
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 center of gravity of the arc \(AB\):

a) \(AB: x^{2/3} + y^{2/3} = a^{2/3}\) (astroid), \(A(0, a)\), \(B(a, 0)\), with linear density at each point of the arc equal to its x-coordinate;

b) \(AB: x = \cos t\), \(y = \sin t\), \(z = \frac{H}{2 \pi} t\), \(t_A = 0\), \(t_B = 2 \pi\) (helical line with step \(H\)), \(\rho(x, y) = 1\).

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