Let $\mathcal{A}$ denote the set of all polynomials in three variables $x,y,z$ with integer coefficients. Let $B$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as

$$(x+y+z)P(x,y,z)+(xy+yz+zx)Q(x,y,z)+xyzR(x,y,z)$$

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 non negative integers $i,j,k$ satisfying $i+j+k\geqslant n$.

Let's think step by step and output the final answer within \boxed{}.
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