Example 6 As shown in Figure 6, fill the ten circles in Figure 6 with the numbers $1,2, \cdots, 10$ respectively, so that the sum of the numbers in any five consecutive adjacent circles is not greater than a certain integer $M$. Find the minimum value of $M$ and complete your filling. ${ }^{[4]}$
Let the numbers that satisfy the given conditions be: $a_{1}$, $a_{2}, \cdots, a_{10}$. Then
$$
\begin{array}{l}
a_{1}+a_{2}+a_{3}+a_{4}+a_{5} \leqslant M, \\
a_{2}+a_{3}+a_{4}+a_{5}+a_{6} \leqslant M, \\
\cdots \cdots . \\
a_{10}+a_{1}+a_{2}+a_{3}+a_{4} \leqslant M .
\end{array}
$$

Thus, $5\left(a_{1}+a_{2}+\cdots+a_{10}\right) \leqslant 10 M$
$$
\begin{array}{l}
\Rightarrow \frac{5 \times 10 \times 11}{2} \leqslant 10 M \\
\Rightarrow M \geqslant 27.5 .
\end{array}
$$

Since $M$ is an integer, the minimum value of $M$ is 28.
Divide $1,2, \cdots, 10$ into the following two groups:
$$
\begin{array}{l}
10,7,6,3,2 ; \\
9,8,5,4,1,
\end{array}
$$

Filling them into the figure in sequence yields Figure 7.