
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
The conclusion "$a^{2}\geqslant 0$" is very useful in mathematics. Sometimes we need to complete the square for algebraic expressions. For example: $x^{2}+4x+5=x^{2}+4x+4+1=\left(x+2\right)^{2}+1$. Since $\left(x+2\right)^{2}\geqslant 0$, $\left(x+2\right)^{2}+1\geqslant 1$, therefore $x^{2}+4x+5\geqslant 1$. Try to solve the following problems using the "completing the square" method:<br/>$(1)$ Fill in the blanks: Because $x^{2}-4x+6=(x $______)$^{2}+$______; so when $x=$______, the algebraic expression $x^{2}-4x+6$ has the smallest/largest (choose "big" or "small") value, and this extremum is ______.<br/>$(2)$ Compare the algebraic expressions $x^{2}-1$ and $2x-3$.

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