4-133 Let $x_{1}, x_{2}, \cdots, x_{7}$ be real numbers satisfying the following equations:
$$
\begin{array}{l}
\sum_{k=1}^{7} k^{2} x_{k}=1, \\
\sum_{k=1}^{7}(k+1)^{2} x_{k}=12, \\
\sum_{k=1}^{7}(k+2)^{2} x_{k}=123,
\end{array}
$$

Find the value of $\sum_{k=1}^{7}(k+3)^{2} x_{k}$.
［Solution］Let $a \cdot k^{2}+b(k+1)^{2}+c \cdot(k+2)^{2}=(k+3)^{2}$. By equating the corresponding coefficients, we get
$$
\left\{\begin{array}{l}
a+b+c=1, \\
2 b+4 c=6, \\
b+4 c=9 .
\end{array}\right.
$$

Solving these, we find $a=1, b=-3, c=3$.
Thus, (1) + (2) $\times(-3)+$ (3) $\times 3$ gives
$$
\sum_{k=1}^{7}(k+3)^{2} x_{k}=1-3 \times 12+3 \times 123=334 .
$$