A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$.