
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

In three-dimensional space, let \( S \) be the region of points \( (x, y, z) \) satisfying \( -1 \leq z \leq 1 \). Let \( S_{1}, S_{2}, \ldots, S_{2022} \) be 2022 independent random rotations of \( S \) about the origin \( (0,0,0) \). The expected volume of the region \( S_{1} \cap S_{2} \cap \cdots \cap S_{2022} \) can be expressed as \( \frac{a \pi}{b} \), for relatively prime positive integers \( a \) and \( b \). Compute \( 100a+b \).

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