
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 line $l$ passes through the origin $O$ with an inclination angle of $\theta _{0}$. The parametric equations of the curve $C$ are $\left\{\begin{array}{l}{x=1+cosα}\\{y=\sqrt{3}+sinα}\end{array}\right.$ ($\alpha$ is the parameter), with the coordinate origin as the pole and the positive $x$-axis as the polar axis. Find:<br/>$(1)$ The polar coordinate equation of the line $l$ and the curve $C$;<br/>$(2)$ When $\theta _{0}\in (\frac{π}{6}$,$\frac{π}{3})$, if the line $l$ intersects the curve $C$ at points $M$ and $N$, find the range of $|OM|+|ON|$.

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