
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 \( i_{1}, i_{2}, \ldots, i_{n} \) be an arbitrary permutation of the elements \( 1, 2, \ldots, n \) and consider the system of inequalities

$$
\begin{aligned}
& x_{1} > x_{2} > \ldots > x_{n-1} > x_{n}, \\
& |x_{i_{1}}| > |x_{i_{2}}| > \ldots > |x_{i_{n-1}}| > |x_{i_{n}}|.
\end{aligned}
$$

Determine how many permutations \( i_{1}, i_{2}, \ldots, i_{n} \) exist for which this system has a solution.

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