$[20$ pts$]$ Convert the following transition graph into a finite automaton by filling in the $($final$)$ table and identifying the start and accept states. You do not have to draw the state transition diagram for the FA$,$ just fill in the table $($note that this is NOT the intermediate table.

$\backslash$table$[[$FA State,a$,$b$],[,,],[,,],[,,],[,,],[,,],[,,],[,,],[,,]]$

Show all the work please!!!