
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A positive integer  $n$  has the property that there are three positive integers  $x, y, z$  such that  $\text{lcm}(x, y) = 180$ ,  $\text{lcm}(x, z) = 900$ , and  $\text{lcm}(y, z) = n$ , where  $\text{lcm}$  denotes the lowest common multiple. Determine the number of positive integers  $n$  with this property.

Present the answer in LaTex format: \boxed{Your answer}
user