What decides what path in the temple of Styx has a bag at the end?

Is there any way to know which path has the bag at the end?

I've done a good bunch of runs that went into the temple of Styx and could never figure out what the logic for where the bag is. I've got it on the any number of paths I go down except first (2nd path chosen and after) on boss paths, normal paths, boon paths, upgrade paths, etc.

So is there any logic for which path is the bag path except random except the first one you choose?

Solution 1:

The number of wings you have to check is random, between two and five inclusive, which means the only guarantee you have is that your first chosen path will not contain the sack.

This is confirmed by Hades speedrunners, who are probably the best authority on the game next to the actual developers.