
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
Over the sides of a square \(ABCD\) with side length \(a\), equilateral triangles \(ABE\), \(BCF\), \(CDG\), and \(DAH\) are constructed outside the square. Additionally, over the diagonal \(AC\), equilateral triangles \(ACM\) and \(ACP\) are constructed, and over the diagonal \(BD\), equilateral triangles \(BDN\) and \(BDQ\) are constructed. Calculate the area of the regions \(EFGH\), \(MNPQ\), and the common area of these two regions.


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