How to add Utility Sink and Washer Drain to existing drain stack

I am trying to add a sink drain and also a washer box. The black lines in the photo show existing infrastructure. I would like to install a sink and washer box. I am assuming I will need to cut into the 3 to 4" section with the cleanout and install some sort of T with a 2" out to the drain for the sink and washer. Where do I put the P trap or traps and also where is it best to vent? Does the vent go between the washer and sink? I also have a 2" pipe on the other side of the wall that could be accessed overhead for a vent. Could someone please help draw this out for me? Thank you!!!

Tim

Cut the sanitary tee for the new drain line in low, maybe 8" or so to the center off of the floor. In the center of the utility sink, turn a 2" combination wye and 1/8th bend up, and stack a 2"x1 1/2"x1 1/2" santee on top of that. The branch of that tee will hook up to the utility sink drain Ptrap, and the top of that tee will be your vent. Take that vent up above the level of both the utility sink and the washing machine box and run it through the wall to the vent. Out of the end of the combination, run the pipe to the area of the washing machine, and install a 2" Ptrap. Add a 24" or so riser to the top of the Ptrap, and attach your washing machine box to the top of that.

Edit: One thing to make sure of is that the drain height for the utility sink will not be too high with this arrangement. Different style sinks need different height rough in heights.