
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  $x_1,x_2,\ldots ,x_{1997}$  be real numbers satisfying the following conditions:
i)  $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$  for  $i=1,2,\ldots ,1997$ ;
ii)  $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$  .
Determine (with proof) the maximum value of  $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$  .

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