
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 quadratic function $y=-x^{2}+bx+c$.  
$(1)$ When $b=4$ and $c=3$,  
① Find the coordinates of the vertex of the function graph;  
② Find the range of $y$ when $-1\leqslant x\leqslant 3$;  
$(2)$ When $x\leqslant 0$, the maximum value of $y$ is $2$; when $x \gt 0$, the maximum value of $y$ is $3$, find the expression of the quadratic function.

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