Advanced AI Prompt for API Gateway Observability and Error Tracking

Act as a API Gateway Architect. Define a comprehensive, high-resolution monitoring suite targeting the REST API [API_ID] currently deployed to the [STAGE_NAME] stage. Generate the exact CloudWatch Metric Math formulas required to calculate real-time 4XX and 5XX error *rates* as a percentage of total requests, constructing Alarms that trigger instantly upon breaching the [ERROR_PERCENTAGE_THRESHOLD]. Set up a brutal, high-severity alarm specifically monitoring the `IntegrationLatency` metric for breaches against the [P99_LATENCY_THRESHOLD]. Detail the exact IAM and execution logging setup required to push API Gateway Access Logs to CloudWatch. Provide an optimized, custom JSON access log format string that explicitly captures `$context.requestId`, `$context.authorizer.principalId`, and `$context.integrationLatency` to guarantee absolute traceability during debugging.