
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 $y=ax^{2}+bx+2$ passes through the points $(2,0)$ and point $C$.  
$(1)$ If the coordinates of point $C$ are $(1,3)$,  
① Find the expression of this quadratic function;  
② Write down the range of $y$ directly when $-1\leqslant x\leqslant 2$.  
$(2)$ If the coordinates of point $C$ are $(1,m)$ and the graph of the function opens upwards, write down the range of $m$ directly.

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