
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 $\overrightarrow{a}=(\sin \pi x,1)$, $\overrightarrow{b}=( \sqrt {3},\cos \pi x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$:
(I) If $x\in[0,2]$, find the interval(s) where $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is monotonically increasing.
(II) Let $P$ be the coordinates of the first highest point and $Q$ be the coordinates of the first lowest point on the graph of $y=f(x)$ to the right of the $y$-axis. Calculate the cosine value of $\angle POQ$.

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