
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 \( R \), if the expression \( \frac{a \cos A + b \cos B + c \cos C}{a \sin B + b \sin C + c \sin A} = \frac{a + b + c}{9R} \), where \( a, b, c \) are the lengths of the sides opposite to angles \( \angle A, \angle B, \angle C \) respectively, determine the measures of the three internal angles of \( \triangle ABC \).

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