
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
The cubes $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ and $E F G H E^{\prime} F^{\prime} G^{\prime} H^{\prime}$ are congruent. The diagonal intersection planes $A C C^{\prime} A^{\prime}$ and $E G G^{\prime} E^{\prime}$ are shared, and these two rectangles can be transformed into each other by a $90^{\circ}$ rotation around their common center. Express the volume of the common part of the cubes in terms of the edge length $A B=a$.

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