
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
In triangle ABC, with opposite sides \(a\), \(b\), and \(c\) corresponding to angles \(\angle A\), \(\angle B\), and \(\angle C\) respectively, the inequality \(\left|\frac{\sin A}{\sin A+\sin B}+\frac{\sin B}{\sin B+\sin C}+\frac{\sin C}{\sin C+\sin A}-\frac{3}{2}\right|<m\) holds, where \(m=\frac{8 \sqrt{2}-5 \sqrt{5}}{6}\) is the least upper bound.

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