
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
1. Suppose \( x_{1}, x_{2}, x_{3} \) are the roots of the polynomial \( a x^{3} + b x^{2} + c x + d \).

Use the identity 

\[ a x^{3} + b x^{2} + c x + d = a (x - x_{1}) (x - x_{2}) (x - x_{3}) \]

to derive formulas that relate the roots and coefficients of this polynomial.


2. Find the roots \( x_{1}, x_{2}, x_{3} \) of the equation

\[ 8 x^{3} - 20 x^{2} - 10 x + 33 = 0 \]

by forming and solving a new cubic equation with roots \( x_{1} + x_{2}, x_{1} + x_{3}, x_{2} + x_{3} \).

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