
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 polar coordinates, the equation of curve C is given by $\rho^2 = \frac{3}{1 + 2\sin^2\theta}$ and the point R has coordinates $(2\sqrt{2}, \frac{\pi}{4})$.
(1) If the pole of the polar coordinate system coincides with the origin of the Cartesian coordinate system, the polar axis coincides with the positive x-axis, and the unit lengths are the same, convert the polar equation of curve C into a Cartesian equation.
(2) Let point P be a moving point on curve C, and rectangle PQRS has PR as its diagonal with one side perpendicular to the polar axis. Find the minimum perimeter of the 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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