Installing a new bathroom setup and I would like some help on how to setup the drain and vent (using Air Admittance Valves).

Old triplex converted to single house, with only one 4" drain/vent column located in back portion of the house. New bathroom on first floor is to be in middle of the house (approx 24 feet away from column).

See attached JPG for sketch.

My question is can I use only one Air Admittance Valve (see my diagram) or do I need 2 has I think? Is my shower to far to use the A.A.V. of the sink/toilet?

Also the toilet will connect to a 3"TY and the T portion will reduce to a 2" to accomodate my sink and Air Admittance Valve. Is that ok?

Is there a potential problem for my toilet and shower? How should my 3" pipe from the toilet connect to the 3" drain? I was thinking of using a standard TY (where the toilet portion connects to the Y part).

Thanks in advance for any suggestion.