
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
In the expression \( x_{1}: x_{2}: \ldots: x_{n} \), brackets are placed to indicate the order of operations, and the result is written as a fraction:

\[
\frac{x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}}{x_{i_{1}} x_{l_{2}} \ldots x_{i_{n-k}}}
\]

(where each of the variables \( x_{1}, x_{2}, \ldots, x_{n} \) is either in the numerator or the denominator). How many distinct expressions can be obtained in this way for all possible ways of placing the brackets?

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