studio: reset Anthropic adapter cursor on auto-continue boundary

Codex P2 on #5549 flagged that the auto-continue branch yields only
a `{"type":"status","text":""}` event between turns; the Anthropic
streaming emitter (`AnthropicStreamEmitter`) and the non-streaming
tool path (`_anthropic_tool_non_streaming`) both ignore `status`
events, so their cumulative-text cursor still holds the previous
turn's full length when the continuation starts streaming. Shorter
continuations get dropped entirely and longer ones lose their prefix.

Treat the empty-text status as an auto-continue boundary in both
paths:

  - `AnthropicStreamEmitter`: close any open text block, open a fresh
    one (matches the `tool_end` reset pattern), and clear `_prev_text`.
  - `_anthropic_tool_non_streaming`: clear `prev_text` so the next
    `content` event's diff baseline is empty.

Non-empty status events (tool progress text) keep their existing
no-op semantics.

studio/backend/core/inference/anthropic_compat.py

@@ -253,9 +253,28 @@ class AnthropicStreamEmitter:
         elif etype == "metadata":
             self._usage = event.get("usage", {})
             return []
-        # status events — no Anthropic equivalent
+        elif etype == "status" and not event.get("text"):
+            # Auto-continue boundary marker emitted by
+            # generate_chat_completion_with_tools — the next "content"
+            # event resets to a fresh cumulative baseline, so close any
+            # open text block and clear the diff cursor. Without this
+            # the next continuation gets diffed against the previous
+            # turn's length (shorter continuations are dropped, longer
+            # ones lose their prefix).
+            return self._handle_boundary()
+        # Other status events (tool progress text) have no Anthropic
+        # equivalent.
         return []
 
+    def _handle_boundary(self) -> list[str]:
+        events = []
+        if self._text_block_open:
+            events.append(self._close_block())
+            self.block_index += 1
+            events.extend(self._open_text_block())
+        self._prev_text = ""
+        return events
+
     def finish(self, stop_reason: str = "end_turn") -> list[str]:
         """Close any open block and emit message_delta + message_stop."""
         events = []

studio/backend/routes/inference.py

@@ -4370,6 +4370,13 @@ async def _anthropic_tool_non_streaming(run_gen, message_id, model_name):
             )
         elif etype == "tool_end":
             prev_text = ""
+        elif etype == "status" and not event.get("text"):
+            # Auto-continue boundary marker: the next content event
+            # restarts the cumulative diff baseline, so reset prev_text
+            # the same way tool_end does. Without this a shorter
+            # continuation gets dropped entirely and a longer one
+            # loses its prefix.
+            prev_text = ""
         elif etype == "metadata":
             usage = event.get("usage", {})