
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
We place a plane on each edge of a unit cube such that the plane forms a $45^{\circ}$ angle with each of the two faces adjacent to the edge and does not intersect the cube. What is the volume of the convex body bounded by these 12 planes?

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