Consider the polynomials \(P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x\) and \(Q(x) = x^{4} - x^{3} - x^{2} - 1.\) Given that \(z_{1},z_{2},z_{3},\) and \(z_{4}\) are the roots of \(Q(x) = 0,\) find \(P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).\)

(第二十一届AIME2 2003 第9题)