
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 $C$ are $\left\{\begin{array}{l}x=2+2\cos\theta\\ y=2\sin\theta\end{array}\right.$ ($\theta$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis. The polar coordinate equation of the line $l$ is $\rho \sin \theta +2\rho \cos \theta =2$. <br/>$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$; <br/>$(2)$ Given the point $P(1,0)$, line $l$ intersects curve $C$ at points $A$ and $B$, find $||PA|-|PB||$.

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