
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
Let the positive real numbers \(c, b, a\) form an arithmetic sequence with \(a > b > c\) and the relationship \(b = a - d, c = a - 2d\) for some positive \(d\). A quadratic defined by \(ax^2 + bx + c\) has two distinct real roots. Find the sum of the roots of this quadratic equation.
A) \(-\frac{2}{3}\)
B) \(-1\)
C) \(-\frac{1}{3}\)
D) \(\frac{1}{3}\)

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