
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 line $l$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter, $0 \leqslant \alpha < \pi$). In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the curve $C$ is given by $\rho=4\cos \theta$.

$(1)$ When $\alpha= \frac {\pi}{4}$, find the polar coordinates of the intersection points of $C$ and $l$;  
$(2)$ Line $l$ intersects curve $C$ at points $A$ and $B$, and the corresponding parameters $t_{1}$ and $t_{2}$ are opposite numbers. Find the value of $|AB|$.

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