
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 vectors $\overrightarrow{a} = (1, \sqrt{3})$ and $\overrightarrow{b} = (\cos{x}, \sin{x})$, let the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 1$.
(1) If $f(x) = 0$, find the set of all solutions for $x$.
(2) If $x \in \left[0, \frac{\pi}{2}\right]$, determine the intervals of monotonicity and the extreme values of $f(x)$.

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