
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
The function $y=A\sin(\omega x+\phi)$ ($A>0$, $\omega>0$, $|\phi|< \frac{\pi}{2}$) reaches its maximum value of 2 at $x= \frac{\pi}{4}$ and its minimum value of -2 at $x= \frac{7\pi}{12}$ within the same period.  
(1) Find the analytical expression of the function $y=f(x)$.  
(2) If $x\in[0,2\pi]$ and $f(x)= \sqrt{3}$, find the value of $x$.  
(3) If the function $f(x)$ satisfies the equation $f(x)=a$ ($1<a<2$), find the sum of all real roots within $[0,2\pi]$.

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