All quest groups (diads or triads) start with the Main, non-repeatable quest, which unlocks the repeatable quests.
The Daily and Public versions of this quest cannot be performed simultaneously.
Quest Type
Needs
Main
Defeat a captive monster
Daily
same as Main
Public
same as Main
Notes
Spoiler Alert! We have hidden this information so that you have the choice to try it on your own, or have the advantage of knowing how the sausage is made...
You cannot attack the target until it starts to move, so just wait a second or so. When the fight starts, hit the monster with any attack.
The demonstration targets CAN drop items (or at least Festival Items) but you need to be at melee-range and ready to loot them instantly when they die, as they fade in less than a second.
Daily Quests are repeatable up to a total (for all Daily Quests) of 10 times per day. The count can be reset to zero by purchasing and using a Daily Quest Ticket from the Item Shop (for 39 ($1.16)† ) or from any Phirius Merchant (for 780 ).
In all other zones they do not reward XP, TP or Gold, only Public Event Points. In all other aspects they are the same as a Daily Quest (except for the infinitely repeatable part and the rewards). Public Event Points contribute towards the completion of Regional Quest objectives, which in turn rewards a number of Energy of Justice commensurate with your amount of participation in the Regional event.
Daily Quest Ticket from the Item Shop (for 39
($1.16)† ) or from any Phirius Merchant (for 780
).
Dual-Function Daily Quest Potion.
Old Bag (hence the common sobriquet, Bag Quests). In Redhill Mountains, each repetition of [67][Public] Freedom Fighters rewards an 
Ancient Memento.