
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 edge of a regular tetrahedron \( ABCD \) is \( a \). Points \( E \) and \( F \) are taken on the edges \( AB \) and \( CD \), respectively, such that the circumscribed sphere around the tetrahedron intersects the line passing through \( E \) and \( F \) at points \( M \) and \( N \). Find the length of segment \( EF \) given that \( ME : EF : FN = 3 : 12 : 4 \).

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