
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 plane rectangular coordinate system $xOy$, the parameter equation of curve $C$ is $\left\{{\begin{array}{l}{x=1+\frac{{2si{n^2}α}}{{co{s^2}α-si{n^2}α}},}\\{y=\frac{{\sqrt{2}sinαcosα}}{{co{s^2}α-si{n^2}α}}}\end{array}}\right.$ ($α$ is the parameter), with the coordinate origin as the pole, and the positive x-axis as the polar axis to establish a polar coordinate system. The polar coordinate equation of the line $l$ is $θ=\frac{π}{6}$ ($ρ∈R$).  
$(1)$ Find the Cartesian equation of $C$ and the rectangular coordinate equation of $l$;  
$(2)$ Find the polar coordinates of the intersection points of $l$ and $C$.

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