
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
In triangle \(ABC\), median \(AM\) is drawn, with point \(O\) being the center of the circumscribed circle and point \(Q\) the center of the inscribed circle. Segments \(AM\) and \(OQ\) intersect at point \(R\), with the ratio \(\frac{OR}{MR}=\sqrt{7} \frac{QR}{AR}\). Find the cosine of the difference between angles \(ABC\) and \(ACB\), given that \(\angle BAC = \frac{\pi}{3}\). Round the answer to the nearest hundredth if necessary.

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