
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
How many ordered triples \((x, y, z)\) of positive integers satisfy \(\text{lcm}(x,y) = 180\), \(\text{lcm}(x,z) = 1000\), and \(\text{lcm}(y,z) = 450\)?
**A)** 6
**B)** 7
**C)** 8
**D)** 9
**E)** 10

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