Algebra Level 5

Say a real number $$r$$ is repetitive if there exist two distinct complex numbers $$z_{1},z_{2}$$ with $$|z_{1}| = |z_{2}| = 1$$ and $$z_{1},z_{2}$$ is not equal to $${−i,i}$$ such that $z_{1}(z_{1}^{3} + z_{1}^{2} + rz_{1} + 1) = z_{2}(z_{2}^{3} + z_{2}^{2} + rz_{2} +1)$ There exist real numbers $$a,b$$ such that a real number $$r$$ is repetitive if and only if $$a < r ≤ b$$. If the value of $$|a| + |b|$$ can be expressed in the form $$\frac{p}{q}$$ for relatively prime positive integers $$p,q$$. Find $$100p +q$$.

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Source: OMO Spring 2016

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