
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
A parabola has the equation \( y = k^2 - x^2 \), for some positive number \( k \). Rectangle \( ABCD \) is drawn with sides parallel to the axes so that \( A \) and \( D \) are the points where the parabola intersects the \( x \)-axis and the vertex \( V \) of the parabola is the midpoint of \( BC \). If the perimeter of the rectangle is 48, what is the value of \( k \)?

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