
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
Given \\(a \in \mathbb{R}\\), the function \\(f(x) = -x^3 + \frac{1}{2}ax^2\\), \\(g(x) = a\ln x\\).
\\((1)\\) Discuss the monotonicity of the function \\(f(x)\\);
\\((2)\\) Let the derivative functions of \\(f(x)\\) and \\(g(x)\\) be \\(f'(x)\\) and \\(g'(x)\\) respectively
\\(①\\) When \\(a=3\\), for \\(\forall x > 1\\), \\(f'(x) \leqslant kg(x)\\) always holds, find the range of real numbers \\(k\\)
\\(②\\) When \\(a > 0\\), if \\(\exists x_0 > 0\\) such that \\(\sqrt{f'(x_0)} \geqslant g'(x_0)\\), find the minimum value of the real number \\(a\\).

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