
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 the line $l:y=kx$ ($k > 0$), and the circles $C_1:(x-1)^2+y^2=1$ and $C_2:(x-3)^2+y^2=1$. If the ratio of the lengths of the two chords intercepted by the line $l$ on the circles $C_1$ and $C_2$ is $3$, then the real number $k=\boxed{\text{\_\_\_\_\_\_\_\_\_\_\_}}$.

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