
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
All three-digit natural numbers with the first digit being odd and greater than 1 are written on the board. What is the maximum number of quadratic equations of the form \(a x^{2}+b x+c=0\) that can be formed using these numbers as \(a\), \(b\), and \(c\), each used no more than once, such that all these equations have roots?

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