Find the minimum in a proper subset with those condtions

Number Theory Level 2

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