
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 $f(x)=\frac{a}{x} +x\ln x$ and $g(x)=x^{3}-x^{2}-3$.

(1) If there exist $x\_1$ and $x\_2$ in $[0,2]$ such that $g(x\_1) - g(x\_2) \geq M$, find the maximum integer $M$ that satisfies the condition;

(2) If for any $s$ and $t$ in $[\frac{1}{2}, 2]$, $f(s) \geq g(t)$ always holds, find the range of values for the real number $a$.

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