Problem:
Let $a_{1}, a_{2}, \ldots, a_{1000}$ be real numbers such that
$$\begin{array}{r} a_{1} \cdot 1+a_{2} \cdot 2+a_{3} \cdot 3+\cdots+a_{1000} \cdot 1000=0 \\ a_{1} \cdot 1^{2}+a_{2} \cdot 2^{2}+a_{3} \cdot 3^{2}+\cdots+a_{1000} \cdot 1000^{2}=0 \\ a_{1} \cdot 1^{3}+a_{2} \cdot 2^{3}+a_{3} \cdot 3^{3}+\cdots+a_{1000} \cdot 1000^{3}=0 \\ \vdots \\ a_{1} \cdot 1^{999}+a_{2} \cdot 2^{999}+a_{3} \cdot 3^{999}+\cdots+a_{1000} \cdot 1000^{999}=0 \\ a_{1} \cdot 1^{1000}+a_{2} \cdot 2^{1000}+a_{3} \cdot 3^{1000}+\cdots+a_{1000} \cdot 1000^{1000}=1 . \end{array}$$
What is the value of $a_{1}$ ?