Hinge (on) (4)
(Other definitions for rely that I've seen before include "All the letters" , "Depend or lean on" , "Depend confidently (on)" , "Run and see" , "Depend, bank" .)
'hinge on' is the definition.
(thesaurus)
This is all the clue.
(Other definitions for depend that I've seen before include "Rest" , "Be contingent upon" , "Rely or hang on" , "Count on or hang from" , "Hang down or be connected to something" .)