
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 points $A(1,1)$ and $B(1,3)$, and circle $C$ with equation ${{(x-a)}^{2}}+{{(y+a-2)}^{2}}=4$, there exists a point $P$ on the circle such that $P{{B}^{2}}-P{{A}^{2}}=32$. Determine the range of values for the x-coordinate of the circle's center, $a$.

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