
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 $xOy$, with the origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. The same unit of length is taken in both coordinate systems. Given that the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=1+\sqrt{3}\cos\theta}\\{y=\sqrt{3}\sin\theta}\end{array}\right.$ ($\theta$ is the parameter), and the parametric equations of line $l$ are $\left\{\begin{array}{l}{x=2+t\cos\alpha}\\{y=1+t\sin\alpha}\end{array}\right.$ ($t$ is the parameter, $\alpha$ is the slope angle of line $l$).<br/>$(1)$ Find the Cartesian equation of curve $C$; when $\alpha =\frac{π}{3}$, find the polar coordinate equation of line $l$;<br/>$(2)$ If curve $C$ and line $l$ intersect at points $M$ and $N$, and $|MN|=\sqrt{10}$, find the slope angle of line $l$.

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