Show a reply's steps, and catch up on a reply whose live push was missed

A reply that works through tools now shows its latest tool step under
"is thinking…" (not its closing note, which is the answer). While a reply
is due, the open conversation checks the session's latest messages every
few seconds (api/admin/bots/session-messages) and replays any it missed
as server:bot-message, so the thread, rail and unread all take it and the
thinking row can't stay up until a reload. MCP and custom tool steps read
in words ("Using Exa search (web search exa)").

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

src/data/app-data.ts

@@ -213,6 +213,16 @@ export const AppData = {
      * bot changes from the database for its own sockets (`liveRelay`).
      */
     LiveRelayPollMs: 1500,
+    /**
+     * While a reply is due in the open conversation, how often the page
+     * checks for new messages itself, in case a live push was missed (the
+     * socket was down) and the thinking row would otherwise stay up.
+     */
+    BotReplyCatchUpMs: 5000,
+    /**
+     * How many of a session's latest messages that check reads.
+     */
+    BotReplyCatchUpMessages: 20,
     /**
      * How much of the conversation a bot sees when it replies.
      */