
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
Given the polar equation of curve $C$ is $\rho =1$, and the parametric equation of the line is $\begin{cases} & x=1+\dfrac{t}{2} \\ & y=2+\dfrac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter).

$(1)$ Write the Cartesian coordinate equations of the line and curve $C$;

$(2)$ Suppose curve $C$ undergoes a scaling transformation $\begin{cases} & x'=2x \\ & y'=y \end{cases}$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x,y)$, find the minimum value of $x+2\sqrt{3}y$.

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