
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
In a regular tetrahedron \(ABCD\), plane \(P\) intersects the edges \(AB\), \(BC\), \(CD\), and \(AD\) at points \(K\), \(L\), \(M\), and \(N\), respectively. The areas of triangles \(AKN\), \(KBL\), and \(NDM\) are \(\frac{1}{5}\), \(\frac{1}{2}\), and \(\frac{1}{5}\) of the area of a face of the tetrahedron, respectively. In what ratio does the plane \(P\) divide the area of face \(BCD\)?

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