The ellipse $x^2+4y^2=4$ and the hyperbola $x^2-m(y+2)^2 = 1$ are tangent. Compute $m.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

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