Draw the state diagram of an FSM which accepts a binary stream and outputs a $1$ whenever it detects a pattern $1111.$ Overlapping sequences are allowed. For example:

Input Stream: $111101111100101$

Output: $000100001100000$

a$.$ Perform state encoding. Show the tabular representations of next state and output functions.

b$.$ Derive minimized Boolean equations for next state and output functions.