
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 the graph of the quadratic function $y=x^{2}+bx+c$ intersects the $y$-axis at point $A\left(0,-3\right)$ and the squares of the abscissas of the two intersection points with the $x$-axis, $x_{1}$ and $x_{2}$, sum up to $15$, find the analytical expression of this quadratic function.<br/>$(2)$ Under the condition in (1), when $b \lt 0$, find the solution set of the one-variable quadratic inequality $ax^{2}-\left(a-b\right)x-c \lt 0\left(a\in R\right)$.

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