
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 $y=ax^{2}-2ax+3 (a\neq 0)$.  
$(1)$ Find the axis of symmetry of the parabola.  
$(2)$ The parabola is shifted $3|a|$ units down along the y-axis. If the vertex of the parabola lies on the x-axis, find the value of $a$.

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