Given the equation \(3x^2 - 4x + k = 0\) with real roots, find the value of \(k\) for which the product of the roots of the equation is a maximum. The original answer is in \(\frac{m}{n}\) format, where \(\frac{m}{n}\) is in its simplest form. Please give the value of \(m + n\).

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