12・111 Given integers $m, n$ satisfying $m, n \in\{1,2, \cdots, 1981\}$ and
$$
\left(n^{2}-m n-m^{2}\right)^{2}=1 \text {. }
$$

Find the maximum value of $m^{2}+n^{2}$.
(22nd International Mathematical Olympiad, 1981)