$(20$ points$)$ Design a finite state machine $($FSM$)$ that detects the sequence $"1101"$ in a serial input stream, allowing overlapping sequences $($e$.$g$.,$ the input $"1101101"$ should detect two sequences$).$ Provide:

$($a$)$ The state diagram of the FSM$.$

$($b$)$ The state table showing the next state and output for each input.

$($c$)$ The Boolean expressions for the next state logic using D flip$-$flops.