
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 $A$ and $B$ are two points on the circle $C: (x-2)^2 + (y-m)^2 = 4$ where $m > 0$, and $|AB| = 2\sqrt{3}$. If there exists a real number $a$ such that the intersection point $P$ of the lines $l_{1}: ax-y=0$ and $l_{2}: x+ay+2a-4=0$ is exactly the midpoint of $AB$, determine the range of the real number $m$.

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