State a version of Rouché's Theorem. Find the number of solutions (counted with multiplicity) of the equation

$$z^{4}=a(z-1)\left(z^{2}-1\right)+\frac{1}{2}$$

inside the open disc $\{z:|z|<\sqrt{2}\}$, for the cases $a=\frac{1}{3}, 12$ and 5 .

[Hint: For the case $a=5$, you may find it helpful to consider the function $\left(z^{2}-1\right)(z-$ 2) $(z-3)$.]