
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 $O$ is the circumcenter of $\triangle ABC$, and $A= \frac{\pi}{3}$, $\frac{\cos B}{\sin C} \overrightarrow{AB}+ \frac{\cos C}{\sin B} \overrightarrow{AC}=2m \overrightarrow{AO}$, find the value of $m$.

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