
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 \( N \) be the least common multiple (LCM) of ten different natural numbers \( a_{1} < a_{2} < a_{3} < \ldots < a_{10} \).

(a) What is the smallest possible value of \( N / a_{3} \)?

(b) Indicate all possible values of \( a_{3} \) in the segment \([1, 1000]\) for which the value \( N / a_{1} \) can reach its minimum value.

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