
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 $[c,a] = 2000$. Additionally, ensure that the lowest power of 2, among $a$, $b$, and $c$, is less than the power of 2 in $[b,c]$ by exactly 1.

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