The following problem is open in the sense that the answer to part (b) is not currently known. Let $n$ be an odd positive integer and let \[ f_n(x,y,z) = x^n + y^n + z^n + (x+y+z)^n. \]$(a)$ Prove that there exist infinitely many values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \]for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. $(b)$ Determine all values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \]for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. (Two integer polynomials are $\emph{congruent modulo 2}$ if every coefficient of their difference is even. A polynomial is $\emph{constant modulo 2}$ if it is congruent to a constant polynomial modulo 2.)