
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 parametric equations of curve $C_1$ are
$$
\begin{cases}
x = 2 + 2\cos \varphi, \\
y = 2\sin \varphi
\end{cases}
(\varphi \text{ as the parameter}).
$$
With the origin $O$ as the pole and the positive x-axis as the polar axis, establish a polar coordinate system. The polar equation of curve $C_2$ is
$$
\rho = 4\sin \theta.
$$

(Ⅰ) Find the general equation of curve $C_1$ and the Cartesian coordinate equation of $C_2$;

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

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