
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 Cartesian coordinate system $(xOy)$ where $O$ is the origin, the positive semi-axis of the $x$-axis is the polar axis, and the polar coordinates of point $P$ are $(4\sqrt{3}, \frac{\pi}{6})$. The polar equation of curve $C$ is ${\rho}^{2} + 4\sqrt{3}\rho \sin \theta = 4$.

(I) Write the Cartesian coordinates of point $P$ and the standard equation of curve $C$.

(II) Suppose $Q$ is a moving point on curve $C$, find the maximum distance from the midpoint $M$ of $PQ$ to the line $l: \begin{cases} & x=3+2t, \\ & y=-2+2t \end{cases} (t \text{ is a parameter})$.

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