
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
Find the centers of mass of the uniform bodies:

a) The hemisphere \( x^{2} + y^{2} + z^{2} \leqslant R^{2}, z \geqslant 0 \)

b) The tetrahedron bounded by the planes \( x + 2y + z = 1 \), and \( x = y = z = 0 \).

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