$\backslash$table$[[,,\backslash$table$[[$Preceding$],[$Activity$]],\backslash$table$[[$Personnel$],[$required$]],],[$A$,$Purchase construction materials,$13$ weeks,$-,3],[$B$,$Purchase catering equipment,,,$],[,$needed for new cafeteria,$3$ weeks,$-,2],[$C$,$Appoint supervisory architect,$2$ weeks,$-,1],[$D$,$Clear site,$3$ weeks,$-,3],[$E$,$Produce final building details,$9$ weeks,C$,$ D$,1],[$F$,$Prepare site for construction,$3$ weeks,E$,2],[$G$,$Construct cafeteria,$14$ weeks,A$,$ F$,5],[$H$,$Install utility services,$3$ weeks,G$,2],[$I$,$Install catering equipment,$5$ weeks,B$,$ H$,2],[$J$,$Decorate internally,$3$ weeks,B$,$ H$,2],[$K$,$Stock cafeteria,$3$ weeks,I, J$,1],[$L$,$Hire catering staff,$8$ weeks,$-,2],[$M$,$Train staff on site,$4$ weeks,I, L$,3]]$

$($a$)$ Identify the critical path.

$($b$)$ Determine float times for all activities.