
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
Let vector $\vec{m}=(\sin x,-1)$, $\vec{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\vec{m}+\vec{n})\cdot\vec{m}$.  
(1) Find the interval of monotonic increase for the function $f(x)$.  
(2) When $x\in(0, \frac{\pi}{2})$, find the range of the function $f(x)$.

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