
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 curve $C_{1}$: $x^{2}+y^{2}+2kx+(4k+10)y+10k+20=0$ ($k\neq -1$), when $k$ takes different values, curve $C$ represents different circles, and the centers of these circles are collinear. The equation of this line is \_\_\_\_\_.

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