
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 equation of a parabola $y=x^{2}-8x+12$.  
$(1)$ Write down the equation of the axis of symmetry and the coordinates of the vertex of the parabola.  
$(2)$ Does the parabola intersect the x-axis? If yes, find the coordinates of the points of intersection between the parabola and the x-axis; if not, explain the reason.

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