The system shown in the image was designed to move $1253$ gal$/$min from Reservoir $1($elevation $=52$ ft$)$ to Reservoir $2($elevation $=146$ ft$),$ via $500$ feet $($total$)$ of pipe with a roughness coefficient of C $=100.$ The pump adds $116$ ft of head to the system. Assuming a constant diameter, find the diameter of the pipe. You may neglect minor losses.