Find the minimum in a proper subset with those condtions

AA is a non-empty proper subset of Z\mathbb Z with at least two elements, and with the following property:

If aa and bb are distinct elements of A,A, then a,-a, a+2b,a+2b, and a+19a+19 are also elements of A.A.

Find minaAa0(a).\min \limits_{a \in A \atop a \neq 0}\big(|a|\big).

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