Add authentication check and logout functionality in app.vue

app/app.vue

@@ -9,6 +9,15 @@
               FABER FINANCES
             </NuxtLink>
           </div>
+          <div class="flex items-center">
+            <button
+              v-if="authenticated"
+              @click="logout"
+              class="text-white hover:text-gray-300 px-3 py-2 text-sm font-medium"
+            >
+              Logout
+            </button>
+          </div>
         </div>
       </div>
     </nav>
@@ -23,4 +32,27 @@ useSeoMeta({
   titleTemplate: "%s - Faber Finances",
   description: "Personal finance and cash flow management system",
 });
+
+const authenticated = ref(false)
+
+// Check authentication status
+onMounted(async () => {
+  try {
+    const { authenticated: isAuth } = await $fetch('/api/auth/check')
+    authenticated.value = isAuth
+  } catch (err) {
+    authenticated.value = false
+  }
+})
+
+// Logout function
+const logout = async () => {
+  try {
+    await $fetch('/api/auth/logout', { method: 'POST' })
+    authenticated.value = false
+    await navigateTo('/login')
+  } catch (err) {
+    console.error('Logout error:', err)
+  }
+}
 </script>