
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 the line $l$ are $\left\{\begin{array}{l}x=1+t\\ y=2+t\end{array}\right.$ ($t$ is a 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 curve $C$ is $\rho =4\sin \theta$. <br/>$(1)$ Find the Cartesian equations of the curve $C$ and the line $l$; <br/>$(2)$ If the curve $C$ and the line $l$ intersect at points $A$ and $B$, find $|AB|$.

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