
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 that the function $f(x)=\sin(x)-2\sqrt{3}\cos(x)$ has a symmetry axis at $x=-\frac{\pi}{6}$, and $f(x_1)+f(x_2)=0$, where the function $f(x)$ is monotonic on $(x_1,x_2)$, find the minimum value of $|x_1+x_2|$.

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