
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 rectangle \(ABCD\). On the rays \(AB\), \(BC\), \(CD\), and \(DA\) starting from points \(A\), \(B\), \(C\), and \(D\) respectively, the points \(A_1\), \(B_1\), \(C_1\), and \(D_1\) are determined such that

$$
\frac{A A_1}{AB} = \frac{B B_1}{BC} = \frac{C C_1}{CD} = \frac{D D_1}{DA} = k > 0.
$$

For what value of \(k\) will the area of the quadrilateral \(A_1 B_1 C_1 D_1\) be minimized?

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