
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
Calculate the volumes of the bodies bounded by the surfaces:
1) \( y = x^2 \), \( y = 1 \), \( x + y + z = 4 \), \( z = 0 \).
2) \( z = y^2 - x^2 \), \( z = 0 \), \( y = \pm 2 \).
3) \( z = 4 - x^2 - y^2 \), \( 2z = 2 + x^2 + y^2 \).
4) \( x^2 + y^2 + z^2 = R^2 \), \( z = a \), \( z = b \); \( R > b > a > 0 \).

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