
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 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 chords intercepted by line $l$ on circles $C_{1}$ and $C_{2}$ is $3$, then the real number $k=$ ______.

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