$(30$ pts$)$ SP$3.$ The state assignment for an asynchronous sequential circuit is shown in Table $3.$

$($a$)$ List all k$-$sets under each column.

$($b$)$ List all Transition pairs and their corresponding transition paths under each k $-$set.

$($c$)$ Form Transition tables for each input, and show all transition paths of transition pairs for all k $-$sets.

$($d$)$ List all critical races and list them.

Table $3.$ Flow table with arbitrary state assignment