Write a C$++$ program that accepts three positive integers representing the lengths of the sides of a triangle. The program should first check whether these three sides form a valid triangle. If the sides do form a valid triangle, classify the triangle into one of the following categories:

Equilateral: All three sides are equal.

Isosceles: Exactly two sides are equal.

Scalene: All three sides are different.

Right$-$angled: The triangle satisfies the Pythagorean theorem:

a$2+$ b$2=$ c$2$

where c is the longest side.

If the given sides do not form a valid triangle, display an error message.