
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 a Cartesian coordinate system, it is known that $A(3,2)$ is a point on circle $C$. After folding the circle twice such that point $A$ coincides with two different points on the circle (points distinct from $A$), the equations of the two fold lines are $x-y+1=0$ and $x+y-7=0$, respectively. If there exists a point $P$ on circle $C$ such that $\angle MPN = 90^{\circ}$, where the coordinates of $M$ and $N$ are $(-m,0)$ and $(m,0)$ respectively, then the set of real number $m$ is \_\_\_\_\_\_.

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