GA flight instructor with student reported an NMAC at CGZ non-towered airport resulting in a go-around to avoid a collision.

Narrative

We were on instrument training mission and were practicing approaches at the TFD stack. It was the third approach that we were doing at the time of event; and there were about three other planes in the stack area for approaches and we were all previously doing Runway X low approaches or touch and go. On the third approach; we were able to join approach altitude right away after going missed and we were shooting the ILS X approach from TFD stack as guided by the stack procedure. There was light traffic and everyone was joining Runway X. Wind was light around 6 knots from 110 based on the weather information we copied before approach briefing. When inbound; I made the radio call for 4nm final Runway X for the option; and then 2nm final Runway X low approach; following the guidance. I didn't realize the previous airplane in the stack was doing circle to XX with touch and go and would be occupying the runway. I was acting under the impression that the circle to land traffic would be doing a full stop taxi back and therefore; would not be a conflict with our course. When we were on the short final; the other traffic announced on the radio 'traffic on final; go-around'. We were just about to go missed at that moment and we initiated go-around and announced that on the radio following the go-around action. The other airplane aborted takeoff; and the head-on conflict was mitigated at the last minute.