
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 the tetrahedron  $ABCD$ ,  $E$  and  $F$  are the midpoints of  $BC$  and  $AD$ ,  $G$  is the midpoint of the segment  $EF$ . Construct a plane through  $G$  intersecting the segments  $AB$ ,  $AC$ ,  $AD$  in the points  $M,N,P$  respectively in such a way that the sum of the volumes of the tetrahedrons  $BMNP$ ,  $CMNP$  and  $DMNP$  to be minimal.

*H. Lesov*

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