A cone-shaped birthday hat has a height that is $60\%$ of its base diameter. If the diameter of the hat is 10 cm, what is the volume of the hat? Use the formula for the volume of a cone, \(V = \frac{1}{3} \pi r^2 h\), and express your answer in terms of \(\pi\). Let's think step by step and output the final answer within \boxed{}.
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