
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 $\overrightarrow{b}$, their included angle is $\theta$, and $|\overrightarrow{a}| = 3$, $|\overrightarrow{b}| = 1$, and $x$ is a positive real number.
(1) If $\overrightarrow{a} + 2\overrightarrow{b}$ is perpendicular to $\overrightarrow{a} - 4\overrightarrow{b}$, find $\tan \theta$;
(2) If $\theta = \frac{\pi}{6}$, find the minimum value of $|x\overrightarrow{a} - \overrightarrow{b}|$ and the corresponding value of $x$, and determine whether $\overrightarrow{a}$ is perpendicular to $x\overrightarrow{a} - \overrightarrow{b}$ in this case.

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