
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 a parabola $y=ax^{2}+bx+c\left(a\neq 0\right)$.<br/>$(1)$ If the vertex of the parabola is $\left(1,10\right)$, and it passes through the point $\left(-1,-2\right)$, find the analytical expression of the parabola;<br/>$(2)$ If the parabola intersects the $x$-axis at points $\left(-1,0\right)$ and $\left(3,0\right)$, and intersects the $y$-axis at point $\left(0,3\right)$, find the analytical expression of the parabola.

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