Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).

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