Let $r_1$, $r_2$, $r_3$ be the (possibly complex) roots of the polynomial $x^3 + ax^2 + bx + \frac{4}{3}$. How many pairs of integers $a$, $b$ exist such that $r_1^3 + r_2^3 + r_3^3 = 0$?

Let's think step by step and output the final answer within \boxed{}.
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