
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
Given an integer  $n \geqslant 3$  the polynomial  $f(x_1, \ldots, x_n)$  with integer coefficients is called *good* if  $f(0,\ldots, 0) = 0$  and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of  $\pi$  of the numbers  $1,\ldots, n$ . Denote by  $\mathcal{J}$  the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where  $m$  is a positive integer and  $q_1,\ldots , q_m$  are polynomials with integer coefficients, and  $p_1,\ldots , p_m$  are good polynomials. Find the smallest natural number  $D{}$  such that each monomial of degree  $D{}$  lies in the set  $\mathcal{J}$ .

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