
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 line $l$: $y = x + m$, where $m \in \mathbb{R}$.

(1) If a circle with center $M(2,0)$ is tangent to line $l$ at point $P$, and point $P$ is on the $y$-axis, find the equation of this circle;  
(2) If the line $l'$ is symmetric to line $l$ with respect to the $x$-axis, does line $l'$ tangent to the parabola $C$: $x^2 = 4y$? Explain your reasoning.

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