feat: add last_injected_context column to chats table (#23798)

Adds a nullable JSONB column `last_injected_context` to the `chats`
table that stores the most recently persisted injected context parts
(AGENTS.md context-file and skill message parts). The column is updated
only when `persistInstructionFiles()` runs — on first workspace attach
or when the agent changes — so there are no redundant writes on
subsequent turns.

Internal fields (`ContextFileContent`, `ContextFileOS`,
`ContextFileDirectory`, `SkillDir`) are stripped at write time so the
column only holds small metadata. No stripping needed on the read path.

<details>
<summary>Implementation notes</summary>

- New migration `000456` adds nullable `last_injected_context JSONB`
column.
- New SQL query `UpdateChatLastInjectedContext` writes the column
without touching `updated_at`.
- `persistInstructionFiles()` strips internal fields from parts via
`StripInternal()` before persisting.
- Sentinel path (no AGENTS.md) persists skill-only parts when skills
exist.
- `codersdk.Chat` exposes `LastInjectedContext []ChatMessagePart`
(omitempty).
- `db2sdk.Chat()` passes through the already-clean data.

</details>

coderd/database/dbauthz/dbauthz.go

@@ -5752,6 +5752,17 @@ func (q *querier) UpdateChatLabelsByID(ctx context.Context, arg database.UpdateC
 	return q.db.UpdateChatLabelsByID(ctx, arg)
 }
 
+func (q *querier) UpdateChatLastInjectedContext(ctx context.Context, arg database.UpdateChatLastInjectedContextParams) (database.Chat, error) {
+	chat, err := q.db.GetChatByID(ctx, arg.ID)
+	if err != nil {
+		return database.Chat{}, err
+	}
+	if err := q.authorizeContext(ctx, policy.ActionUpdate, chat); err != nil {
+		return database.Chat{}, err
+	}
+	return q.db.UpdateChatLastInjectedContext(ctx, arg)
+}
+
 func (q *querier) UpdateChatLastModelConfigByID(ctx context.Context, arg database.UpdateChatLastModelConfigByIDParams) (database.Chat, error) {
 	chat, err := q.db.GetChatByID(ctx, arg.ID)
 	if err != nil {

coderd/database/dbauthz/dbauthz_test.go

@@ -1204,6 +1204,19 @@ func (s *MethodTestSuite) TestChats() {
 		dbm.EXPECT().UpdateChatMCPServerIDs(gomock.Any(), arg).Return(chat, nil).AnyTimes()
 		check.Args(arg).Asserts(chat, policy.ActionUpdate).Returns(chat)
 	}))
+	s.Run("UpdateChatLastInjectedContext", s.Mocked(func(dbm *dbmock.MockStore, faker *gofakeit.Faker, check *expects) {
+		chat := testutil.Fake(s.T(), faker, database.Chat{})
+		arg := database.UpdateChatLastInjectedContextParams{
+			ID: chat.ID,
+			LastInjectedContext: pqtype.NullRawMessage{
+				RawMessage: json.RawMessage(`[{"type":"text","text":"test"}]`),
+				Valid:      true,
+			},
+		}
+		dbm.EXPECT().GetChatByID(gomock.Any(), chat.ID).Return(chat, nil).AnyTimes()
+		dbm.EXPECT().UpdateChatLastInjectedContext(gomock.Any(), arg).Return(chat, nil).AnyTimes()
+		check.Args(arg).Asserts(chat, policy.ActionUpdate).Returns(chat)
+	}))
 	s.Run("UpdateChatLastReadMessageID", s.Mocked(func(dbm *dbmock.MockStore, faker *gofakeit.Faker, check *expects) {
 		chat := testutil.Fake(s.T(), faker, database.Chat{})
 		arg := database.UpdateChatLastReadMessageIDParams{