(1) $a_{1}, a_{2}, a_{3}, \cdots$ is an arithmetic sequence, where $a_{1}>0, S_{n}$ represents the sum of the first $n$ terms. If $S_{3}=S_{11}$, and among $S_{1}, S_{2}, S_{3}, \cdots$, the maximum number is $S_{k}$, then $k=$ $\qquad$ .
(1) 7 Hint: Let the common difference be $d$, then $a_{n}=a_{1}+(n-1) d$, from $S_{3}=S_{11}$ we get
$$
d=-\frac{2}{13} a_{1}<0,
$$

which means
$$
a_{n}=a_{1}+(n-1) \cdot \frac{-2}{13} a_{1}=\frac{a_{1}}{13}(15-2 n),
$$

thus the sequence is monotonically decreasing, and the largest positive integer $n$ for which $a_{n} \geqslant 0$ is $n=7$.