feat(core): bettery save mode (#12996)

<!-- This is an auto-generated comment: release notes by coderabbit.ai
-->

## Summary by CodeRabbit

* **New Features**
* Introduced a battery save mode that can pause background sync and
processing to conserve energy.
* Battery save mode is automatically enabled when the app window loses
focus and disabled when the window regains focus or is interacted with.
  * Available in both web and desktop (Electron) versions.

* **Improvements**
* Added user activity detection to optimize background operations for
better battery efficiency.

<!-- end of auto-generated comment: release notes by coderabbit.ai -->

packages/common/nbstore/src/worker/client.ts

@@ -86,6 +86,21 @@ export class StoreManagerClient {
       connection.dispose();
     });
   }
+
+  enableBatterySaveMode() {
+    this.connections.forEach(connection => {
+      connection.store.enableBatterySaveMode().catch(err => {
+        console.error('error enabling battery save mode', err);
+      });
+    });
+  }
+  disableBatterySaveMode() {
+    this.connections.forEach(connection => {
+      connection.store.disableBatterySaveMode().catch(err => {
+        console.error('error disabling battery save mode', err);
+      });
+    });
+  }
 }
 
 export class StoreClient {
@@ -113,6 +128,13 @@ export class StoreClient {
   readonly blobFrontend: BlobFrontend;
   readonly awarenessFrontend: AwarenessFrontend;
   readonly indexerFrontend: IndexerFrontend;
+
+  enableBatterySaveMode(): Promise<void> {
+    return this.client.call('sync.enableBatterySaveMode');
+  }
+  disableBatterySaveMode(): Promise<void> {
+    return this.client.call('sync.disableBatterySaveMode');
+  }
 }
 
 class WorkerDocStorage implements DocStorage {