Show that there exists $w \in \mathbb C$ such that $a w^2 - w +1 =0$ and $| w - 1| \leq 1$

Question

$\forall a \in \mathbb{C}$ prove that there exists $w \in \mathbb{C}$ such that $$aw^2 - w + 1 = 0$$ $$|w - 1| ≤ 1$$

My attempt was to use the Gershgorin circle Theorem on a matrix $A \in \mathbb{C}^{2 \times 2}$ such that $p_A(z) = az^2 - z + 1$

Then the solutions of the equation

$$az^2 - z + 1 = 0$$

Would be the eigenvalues of $A$

unfortunately I haven't managed to complete this proof.