
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 placed two unit cubes so that one pair of their space diagonals have common endpoints, and the two cubes are rotated by $60^{\circ}$ around this axis relative to each other.

What is the volume they occupy together, and what is the surface area of the solid formed by their union?

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