Given the parabola $y=x^{2}$. The line $y=\frac{1}{4}+ax$ intersects this parabola in the points $B(\frac{a-\sqrt{a^{2}+1}}{2},\frac{-2a\sqrt{a^{2}+1}+2a^{2}+1}{4})$ and $D(\frac{\sqrt{a^{2}+1}+a}{2},\frac{2a\sqrt{a^{2}+1}+2a^{2}+1}{4})$. The line $y=\frac{1}{4}-\frac{x}{a}$ intersects this parabola in the points $A(-\frac{\sqrt{a^{2}+1}+1}{2a},\frac{2\sqrt{a^{2}+1}+a^{2}+2}{4a^{2}})$ and $C(\frac{\sqrt{a^{2}+1}-1}{2a},\frac{-2\sqrt{a^{2}+1}+a^{2}+2}{4a^{2}})$. The difference $m=\frac{a^{2}+1}{a}$. The area of $ABCD$ is $\frac{1}{2}m^{2}$.