
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
Given a cube $ABCDEFGH$ of edge length $s$, a plane cuts through vertex $D$ (treated as the origin) and the midpoints $M$ and $N$ of edges $\overline{AB}$ and $\overline{CG}$, respectively, extending diagonally across the cube from $AB$ to $CG$. Determine the volume of the larger solid formed from this division.

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