Example 9 In a sequence of numbers $a_{1}, a_{2}, \cdots, a_{n}$, the sum of any five consecutive terms is negative, while the sum of any nine consecutive terms is positive. What is the maximum value of $n$? Prove your conclusion. [5]
$(2015$, National Junior High School Mathematics Invitational Competition)
When there are 13 terms, arrange them as follows:
$$
\begin{array}{l}
a_{1}, a_{2}, \cdots, a_{9} ; \\
a_{2}, a_{3}, \cdots, a_{10} ; \\
a_{3}, a_{4}, \cdots, a_{11} ; \\
a_{4}, a_{5}, \cdots, a_{12} ; \\
a_{5}, a_{6}, \cdots, a_{13} .
\end{array}
$$

Consider calculating the sum $M$ of all numbers in the table.
By the given condition, the sum of the numbers in each row is positive, thus, $M>0$.
Also, the sum of the numbers in each column is negative, thus, $M<0$.
This is a contradiction.
This indicates that the sequence satisfying the requirement has at most 12 terms. Construct the following sequence:
$$
\begin{array}{l}
-4,-4,-4,15,-4,-4,-4,-4,15, \\
-4,-4,-4 .
\end{array}
$$

Verification shows that these 12 terms meet the requirements of the problem.
Therefore, the maximum value of $n$ that satisfies the problem's requirements is 12.