How many ordered triples $(x, y, z)$ of positive integers satisfy the following conditions?
- $\text{lcm}(x, y) = 72$
- $\text{lcm}(x, z) = 600$
- $\text{lcm}(y, z) = 900$

Find the number of such ordered triples.

Let's think step by step and output the final answer within \boxed{}.
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