
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 established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho =2$, and the vertices of the positive triangle $ABC$ are all on $C_{1}$. The coordinates of points $A$, $B$, and $C$ are arranged in counterclockwise order, and the coordinates of point $A$ are $(2,0)$.

(I) Find the rectangular coordinates of points $B$ and $C$;

(II) Let $P$ be any point on the circle $C_{2}$: ${{x}^{2}}+{{(y+{{\sqrt{3}}})}^{2}}=1$, find the range of values for $|PB{{|}^{2}}+|PC{{|}^{2}}$.

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