
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 circle $C$ that passes through the point $(1,0)$, and its center lies on the positive semi-axis of the $x$-axis. The length of the chord intercepted by the circle $C$ on the line $l$: $y=x-1$ is $2\sqrt{2}$. Determine the equation of the line passing through the center of the circle and perpendicular to line $l$.

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