Example 2 Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=2, a_{n}=2^{2 n} a_{n-1}+2^{n^{2}} n \quad (n=2,3, \cdots) \text {. }
$$

Find the general term $a_{n} \ (n=1,2, \cdots)$.
(Sixth Northern Mathematical Olympiad Invitational Competition)