
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
The equation of the line that passes through the center of the circle $x^{2}-2x+y^{2}=0$ and is parallel to the line $x+2y=0$ is $y=\frac{-1}{2}x+c$. Find the value of $c$ such that this line passes through the center of the circle.

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