
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 gravity:

a) of a hemisphere \(z=\sqrt{R^{2}-x^{2}-y^{2}}\), with surface density \(\rho = k(x^{2}+y^{2})\), where \(k\) is a constant;

b) of a cone \( a z^{2}=b^{2}(x^{2}+y^{2}), 0 \leqslant z \leqslant b \), with density \(\rho = \rho_0\), where \(\rho_0\) is a constant.

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