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This is one of the easier Eidolon battles in Final Fantasy XIII (Edit- According to the comments below, this is not true. I thought it was easy myself, maybe I got lucky?). Hecatoncheir will begin his battles by posing for about 30 seconds, and then will launch into a devastating combo that will pretty much kill whoever he targets outright- even if they are on full health. This means that, like it or not, this Eidolon battle will be dictated by Hecatoncheir’s actions.
While he is using looming wrath, hit Hecatoncheir with everything you have using the slash and burn paradigm to get his Gestalt gauge up. Once he slips into the combo attack, slip into the stumbling block paradigm to ensure that this eidolon doesn’t wipe out a party member (Remember, Fang’s sentinel role reduces incoming damage). You will gain some points on the gestalt gauge by inflicting debuffs on Hecatoncheir.
After that, at least one of your party members will be in pretty poor shape, and the other will probably have a few battle wounds as well. Switch to life guard to heal up while still building up your gestalt gauge a little. If you can, switch back to slash and burn when everyone’s looking in good health.
From now on, you’ll need to make use of the stumbling block or life guard paradigms to defend against Hecatoncheir’s attacks, the former for building up the gestalt gauge, the latter for making sure everyone survives. Switch to relentless assault when you can to help build the gauge up more quickly, but remember that you can’t win if you’re dead!
However you play this Eidolon battle, it’s probable that the doom counter will be in triple digits before you win, so don’t panic when it ticks down- just keep fighting hard! Remember, debuffs will help build the gestlat gauge, so it’s not all about the DPS.