Consider the polynomials \( P(x) = x^3 + 2x^2 + 3x + 4 \) and \( Q(x) = x^3 - 4x^2 + 5x - 6 \). Suppose \( \alpha, \beta, \gamma \) are the roots of \( P(x) \) and \( \delta, \epsilon, \zeta \) are the roots of \( Q(x) \). Find the value of \( \alpha + \beta + \gamma + \delta + \epsilon + \zeta \). Let's think step by step and output the final answer within \boxed{}.
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