jmc

We know that $a_n+a_{n+1}+a_{n+2}=n$ and $a_{n-1}+a_n+a_{n+1}=n-1.$ Subtracting these equations, we get $$a_{n+2}-a_{n-1}=1,$$so $a_{n+2}=a_{n-1}+1.$

Hence, the terms $$a_1=2007,a_4,a_7,a_{10},\dots ,a_{1000}$$form an arithmetic sequence with common difference 1. The common difference of 1 is added $\frac{1000-1}{3}=333$ times, so $a_{1000}=2007+333=\boxed{2340}.$