Holiday Contest 2025 - Guts Round - Set 8 Problem 2

Call an integer a bad if there exists another integer b such that a \equiv b^2 \pmod{44}. Numbers are randomly drawn from \{1,2,\dots,2024\} (without replacement) until a bad number is drawn. Say we draw K cards. If the expected value of K can be written as \dfrac{a}{b}, where a and b are relatively prime positive integers, find a+b