In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), please find the value of \(k\).

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