
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 solids bounded by the following surfaces:

a) The cylinder \( x^{2} + y^{2} = 1 \) and the planes \( z = 0 \) and \( x + y + z = 2 \);

b) The planes \( x + y + z = 2 \), \( 3x + y = 2 \), \( 3x + 2y = 4 \), \( z = 0 \), and \( y = 0 \);

c) The elliptical paraboloid \( z = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \) and the planes \( x = -1 \), \( x = 1 \), \( y = -1 \), and \( y = 1 \);

d) The sphere \( x^{2} + y^{2} + z^{2} = 2R^{2} \), the cylinder \( x^{2} + y^{2} = R^{2} \), and the plane \( z = 0 \);

e) The cone \( x^{2} + y^{2} = z^{2} \), the cylinder \( x^{2} + y^{2} = 2y \), and the plane \( z = 0 \).

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