
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 $C:y^{2}=2px(p > 0)$ and the line $x-\sqrt{2}y+4=0$ are tangent to each other.

   $(1)$ Find the equation of this parabola;

   $(2)$ On the positive half of the $x$-axis, is there a specific point $M$ such that a moving line $l$ passing through this point intersects the parabola $C$ at points $A$ and $B$, making $\frac{1}{|AM|^{2}}+\frac{1}{|BM|^{2}}$ a constant value? If such a point exists, find the coordinates of point $M$; if not, explain why.

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