What is the best upper and lower bound you can give on the asymptotic runtime of

the above algorithm,

$($a$)$ assuming LPWC is true? $($Use O$,,\backslash$Theta as appropriate.$)$

$($b$)$ assuming LPWC is false? $($Use O$,,\backslash$Theta as appropriate.$)$

$($c$)$ assuming LPWC remains unknown? $($Use O$,,\backslash$Theta as appropriate.$)$