
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
Let us denote $S$ and $S_{1}$ as the midpoints of the edges $AD$ and $B_{1}C_{1}$, respectively. The rotated cube is denoted as $A^{\prime} B^{\prime} C^{\prime} D^{\prime} A_{1}^{\prime} B_{1}^{\prime} C_{1}^{\prime} D_{1}^{\prime}$. The common part of the original cube and the rotated cube is a polyhedron composed of a regular quadrangular prism $E F G H E_{1} F_{1} G_{1} H_{1}$ and two regular quadrangular pyramids $S E F G H$ and $S_{1} E_{1} F_{1} G_{1} H_{1}$.

The side length of the base of each pyramid is 1 and the height is $\frac{1}{2}$, so its volume is $\frac{1}{6}$. The volume of the prism is $\sqrt{2}-1$.

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