Assume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) polynomial in $F$ of the form
$$
x^{2}+b x+c, \quad b, c \in F
$$
For how many elements $k$ in $F$, $p(x)+k$ is irreducible in $F$? How many elements are there when $m=7$.

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