
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 $C: x^{2}+y^{2}=r^{2}(r \gt 0)$, if there is exactly one chord passing through the fixed point $P(1,1)$ with a chord length of $2$, then $r$ can be ______$.(Just need to write one of the values, if multiple answers are provided, score according to the first answer.)$

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