
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
Points \( M \) and \( N \) lie on the sides \( AD \) and \( BC \) respectively of the rhombus \( ABCD \), such that \( DM: AM = BN: NC = 2:1 \). Find \( MN \) if it is known that the side of the rhombus is \( a \) and \( \angle BAD = 60^\circ \).

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