Link multiple sheets with data entered into one propagating to others w/different color

Hi All, I'd appreciate feedback on this.

I have sheets A,B,C in a workbook. This is for appointment scheduling of 3 doctors (Doctor A, Doctor B, Doctor C) sharing office rooms.
- Anything entered into A should get cell fill color Red and propagate to cells in sheets B and C using Red fill color.
- Anything entered into B should get cell fill color yellow and propagate to cells in sheets A and C using yellow fill color.
- Anything entered into C should get cell fill color green and propagate to cells in sheets A and B using green fill color.
- I can delete any of the entered data in any sheet and it will be deleted from all sheets and clear the fill color.

I know I can link cells in different sheets but not sure if this is used for copying v.s. just formula. I can also color code regions in each sheet based on conditional formatting (if cell has data make green) but don't know how this would work if each sheet has a different color. i.e.

please see attached xlsx sheet sample.



Any help here appreciated !!

Thank you so much @ AMvichopalacios

this is what I needed! My only issue now is that
if I merge two cells for say "Doctor A", only the 1st cell data and red fill color
is carried over to other sheets. If you point me to the right direction
I can fiddle more with the VB script.

Thanks again!

This is awesome help @ vichopalacios. I will work on it and let you know.

This is what i will do as I don't have too much experience with VB. I will take your advise. TY.