ci: deploy fails because of missing public/ dir

Problem:
Since moving to `peaceiris/actions-gh-pages` #401
there is a new problem:

    /home/runner/actions_github_pages_1764026989086
      cp: no such file or directory: /home/runner/work/neovim.github.io/neovim.github.io/public/*
      cp: no such file or directory: /home/runner/work/neovim.github.io/neovim.github.io/public/.*

The old way magically worked because `actions/upload-pages-artifact` is
internally handled by `actions/deploy-pages`. There is no complementary
`actions/download-pages-artifact` so we have to do some other approach.

Solution:
Do the build in the deploy step, avoid uploading/downloading artifacts.

Author: justinmk

.github/actions/build.yaml

@@ -0,0 +1,26 @@
+name: "Build Hugo Site"
+description: "Shared Hugo build logic"
+
+env:
+  HUGO_VERSION: "0.152.2"
+
+runs:
+  using: "composite"
+  steps:
+    - name: Install Hugo CLI
+      run: |
+        wget -O ${{ runner.temp }}/hugo.deb \
+          https://github.com/gohugoio/hugo/releases/download/v${{ env.HUGO_VERSION }}/hugo_extended_${{ env.HUGO_VERSION }}_linux-amd64.deb \
+        && sudo dpkg -i ${{ runner.temp }}/hugo.deb
+    - name: Checkout
+      uses: actions/checkout@v6
+      with:
+        submodules: recursive
+        fetch-depth: 0
+    - name: Build with Hugo
+      env:
+        # For backward compatibility with Hugo modules
+        HUGO_ENVIRONMENT: production
+        HUGO_ENV: production
+      run: |
+        hugo --gc --minify --baseURL "$BASE_URL"

.github/workflows/build.yaml

@@ -0,0 +1,16 @@
+# Builds but does not deploy, for pull request validation.
+
+name: build
+
+on:
+  # Runs on pull requests
+  pull_request:
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+    steps:
+      - name: Build site
+        uses: ./.github/actions/build-site
+        env:
+          BASE_URL: "https://example.com/"

.github/workflows/deploy.yaml

@@ -0,0 +1,51 @@
+# Builds and deploys the site to GitHub Pages.
+# Does NOT run on pull requests.
+
+name: build-and-deploy
+
+on:
+  # Runs on pushes targeting the default branch
+  push:
+    branches:
+      - master
+
+  # Allows you to run this workflow manually from the Actions tab
+  workflow_dispatch:
+
+# Sets permissions of the GITHUB_TOKEN to allow deployment to GitHub Pages
+permissions:
+  contents: read
+  pages: write
+  id-token: write
+
+# Allow only one concurrent deployment, skipping runs queued between the run in-progress and latest queued.
+# However, do NOT cancel in-progress runs as we want to allow these production deployments to complete.
+concurrency:
+  group: "pages"
+  cancel-in-progress: false
+
+# Default to bash
+defaults:
+  run:
+    shell: bash
+
+jobs:
+  # Deployment job
+  build-and-deploy:
+    runs-on: ubuntu-latest
+    permissions:
+      contents: write
+    # environment:
+    #   name: github-pages
+    #   url: ${{ steps.deployment.outputs.page_url }}
+    steps:
+      - name: Build site
+        uses: ./.github/actions/build-site
+        env:
+          BASE_URL: "${{ steps.pages.outputs.base_url }}/"
+      - name: Deploy
+        id: deployment
+        uses: peaceiris/actions-gh-pages@v4
+        with:
+          publish_dir: ./public
+          github_token: ${{ secrets.GITHUB_TOKEN }}