
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

Find all non-negative real solutions to the following system of equations:

\[ x_2^2 + x_1 x_2 + x_1^4 = 1 \]
\[ x_3^2 + x_2 x_3 + x_2^4 = 1 \]
\[ x_4^2 + x_3 x_4 + x_3^4 = 1 \]
\[ \vdots \]
\[ x_{1999}^2 + x_{1998} x_{1999} + x_{1998}^4 = 1 \]
\[ x_1^2 + x_{1999} x_1 + x_1^4 = 1 \]

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