
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
The non-zero integers \( a, b, c \) are three consecutive terms of a decreasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers \( a, 2b, 4c \), taken in any order, have two roots. Find the greatest possible difference of the progression values under these conditions and the corresponding values of \( a, b, c \).

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