
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
Among the natural numbers \(a_1, \ldots, a_k\), none are the same, and the difference between the largest and smallest is less than 1000. What is the maximum possible \(k\) for which all quadratic equations \(a_i x^2 + 2 a_{i+1} x + a_{i+2} = 0\), where \(1 \leq i \leq k-2\), have no roots?

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