
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 polynomial \( g(x) = x^2 + ax + b \), which has exactly one root. Find the coefficients \( a \) and \( b \), given that the polynomial \( g\left(x^5 + 2x - 1\right) + g\left(x^5 + 3x + 1\right) \) also has exactly one root.

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