
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]


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Find a third-degree polynomial with the leading coefficient of 1, given that its three roots \( x_{1}, x_{2}, x_{3} \) satisfy the equations: \( \alpha = x_{1} + x_{2}, \beta = x_{1} + x_{3}, \gamma = x_{2} + x_{3} \), where \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^{3} - 10x^{2} + 31x - 29 \).

Example of answer notation: \( x^{3} - 2x^{2} + 33x - 4 \).

The symbol \( \wedge \) denotes exponentiation. Write the terms in the same order as shown in the example.

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