In triangle \(ABC\), \(\angle BAC = 60^\circ\), \(AB = 3a\), and \(AC = 4a\), where \(a > 0\). Let \(M\) be a point on segment \(AB\) such that \(AM = \frac{1}{3} AB\), and \(N\) be a point on side \(AC\) such that \(AN = \frac{1}{2} AC\). Let \(I\) be the midpoint of \(MN\). Determine the length of \(BI\). The original answer is in the form \(\frac{ka\sqrt{m}}{n}\), please give the value of \(k + m + n\).

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