
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: $(x+2)^2+(y-b)^2=3$ ($b>0$) that passes through the point $(-2+\sqrt{2}, 0)$, and a line $l: y=x+m$ ($m \in \mathbb{R}$).

(1) Find the value of $b$;  
(2) If line $l$ is tangent to circle C, find the value of $m$;  
(3) If line $l$ intersects circle C at points M and N, and $OM \perp ON$ ($O$ is the origin), find the value of the real number $m$.

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