
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
Inside a regular triangular pyramid, there is a vertex of a trihedral angle, all plane angles of which are right angles, and the bisectors of these plane angles pass through the vertices of the base. In what ratio does the surface of this angle divide the volume of the pyramid if each face of the pyramid is divided into two equal parts by it?

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