Find the vertical asymptote of the rational function \( f(x) = \frac{x^2 + 5x + 6}{x^2 - 4} \). Then, determine the value of \( k \) such that the quadratic equation \( x^2 + kx + 16 = 0 \) has a root at the vertical asymptote. Let's think step by step and output the final answer within \boxed{}.
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