14. 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.
14. If there are 13 numbers in total, they can be arranged as follows:
$$
\begin{array}{l}
a_{1}, a_{2}, a_{3}, \cdots, a_{8}, a_{9}, \\
a_{2}, a_{3}, a_{4}, \cdots, a_{9}, a_{10}, \\
a_{3}, a_{4}, a_{5}, \cdots, a_{10}, a_{11}, \\
a_{4}, a_{5}, a_{6}, \cdots, a_{11}, a_{12}, \\
a_{5}, a_{6}, a_{7}, \cdots, a_{12}, a_{13} .
\end{array}
$$

From the given information, the sum of the numbers in each row is positive, thus the sum of all numbers in the table is positive; on the other hand, the sum of the numbers in each column is negative, thus the sum of all numbers in the table is negative. This is a contradiction. This indicates that the sequence satisfying the requirements can have at most 12 terms.
Consider the following sequence of numbers:
$$
\begin{array}{l}
-4,-4,-4,15, -4,-4,-4,-4,15, \\
-4, -4, -4,
\end{array}
$$

This sequence satisfies the requirements of the problem and has 12 terms, so the maximum value of \( n \) that satisfies the requirements is 12.