
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 \( n > 1 \) be an integer. Find, with proof, all sequences \( x_{1}, x_{2}, \ldots, x_{n-1} \) of positive integers with the following three properties:
(a) \( x_{1} < x_{2} < \cdots < x_{n-1} \);
(b) \( x_{i} + x_{n-i} = 2n \) for all \( i = 1, 2, \ldots, n-1 \);
(c) given any two indices \( i \) and \( j \) (not necessarily distinct) for which \( x_{i} + x_{j} < 2n \), there is an index \( k \) such that \( x_{i} + x_{j} = x_{k} \).

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