:
1. The center of the circle lies within the convex quadrilateral.
2. The maximum side length is \( a \), and the minimum side length is \( \sqrt{4-a^{2}} \).

Tangents \( L_{A}, L_{B}, L_{C}, L_{D} \) are drawn at points \( A, B, C, D \) respectively to the circle \( \Gamma \). These tangents intersect as follows: \( L_{A} \) intersects \( L_{B} \) at \( A' \), \( L_{B} \) intersects \( L_{C} \) at \( B' \), \( L_{C} \) intersects \( L_{D} \) at \( C' \), and \( L_{D} \) intersects \( L_{A} \) at \( D' \).

Find the maximum and minimum values of the ratio of the areas \(\frac{S_{\text{quadrilateral } A'B'C'D'}}{S_{\text{quadrilateral } ABCD}}\).

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