
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 that $P(x_{0}, y_{0})$ is any point on the circle $C: (x-t)^{2} + (y-s)^{2} = r^{2}$ where $r > 0$. When $a \neq b$, the value of $|x_{0}-y_{0}+a|+|x_{0}-y_{0}+b|$ is independent of $x_{0}$ and $y_{0}$. Which of the following conclusions is correct?
- $(1)$ When $|a-b| = 2\sqrt{2}r$, the locus of point $(t, s)$ is a straight line.
- $(2)$ When $|a-b| = 2\sqrt{2}$, the maximum value of $r$ is $1$.
- $(3)$ When $r = \sqrt{2}$ and $b = 2$, the range of values for $a$ is $a \geq 6$.

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