
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 that $P(2,0)$ is a point inside the circle $C$: $x^{2}+y^{2}-2x+2my+m^{2}-7=0 (m > 0)$, and the line $AB$ passing through point $P$ intersects circle $C$ at points $A$ and $B$. If the maximum area of $\triangle ABC$ is $4$, then the range of the positive real number $m$ is $\_\_\_\_\_\_\_\_\_.$

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