
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 $\overrightarrow{m}=({\sqrt{3}\cos x, \cos x}), \overrightarrow{n}=({\sin x, \cos x})$, and the function $f(x)=\overrightarrow{m} \cdot \overrightarrow{n}$:
- $(1)$ Find the interval where the function $f(x)$ is monotonically increasing.
- $(2)$ If $x \in [-\frac{\pi}{8}, \frac{\pi}{2}]$, determine the range of the function $f(x)$.

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