[FEATURE]: Add the Perplexity Agent API as a provider

[kesku]

Feature hasn't been suggested before.

• I have verified this feature I'm about to request hasn't been suggested before.

Describe the enhancement you want to request

https://docs.perplexity.ai/docs/agent-api/quickstart

uses/v1/responses

[github-actions]

This issue might be a duplicate of existing issues. Please check:

• [FEATURE]: Use Responses api for providers that support it #10601: Feature request to use the Responses API (/v1/responses) for providers that support it
• [FEATURE]: enable responses API in openai compatible provider #9628: Feature request to enable the Responses API in OpenAI-compatible providers
• [FEATURE]: Support Responses API message phases (commentary/final_answer) for GPT-5.3-Codex & later #15528: Feature request for Responses API message phases support

The Perplexity Agent API's use of /v1/responses overlaps significantly with these existing requests. Please review them and consider whether your request is already covered or can be addressed there.