Example 2 If $z \in \mathbf{C}$, satisfies $3 z^{6}+2 \mathrm{i} z^{5}-2 z-3 \mathrm{i}=0$, find $|z|$.
Solve the equation which can be transformed into
$$
z^{5}=\frac{2 z+3 \mathrm{i}}{3 z+2 \mathrm{i}} .
$$

Let $z=a+b \mathrm{i}(a, b \in \mathbf{R})$, then
$$
|z|^{5}=\sqrt{\frac{4\left(a^{2}+b^{2}\right)+12 b+9}{9\left(a^{2}+b^{2}\right)+12 b+4}} .
$$

If $a^{2}+b^{2}>1$, then the left side of equation (1) $=|z|^{5}>1$, while the right side $<1$, a contradiction.
Thus, we must have $a^{2}+b^{2}=1$, i.e., $|z|=1$.