
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 $C$ with its vertex at the origin and focus on the $x$-axis. There is a point $P(4, m)$ on the parabola, and the distance between $P$ and the focus is $6$.

(1) Find the equation of the parabola $C$.
(2) If parabola $C$ intersects with 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}
user