
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
It is known that the polynomial \( f(x) = 8 + 32x - 12x^2 - 4x^3 + x^4 \) has 4 distinct real roots \(\{x_{1}, x_{2}, x_{3}, x_{4}\}\). The polynomial of the form \( g(x) = b_{0} + b_{1} x + b_{2} x^2 + b_{3} x^3 + x^4 \) has roots \(\{x_{1}^2, x_{2}^2, x_{3}^2, x_{4}^2\}\). Find the coefficient \( b_{1} \) of the polynomial \( g(x) \).

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