
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 $\rho =1$, with the pole as the origin and the positive half-axis of the $x$-axis as the polar axis, a Cartesian coordinate system is established. The parametric equation of line $\ell$ is $\begin{cases}x=1+ \frac{1}{2}t \\ y=2+ \frac{3}{2}t\end{cases}$ (where $t$ is the parameter).

$(1)$ Write the Cartesian equations for line $\ell$ and curve $C$;

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

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