$(40$ pts$.)$ Given the following intersection geometry, volumes, and information,

$(1)$ Determine the minimum number of phases needed for the pretimed signal control.

$(2)$ Draw the phase diagram showing the movements for each phase $($indicate clearly protected vs$.$ permitted movements$).$

$(3)$ Determine the critical lane group for each phase.

$(4)$ Determine the optimal cycle length to minimize delay $($i$.$e$.,$ Webster's method$).$

$(5)$ Compute the yellow and all$-$red intervals for each phase.

$(6)$ Compute the effective and actual green times for each phase.

$(7)$ Determine whether the actual green times are greater than their respective minimum pedestrian green times. Pedestrians should be served during the phases that serve the vehicular through movements.