
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
Let \([r,s]\) denote the least common multiple of positive integers \(r\) and \(s\). Find the number of ordered triples \((a,b,c)\) of positive integers for which \([a,b] = 500\), \([b,c] = 1000\), and \([c,a] = 1000\).

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