I would put some pea gravel down (~2-3") and then sand (2-3"). Or just the sand. if you can afford it I'd also cover all or part of the run, old corrugated metal roofing works great. since you get snow, the roof would have to be strong and well supported.
At my last house I also had success with using bark nuggets - the big flat ones (2-3" around x 1/2" thick). I tried a lot of other mulch/bark type products and the big nuggets worked best. Only thing I didn't like was that when cleaning the run it was hard to separate the bark from the poop. I like to harvest the poop for my compost bin - the sand is perfect for that as it's like a big kitty litter box.
good luck!
At my last house I also had success with using bark nuggets - the big flat ones (2-3" around x 1/2" thick). I tried a lot of other mulch/bark type products and the big nuggets worked best. Only thing I didn't like was that when cleaning the run it was hard to separate the bark from the poop. I like to harvest the poop for my compost bin - the sand is perfect for that as it's like a big kitty litter box.
good luck!