Support chat participants in agent mode (#256113)

* Allow participants in any mode
Fix #255921

* Get rid of unneeded context keys

* Preserve modes for default agents

* Fix tests

* Fix test

extensions/vscode-api-tests/package.json

@@ -75,6 +75,9 @@
             "name": "hello",
             "description": "Hello"
           }
+        ],
+        "modes": [
+          "agent", "ask", "edit"
         ]
       },
       {