
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
Read the following material and answer the questions below:<br/>By using the complete square formula $\left(a\pm b\right)^{2}=a^{2}\pm 2ab+b^{2}$, appropriate transformations can be made to $a^{2}+b^{2}$ through completing the square, such as:<br/>$a^{2}+b^{2}=\left(a+b\right)^{2}-2ab$ or $a^{2}+b^{2}=\left(a-b\right)^{2}+2ab$, thus solving some problems.<br/>Example: Given $a+b=5$, $ab=3$, find the value of $a^{2}+b^{2}$.<br/>Solution: $a^{2}+b^{2}=\left(a+b\right)^{2}-2ab=5^{2}-2\times 3=19$.<br/>Solve the following problems based on the understanding of the example:<br/>$(1)$ Given $a-b=2$, $ab=3$, find the value of $a^{2}+b^{2}$;<br/>$(2)$ If $a+\frac{1}{a}=6$, find the value of ${a}^{2}+\frac{1}{{a}^{2}}$;<br/>$(3)$ If $n$ satisfies $\left(n-2024\right)^{2}+\left(2023-n\right)^{2}=1$, find the value of the expression $\left(n-2024\right)\left(2023-n)$.

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