
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 set of circles $C_{k}$: $(x-k+1)^{2}+(y-3k)^{2}=2k^{4}$ $(k\in\mathbb{N}^{*})$, if the centers of these circles lie on a fixed line $l$, then the equation of this line is __________.

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