
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
Let \\(f(x)= \dfrac{a}{x}+x\ln x\\), \\(g(x)=x^{3}-x^{2}-3\\).

\\((1)\\) If there exist \\(x_{1}\\), \\(x_{2} \in [0,2]\\) such that \\(g(x_{1})-g(x_{2}) \geqslant M\\) holds, find the maximum integer \\(M\\) that satisfies the condition;

\\((2)\\) If for any \\(s\\), \\(t \in \left[ \frac{1}{2},2 \right]\\), it holds that \\(f(s) \geqslant g(t)\\), find the range of the real number \\(a\\).

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