
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
In the plane rectangular coordinate system $xOy$, the parabola $y=a\left(x-3\right)^{2}-1$ passes through the point $\left(2,1\right)$.  
$(1)$ Find the expression of this parabola;  
$(2)$ After moving this parabola up by ______ units, the resulting parabola has only one point of intersection with the $x$-axis.

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