Consider the sequence defined by the recursive relationship \( a_{k+1} = a_{k-1} - \frac{3}{a_k} \) for \( k = 1, 2, \dots, m-1 \) with initial conditions \( a_0 = 2 \) and \( a_1 = 5 \). Determine the value of \( m \) such that \( a_m = 0 \). Let's think step by step and output the final answer within \boxed{}.
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