
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 rectangular coordinate system $(xOy)$, the parametric equations of curve $C_1$ are given by $\begin{cases} x=2+2\cos{\varphi}, \\ y=2\sin{\varphi} \end{cases}$ where $\varphi$ is a parameter. Curve $C_2$ has a polar coordinate equation of $\rho =4\sin{\theta}$ with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(1) Find the Cartesian equation of $C_1$ and the rectangular coordinate equation of $C_2$.

(2) The polar coordinate equation of curve $C_3$ is given by $\theta =\alpha (0 < \alpha < \pi, \rho \in \mathbb{R})$. Point $A$ is the intersection of curves $C_3$ and $C_1$, and point $B$ is the intersection of curves $C_3$ and $C_2$. Both points $A$ and $B$ are distinct from the origin $O$. Given that $|AB|=4\sqrt{2}$, find the value of the real number $\alpha$.

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