Flight Instructor with student reported a NMAC while entering the traffic pattern at AUO non-towered airport. Flight Instructor took evasive action to avoid a collision.

Narrative

While attempting to enter the traffic pattern at AUO; my student and I experienced a Near Mid Air Collision. We made our radio calls at 8; 4 and 3 miles away from the airport. I heard that there was an airplane taking off Runway 36 (our intended runway) and they were southwest bound which could cause a traffic conflict if they decided to exit on the downwind or immediately turned to the southwest. It is not in our company policy to make the 4-mile call; but at the time; I saw the airplane was then turning cross wind which could cause a traffic conflict. As we began to make the 3-mile call; the aircraft departing the cross wind continued their turn towards the southwest and climbed in the direction we were headed for entering the traffic pattern. Soon thereafter; the TCAS advised me that there was traffic 200 ft. below us and appeared to be continuing in our direction. At that time; I promptly took the controls from my student and did a climbing 180 degree turn to avoid the traffic. After the traffic passed; I asked if they had us in sight. The traffic said they did have us in sight. I am unsure if they meant they had us in sight the entire time or when I asked if they had us in sight. However; I do not believe the traffic ever had us in sight as they not only continued their climb; but also; they turned into us. I believe there are multiple ways to prevent this problem from happening in the future. There is a lot of flight training at AUO across multiple flight schools. I believe if the flight schools discussed their procedures with each other; they would be more aware of what the other schools are planning to do. Also; I believe the need for AUO to have a Control tower would greatly alleviate the chaos that happens in traffic patterns or nearby.