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Denote by $\mathbb{Z}_{29}$ the set of all congruence classes modulo $29$. Consider the following quadratic congruence:

$$ x^2 + x + 1 \equiv 0 \ \mathrm{mod} \ 29 $$

Which of the following statements are correct? Select ALL that apply.

$x\equiv 1 \ \mathrm{mod} \ 29$ is a solution.

The equation has exactly $2$ distinct solutions in $\mathbb{Z}_{29}$.

The equation has exactly one solution in $\mathbb{Z}_{29}$.

The equation does not have any solution in $\mathbb{Z}_{29}$.