
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
Let \( M \) and \( N \) be the points of intersection of the medians of the faces \( ABD \) and \( BCD \) of the tetrahedron \( ABCD \). Find the length \( MN \) if it is known that \( AC = a \).

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