Ugh. Yeah. This one is rough. Unfortunately there's no real puzzle linchpin here, it's just a matter of dancing the goblins around. I'll write some tips for surviving, but I also posted a demo in case you want to just be done with this room.
First, it helps to try to keep the number of goblins that are "
out"
as low as possible. So you can use that as a bit of a gauge of if you are being efficient enough. You don't really want more than 2 or 3 goblins out for more than a few turns.
At the end I was able to survive with four goblins out when the roach escaped. So 4 is manageable (though tricky).
I started the room with my sword pointing North and moving south immediately I was able to kill the first two goblins on yellow doors fairly quickly.
Other than that, it's a lot of guess-and-check. The attached demo isn't very optimal, but hopefully helps a bit.
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