
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 the circumradius of triangle \( \triangle ABC \) is \( R \), and 
\[
\frac{a \cos A + b \cos B + c \cos C}{a \sin B + b \sin C + c \sin A} = \frac{a + b + c}{9R},
\]
where \( a, b, \) and \( c \) are the lengths of the sides opposite to angles \( A, B, \) and \( C \) respectively. Find the measures of the three interior angles of \( \triangle ABC \).

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