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actuality of inescapable composition of polynomials in two completely different finite fields

At each $ninmathbb N$ is there $g_{2,1}^{(n)},dots,g_{2,ok}^{(n)}inmathbb F_2[x_1,dots,x_n]$ the place $ok=mathsf{poly}(n)$ and $g_3^{(n)}inmathbb F_3[x_1,dots,x_k]$ of diploma $O(1)$ satisfying ($g_i=g_i^{(n)}$ under) $forall(x_1,dots,x_n)in{0,1}^n$
$$g_3(g_{2,1}(x_1,dots,x_n),dots,g_{2,ok}(x_1,dots,x_n))=0iff2sum_{i=1}^nx_ileq n?$$

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