
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 curve $C$ is $\begin{cases} x=1+2\cos \alpha \\ y=2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). Taking the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\theta= \frac {\pi}{4} (\rho\in\mathbb{R})$.

$(1)$ Find the polar equation of curve $C$ and the Cartesian equation of line $l$;  
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find $|AB|$.

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