
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 $xOy$, with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho= \frac {2\cos \theta}{1-\cos ^{2}\theta }$, and the parametric equation of line $l$ is $\begin{cases} x=2+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter).

$(1)$ If $\alpha= \frac {3\pi}{4}$, find the general equation of $l$ and the Cartesian coordinate equation of $C$;  
$(2)$ If $l$ intersects $C$ at two distinct points $A$ and $B$, and $P(2,1)$ is the midpoint of $AB$, find $|AB|$.

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