
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
One side of a triangle is divided into segments $d, e,$ and $f$ by the trisectors of the opposite angle. Calculate the angle opposite to this side for the following instances of $d, e,$ and $f$:
I. $1 ; 1 ; 2$
II. $11 ; 8 ; 11$
III. $4 ; 3 ; 5$.

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