HMMT 2019 U24

Let $S$ be the set of all positive factors of 6000 . What is the probability of a random quadruple $(a, b, c, d) \in S^{4}$ satisfies

$$\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d))=\operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d)) ?$$