
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 equation of curve $C_1$ is $$\begin{cases} x=1+\cos\alpha \\ y=\sin\alpha \end{cases}$$ (where $\alpha$ is the parameter). Taking the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, we establish a polar coordinate system. The polar equation of curve $C_2$ is $\rho=4\sin\theta$.  
(Ⅰ) If $A$ and $B$ are the common points of curves $C_1$ and $C_2$, find the slope of line $AB$;  
(Ⅱ) If $A$ and $B$ are moving points on curves $C_1$ and $C_2$ respectively, when $|AB|$ takes the maximum value, find the area of $\triangle AOB$.

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