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

(1) When $a=2$, find the equation of the tangent line to the curve $y=f(x)$ at $x=1$;

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

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

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