I have problems with solving this question. I have added an image about it. I am pretty new to finite state machines so its hard for me to figure this state transition table out.

P.3 Given the following finite state machine: a)Find the shortest synchronizing sequence (if it does not exist find the shortest homing sequence) c)Find the shortest distinguishing sequence if any exists. d)Construct a fault detection experiment (checking sequence) to verify the existence of four distinct states and the state transitions