
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
After students learned about the golden ratio of a line segment, Teacher Zeng proposed a new definition: Point $C$ is a point on the line segment $AB$. If $\frac{BC}{\sqrt{n}AC}=\frac{\sqrt{n}AC}{AB}=k_{n}$, then point $C$ is called the "near $A$, $n$-order golden section point" of line segment $AB". For example, if $\frac{BC}{\sqrt{2}AC}=\frac{\sqrt{2}AC}{AB}=k_{2}$, then point $C$ is called the "near $A$, $2$-order golden section point" of line segment $AB$; if $\frac{BC}{\sqrt{3}AC}=\frac{\sqrt{3}AC}{AB}=k_{3}$, then point $C$ is called the "near $A$, $3$-order golden section point" of line segment $AB$. If point $C$ is the "near $A$, $4$-order golden section point" of line segment $AB$, then $k_{4}=$____.

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