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)