
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
The vertices \( M \) and \( N \) of the equilateral triangle \( BMN \) lie on the sides \( AD \) and \( CD \) of the square \( ABCD \) with side length \( a \), respectively. Find the length of \( MN \).

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