
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
Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all \( n \)-tuples \(\left(x_{1}, \ldots, x_{n}\right)\), satisfying \( x_{i} \geq 0 \) and \(\sum_{i=1}^{n} x_{i} = 1 \).

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