Design a chronometer that starts from $1$ and counts up when a START input entered. The chronometer can count time until $(4)10.$ When PAUSE input is entered, the time value does not increase, and it can only resume counting with clock if START is entered again. If PAUSE and START are pressed at the same time, it goes to $1$ directly, basically resets the chronometer. When the chronometer reaches $4,$ it rolls back to $1$ and outputs LAP $=1($otherwise$,$ LAP $=0).$ Design the sequential logic circuit for this chronometer. Each part below is worth $25$ points. $($Hint: Choose number of state bits wisely$)$

$($a$)$ Construct the state table including the present and next states as well as inputs and outputs.

$($b$)$ Find the simplest state equations of this circuit.

$($c$)$ Design the state diagram for this circuit.

$($d$)$ Draw the logic circuit that implements the described sequential logic operations.