
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 largest value the expression   $$  \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j}  $$  may achieve, as  $ x_1,x_2,x_3,x_4 $  run through the non-negative real numbers, and add up to  $ 1. $  Find also the specific values of this numbers that make the above sum achieve the asked maximum.

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