Example 5 Figure 3 is a five-ring composed of circles. Please fill in the numbers $1,2, \cdots, 9$ in the 9 blank spaces of the five-ring, so that the sum of the numbers filled in each circle is equal. Find the maximum and minimum values of this equal sum. ${ }^{[3]}$
Let the sum of the numbers in Figure 3 be $S$. Then
$$
\begin{aligned}
5 S= & (a+b)+(b+c+d)+ \\
& (d+e+f)+(f+g+h)+(h+i) \\
= & (a+b+c+d+e+f+g+h+i)+ \\
& (b+d+f+h) \\
= & (1+2+\cdots+9)+(b+d+f+h) \\
= & 45+(b+d+f+h) .
\end{aligned}
$$

Thus, $b+d+f+h$ must be divisible by 5.
The maximum value of $b+d+f+h$ is $6+7+8+9=30$, and the minimum value is $1+2+3+4=10$, so the possible values of $b+d+f+h$ are $10, 15, 20, 25, 30$.
(1) When $b+d+f+h=30$, $S=15$, i.e.,
$$
b+d+f+h=6+7+8+9 \text {, }
$$

In this case, at least one circle containing 9 has a sum greater than 15, so $S=15$ is not valid.
$$
\begin{array}{l}
\text { (2) When } b+d+f+h=25 \text {, } S=14 \text {, in this case, } \\
b+d+f+h=8+7+6+4 \\
=9+7+5+4=9+7+6+3 \\
=9+8+6+2,
\end{array}
$$

As shown in Figure 4, there exists a way to fill the numbers such that
$$
b+d+f+h=9+7+6+3 \text {. }
$$

Therefore, the maximum value of $S$ is 14.
(3) When $b+d+f+h=1+2+3+4=10$, $S=11$, in this case, there exists a way to fill the numbers as shown in Figure 5. Therefore, the minimum value of $S$ is 11.