Call admissible a set $A$ of integers that has the following property: If $x,y\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\in A$ for every integer $k.$ Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. Please provide the sum of the pair $(m, n)$.

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