### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

LiteLLM uses prompt_factory function (https://github.com/BerriAI/litellm/blob/69b0dd2da072c2ae0dd0c41ed969cbebf8e36828/litellm/llms/triton/completion/transformation.py#L199) in triton model requests handler.

Looks like all extra params are send to Triton instead, and triton by itself doesnt do any chat templating.

Problem is, it looks like there is no way to pass in parameters related to chat template when making a request. For example: user might want to disable model thinking to use less tokens in the output.

### Steps to Reproduce

```py import requests

API_KEY="sk-..." LLM_MODEL="qwen3.6-27b" query_text = "test?" LLM_REQUEST_TIMEOUT=1000 url = "http://localhost:4000/project-api/litellmqwe/litellm-webui/chat/completions"

r = requests.post( url, headers={ "Content-Type": "application/json", "Authorization": f"Bearer {API_KEY}", }, json={ "model": LLM_MODEL, "messages": [{"role": "system", "content": query_text}], "max_tokens": 10000, …