feat: Add full text search over chat messages (#27126)

Closes CODAGT-721
Closes CODAGT-722
Closes CODAGT-723
Closes CODAGT-724
Closes CODAGT-725

This PR adds the database and API pieces necessary to support full-text
chat message search.

- Adds required chat schema for full-text search
- Adds dbpurge job to populate search_tsv in the background
- Adds `search` parameter to GetChats query
- Adds `search` filter to `searchquery.Chats`
- Wires chat search filter into chats API

> Implemented by Coder Agents, reviewed and tested by a human.

coderd/database/querier.go

@@ -67,6 +67,12 @@ type sqlcQuerier interface {
 	// created_at ASC flows through to dbpurge's digest truncation; see
 	// buildDigestData in dbpurge.go for the tradeoff rationale.
 	AutoArchiveInactiveChats(ctx context.Context, arg AutoArchiveInactiveChatsParams) ([]AutoArchiveInactiveChatsRow, error)
+	// Backfills chat_messages.search_tsv for pending rows, newest first.
+	// The WHERE clause must match the predicate of
+	// idx_chat_messages_search_tsv_pending exactly so the partial index
+	// serves this query.
+	// NULL means "pending", empty tsvector means "backfilled, no text".
+	BackfillChatMessagesSearchTsv(ctx context.Context, batchSize int32) (int64, error)
 	BackoffChatDiffStatus(ctx context.Context, arg BackoffChatDiffStatusParams) error
 	// Deletes heartbeat rows for the supplied (chat_id, runner_id) pairs.
 	BatchDeleteChatHeartbeats(ctx context.Context, arg BatchDeleteChatHeartbeatsParams) (int64, error)
@@ -80,6 +86,9 @@ type sqlcQuerier interface {
 	// Calculates the telemetry summary for a given provider, model, and client
 	// combination for telemetry reporting.
 	CalculateAIBridgeInterceptionsTelemetrySummary(ctx context.Context, arg CalculateAIBridgeInterceptionsTelemetrySummaryParams) (CalculateAIBridgeInterceptionsTelemetrySummaryRow, error)
+	// Reports whether search text tokenizes to an empty tsquery (e.g. '!!!').
+	// Used to reject input that would silently match nothing.
+	ChatSearchQueryIsEmpty(ctx context.Context, search string) (bool, error)
 	ClaimPrebuiltWorkspace(ctx context.Context, arg ClaimPrebuiltWorkspaceParams) (ClaimPrebuiltWorkspaceRow, error)
 	CleanTailnetCoordinators(ctx context.Context) error
 	CleanTailnetLostPeers(ctx context.Context) error