
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
Let \( x_{1}, x_{2}, x_{3}, x_{4} \) be non-negative real numbers satisfying the equation:
\[ 
x_{1} + x_{2} + x_{3} + x_{4} = 1 
\]
Find the maximum value of \( \sum_{1 \leq i < j \leq 4}(x_{i} + x_{j}) \sqrt{x_{i} x_{j}} \), and determine the values of \( x_{1}, x_{2}, x_{3}, x_{4} \) that achieve this maximum value.

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