Hello yayiz
Here is what I usually do when a game doesn’t start with steam automatically:
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I had the same issue with CS:GO for some time. Restarting Steam one or two times solved the problem most the time. Maybe restart your computer.
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Check ProtonDB
Entry for Far Cry 3: https://www.protondb.com/app/220240
Try the recommended settings or launch parameters (right click on game in your library → properties → general) -
Try different Proton versions (right click on the game in your library → properties → compatibility)
Since the ProtonDB entries for older Proton versions are quite positive, you can try Proton 5 first. -
Google search
Hope you can find a solution.