Advanced AI Prompt for AWS API Gateway WebSocket Chat Architecture

Act as a Serverless Network Architect. Design a highly scalable, bidirectional WebSocket architecture utilizing AWS API Gateway specifically engineered for a real-time chat application. The system must support a strict [CONCURRENCY_LIMIT] of 100,000 active connections. Detail the exact Lambda route handlers required for the '$connect', '$disconnect', and '$default' routes. You must architect the connection state-management layer utilizing [DB_BACKEND] (e.g., DynamoDB) to map connection IDs to user IDs. Explain the programmatic implementation of the [AUTH_MECHANISM] using a Lambda Authorizer before the '$connect' route fires, and provide the exact Boto3 code required to dispatch a payload from the backend to a specific connected client via the 'ApiGatewayManagementApi'.