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

If \(a\) and \(b\) are distinct elements of \(A,\) then \(-a,\) \(a+2b,\) and \(a+19\) are also elements of \(A.\)

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

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