7. Let $n$ be a given positive integer. Find the largest integer $k$ such that there exist sets
$$
\begin{array}{l}
A=\left\{x_{1}, x_{2}, \cdots, x_{k}\right\}, \\
B=\left\{y_{1}, y_{2}, \cdots, y_{k}\right\}, \\
C=\left\{z_{1}, z_{2}, \cdots, z_{k}\right\}
\end{array}
$$

satisfying for any $j(1 \leqslant j \leqslant k)$,
$$
x_{j}+y_{j}+z_{j}=n .
$$
(Supplied by Li Shenghong)
7. From the conditions, we have
$$
\begin{array}{l}
k n=\sum_{i=1}^{k}\left(x_{i}+y_{i}+z_{i}\right) \\
\geqslant 3 \sum_{i=0}^{k-1} i=\frac{3 k(k-1)}{2} .
\end{array}
$$

Therefore, \( k \leqslant\left[\frac{2 n}{3}\right]+1 \).
Below is an example for \( k=\left[\frac{2 n}{3}\right]+1 \).
If \( n=3 m \), for \( 1 \leqslant j \leqslant m+1 \), let \( x_{j}=j-1 \), \( y_{j}=m+j-1 \), \( z_{j}=2 m-2 j+2 \); for \( m+2 \leqslant j \leqslant 2 m+1 \), let \( x_{j}=j-1 \), \( y_{j}=j-m-2 \), \( z_{j}=4 m-2 j+3 \).

If \( n=3 m+1 \), for \( 1 \leqslant j \leqslant m \), let \( x_{j}=j-1 \), \( y_{j}=m+j \), \( z_{j}=2 m-2 j+2 \); for \( m+1 \leqslant j \leqslant 2 m \), let \( x_{j}=j+1 \), \( y_{j}=j-m-1 \), \( z_{j}=4 m-2 j+1 \), and \( x_{2 m+1}=m \), \( y_{2 m+1}=2 m+1 \), \( z_{2 m+1}=0 \).

If \( n=3 m+2 \), for \( 1 \leqslant j \leqslant m+1 \), let \( x_{j}=j-1 \), \( y_{j}=m+j \), \( z_{j}=2 m-2 j+3 \); for \( m+2 \leqslant j \leqslant 2 m+1 \), let \( x_{j}=j \), \( y_{j}=j-m-2 \), \( z_{j}=4 m-2 j+4 \), and \( x_{2 m+2}=2 m+2 \), \( y_{2 m+2}=m \), \( z_{2 m+2}=0 \).

In summary, the maximum value of \( k \) is \( \left[\frac{2 n}{3}\right]+1 \).