Add AI agent monitoring and structured logs to Sentry example

[astrobot-houston]

Originally proposed by @sergical in #180. Their description is reproduced verbatim below.

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Summary

• Extends the examples/sentry/ example from error-only reporting to full Sentry observability using AI Agent Monitoring conventions
• Adds gen_ai.invoke_agent → gen_ai.chat → gen_ai.execute_tool span hierarchy with proper parent-child nesting, real wall-clock durations, aggregated token usage, cost attributes, and conversation ID tracking
• Forwards all ctx.log.* calls to Sentry.logger with flue.run_id / flue.agent correlation attributes
• Enables streamGenAiSpans: true so gen_ai spans are sent as standalone envelopes
• Adds optional SENTRY_AI_RECORD_INPUTS / SENTRY_AI_RECORD_OUTPUTS env vars for privacy-controlled prompt/response capture
• Adds a new chat.ts agent that makes a real LLM call to exercise the full pipeline
• Bumps @sentry/node to ^10.53.0 for streamGenAiSpans, enableLogs, and setConversationId support

Verified in Sentry

Tested against the flue-sentry-example project in sentry-developer-experience org. Confirmed:

• invoke_agent spans with gen_ai.agent.name, gen_ai.request.model, gen_ai.usage.*, gen_ai.conversation.id
• chat child spans with per-turn model, token usage, cost, and real duration (~1s, not 0ms)
• Proper parent-child span nesting (http.server → invoke_agent → chat)
• Structured logs in Sentry Logs view with correlation attributes
• Error capture unchanged for boom and explicit agents

Test plan

• npx tsc --noEmit passes (strict mode)
• pnpm exec flue dev --target node builds and serves all 4 agents
• curl /agents/chat/test1 returns LLM response with usage
• curl /agents/hello/test1 returns success (agent span + logs, no errors)
• curl /agents/boom/test1 returns error (captureException + error span)
• curl /agents/explicit/test1 returns 200 with error logs captured
• Spans verified in Sentry via MCP: correct hierarchy, attributes, durations
• Logs verified in Sentry: messages, severity levels, correlation attrs

🤖 Generated with Claude Code

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Original implementation from #180 by @sergical

[sergical]

👋 hi, Serge here, DevEx engineer at Sentry. I was using Flue to setup an issue triage bot on our docs and stumbled upon some examples and wanted to extend the Sentry one to better reflect the full feature scope of the product (that is included in all plans, none of the additions were "extra features"

• Tracing - using gen_ai convention from Otel and our Agent Monitoring product
• Sentry.logger - our new approach that is better DX than breadcrumbs since it's not just associated with an issue, these are standalone but trace connected across errors and tracing

Let me know if there are any questions! Loving Flue so far