
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 $xOy$, a line $l$ passes through the point $(1,0)$ with an inclination angle of $\alpha$. Taking the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho= \frac {8\cos \theta}{1-\cos ^{2}\theta }$.  
$(1)$ Write the parametric equation of line $l$ and the Cartesian equation of curve $C$;  
$(2)$ If $\alpha= \frac {\pi}{4}$, suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the area of $\triangle AOB$.

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