
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
By using the complete square formula, the polynomial $ax^{2}+bx+c\left(a\neq 0\right)$ can be transformed into the form $a\left(x+h\right)^{2}+k$. We call this transformation method the "completing the square" method. We have learned to solve quadratic equations using the completing the square method. In addition, the completing the square method can also be used to solve maximum and minimum problems of quadratic trinomials. Read the following material and answer the following questions:<br/>Material: For solving maximum and minimum problems of quadratic trinomials, consider the following example:<br/>$x^{2}-2x+3=x^{2}-2x+1^{2}-1^{2}+3=\left(x-1\right)^{2}+2$. Since $\left(x-1\right)^{2}\geqslant 0$, then $\left(x-1\right)^{2}+2\geqslant 2$, so when $x=1$, the minimum value of the original expression is $2$.<br/>Answer the following questions:<br/>$(1)$ Find the minimum value of $x^{2}-4x-3$;<br/>$(2)$ If a real number $n$ satisfies $n=6-x$, find the maximum value of $\sqrt{x^2-2n^2}$.

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