
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 \( |a|>1 \):
\[
\begin{aligned}
x_{1}^{2} &= a x_{2} + 1, \\
x_{2}^{2} &= a x_{3} + 1, \\
& \vdots \\
x_{999}^{2} &= a x_{1000} + 1, \\
x_{1000}^{2} &= a x_{1} + 1.
\end{aligned}
\]

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