Example 1 Let $P(x)$ be a polynomial of degree $2n$, such that
$$
\begin{array}{l}
P(0)=P(2)=\cdots=P(2 n)=0, \\
P(1)=P(3)=\cdots=P(2 n-1)=2, \\
P(2 n+1)=-6 .
\end{array}
$$

Determine $n$ and $P(x)$.