
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
Someone measured the combined length of the two arcs on the circumcircle of triangle $ABC$, where the points are projected onto the line $AB$, and the projection lies on one of the extensions of the segment $AB$. They performed the same measurement for the lines $BC$ and $CA$, and stated that the ratio of the three measured lengths is $1: 2: 3$. Based on this, what can we say about the angles of the triangle?

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