
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
Let the real number \( r \) satisfy \( r^{3} - r - 1 = 0 \), and let \( r + \sqrt{2} \) be a root of the polynomial \( P(x) = 0 \). If the leading coefficient of \( p(x) \) is 1, and \( p(x) \) is the polynomial of least degree with integer coefficients that satisfies the conditions, then \( p(x) = \) _____.

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