Native support for Response API

[kowyo]

Summary

Add native support for OpenAI Response API format, supporting both Anthropic and OpenAI SDKs natively.

Motivation

While Anthropic models work well with the current Anthropic-endpoint format, OpenAI models do not perform as expected when using the same format. Many providers (Google, open-source models) have great OpenAI format support, but current implementations through proxies like LiteLLM and OpenRouter have limitations:

• LiteLLM does not map output_config.effort to reasoning.effort
• OpenRouter BYOK has mapping issues despite free usage benefits

Supporting both SDKs natively will provide better performance and reliability.

Proposed Solution

Refactor the codebase to support both SDKs natively:

• Add native OpenAI SDK integration alongside existing Anthropic SDK
• Handle format differences properly for each provider
• Support OpenAI-specific features like reasoning.effort correctly

Acceptance Criteria

• Native OpenAI SDK support implemented
• Anthropic SDK continues to work as before
• Proper handling of reasoning/effort parameters
• Documentation for SDK selection and configuration

Additional Context

This will require a major refactor. Tested alternatives:

• LiteLLM (does not map output_config.effort to reasoning.effort)
• OpenRouter BYOK (maps output_config.effort to verbosity)

Google and many open-source models support OpenAI format well.