
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 a polar coordinate system, the equation of curve \(C\) is given by \({ρ}^{2}= \frac{3}{1+2{\sin }^{2}θ}\). Consider the point \(R\left(2 \sqrt{2}, \frac{\pi}{4}\right)\).

\(1.\) Establish a Cartesian coordinate system with the pole as the origin and the polar axis as the positive \(x\)-axis, and convert the polar equation of curve \(C\) to a Cartesian equation. Also, convert the polar coordinates of point \(R\) to Cartesian coordinates;

\(2.\) Let \(P\) be a point on curve \(C\), and \(PQRS\) be a rectangle with \(PR\) as the diagonal and one side perpendicular to the \(x\)-axis. Find the minimum perimeter of rectangle \(PQRS\), and the Cartesian coordinates of point \(P\) at this minimum.

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