
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 $f(x)=\sqrt{3}\sin\omega x-\cos\omega x$ with $\omega>0$ and a symmetry axis at $x=\frac{\pi}{3}$, find the range of real number $m$ when $\omega$ takes the smallest positive value, such that the line $x+my+c=0$ is tangent to the graph of the function $f(x)$.

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