
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 Cartesian coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=a+\frac{1}{2}t,}\\{y=2a+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ ($t$ is the parameter, $a\in R$). Establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\cos \theta $.  
$(1)$ Write down the general equation of line $l$ and the Cartesian equation of curve $C$.  
$(2)$ If there is exactly one point on curve $C$ that is at a distance of $\sqrt{3}-1$ from line $l$, find the value of the real number $a$.

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