
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 circle $C: x^{2}+y^{2}-2x-2\sqrt{3}y+3=0$, and point $A(0,m)$ ($m > 0$), where point $A$ and point $B$ are symmetric about the $x$-axis. If there exists a point $M$ on circle $C$ such that $\vec{AM} \cdot \vec{BM}=0$, find the coordinates of point $M$ when $m$ is at its maximum value.

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