Restriction of trivariate polynomial to $1$ variable

Question

Let $p(x,y,z): \mathbb{F}^3 \to \mathbb{F}$ be a trivariate polynomial of degree $d \ll |\mathbb{F}|$. We choose uniformly at random an affine $1$-dimentional space $\ell = \{(a_1,a_2,a_3)t + (b_1,b_2,b_3) : t \in \mathbb{F}\}$. What is a good bound on the probability that the restriction of $p$ to $\ell$ (as a univariate polynomial) is of degree strictly less than $d$?

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