The Biggest Degree

Algebra Level 4

Find the largest possible degree \(n \le 1000\) of a polynomial \(p(x)\) such that

\(p(i) = i \,\) for every integer \(i\) with \(1 \le i \leq n;\)
\(p(-1) = 1671;\)
\(p(0)\) is an integer (not necessarily 0).