
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
Given the function $y=x+\frac{1}{x}$ defined on the interval $[1, 2]$, with endpoints $A$ and $B$ of its graph, and $M(x, y)$ is any point on the graph of $f(x)$, where $x=\lambda \cdot 1 + (1-\lambda)\cdot 2$ ($\lambda \in \mathbb{R}$). The vector $\overrightarrow{ON} = \lambda \overrightarrow{OA} + (1-\lambda) \overrightarrow{OB}$, where $O$ is the origin. If the inequality $|\overrightarrow{MN}| \leq k$ always holds, then determine the range of values for the real number $k$.

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