Example 4 A four-digit number has the sum of the squares of its first and last digits equal to 13, and the sum of the squares of its middle two digits equal to 85. If 1089 is subtracted from this number, the result is the same four digits written in reverse order. What is the original four-digit number? $\qquad$
(1994, Dongfang Airlines Cup - Shanghai Junior High School Mathematics Competition)
Solution: Let the original four-digit number be $\overline{a b c d}$. According to the problem, we have
$$
\begin{array}{l}
a^{2}+d^{2}=13, \\
b^{2}+c^{2}=85, \\
\overline{a b c d}-1089=\overline{d c b a} .
\end{array}
$$

Analyzing from the first digit, it is easy to know
$$
a-d=1 \text {. }
$$

From (1) and (4), we get
$$
a=3, d=2 \text {. }
$$

Therefore, $\overline{3 b c 2}-1089=\overline{2 c b 3}$.
Performing subtraction in column form, we get
$$
b+8+1=c+10 \text {. }
$$

Thus, $b-c=1$.
From (2) and (5), we get
$$
b=7, c=6 \text {. }
$$

Therefore, the original four-digit number is 3762.