
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
$(1)$ Given that $a$, $b$, and $c$ are real numbers, and $\sqrt{a-2}+|b+1|+\left(c+2\right)^{2}=0$, find the roots of the equation $ax^{2}+bx+c=0$ in terms of $x$. <br/>$(2)$ Given that the graph of the quadratic function $y=ax^{2}+bx+c$ passes through the points $A\left(-1,0\right)$, $B\left(0,-3\right)$, and $C\left(3,0\right)$, find the analytical expression of this quadratic function.

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