
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 a triangle \( \triangle ABC \) with a circumradius \( R \), it is known that:

\[ \frac{a \cos \alpha + b \cos \beta + c \cos \gamma}{a \sin \beta + b \sin \gamma + c \sin \alpha} = \frac{a + b + c}{9R} \]

where \( a, b, c \) are the side lengths of \( \triangle ABC \), and \( \alpha, \beta, \gamma \) are the angles opposite these sides respectively. Find the angles \( \alpha, \beta, \gamma \).

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