
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 \(x_{1}: x_{2}: \cdots: x_{n}\), use parentheses to indicate the order of operations, and the result can be written in fractional form:
$$
\frac{x_{i_{1}} x_{i_{2}} \cdots x_{i_{k}}}{x_{j_{1}} x_{j_{2}} \cdots x_{j_{n-k}}}
$$
(At the same time, each letter in \(x_{1}, x_{2}, \cdots, x_{n}\) may appear in the numerator or the denominator.) How many different fractions can be obtained by placing parentheses in all possible ways?

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