
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
Squares  $ABCD$  and  $AEFG$  each with side length  $12$  overlap so that  $\vartriangle AED$  is an equilateral triangle as shown. The area of the region that is in the interior of both squares which is shaded in the diagram is  $m\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m + n$ .
![Image](https://cdn.artofproblemsolving.com/attachments/c/2/a2f8d2a090a6342610c43b3fed8a87fa5d7f03.png)

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