
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 equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.  
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur.

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