Lot's of good suggestions so far.
Since you are doing true back country skiing in Japan, you really need a light weight ski. I would suggest you keep it at 1900 grams or less. The max weight ski I would suggest is the Line Sick Day 114, at 1920 grams. Go to Blister review site and look at this years buyers guide on line. Look at their 50/50 category and there are a number of great skis that would work for you. A couple I would suggest are the Atomic Bent Chetler 120, 1720 g; the Moment Wildcat tour 116. 1800 g; and the 4FRNT Raven 104, 1750 g. These will be good in Japan, especially with a shift binding on them. They will also work well as a powder ski in Utah. You should get boots with a walk mode and pins, I would stay with the grip walk or WTR soles (or changwable soles) so you can use the boots in alpine spec bindings. That is what I did (WTR) and I really like the light weight touring boot for soft snow days, inbounds, especially if I am going in the glades. I also use them for coaching my race team, as the walk mode is great when standing around.
As others have said, you need to decide on touring gear versus downhill gear. You will get a lot more enjoyment from your trip if you are on lightweight touring gear, especially the bindings and boots. You want the "downhill oriented" end of the touring spectrum. Renting maybe your best option. Then you can get whatever wide ski you want for Utah (or a one ski quiver type if that is your desire) that is ideal for your skiing there. You won't have to invest in new boots and can stick with lower priced downhill bindings. I bought a pair of shifts and love them, but they were $500 versus $200 for downhill bindings. I occasionally travel to Japan and have considered a ski trip there. I would be more side country focused, such as the trips Matt runs. I wan't going to shell out the $$ for a trip till I had the right gear for it. I do now and I am glad I bought it. It made a sizeable dent in my wallet, but "you can't take it with you" and my buddy keeps telling me the boys with the most toys win !!!