
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 equation of the circle $C$ is $x^{2}+y^{2}-2mx+2my=4-2m^{2}$.
$(1)$ Find the general equation of the straight line where the center of circle $C$ is located;
$(2)$ If the length of the chord intercepted by the line $l: x-y+4=0$ on circle $C$ is $2\sqrt{2}$, find the value of real number $m$;
$(3)$ If $A$ and $B$ are two moving points on circle $C$ and point $P(\sqrt{2}, \sqrt{2})$ is fixed, when the maximum value of $\angle APB$ is $90^{\circ}$, find the value of real number $m$.

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