
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 line $l$ is given by the parametric equations $\begin{cases} x=-\frac{1}{2}t, \\ y=3+\frac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter). With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho =4\sin \left( \theta +\frac{\pi }{3} \right)$.

    (1) Find the Cartesian equation of curve $C$;

    (2) Let the polar coordinates of point $M$ be $\left( 3,\frac{\pi }{2} \right)$, and the intersection points of line $l$ and curve $C$ be $A$ and $B$. Calculate the value of $|MA|+|MB|$.

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