
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
Find the minimum value of the algebraic expression $x^{2}+4x-5$.  
Solution:  
Since $x^{2}+4x-5=x^{2}+4x+4-4-5=\left(x+2\right)^{2}-9$,  
and $\left(x+2\right)^{2}\geqslant 0$,  
thus $\left(x+2\right)^{2}-9\geqslant -9$,  
so when $x=-2$, the algebraic expression $x^{2}+4x-5$ has a minimum value of $-9$.  
Following the above method, complete the following questions:  
$(1)$ Find the minimum value of the algebraic expression $x^{2}-3x+2023$;  
$(2)$ Find the maximum value of the algebraic expression $-2x^{2}+x+3$.

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