
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
Find all quadratic trinomials \(a x^{2}+b x+c\) with real coefficients \(a, b,\) and \(c\) such that if any one of the three coefficients is increased by 1, the resulting quadratic trinomial will have exactly one root.

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