TR-MO 2006 Problem 3

Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where $$P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.$$