
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, whose first digits are 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 created by using these numbers as \( a \), \( b \), and \( c \), each no more than once, such that all these equations have roots?

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