22. (12 points) Let
$$
f(x)=\frac{a}{x}+x \ln x, \quad g(x)=x^{3}-x^{2}-3 \text {. }
$$
(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
$$
g\left(x_{1}\right)-g\left(x_{2}\right) \geqslant M
$$

holds, find the maximum integer $M$ that satisfies the above condition;
(3) If for any $s, t \in \left[\frac{1}{2}, 2\right]$ we have
$$
f(s) \geqslant g(t)
$$

holds, find the range of values for the constant $a$.