
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 nonzero integers \( a, b, c \) are three consecutive terms of a decreasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers \( 2a \), \( 2b \), \( c \) taken in any order, have two roots. Find the largest possible value of the common difference of the progression under these conditions and the corresponding numbers \( a, b, c \).

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