
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
Solve the following system of equations, in which  $a$  is a given number satisfying  $|a| > 1$ :

 $\begin{matrix} x_{1}^2 = ax_2 + 1  x_{2}^2 = ax_3 + 1  \ldots  x_{999}^2 = ax_{1000} + 1  x_{1000}^2 = ax_1 + 1  \end{matrix}$ 

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