refactor(app): rewrite application with Nuxt 4, Pinia, and TailwindCSS

This commit replaces the original vanilla JavaScript implementation with a modern frontend stack. The entire application has been rewritten to improve maintainability, developer experience, and leverage modern
web technologies.

Key changes include:
- Replaced plain HTML, CSS, and JS with a Nuxt 4 project structure.
- Migrated state management from a global state object to a Pinia store for better organization and reactivity.
- Rebuilt the UI with Vue 3 components and styled with TailwindCSS.
- Changed the primary persistence mechanism from manual file I/O to automatic LocalStorage saving, while retaining import/export functionality.
- All core features, including drag-and-drop, task management, and the import/merge diffing logic, have been ported to the new architecture.

public/sample-board.json

@@ -0,0 +1,36 @@
+{
+  "categories": [
+    { "uuid": "c-urgent", "title": "Urgent", "color": "ff3b30" },
+    { "uuid": "c-normal", "title": "Normal", "color": "34c759" }
+  ],
+  "stages": [
+    { "uuid": "s-todo", "title": "Todo", "tasks": ["t-sample-1"] },
+    { "uuid": "s-doing", "title": "Doing", "tasks": [] },
+    { "uuid": "s-done", "title": "Done", "tasks": ["t-sample-2"] }
+  ],
+  "tasks": [
+    {
+      "uuid": "t-sample-1",
+      "title": "Example: Set up project",
+      "description": "Try adding a stage, a task, and drag it.",
+      "category": "c-normal",
+      "steps": [
+        { "details": "Create repository", "done": true },
+        { "details": "Add README", "done": false }
+      ]
+    },
+    {
+      "uuid": "t-sample-2",
+      "title": "Example: Celebrate",
+      "description": "Mark a task done to see step stats.",
+      "category": "c-urgent",
+      "steps": [
+        { "details": "Share with team", "done": true }
+      ]
+    }
+  ],
+  "layout": {
+    "columns": [["s-todo"], ["s-doing"], ["s-done"]]
+  }
+}
+