Adding a new switch for recessed lights to existing 2 gang box

Hello Everyone. I am doing some remodeling and I am adding recessed lights to the living room where there is currently only one light source - from a ceiling fan - in the middle of the room. It has always been quite dark in the room so adding lights will greatly improve the lighting, (and my mood :vs_laugh: ). I am adding multiple LED recessed lights, strung together around the perimeter of the room. The current electrical source is a 2 gang box that has 4 cables coming into it. Two cables are 12/3 and 2 cables are 12/2.

The 12/3 cables run the ceiling fan motor and lights from 2 different switches and 2 different locations: two switches from one entry/exit point into the living room, and the second 2 switch set from another entry/exit point in the living room.

The 12/2 cables: one comes from the breaker box, the second goes to an outlet close to the switch box.

I have already replaced the 2 gang box with a 3 gang box and I have added the new (12/2) wire from the living room ceiling (recessed lights) to the switch box.

I need to know how to connect the wires to provide power to the new recessed lights.

I am adding a diagram of what the wiring looks like, as they are connected in the existing configuration.

Thank you in advance for your knowledge and help in making these connections.