
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
Suppose that \( A B C D \) is a square with a side length of 4 and that \( 0 < k < 4 \). Let points \( P, Q, R \), and \( S \) be on \( B C, C D, D A \), and \( A B \), respectively, such that

\[ \frac{B P}{P C} = \frac{C Q}{Q D} = \frac{D R}{R A} = \frac{A S}{S B} = \frac{k}{4 - k} \]

What is the value of \( k \) which minimizes the area of quadrilateral \( P Q R S \)?

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