
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 two non-collinear vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, such that $|\overrightarrow {a}|=3$, $|\overrightarrow {b}|=1$, and the angle between them is $\frac{\pi}{3}$. Find the minimum value of $|x\overrightarrow {a}-\overrightarrow {b}|$ and the corresponding real number $x$, and determine whether the vector $\overrightarrow {a}$ is perpendicular to the vector $x\overrightarrow {a}-\overrightarrow {b}$ in this case.

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