And here's a super-simple version, because why not.

import Data.List (elemIndex, elemIndices)  
import Data.Map qualified as Map  
import Data.Maybe (fromJust)  
import Data.Set qualified as Set  

main = do  
  (start : rows) <- lines <$> readFile "input07"  
  let splitsByRow =  
        zipWith  
          ( \row beams ->  
              Set.intersection (Map.keysSet beams)  
                . Set.fromDistinctAscList  
                $ elemIndices '^' row  
          )  
          rows  
          beamsByRow  
      beamsByRow =  
        scanl  
          ( \beams splits ->  
              let unsplit = beams `Map.withoutKeys` splits  
                  split = beams `Map.restrictKeys` splits  
                  splitLeft = Map.mapKeysMonotonic pred split  
                  splitRight = Map.mapKeysMonotonic succ split  
               in Map.unionsWith (+) [unsplit, splitLeft, splitRight]  
          )  
          (Map.singleton (fromJust $ elemIndex 'S' start) 1)  
          splitsByRow  
  print . sum $ map Set.size splitsByRow  
  print . sum $ last beamsByRow