
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
If the function $f(x)=2\sin(\omega x+\frac{\pi}{3})(\omega>0)$, and $A(\alpha,2)$, $B(\beta,0)$ are two points on the graph of the function $f(x)$, and the minimum value of $|AB|$ is $\sqrt{4+\frac{{\pi}^{2}}{4}}$, then the value of $f(\frac{5\pi}{6})$ is ______.

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