Unique Absolute Difference Permutation

You are given an integer n. Your task is to construct a permutation of the numbers from 1 to n such that the absolute differences between every pair of adjacent numbers are unique. In other words, if the permutation is p = [p₁, p₂, ..., p_n], then the set of differences {|p₂ - p₁|, |p₃ - p₂|, ..., |p_n - p_n-1|} must contain only distinct values. Note that if no such permutation exists, you should output an empty list.

For example:

• For n = 1, output: 1.
• For n = 2, output: [] (no valid permutation exists under the problem constraints).
• For n = 3, one acceptable answer is: 1 3 2.
• For n = 4, one acceptable answer is: 1 3 2 4 or 2 4 1 3.
• For n = 5, one acceptable answer is: 1 4 2 5 3 (other variations such as 3 1 4 2 5 or 1 5 2 4 3 are also valid).

The uniqueness condition can be formally stated as follows: $$\{d_i = |p_{i+1} - p_i|: 1 \le i \le n-1\} = \{1, 2, \dots, n-1\}.$$

inputFormat

The input consists of a single integer n provided via standard input.

Format: A single line containing the integer n (1 ≤ n ≤ 10⁵).

outputFormat

Output the permutation as space‐separated integers on one line. If no valid permutation exists, output [] (without any extra spaces or characters).

## sample

1

1

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