
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 the vertex of a parabola is at the origin and its focus is on the $x$-axis. It is also known that the distance from a point $A(4,m)$ on the parabola to the focus is $6$.

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

$(2)$ If the equation of this parabola intersects with the line $y=kx-2$ at two distinct points $A$ and $B$, and the $x$-coordinate of the midpoint of $AB$ is $2$, find the value of $k$.

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