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Let $a + a r_{1} + a r_{1}^{2} + a r_{1}^{3} + \dots$ and $a + a r_{2} + a r_{2}^{2} + a r_{2}^{3} + \dots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_{1} + \frac{1}{2}$ , and the sum of the second series is $r_{2} + \frac{1}{2}$ . What is $r_{1} + r_{2}$ ?