Light aircraft instructor pilot reported an NMAC in the pattern at LYH when there was confusion in the cockpit regarding sequencing.

Narrative

I was training with a pre-private student in the LYH pattern. We were on the right downwind and there was another traffic on long final. Student was flying the aircraft and doing radios. The tower controller instructed us to report the traffic on final in sight. The student reported that he had the traffic in sight and was cleared to land number 2 behind the traffic. I did not have the traffic in sight at the moment and asked the student to confirm the traffic with me. The student stated that the traffic was well past us getting ready to land as he started turning base. At this point; I thought it wouldn't be a good idea to start turning into the final approach until both the student and I had the traffic in sight. I stated that I will take flight controls when the tower controller instructed us to make a right 360 turn back onto the downwind because we have flown into the path of the aircraft we were supposed to follow. I initiated a right climbing turn to avoid the traffic and got established back on right downwind. Contributing factors to this event was confusion between the student and I on where the traffic was; lack of situational awareness on where other aircraft were positioned in the pattern; and some miscommunication with the student. The student thought he had traffic in sight when he actually didn't. To prevent events such as this; we need to make sure that we actually have the traffic in sight and to keep situational awareness. I became aware [later] that the distance with the other aircraft during this event was approximately 400ft.