
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
Find the number of ordered triples  $(a, b)$  of positive integers with  $a < b$  and  $100 \leq a, b \leq 1000$  satisfy  $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$ ?

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