
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
Given the parametric equation of curve $C_1$ is $\begin{cases} x = 2\cos \varphi, \\ y = 3\sin \varphi \end{cases}$ (where $\varphi$ is the parameter), and by setting up a polar coordinate system with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho = 2$. The vertices of the square $ABCD$ are on curve $C_2$, and points $A$, $B$, $C$, $D$ are arranged in counterclockwise order, with the polar coordinates of point $A$ being $(2, \frac{\pi}{3})$.

(1) Find the Cartesian coordinates of points $A$, $B$, $C$, $D$;

(2) Let $P$ be any point on $C_1$. Calculate the range of values for $|PA|^2 + |PB|^2 + |PC|^2 + |PD|^2$.

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