A simple solution would be flagging the gate as cleared before the quest starts
Or remembering what you did on your own. Seems pretty simple
if you do all paths together yes, it's simple; if you do one path per day, it's a little more difficult to remember wich gate you have already opened. Sure, you can do a screenshot, or keep note on a paper, but having a way to actually see what you still need wouldn't be a bad thing, in my opinion
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I started on went half way through and ran out of energy. Later, when I came to finish the quest I accidentally backed out. Wasted a ticket.
This should help ya