Neither Wayout nor Capture the flag, it's older prettier brother uses raycasting.
Sources are out and there's some more info in AtariAge forums where some maniacs ripped it apart and even made it faster using more current tricks.
https://atariage.com/forums/topic/320414-smooth-3d-movement-...
I once walked out of a John Carmack SIGGRAPH talk when he opened with claiming he invented the FPS because of Ultima Underworld, The Eidolon, and Way Out...
Like him otherwise, but nope he didn't invent it...
Hot take: The Eidolon is more impressive than this effort and it works with the technological limitations of the platform rather than against it like this does.
All those LucasArts games were stunning for the time and Rescue At Fractalus has a genuinely terrifying experience within it not replicated IMO until the first time you hear "Anytime..." in Alien vs Predator...
Good intro to raycasting for those interested: https://lodev.org/cgtutor/raycasting.html