Refactor workflow routing and greeting management in Brain classes

- Update WorkflowBrain to handle greeting playback more effectively, ensuring that the initial greeting completes before transitioning to the first node.
- Introduce new methods for managing greeting states and conditions, enhancing the interaction flow for user turns.
- Refactor WorkflowLLMRouter to improve routing logic and ensure proper handling of conditional paths.
- Enhance tests to verify the correct behavior of greeting management and routing under various scenarios, including waiting for audio playback to finish.
- Update frontend components to reflect changes in edge handling and improve user experience in workflow configurations.

backend/services/brains/base.py

@@ -101,8 +101,16 @@ class BaseBrain:
     async def setup(self, cfg: AssistantConfig, runtime: BrainRuntime) -> None:
         """Register tools and initialize per-call orchestration."""
 
-    async def on_connected(self) -> None:
-        """Handle a connected client after the common greeting is queued."""
+    async def on_connected(self, *, greeting_pending: bool = False) -> None:
+        """Handle a connected client before an optional greeting is played.
+
+        ``greeting_pending`` lets an orchestration brain prepare its initial
+        state without starting node-entry speech while the shared greeting is
+        still playing.
+        """
+
+    async def on_greeting_finished(self) -> None:
+        """Continue startup after the shared greeting has actually played."""
 
     def prepare_greeting_context(
         self,
@@ -166,7 +174,9 @@ class Brain(Protocol):
 
     async def setup(self, cfg: AssistantConfig, runtime: BrainRuntime) -> None: ...
 
-    async def on_connected(self) -> None: ...
+    async def on_connected(self, *, greeting_pending: bool = False) -> None: ...
+
+    async def on_greeting_finished(self) -> None: ...
 
     def prepare_greeting_context(
         self,