Avoiding four-term progressions from finitely many starts

For every finite set $A\subset\mathbb{Z}$, we construct a bijection $p:\mathbb{N}_0\to\mathbb{Z}$ beginning with $0,1$ that contains no four-term arithmetic progression, in occurrence order, whose first value belongs to $A$. More generally, for each $n\ge2$ the enumeration can begin with $0,1,3,\ldo