$(20$ points$)$ A signalized intersection operates a pretimed three$-$phase signal. The saturation flow rate is $1500vph.$ Flows on critical lane groups are indicated in the figure. Lost times for phases $1,2,$ and $3$ are $3,4,4$ seconds.

$(1).$ Find the minimum cycle length that matches this demand.

$(2).$ Determine phase splits for phase $2$ assuming approach speed $30mph,$ level road, intersection width $60ft,$ and average vehicle length $20ft.$