Codeforces Round 638 (Div. 2) |
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D. Phoenix and Science

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputPhoenix has decided to become a scientist! He is currently investigating the growth of bacteria.

Initially, on day $$$1$$$, there is one bacterium with mass $$$1$$$.

Every day, some number of bacteria will split (possibly zero or all). When a bacterium of mass $$$m$$$ splits, it becomes two bacteria of mass $$$\frac{m}{2}$$$ each. For example, a bacterium of mass $$$3$$$ can split into two bacteria of mass $$$1.5$$$.

Also, every night, the mass of every bacteria will increase by one.

Phoenix is wondering if it is possible for the total mass of all the bacteria to be exactly $$$n$$$. If it is possible, he is interested in the way to obtain that mass using the minimum possible number of nights. Help him become the best scientist!

Input

The input consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.

The first line of each test case contains an integer $$$n$$$ ($$$2 \le n \le 10^9$$$) — the sum of bacteria masses that Phoenix is interested in.

Output

For each test case, if there is no way for the bacteria to exactly achieve total mass $$$n$$$, print -1. Otherwise, print two lines.

The first line should contain an integer $$$d$$$ — the minimum number of nights needed.

The next line should contain $$$d$$$ integers, with the $$$i$$$-th integer representing the number of bacteria that should split on the $$$i$$$-th day.

If there are multiple solutions, print any.

Example

Input

3 9 11 2

Output

3 1 0 2 3 1 1 2 1 0

Note

In the first test case, the following process results in bacteria with total mass $$$9$$$:

- Day $$$1$$$: The bacterium with mass $$$1$$$ splits. There are now two bacteria with mass $$$0.5$$$ each.
- Night $$$1$$$: All bacteria's mass increases by one. There are now two bacteria with mass $$$1.5$$$.
- Day $$$2$$$: None split.
- Night $$$2$$$: There are now two bacteria with mass $$$2.5$$$.
- Day $$$3$$$: Both bacteria split. There are now four bacteria with mass $$$1.25$$$.
- Night $$$3$$$: There are now four bacteria with mass $$$2.25$$$.

$$$ $$$

In the second test case, the following process results in bacteria with total mass $$$11$$$:

- Day $$$1$$$: The bacterium with mass $$$1$$$ splits. There are now two bacteria with mass $$$0.5$$$.
- Night $$$1$$$: There are now two bacteria with mass $$$1.5$$$.
- Day $$$2$$$: One bacterium splits. There are now three bacteria with masses $$$0.75$$$, $$$0.75$$$, and $$$1.5$$$.
- Night $$$2$$$: There are now three bacteria with masses $$$1.75$$$, $$$1.75$$$, and $$$2.5$$$.
- Day $$$3$$$: The bacteria with mass $$$1.75$$$ and the bacteria with mass $$$2.5$$$ split. There are now five bacteria with masses $$$0.875$$$, $$$0.875$$$, $$$1.25$$$, $$$1.25$$$, and $$$1.75$$$.
- Night $$$3$$$: There are now five bacteria with masses $$$1.875$$$, $$$1.875$$$, $$$2.25$$$, $$$2.25$$$, and $$$2.75$$$.

$$$ $$$

In the third test case, the bacterium does not split on day $$$1$$$, and then grows to mass $$$2$$$ during night $$$1$$$.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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