
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 $\overrightarrow{a}=(\cos x+\sin x,2\sin x)$ and $\overrightarrow{b}=(\cos x-\sin x,\cos x)$. Let $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$.
(I) Find the smallest positive period of $f(x)$;
(II) Find the interval(s) on which $f(x)$ is monotonically increasing in $\left[\frac{\pi}{4},\frac{3\pi}{4}\right]$.

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