
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 that circle $C$ is symmetric about the $y$-axis, passes through the focus of the parabola $y^{2}=4x$, and is divided into two arc lengths with a ratio of $1:2$ by the line $y=x$.

(I) Find the equation of circle $C$.

(II) If the center of circle $C$ is below the $x$-axis, a line $l$ is drawn through point $P(-1,2)$ and is tangent to circle $C$. Find the equation of line $l$.

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