Would you be so kind as to provide me with a hint for a question that I can't solve?

It is supposed to be more or less easy, but I don't see what the quick way to settle is. Let me thank you in advance for your help.

**Question.** How many $4$-tuples $(a,b,c,d) \in \mathbb{N}^{4}$ are there satisfying the following constraints $$\mathrm{lcm}(a,b,c) = \mathrm{lcm}(b,c,d)=\mathrm{lcm}(c,d,a)= \mathrm{lcm}(d,a,b)=2^{5}3^{4}5^{3}7^{2}$$?