
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
Coordinate Systems and Parametric Equations

Given curve $C\_1$: $\begin{cases}x=12\cos \theta \ y=4\sin \theta\end{cases} (\text{parameter }R )$, with the coordinate origin $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of curve $C\_2$ is $\rho= \frac{3}{\cos \left(\theta+ \frac{\pi}{3}\right) }$, and the polar coordinates of point $Q$ are $\left(4 \sqrt{2}, \frac{\pi}{4}\right)$

(1) Convert the polar coordinate equation of curve $C\_2$ to a rectangular coordinate equation and find the rectangular coordinates of point $Q$;

(2) Let $P$ be a point on curve $C\_1$, find the minimum value of the distance from the midpoint $M$ of $PQ$ to a point on curve $C\_2$.

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