
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 polar coordinate system, the polar equation of curve $C$ is given by $\rho^{2}= \frac{3}{1+2\sin ^{2}\theta }$, and there is a point $R(2 \sqrt{2}, \frac{\pi}{4})$.

(1) If the pole of the polar coordinate system coincides with the origin of the rectangular coordinate system, the polar axis is coincident with the positive semi-axis of $x$, and the units of length are the same, convert the polar equation of curve $C$ to a rectangular equation.

(2) Let point $P$ be a moving point on curve $C$, and rectangle $PQRS$ has $PR$ as its diagonal, with one side of the rectangle perpendicular to the polar axis. Determine the minimum value of the perimeter of rectangle $PQRS$ and the rectangular coordinates of point $P$ at that time.

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