
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 $\overrightarrow{m}=(-2\sin (\pi-x),\cos x)$ and $\overrightarrow{n}=( \sqrt {3}\cos x,2\sin ( \frac {\pi}{2}-x))$, and the function $f(x)=1- \overrightarrow{m}\cdot \overrightarrow{n}$.
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ When $x\in[0,\pi]$, find the intervals where $f(x)$ is monotonically increasing.

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