
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
Determine the smallest positive real number \( k \) with the following property. Let \(ABCD\) be a convex quadrilateral, and let points \(A_1\), \(B_1\), \(C_1\), and \(D_1\) lie on sides \(AB\), \(BC\), \(CD\), and \(DA\), respectively. Consider the areas of triangles \(AA_1D_1\), \(BB_1A_1\), \(CC_1B_1\), and \(DD_1C_1\); let \(S\) be the sum of the two smallest ones, and let \(S_1\) be the area of quadrilateral \(A_1B_1C_1D_1\). Then we always have \(kS_1 \geq S\).

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