
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 two vertices \(P\) and \(R\) of a regular pentagon, we can identify the other vertices using the following process involving 8 circles drawn exclusively with a compass. The first circle \(k_{1}\) is centered at \(P\) with radius \(PR\), denoted as \(k_{1}=P(PR)\). The second circle \(k_{2}\) is centered at \(R\) with radius \(PR\), intersecting \(k_{1}\) at points \(X\) and \(Y\). Circle \(k_{3}\) is centered at \(X\) with radius \(XY\), intersecting \(k_{1}\) at \(Y\) and \(Z\). Circle \(k_{4}\) is centered at \(P\) with radius \(XY\), intersecting \(k_{3}\) at \(V\) and \(W\). Circle \(k_{5}\) is centered at \(V\) with radius \(RZ\), and circle \(k_{6}\) is centered at \(W\) with radius \(RZ\), intersecting at \(U\) and \(T\). 

What could be the centers and radii of the remaining two circles?

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