
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 the coordinates of the vertex of the quadratic function $y=ax^{2}+bx+c$ are $\left(1,-3\right)$, and it passes through the point $(2, -\frac{5}{2})$.  
$(1)$ Find the expression of this quadratic function.  
$(2)$ If the graph of this quadratic function intersects the line $y=m$ (where $m$ is a constant) at points $A$ and $B$ with $AB=6$, then $m=\_\_\_\_\_\_$.  
$(3)$ When $-3 \lt x \lt 3$, the range of $y$ is ______.

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