
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+\left(y-2\right)^{2}=4$. Taking the origin $O$ as the pole and the non-negative x-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is $\rho =4\cos \theta $.<br/>$(1)$ Find the Cartesian equation of curve $C_{2}$;<br/>$(2)$ Given that the polar coordinate equation of curve $C_{3}$ is $\theta =\alpha $, $0 \lt \alpha\ \ \lt \pi $, $\rho \in R$, point $A$ is the intersection point of curves $C_{3}$ and $C_{1}$, point $B$ is the intersection point of curves $C_{3}$ and $C_{2}$, and both $A$ and $B$ are different from the origin $O$, and $|AB|=4\sqrt{2}$, find the value of the real number $a$.

Present the answer in LaTex format: \boxed{Your answer}
user