
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\ge S$ .

*Author: Zuming Feng and Oleg Golberg, USA*

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