
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
Determine the coefficients \(a\), \(b\), and \(c\) such that the roots of the equation \(a x^{2} + 6x + c = 0\) are denoted by \(x_{1}\) and \(x_{2}\). The numbers \(a, x_{1}, b, x_{2}, c\) should form an arithmetic sequence in this order.

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