
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 parameter equation of the line $l$ is $\left\{\begin{array}{c}x=2+\frac{\sqrt{2}}{2}t\\ y=1+\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sqrt{2}\sin\left(\theta+\frac{π}{4}\right)$.  
$(Ⅰ)$ Find the Cartesian equation of the curve $C$;  
$(Ⅱ)$ Let point $P(2,1)$, the intersection points of line $l$ and curve $C$ be $A$ and $B$, find the value of $|PA|^2+|PB|^2$.

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