You are tasked with creating a production$-$ready Docker solution for a multi$-$service web application. The application consists of: Node.js backend API, React.js frontend, PostgreSQL database $.$The requirements are as follows:

a$)$ Use multi$-$stage builds to optimize the Docker image size for both the backend and frontend.

b$)$ The frontend should be served through an NGINX web server inside a container.

c$)$ The Node.js API should use environment variables $($for database connection, API keys, etc.$)$ that can be changed without modifying the Docker image.

d$)$ Use Docker's networking to allow the backend, frontend, and database to communicate with each other without exposing internal services to the host.

e$)$ Ensure that PostgreSQL data persists across container restarts.

f$)$ Create a Docker Compose setup to orchestrate the multi$-$container environment

For the above:

$1.$ Write the Dockerfile for both the backend $($Node$.$js$)$ and frontend $($React$.$js$)$ using multi$-$stage builds.

$2.$ Write the Docker Compose file to orchestrate the services $($backend$,$ frontend, PostgreSQL, and NGINX$).$

$3.$ Ensure that the setup allows developers to run the system in development mode $($with live reloading for the frontend and backend$)$ and in production mode $($with optimized images and no live reloading$).$

$4.$ Explain how you would handle logging and monitoring for these containers in production