
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 a parabola passes through two points $A(1,0)$, $B(0,-3)$, and its axis of symmetry is the line $x=2$.
$(1)$ Find the equation of this parabola.
$(2)$ If point $C$ is a point on the parabola and the distance from $C$ to the $x$-axis is $3$, find the coordinates of point $C$.

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