All spores on a single island must be destroyed within a single round. Doing so also resets the Tentacle's timer so you aren't being attacked while doing this quest.

After all, the spores have been destroyed, progress to the next round where you will need to repeat the process, once destroyed, you will need to repeat it for a final time totaling three islands with spores on them. Afterward flip the round and head to the Sea Caves where you must destroy all the spores in the cave, completing the quest and rewarding you with loot.

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