Find explicitly a polynomial $p$ of degree $\leqslant 3$ such that

$$\sup _{x \in[-1,1]}\left|x^{4}-p(x)\right| \leqslant \sup _{x \in[-1,1]}\left|x^{4}-q(x)\right|$$

for every polynomial $q$ of degree $\leqslant 3$. Justify your answer.