Support Responses API in DIAL

[sdryapko]

Name and Version

development

What is the problem this feature will solve?

List of requirements

• Support the responses api for all our existing applications and models
• Make conversion from chat/completion to responses api possible
• Make conversion from responses api to chat/completion possible (with restrictions)
• We don't support background flag and previous_response_id in first iteration, going to support in the next phase

What is the feature you are proposing to solve the problem?

Design is attached to the issue.

responses_api.md

What alternatives have you considered?

No response

[sdryapko]

DIAL OpenAI Adapter Response API exposure: epam/ai-dial-adapter-openai#404