
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 $O: x^{2}+y^{2}=r^{2}(r \gt 0)$ and a line $l: x-y+4=0$, the distance from the center of the circle $O$ to the line $l$ is ______; if there are exactly two points on the circle $O$ that are at a distance of $\sqrt{2}$ from the line $l$, write down a value of $r$ that satisfies this condition, $r=\_\_\_\_\_\_$.

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