
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 rectangular coordinate system, point $A(3,3)$ is on circle $⊙C$. The circle is folded twice, causing point $A$ to coincide with two distinct points on the circle (other than point $A$ itself). The equations of the creases from the two folds are $x-y+1=0$ and $x+y-7=0$, respectively. If there is a point $P$ on $⊙C$ such that $∠MPN=90^{\circ}$, where $M$ and $N$ have coordinates $(-m,0)$ and $(m,0)$, respectively, determine the maximum value of $m$.

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