
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, the axis of symmetry is the x-axis, and the focus is on the line $x-2y-4=0$.  
$(1)$ Find the equation of the parabola.  
$(2)$ If the abscissa of point $A$ on the parabola is $2$, find the distance from point $A$ to the focus of the parabola.

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