fix(health): fix chatgpt provider Test Connection endpoint

Two bugs caused the LiteLLM UI "Test Connection" button to always fail
for chatgpt provider models (e.g. chatgpt/gpt-5.6-sol, gpt-5.6-terra):

1. litellm/main.py — ahealth_check() mode auto-detection uses the
   model_cost map, but unlisted chatgpt models (any model not explicitly
   registered) fell through to the "chat" fallback. The chatgpt provider
   routes exclusively through the Responses API; sending a Chat
   Completions messages payload caused the backend to reject with:
   ChatgptException - {"detail": "Input must be a list"}.
   Fix: after get_llm_provider() resolves custom_llm_provider=="chatgpt"
   and mode is still None, default to "responses" instead of "chat".

2. litellm/litellm_core_utils/health_check_helpers.py — the "responses"
   mode handler passed input=prompt (a plain string) to litellm.aresponses().
   The Responses API requires input to be a list of message objects, not a
   bare string. The ChatGPT/Codex backend rejected this with:
   ChatgptException - {"error": {"message": "Invalid type for 'input[0]':
   expected an input item, but got a string instead."}}
   Fix: pass input=[{"role": "user", "content": prompt or "test"}].

Both fixes were validated end-to-end against a live chatgpt provider
deployment (gpt-5.6-sol, gpt-5.6-terra, two seats each — all returned
status: success after the patch).

litellm/litellm_core_utils/health_check_helpers.py

@@ -283,7 +283,7 @@ class HealthCheckHelpers:
             ),
             "responses": lambda: litellm.aresponses(
                 **_filter_model_params(model_params=model_params),
-                input=prompt or "test",
+                input=[{"role": "user", "content": prompt or "test"}],
             ),
             "anthropic_messages": lambda: litellm.anthropic_messages(
                 **{

litellm/main.py

@@ -8812,6 +8812,10 @@ async def ahealth_check(
         )
 
         model_params["cache"] = {"no-cache": True}  # don't used cached responses for making health check calls
+        # chatgpt provider uses the Responses API exclusively; default to "responses" so the
+        # health check doesn't send a Chat Completions payload that the backend rejects.
+        if mode is None and custom_llm_provider == "chatgpt":
+            mode = "responses"
         mode = mode or default_health_check_mode(
             requested_model=requested_model, model=model, custom_llm_provider=custom_llm_provider
         )