\section*{Problem 1 - 331021}

Determine whether there exists a four-digit square number \(q\) with the following properties (1), (2)! If such numbers exist, find all such square numbers!

(1) All four digits of \(q\) are less than 7.

(2) If each digit of \(q\) is increased by 3, the resulting four-digit number is also a square number.