use variables for setting path: key when using actions/cache ?

i'm attempting to cache the homebrew packages for my repo using the actions/cache github action in my workflow file. i use different runners for my project, ie. the arm64 macos runner, and the legacy amd64 / intel macos runners. with homebrew installed on macos arm, the default prefix ie. $(brew --prefix) is /opt/homebrew whereas on intel macs, the default homebrew prefix is /usr/local

i attempted to use the below step in my workflow file to setup some vars to be used throughout my workflow however i don't think this action actions/cache supports using variables stored in the GITHUB_ENV from what i have been trying.

  - name: set brew prefix & cache paths
        id: set-brew-prefix-and-cache-paths
        run: |
           bp=$(brew --prefix)
           hbc="$bp/cache"
           echo "bp=\"$bp\"" >> "$GITHUB_OUTPUT"
           echo "hbc=\"$hbc\"" >> "$GITHUB_OUTPUT"
           echo "bp=\"$bp\"" >> "$GITHUB_ENV"
           echo "hbc=\"$hbc\"" >> "$GITHUB_ENV"

  - name: save homebrew install
        uses: actions/cache@v2
        env:
          bp: ${{ steps.set-brew-prefix-and-cache-paths.outputs.bp }}
        with:
          #----
          # NO WORK!
          # path: $(brew --prefix)
          #----
          # NO WORK!
          path: $bp
          #----
          # NO WORK!
          # path: ${{ env.bp }} # does not properly save /opt/homebrew on macos arm64
          #----
          # key: ${{ env.cache-key }}
          # key: ${{ env.envHash }}-freecad-deps-${{ runner.os }}-brew
          # key: homebrew-packages-${{ runner.os }}-${{ github.run_id }}
          # key: ${{ steps.cache-brewdeps-restore.outputs.cache-primary-key }}
          key: ${{ runner.os }}-brewdeps

the only way i've been able to properly save a cache is to use a literal path ie. /opt/homebrew

does this action not support the use of variables when setting the path: key?

Start with the workflow step shown in the report, especially the actions/cache@v2 path input and the set-brew-prefix-and-cache-paths outputs and GITHUB_ENV writes. Compare how those values are consumed on ARM64 and Intel macOS runners; done means establishing whether variable-based paths are supported and documenting the resulting behavior.