Let \(ABC\) be a triangle such that \(|AB| = 8\) and \(|AC| = 2|BC|\). What is the largest value of the altitude from side \([AB]\)? The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).

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