
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 in the Cartesian coordinate system, the parabola $y=ax^{2}+bx+c$ passes through three points $A(-1,0)$, $C(0,3)$, and $B(2,-3)$.  
$(1)$ Find the expression of the parabola.  
$(2)$ Find the axis of symmetry and the coordinates of the vertex of the parabola.

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