
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
Non-zero integers \( a, b, c \) are three consecutive terms in an increasing arithmetic progression. Each of the six quadratic equations, which have coefficients consisting of the numbers \( a, b, 2c \) taken in any order, have two distinct roots. Find the smallest possible value of the common difference of the progression under these conditions and the corresponding values of \( a, b, \) and \( c \).

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