
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 vectors $\overrightarrow{a} = (1 + \log_{2}x, \log_{2}x)$ and $\overrightarrow{b} = (\log_{2}x, 1)$,
(1) Find the value of the real number $x$ if $\overrightarrow{a} \perp \overrightarrow{b}$;
(2) Find the range of the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$, where $x \in [\frac{1}{4}, 2]$.

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