Carry conversations into new sessions with a visible summary

- Bots get continue_in_new_session: the bot writes a title and summary,
  the session closes for good (read-only) and the new one opens with the
  summary as a "Carried over from the last session" message that links
  back to the old session.
- Continue in new session (conversation menu, sessions modal) and the
  automatic rollover at the session length do the same; New session now
  starts a fresh, unlinked session.
- Session length is an app setting in Settings > General.
- ?session_id= opens a session; a carried-over one shows a bar linking to
  where the conversation went on instead of the composer.
- The thread can open the previous session above the current one (shown
  only, never in the prompt), and replies get search_past_sessions and
  read_past_session.
- Sessions are named from their opening messages, not their latest, and
  a carried summary is labelled as the earlier session's.

Co-Authored-By: Claude Opus 5.5 <[email protected]>

src/functions/backend/db/bot-sessions/has-open-session-job.ts

@@ -0,0 +1,20 @@
+import BunSQLite from "@moduletrace/bun-sqlite";
+
+type Params = {
+    session_id: number | string;
+};
+
+/**
+ * Whether the bot still has work queued or running in a session, whose
+ * answer lands there: a session isn't carried over meanwhile.
+ */
+export default async function hasOpenSessionJob({ session_id }: Params) {
+    return Boolean(
+        (
+            await BunSQLite.sql({
+                sql: `SELECT id FROM bots_queue WHERE session_id = ? AND status IN ('pending', 'processing') LIMIT 1`,
+                values: [session_id],
+            })
+        ).payload?.[0],
+    );
+}