A traffic light controller receives a $4-$bit input that changes every $5$ seconds. The input sequence is a simple

counter that counts from binary $0$ to binary $15$ and then starts again from binary $0.$ These signals go to $4$ outputs,

NSH$,$ NSL$,$ EWH, EWL. The first two outputs NSH and NSL respectively represent the high and low bits that

are fed to the lamps that face in the North$-$south direction. The other two outputs EWH and EWL respectively

represent the high and low bits that are fed to the lamps that face in the East$-$West direction. The following table

indicates how setting the high and low bits determines the color of the lamp:

High Low Lamp Color

$00$ Green

$01$ Yellow

$10$ Red

$11$ Red

Each light should be green for $30$ seconds, yellow for $5$ seconds, and then red for $45$ seconds. There should

be two $5$ second intervals when both lights are red. Assume that for the interval where the input is $0000,$ the

North$-$South light has just turned green, and the East$-$West light has been red for $5$ seconds already.

Design the circuitry that feeds from ABCD to the two outputs. You may represent your answer as algebraic

equations $($make sure to simplify$).$