
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 parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=6+\sqrt{3}t}\\{y=-t}\end{array}}\right.$ (where $t$ is a parameter). Using the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of curve $C$ is $ρ=2cosα$ ($0<α<\frac{π}{2}$).  
$(Ⅰ)$ Find the polar coordinate equation of the line $l$ and the rectangular coordinate equation of curve $C$.  
$(Ⅱ)$ If the line $l_{1}$ passing through the pole $O$ intersects $l$ at point $M$ and intersects $C$ at point $N$, find the minimum value of $\frac{{|{OM}|}}{{|{ON}|}}$.

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