In my old age, I can honestly say I’ve come across a lot of wildlife while running/backpacking - lots of black bears, a few brown bears, bobcats, moose, etc., but nothing scares the heebie-jeebies out of me like a big cat. They never announce their arrival, and once they engage you they usually don’t stop, as opposed to a brown bear, if you’re fortunate enough to back off (been there...). I have read where it’s best to try and “look big” - however that works - and just scream, yell, just make a lot of noise. I do know they have a natural aversion to dogs, and if I lived in the northwest I probably wouldn’t go anywhere in the mountains without a good sized dog (long time owner of airedales).
I run trails in MT. My dog has treed big cats twice. Out here, people hunt mountain lions with dogs and the cats are very wary of them. I generally run with a group and have bear spray and one companion has a pistol.