P7071 [CSP-J 2020] Excellent Decomposition

In general, a positive integer can be decomposed into the sum of several positive integers. For example, $1 = 1$, $10 = 1 + 2 + 3 + 4$, and so on. For a specific decomposition of a positive integer $n$, we call it “excellent” if and only if, under this decomposition, $n$ is split into several **distinct** powers of $2$ with **positive integer** exponents. Note that a number $x$ can be written as a power of $2$ with a positive integer exponent if and only if $x$ can be obtained by multiplying $2$ together a positive integer number of times. For example, $10 = 8 + 2 = 2^3 + 2^1$ is an excellent decomposition. However, $7 = 4 + 2 + 1 = 2^2 + 2^1 + 2^0$ is not an excellent decomposition, because $1$ is not a power of $2$ with a positive integer exponent. Now, given a positive integer $n$, you need to determine whether there exists an excellent decomposition among all decompositions of this number. If it exists, output a specific decomposition scheme.

The input contains only one line with an integer $n$, which is the number to be checked.

If there exists an excellent decomposition among all decompositions of this number, output each number in the decomposition from large to small, with a single space between adjacent numbers. It can be proven that after fixing the order of the numbers in the decomposition, the scheme is unique. If no excellent decomposition exists, output `-1`.

### Explanation for Sample 1 $6 = 4 + 2 = 2^2 + 2^1$ is an excellent decomposition. Note that $6 = 2 + 2 + 2$ is not an excellent decomposition, because the $3$ numbers in the decomposition are not all distinct. --- ### Constraints - For $20\%$ of the testdata, $n \le 10$. - For another $20\%$ of the testdata, $n$ is guaranteed to be odd. - For another $20\%$ of the testdata, $n$ is guaranteed to be a power of $2$ with a positive integer exponent. - For $80\%$ of the testdata, $n \le 1024$. - For $100\%$ of the testdata, $1 \le n \le 10^7$. Translated by ChatGPT 5