
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 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 lengths of the sides of \( \triangle ABC \), and \( \alpha, \beta, \gamma \) are the measures of angles \( \angle A, \angle B, \angle C \) respectively. Determine \( \alpha, \beta, \gamma \).

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