
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: $(x-a)^2+(y-b)^2=2$, with its center C on the curve $y= \frac{1}{x}$ (where $x \in [1,2]$). Find the value of $ab$ and the range of lengths of the segment intercepted by the line $l: x+2y=0$ on circle C.

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