
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 the parabola $L: x^{2}=2py(p > 0)$ and the point $M(2,2)$, if there exist two different points $A$ and $B$ on the parabola $L$ satisfying $\overrightarrow{AM}+ \overrightarrow{BM}=0$.  
$(1)$ Find the range of the real number $p$;  
$(2)$ When $p=2$, determine whether there exists a point $C$ on the parabola $L$, different from $A$ and $B$, such that the circle passing through points $A$, $B$, and $C$ has the same tangent line at point $C$ as the parabola $L$. If such a point exists, find the coordinates of point $C$; if not, explain why.

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