
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 square has a side length of $4$. Within this square, two equilateral triangles are placed such that one has its base along the bottom side of the square, and the other is rotated such that its vertex touches the midpoint of the top side of the square, and its base is parallel to the bottom of the square. Find the area of the rhombus formed by the intersection of these two triangles.

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