
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
The expression \( x_{1} : x_{2} : x_{3} : \ldots : x_{n} \) has a definite value only when parentheses are used to indicate the order of divisions. The expression can be rewritten in the form

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

where \( i_{1}, i_{2}, \ldots, i_{k} \) and \( j_{1}, j_{2}, \ldots, j_{n-k} \) are any permutation of the indices \( 1, 2, \ldots, n \) with \( i_{1} < i_{2} < \ldots < i_{k} \) and \( j_{1} < j_{2} < \ldots < j_{n-k} \).

For example,

\[ \left(x_{1} : \left(x_{2} : x_{3}\right)\right) : x_{4} = \frac{x_{1} x_{3}}{x_{2} x_{4}}, \quad \left(\left(x_{1} : x_{2}\right) : x_{3}\right) : x_{4} = \frac{x_{1}}{x_{2} x_{3} x_{4}} \]

How many distinct fractions of this form can be obtained by placing the parentheses in different ways?

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