
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 a parabola with its vertex at the origin and its focus on the $x$-axis, it is also known that a point $A$ on the parabola has an $x$-coordinate of $3$, and the distance from point $A$ to the focus is $4$.

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

$(2)$ If the parabola intersects with the line $y=kx-1$ at two distinct points $A$ and $B$, and the $x$-coordinate of the midpoint of $AB$ is $1$, find the value of $k$.

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