
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 a circle $C$ with radius $2$ and center on the line $y=x+2$.

(1) When the circle $C$ passes through the point $A(2,2)$ and is tangent to the $y$-axis, find the equation of circle $C$.

(2) Given $E(1,1)$, $F(1,3)$, if there exists a point $Q$ on circle $C$ such that $|QF|^2 - |QE|^2 = 32$, find the range of the abscissa $a$ of the center of the circle.

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