
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 $L$: $y=x+m$, $m∈R$.

(1) A circle with center at point $M(2,0)$ is tangent to line $L$ at point $P$, and point $P$ is on the $y$-axis. Find the equation of circle $M$.

(2) If line $L$ is symmetric about the $x$-axis and the chord length intercepted by line $L'$ and the circle $M$ from condition (1) is $4$. Find the value of $m$.

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