
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 graph of the quadratic function $y=x^{2}+\left(a-2\right)x+3$ passes through the point $\left(2,3\right)$.  
$(1)$ Find the expression of this quadratic function;  
$(2)$ When $0 \lt x \lt 3$, the range of $y$ is ______;  
$(3)$ Given points $P(m-1$,$y_{1})$ and $Q(m$,$y_{2})$ on the graph of this quadratic function, if $y_{1} \gt y_{2}$, directly write down the range of $m$.

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