gh-36636: Deploy live doc on push to develop 
    
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Live doc is built separately from the usual doc preview on push to
`develop`, and deployed to Netlify.

Sample deployment: https://deploy-livedoc--
sagemath.netlify.app/html/en/a_tour_of_sage/

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### ⌛ Dependencies

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- #36601

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URL: #36636
Reported by: Kwankyu Lee
Reviewer(s): Kwankyu Lee, Matthias Köppe

Author: Release Manager

.github/workflows/doc-build.yml

@@ -197,3 +197,36 @@ jobs:
         with:
           name: docs
           path: docs.zip
+
+      - name: Build live doc
+        id: buildlivedoc
+        if: (success() || failure()) && steps.copy.outcome == 'success' && github.repository == 'sagemath/sage' && github.ref == 'refs/heads/develop'
+        run: |
+          set -ex
+          export SAGE_USE_CDNS=yes
+          export SAGE_LIVE_DOC=yes
+          export SAGE_JUPYTER_SERVER=binder:sagemath/sage-binder-env/dev
+          make doc-clean doc-uninstall
+          ./config.status && make sagemath_doc_html-no-deps
+        working-directory: ./worktree-image
+        env:
+          MAKE: make -j2 --output-sync=recurse
+          SAGE_NUM_THREADS: 2
+
+      - name: Copy live doc
+        id: copylivedoc
+        if: (success() || failure()) && steps.buildlivedoc.outcome == 'success'
+        run: |
+          set -ex
+          mkdir -p ./livedoc
+          cp -r -L /sage/local/share/doc/sage/html ./livedoc
+          cp  /sage/local/share/doc/sage/index.html ./livedoc
+          zip -r livedoc.zip livedoc
+
+      - name: Upload live doc
+        if: (success() || failure()) && steps.copylivedoc.outcome == 'success'
+        uses: actions/upload-artifact@v3
+        with:
+          name: livedoc
+          path: livedoc.zip
+

.github/workflows/doc-publish.yml

@@ -90,5 +90,48 @@ jobs:
 
       - name: Report deployment url
         run: |
-          echo "::notice::The documentation has being automatically deployed to Netlify. %0A ✅ Preview: ${{ steps.deploy-netlify.outputs.NETLIFY_URL }}"
+          echo "::notice::The documentation has been deployed - ${{ steps.deploy-netlify.outputs.NETLIFY_URL }}"
+
+  publish-live-doc:
+    runs-on: ubuntu-latest
+    if: github.event.workflow_run.conclusion == 'success' && github.event.workflow_run.head_branch == 'develop'
+    steps:
+      - name: Download live doc
+        uses: actions/[email protected]
+        with:
+          script: |
+            var artifacts = await github.rest.actions.listWorkflowRunArtifacts({
+               owner: context.repo.owner,
+               repo: context.repo.repo,
+               run_id: ${{github.event.workflow_run.id }},
+            });
+            var matchArtifact = artifacts.data.artifacts.filter((artifact) => {
+              return artifact.name == "livedoc"
+            })[0];
+            var download = await github.rest.actions.downloadArtifact({
+               owner: context.repo.owner,
+               repo: context.repo.repo,
+               artifact_id: matchArtifact.id,
+               archive_format: 'zip',
+            });
+            var fs = require('fs');
+            fs.writeFileSync('${{github.workspace}}/livedoc.zip', Buffer.from(download.data));
+
+      - name: Extract live doc
+        run: unzip livedoc.zip -d doc && unzip doc/livedoc.zip -d doc/doc
+
+      - name: Deploy to Netlify
+        id: deploy-netlify
+        uses: netlify/actions/cli@master
+        with:
+          args: deploy --dir=doc/doc/livedoc --message ${NETLIFY_MESSAGE} --alias ${NETLIFY_ALIAS}
+        env:
+          NETLIFY_AUTH_TOKEN: ${{ secrets.NETLIFY_AUTH_TOKEN }}
+          NETLIFY_SITE_ID: ${{ secrets.NETLIFY_SITE_ID }}
+          NETLIFY_MESSAGE: Deployed live doc
+          NETLIFY_ALIAS: deploy-livedoc
+
+      - name: Report deployment url
+        run: |
+          echo "::notice::The live documentation has been deployed - ${{ steps.deploy-netlify.outputs.NETLIFY_URL }}"