
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)^2 + y^2 = 4, line l_1: y = √3x, and line 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, determine the value of k.

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