
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 $f(x)=\sqrt{3}\sin(\omega x+\varphi)+2\sin^2(\frac{\omega x+\varphi}{2})-1 (\omega>0, 0<\varphi<\pi)$ is an odd function, and the distance between adjacent pairs of symmetry axes of the graph of $f(x)$ is $\frac{\pi}{2}$.  
$(1)$ Find the analytical expression and the interval of monotonic decrease of $f(x)$;  
$(2)$ Given that $f(x)$ is in the interval $[-\frac{\pi}{6}, \frac{5\pi}{6}]$, find the sum of all roots of the equation $2f^2(x)+\sqrt{3}f(x)-3=0$.

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