
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
Let the function $y=f(x)$ be defined on the interval $[x_1, x_2]$ with its graph's endpoints being $A(x_1, y_1)$ and $B(x_2, y_2)$. Let $M(x, y)$ be any point on the graph of $f(x)$, where $x=\lambda x_1 + (1-\lambda)x_2$ ($\lambda \in \mathbb{R}$), and $\overrightarrow{ON} = \lambda \overrightarrow{OA} + (1-\lambda) \overrightarrow{OB}$. If the inequality $|\overrightarrow{MN}| \leq k$ always holds, then the function $f(x)$ is said to have a "k-order linear approximation" on $[x_1, x_2]$. If the function $y=\sqrt{x}$ and $y=3x$ have exactly one "k-order linear approximation" on $[0, 1]$, then the range of values for the real number $k$ is \_\_\_\_\_\_.

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