
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 $C$: $y^{2}=2px(p > 0)$ with focus $F$, the distance from the point on the parabola with an abscissa of $\dfrac{1}{2}$ to the vertex of the parabola is equal to the distance from this point to the directrix of the parabola.

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

$(2)$ Suppose the line $x-my-6=0$ intersects the parabola $C$ at points $A$ and $B$. If $\angle AFB=90^{\circ}$, find the value of the real number $m$.

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