You are tasked with designing an enhanced grading system for a university course. The system should take a student's numerical score and a boolean value indicating if extra credit was received. The output should be the letter grade with a potential increase if extra credit was received. The criteria are:

A: $90-100$

B: $80-89$

C: $70-79$

D: $60-69$

F: $0-59$

If the student received extra credit $($true$),$ their grade should be increased by one letter $($e$.$g$.,$ B becomes A$,$ C becomes B$,$ etc.$).$ Write a C$++$ program that prompts the user to input a score and whether extra credit was received. Use an if$-$else statement to determine and print the final letter grade.