Let $ABC$ be an acute triangle with $\angle ABC = 60^{\circ}$. Suppose points $D$ and $E$ are on lines $AB$ and $CB$, respectively, such that $\triangle CDB$ and $\triangle AEB$ are equilateral triangles. Given that the positive difference between the perimeters of $\triangle CDB$ and $\triangle AEB$ is $60$ and $DE = 45$, what is the value of $AB \cdot BC?$

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