ci: Use biome ci in lib-check format (#4319)

Co-authored-by: Link <link20050703@gmail.com>

.github/workflows/lib-check.yml

@@ -60,18 +60,13 @@ jobs:
           node-version: 20
           cache: yarn
       - run: yarn install --immutable
-      - name: Turbo Cache
-        id: turbo-cache
-        uses: actions/cache@v4
-        with:
-          path: .turbo
-          key: ${{ runner.os }}-turbo-lint-${{ github.sha }}
       - name: Build type cache
-        if: steps.turbo-cache.outputs.cache-hit != 'true'
         uses: actions/cache@v4
         with:
           path: .turbo
           key: ${{ runner.os }}-turbo-build:type-${{ github.sha }}
+      - name: Build Types
+        run: yarn build:type --cache-dir=".turbo"
       - name: Check Formatting
         run: yarn check:all