Office Space: Charting A Path

Repeat after me: One cannot solve everything in one meeting.
This is especially true with problems are multi-faceted and/or there are complicated personalities involved.
What you need is a plan. How to see the board before you and chart your path. I do a lot of embroidery, blackwork embroidery specifically. And in blackwork, you have to plan how you will complete the elements of your pattern and work your way out of there. If you don't visualize ahead you can find yourself trapped somewhere with nowhere to place the threads that will lead you out cleanly.
This happens in conversations as well.
The more complex the issues and the more contentious the parties involved, the more you have to think about how to get into the individual elements and get back out cleanly to the next. You cannot control how others react to the world. But, you can both prepare yourself for your best entry and be decisive about the path you wish to travel.