
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 $\overrightarrow{a}=(1,\sin x)$, $\overrightarrow{b}=(\cos (2x+\frac{\pi }{3}),\sin x)$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}-\frac{1}{2}\cos 2x$;

(1) Find the analytical expression of function $f(x)$ and its monotonically increasing interval;

(2) When $x\in [0,\frac{\pi }{3}]$, find the range of function $f(x)$.

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