
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 line $l$ is 
$$
\begin{cases} 
x=1+ \frac { \sqrt {2}}{2}t \\
y=2+ \frac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter, } t\in\mathbb{R}).
$$
In the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=4\sin \theta$.  
(Ⅰ) Find the equation of circle $C$ in Cartesian coordinates;  
(Ⅱ) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If the coordinates of point $P$ are $(1,2)$, find $|PA|+|PB|$.

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