
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
(1) Given that the center of circle C is the intersection point of $x-y+1=0$ and the x-axis, and it is tangent to the line $x+y+3=0$, find the standard equation of circle C;  
(2) If point P $(x, y)$ is on the circle $x^2+y^2-4y+3=0$, find the maximum value of $\frac{y}{x}$.

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