
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
Two identical cubes with edge length \( a \) share a common segment \( EF \), the ends of which are midpoints of two opposite edges not belonging to the same face. One of the cubes is obtained by rotating the other around the line \( \boldsymbol{EF} \) by \(90^{\circ}\). Find the volume of the common part of these cubes.

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