# Find the minimum in a proper subset with those condtions

$$A$$ 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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