
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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