Considering how late game it appears and how everyone who likes it up the butt wouldn't really wait that long to discover the joys of anal. You can already buy Thicc&Shake at vending machines at Tavros and Mhen'ga to get the perk early on as well. Considering how Tavros is supposed to be a base for many Rushers and includes characters like Sera and Aliss means that it works well with the bad end and also opens up some posibilities for scenes where you bring people and have them use it or meeting a new character by walking in on them if it's not the first time you've used the machine. My idea for its location is the northernmost tile of the residential deck. When you step on the tile there is a chance for you notice the empty east apartment's door slightly ajar and with a suspicious lack of latch to hold it closed. This would change KnockEast to Investigate until you check it out giving you a scene where you look around the apartment to find it stripped of all its amenities except for a suspicious machine in the center if the empty room and letting you walk in and out with the directional buttons. The room's background text would describe the machine and how it faces away from the door, and change slightly to refer to it as a familiar machine instead of mysterious once you've used it. I'm going to try and write this so if anyone has any suggestions I'm going to check back here later. Even if something like this doesn't get in I think we can all agree that Tavros is a significantly better place to house the Buttslitinator than Uveto.
