Determine a constant $k$ such that the polynomial$$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$is divisible by $x+y+z$.

Determine a constant $k$ such that the polynomial$$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$is divisible by $x+y+z$.