Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of $105$ free throws between them, with each person taking at least one free throw. If Andrew made exactly $\frac{1}{3}$ of his free throw attempts and Beatrice made exactly $\frac{3}{5}$ of her free throw attempts, what is the highest number of successful free throws they could have made between them?

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