
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 $\overrightarrow{a} = (\sin(2x - \frac{\pi}{3}), 1)$ and $\overrightarrow{b} = (\sqrt{3}, -1)$, with the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$.
1. Find the period and the interval where $f(x)$ is monotonically decreasing.
2. If $x \in [0, \frac{\pi}{2}]$, find the range of $f(x)$.

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