
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 the image, there is a square \(A B C D\) with a side length of \(10 \, \text{cm}\) and a smaller square \(E F G H\), whose vertices \(E, F, G, H\) lie on the diagonals \(A C, B D\) of the square \(A B C D\) (see the image). The area that lies inside the square \(A B C D\) and outside the square \(E F G H\) is shaded. A line \(p\), which is parallel to \(A B\) at a distance of \(6.5 \, \text{cm}\) and passes through the square \(E F G H\), divides the shaded area into two parts. The area of one of these parts is \(13.8 \, \text{cm}^2\) larger than the area of the other. Calculate the side length of \(E F\).

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