
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 line $l$ passes through point $P(0,1)$ and has a slope of $1$. In the polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho =2\sin \theta +2\cos \theta$.

(Ⅰ) Find the parametric equation of line $l$ and the Cartesian equation of curve $C$;

(Ⅱ) If the intersection points of line $l$ and curve $C$ are $A$ and $B$, find the value of $|PA|+|PB|$.

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