
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
Nonnegative reals  $x_1$ ,  $x_2$ ,  $\dots$ ,  $x_n$  satisfies  $x_1+x_2+\dots+x_n=n$ . Let  $||x||$  be the distance from  $x$  to the nearest integer of  $x$  (e.g.  $||3.8||=0.2$ ,  $||4.3||=0.3$ ). Let  $y_i = x_i ||x_i||$ . Find the maximum value of  $\sum_{i=1}^n y_i^2$ .

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