
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 planar vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ satisfy $|\vec{a}|=3$, $|\vec{b}|=4$, and $|\vec{a}-\vec{b}|=2 \sqrt{3}$. Find the minimum value of the function $\mathrm{f}(\mathrm{x})=\left|\mathrm{x} \overrightarrow{\mathrm{a}}+\frac{1}{\mathrm{x}} \overrightarrow{\mathrm{b}}\right|$ $(x \neq 0)$.

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