
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
The coefficients of the quadratic polynomials \( f_{i}(x) = x^{2} + b_{i} x + c_{i} \) satisfy the equations \( \frac{b_{i+1}}{b_{i}} = 2 \) and \( c_{i} = -32 \cdot b_{i} - 1024 \) for \( i = 1,2, \cdots \). It is known that the roots of the polynomial \( f_{1}(x) \) are 32 and -31. 

a) Find the roots of the quadratic polynomial \( f_{12} \).

b) Find the roots of the quadratic polynomial \( f \).

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