Guild Tasks(Quests) features three guilds that offer "Tasks" that, once completed, have some very nice rewards. Each task is different and non-repeatable. Many of the rewards are similar -- Gold and Blueprints especially. Some quests, however, are the only source for "Daily" or "Weekly" magazines that will allow the team to obtain additional drops and splits from monsters.
Battle Chain is a running tally of defeated monsters. The tally resets to 0 upon exiting to the world map, upon leaving a dungeon by completing a quest, or upon being ambushed. The higher the chain, the more stat gains and loot the party will receive from battles; the maximum displayable chain seems to be 9999.
Battle Rank (BR) is the Last Remnant's version of a leveling scheme. Unlike traditional RPGs, where character level translates directly into character strength, BR is a measure of battlefield performance that boosts enemy difficulty and enables certain team improvements as it rises. It can be viewed after battle, or on the PC version's party menu screen.
You can basically find any information you need for this game from the wikia..it's pretty awesome.
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