Let $N$ be the value of the following expression. $$\sum_{k=0}^{n-1}\left(\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\right)\left(\binom{n}{k+1}+\binom{n}{k+2}+\cdots+\binom{n}{n}\right)$$
Show $$N=\frac{n}{2}\binom{2n}{n}$$
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