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The DPS legendary ring effect should be used at the start of the encounter, to kill the first Doom Lord quickly, and then on CD from there. Fel Imps and Dread Infernals remain priority targets during the Mannoroth fight, so it may be worthwhile holding the ring effect a few seconds until a new wave of adds spawns.
The healer legendary ring effect should be used to heal through Fel Hellstorm damage or whenever the raid dips low from mechanics such as Mannoroth's Gaze or Empowered Shadowforce.
The tank legendary ring effect should be used to mitigate damage during the Glaive Combo abilities; this will help both tanks survive the powerful attacks.