
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
Given the quadratic equation \(ax^{2} + bx + c = 0\), if we substitute \(y + k\) for \(x\), we obtain a new quadratic equation in terms of \(y\). Determine \(k\) such that the product of the roots of the two equations is equal. Additionally, compute the sum of all four roots of the two equations in this case.

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