设 $a_i \neq a_i(i \neq j ; i, j=1,2, \cdots, n)$ ，
$$
\boldsymbol{A}=\left(\begin{array}{ccccc}
1 & 1 & 1 & \cdots & 1 \\
a_1 & a_2 & a_3 & \cdots & a_n \\
a_1^2 & a_2^2 & a_3^2 & \cdots & a_n^2 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & \cdots & a_n^{n-1}
\end{array}\right), \boldsymbol{X}=\left(\begin{array}{r}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_n
\end{array}\right), B=\left(\begin{array}{c}
1 \\
1 \\
1 \\
\vdots \\
1
\end{array}\right)
$$
则线性方程组 $A^T \boldsymbol{X}=B$ 的解是