
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
In the Cartesian coordinate system $xOy$, the line $x+y+3\sqrt{2}+1=0$ is tangent to circle $C$, and the center of circle $C$ is at $(1, -2)$.
(Ⅰ) Find the equation of circle $C$;
(Ⅱ) If the line $y=kx+1$ does not intersect circle $C$, find the range of values for $k$;
(Ⅲ) If the line $y=x+m$ intersects circle $C$ at points $M$ and $N$, and $OM \perp ON$, find the value of $m$.

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