
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
The circle $C: (x-2)^2+y^2=4$, and the lines $l_1: y=\sqrt{3}x$, $l_2: y=kx-1$. If the ratio of the lengths of the chords intercepted by circle $C$ on lines $l_1$ and $l_2$ is $1:2$, then the value of $k$ is $\boxed{\frac{1}{2}}$.

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