Problem: You are tasked with implementing a simple two$-$factor authentication system using Object$-$Oriented Programming in Python. The system should allow users to register with a username and password, log in$,$ and verify their identity using a randomly generated two$-$factor authentication code.

Requirements:

$1.$ User Registration: Implement a method to register new users. If a user attempts to register with a username that already exists, the system should raise an exception indicating that the username is taken.

$2.$ Login Functionality: When a user logs in with a correct username and password, the system should generate and print a random six$-$digit two$-$factor authentication code. $3.$ Two$-$Factor Code Verification: After the user receives the code, they should input it into the system. The system should verify if the entered code matches the generated one. If the code is correct, the user is authenticated successfully; otherwise, an error message should be displayed.

$4.$ Error Handling: Ensure proper error handling for cases such as invalid usernames, incorrect passwords, and mismatched two$-$factor codes.

Example Usage:

$$

A user named "john$\_$doe" registers with the password "securepassword."

The user logs in and receives a two$-$factor authentication code.

The user enters the code, and the system verifies it$.$

Based on the above problem, the following Python program has been written, which produces the correct outcome.