vercel/v0-sdk — reverse-engineered prompt

Build me a TypeScript monorepo for a v0 Platform API SDK that feels ready to try right away. I want a core SDK for creating and managing AI chat conversations, projects, integrations, and related API actions, plus a small React package for rendering v0 content, an AI tools package for agent style workflows, and a simple create app starter so someone can spin up a demo quickly.

Please include a few working examples, one super simple prompt in and app out demo, one classic clean chat style interface, one more complete clone with auth and a database feel, and one React example with a couple of themes. It should use a V0_API_KEY env variable, have a clean developer experience, and include build, test, lint, format, and type check scripts across the workspace.

Keep it in beta friendly shape with solid docs in the README, sensible defaults, and generated SDK support from the API spec if that fits. Look up the current v0 docs online if you need to.