
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 the function $f(x)=2\sin x \sin\left(x+\frac{\pi}{3}\right)+\cos 2x$.  
$(Ⅰ)$ Find the intervals where $f(x)$ is monotonically increasing and its maximum value.  
$(Ⅱ)$ If the function $g(x)=f(x)-a$ has exactly two zeros for $x\in\left[0,\frac{\pi}{2}\right]$, find the range of real number $a$.

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