Let \(ABCD\) be a circumscribed quadrilateral such that \(m(\widehat{A})=m(\widehat{B})=120^\circ\), \(m(\widehat{C})=30^\circ\), and \(|BC|=2\). Find the length of \(|AD|\). The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m.

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