fix: Set the Server session initialization state immediately after respond… (#1478)

Co-authored-by: Max Isbey <224885523+maxisbey@users.noreply.github.com>

src/mcp/server/session.py

@@ -163,6 +163,7 @@ class ServerSession(
                             )
                         )
                     )
+                self._initialization_state = InitializationState.Initialized
             case types.PingRequest():
                 # Ping requests are allowed at any time
                 pass