
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 \(\mathcal{A}\) denote the set of all polynomials in three variables \(x, y, z\) with integer coefficients. Let \(\mathcal{B}\) denote the subset of \(\mathcal{A}\) formed by all polynomials which can be expressed as
\[ (x + y + z) P(x, y, z) + (xy + yz + zx) Q(x, y, z) + xyz R(x, y, z) \]
with \(P, Q, R \in \mathcal{A}\). Find the smallest non-negative integer \(n\) such that \(x^{i} y^{j} z^{k} \in \mathcal{B}\) for all nonnegative integers \(i, j, k\) satisfying \(i + j + k \geqslant n\).

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