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For this achievement, you need to defeat this boss without killing any Zandalari Dinomancers.  These are oversized troll mobs that spawn after a few waves of adds at each gate.  When they hit 50% health, they turn into a dinosaur and drop an orb that will make Horridon smash the gate when picked up, and add waves will stop come from that door and instead start from the next one.

One easy method of getting this achievement is to three tank the fight.  Once each dinomancer is brought to 50% health (so it will drop that needed orb), an offtank can taunt them to remove them from the fray and hold them, alive, off to the side until the end of the fight.  Just be careful not to accidentally kill them!

You do not need to smash all the gates to complete this achievement; if your raid desires and has the DPS, you can burn Horridon at an earlier door.