
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
In trapezoid \( \mathrm{ABCD} \), \( CD = 2 AB \). Points \( E \) and \( F \) are the midpoints of \( AD \) and \( AB \), respectively. If the area of triangle \( CDG \) minus the area of quadrilateral \( AEGF \) is equal to \( \frac{k}{24} \) square centimeters (where \( \mathbf{k} \) is a positive integer), and for the area of trapezoid \( \mathrm{ABCD} \) to be an integer, what is the minimum value of \( k \) ?

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