## Background

The AI Gateway provider was missing the ability to look up detailed information about specific generations after they were created. Users needed a way to retrieve cost, token usage, latency, and provider details for individual generations using their IDs.

## Changes

This PR adds generation lookup functionality to the AI Gateway provider:

- **New `getGenerationInfo()` method**: Allows fetching detailed information about a specific generation by its ID
- **Generation ID access**: Generation IDs are now available in `providerMetadata.gateway.generationId` on both `generateText` and `streamText` responses
- **Comprehensive generation data**: Returns cost breakdown, token usage, latency metrics, provider details, and more
- **Error handling**: Proper error handling for authentication, server errors, and malformed responses
- **Documentation**: Added complete documentation with examples showing how to use the new functionality
- **Example implementation**: Added a working example demonstrating the feature

The `getGenerationInfo()` method returns detailed information including:
- Cost breakdown (total cost, upstream inference cost for BYOK)
- Token usage (prompt, completion, reasoning, cached tokens)
- Performance metrics (latency, generation time)
- Provider and model information
- Generation metadata (streaming, finish reason, etc.)

## Manual Verification

Tested the feature by:
1. Making a generation request with the gateway provider
2. Extracting the generation ID from the provider metadata
3. Using `getGenerationInfo()` to fetch detailed generation information
4. Verifying all returned fields contain expected data types and values
5. Testing error scenarios (invalid IDs, authentication failures)

## Checklist

- [x] Tests have been added / updated (for bug fixes / features)
- [x] Documentation has been added / updated (for bug fixes / features)
- [x] A _patch_ changeset for relevant packages has been added (for bug fixes / features - run `pnpm changeset` in the project root)
- [x] I have reviewed this pull request (self-review)