Given the infinite geometric series \(\frac{a}{b} + \frac{a}{b^2} + \frac{a}{b^3} + \cdots = 6\), find the value of the infinite geometric series \(\frac{2a}{b} + \frac{2a}{b^2} + \frac{2a}{b^3} + \cdots\). Let's think step by step and output the final answer within \boxed{}.
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