A disk of radius R with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$$
V(z)=\frac{\sigma}{2 \epsilon_0}\left(\sqrt{R^2+z^2}-z\right)
$$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$-axis with $z=z 0$ and $z 0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \varepsilon_0}{q \sigma}$. Which of the following statement(s) is(are) correct?
Select ALL correct options:
A
For $\beta=\frac{1}{4}$ and $z_0=\frac{25}{7} R$, the particle reaches the origin.
B
For $\beta=\frac{1}{4}$ and $z_0=\frac{3}{7} R$, the particle reaches the origin.
C
For $\beta=\frac{1}{4}$ and $z_0=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_0$
D
For $\beta>\frac{1}{4}$ and $z_0>0$, the particle always reaches the origin.
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