In number theory, Goldbach's conjecture states that every even integer greater than $2$ is the sum of two prime numbers. A weaker version of this conjecture states that every odd number greater than $5$ is the sum of three prime numbers.
Here is an ultra weak version of Goldbach's conjecture: every integer greater than $11$ is the sum of $textbf(s i x)$ prime numbers. Can you help to verify or disprove this conjecture?
The first line of the input gives the number of test cases, $T$ ($1 <= T <= 200$). $T$ test cases follow.
Each test case contains one integer $N$ ($1 <= N <= 10^(12)$).
For each test case, output "_Case x:_" first, where _x_ is the test case number (starting from $1$). If the solution exist, output six prime numbers separated by spaces; otherwise output "IMPOSSIBLE" (quotes for clarity) when the solution does not exist. When the solution exists, any valid solution is acceptable.
## Input
The first line of the input gives the number of test cases, $T$ ($1 <= T <= 200$). $T$ test cases follow.Each test case contains one integer $N$ ($1 <= N <= 10^(12)$).
## Output
For each test case, output "_Case x:_" first, where _x_ is the test case number (starting from $1$). If the solution exist, output six prime numbers separated by spaces; otherwise output "IMPOSSIBLE" (quotes for clarity) when the solution does not exist. When the solution exists, any valid solution is acceptable.
[samples]
**Definitions**
Let $ n, m \in \mathbb{Z} $ denote the number of stations and public railway edges, respectively.
Let $ A = (A_1, A_2, \dots, A_n) \in \{0,1\}^n $, where $ A_i = 0 $ if station $ i $ belongs to JR West, and $ A_i = 1 $ if JR East.
Let $ G = (V, E) $ be the undirected graph of public railways, with $ V = \{1, 2, \dots, n\} $ and $ E $ of size $ m $, representing bidirectional public edges.
Private railways exist between all pairs of stations within the same JR region (i.e., complete subgraphs on $ \{i \mid A_i = 0\} $ and $ \{i \mid A_i = 1\} $).
**Constraints**
1. $ 1 \le n \le 10^5 $, $ 0 \le m \le 10^5 $
2. $ G $ is connected.
3. No duplicate public edges.
**Objective**
For each station $ i \in \{1, \dots, n\} $, compute:
$$
S_i = \sum_{j=1}^{n} D(i, j)
$$
where $ D(u, v) $ is the minimum number of tickets required to travel from $ u $ to $ v $, with the following rules:
- Traveling along a **private railway** (same JR region) costs **0 tickets**.
- Traveling along a **public railway** costs **1 ticket**.
- Switching regions via a public edge incurs 1 ticket.
- The cost of a path is the number of public edges used.
Thus, $ D(u, v) $ is the shortest path in $ G $, where edge weights are 1 for public edges and 0 for private edges (implicitly complete within regions).