A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$V\left(z\right)=\frac{\sigma }{2{ϵ}_0}\left(\sqrt{R^2+z^2}-z\right)$ 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 $\overrightarrow{F}=-c\hat{k}$ with $c>0$. Let $\beta =\frac{2c{\epsilon }_0}{q\sigma }$. Which of the following statement(s) is(are) correct?

1. A For $\beta =\frac{1}{4}$ and $z_0=\frac{25}{7}R$, the particle reaches the origin.
2. B For $\beta =\frac{1}{4}$ and $z_0=\frac{3}{7}R$, the particle reaches the origin.
3. C For $\beta =\frac{1}{4}$ and $z_0=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_0$
4. D For $\beta >1$ and $z_0>0$, the particle always reaches the origin.