| Carlos is hoping to strike it rich with his new consulting firm, Carlos | |
| Structures Industries (CSI). With a solid mission statement and a can-do | |
| attitude, he's sure to succeed in the wide world of bespoke data structures. | |
| _ "Our vision is to synergistically leverage our core competencies to deliver | |
| competitive market solutions that assertively meet our clients' needs." _ | |
| His first client, the well-known translation firm Treehouse, is looking to | |
| update their logo. Treehouse wants their logo to be a rooted tree with **N** | |
| nodes numbered 1 through **N**. They have a rough idea of what the tree should | |
| look like, but they want Carlos to finish fleshing it out. In particular, they | |
| have **M** requirements, the _i_th of which states that the [lowest common | |
| ancestor](https://en.wikipedia.org/wiki/Lowest_common_ancestor) of nodes | |
| **Xi** and **Yi** must be node **Zi**. | |
| Carlos's goal is to find any valid tree consistent with all of these | |
| requirements if possible, or to determine that no such tree exists. | |
| ### Input | |
| Input begins with an integer **T**, the number of tree designs. For each | |
| design, there is first a line containing the space-separated integers **N** | |
| and **M**. Then **M** lines follow, the _i_th of which contains the space- | |
| separated integers **Xi**, **Yi**, and **Zi**. | |
| ### Output | |
| For the _i_th design, print a line containing "Case #_i_: " followed by a | |
| description of a valid rooted tree if possible, or the string "Impossible" if | |
| no valid tree exists. A tree description is **N** space-separated integers, | |
| the _j_th of which is node _j_'s parent (or 0 if node _j_ is the root). | |
| ### Constraints | |
| 1 ≤ **T** ≤ 100 | |
| 2 ≤ **N** ≤ 60 | |
| 1 ≤ **M** ≤ 120 | |
| 1 ≤ **Xi**, **Yi**, **Zi** ≤ **N** | |
| **Xi** ≠ **Yi** | |
| ### Explanation of Sample | |
| In the first case, the LCA of nodes 1 and 2 in the chosen tree is 3, as | |
| required. This is the only valid output for this case. | |
| In the third case, "2 3 0 3" would also be accepted, while no other outputs | |
| would be. | |
| In the fifth and sixth cases, multiple possible outputs would be accepted. | |