
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:

$$
\begin{aligned}
& x_{1} + x_{1} x_{2} = 1 \\
& x_{2} + x_{2} x_{3} = 1 \\
& \vdots \\
& x_{99} + x_{99} x_{100} = 1 \\
& x_{100} + x_{100} x_{1} = 1
\end{aligned}
$$

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