Fix JSONL patch format in /docs and add API route documentation

[reiyi-lai]

Issue

1. JSONL patch format is not documented - structure expected by useUIStream's internal applyPatch function not specified in the README or docs.
2. apps/web/app/docs/streaming/page.tsx shows /root/children paths which silently fail, as the applyPatch function at packages/react/src/hooks.ts only supports /root and /elements/{key} paths.
3. Renderer uses wrong prop name - README example shows components={registry} but should be registry={registry}.

Based on packages/react/src/hooks.ts:

• The patch format uses /root and /elements/{key} paths, not JSON Pointer paths like /children/-
• send(prompt) takes a string directly, not { prompt }
• The hook returns isStreaming (not status) and clear() (not reset())

Changes

1. README.md:
Added "API Route Integration" section with:

• JSONL patch format specification
• useUIStream hook reference
• Complete API route example (Next.js + Vercel AI SDK)
• Context provider for Renderer component
• Fixed Renderer prop: components → registry

2. apps/web/app/docs/streaming/page.tsx:

• Fixed incorrect path: /root/children → /elements/{key} with children array

[vercel]

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[vercel]

Additional Suggestions:

1. Streaming documentation shows incorrect JSON patch path format (/root/children) that silently fails in the actual implementation

2. README example uses incorrect prop name 'components' instead of 'registry' for the Renderer component

[zukilover]

Reading through #37 , it seems like we could simplify the consumer API here. Would it make sense to move the system prompt generation inside generateCatalogPrompt so the user doesn't have to provide it explicitly?