
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
Three mutually perpendicular lines intersect at a point \( O \). Points on one of these lines that are at a distance of 1 unit from \( O \) are denoted by \( X_{1} \) and \( X_{1}^{\prime} \), and points 2 units from \( O \) are denoted by \( X_{2} \) and \( X_{2}^{\prime} \). Similarly, points on the other two lines are labeled \( Y_{1}, Y_{1}^{\prime}, Y_{2}, Y_{2}^{\prime} \) and \( Z_{1}, Z_{1}^{\prime}, Z_{2}, Z_{2}^{\prime} \). Planes are formed with every possible combination of a point with a 1-index from one line and a 2-index point from each of the other two lines, for example, passing through \( X_{1}^{\prime} \), \( Y_{2} \), and \( Z_{2}^{\prime} \).

Determine the volume of the solid enclosed by these planes.

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