| With a successful commission under his belt, Carlos has really made a splash |
| with his consulting firm, Carlos Structures Industries. His next customer is |
| Texas Instruments, the well-known manufacturer of graphing calculators. |
|
|
| _ "In today's competitive environment, you need an edge. I can make sure that |
| your newest graphing calculator is chock-full of the best modern graphs." _ |
|
|
| Carlos's pitch seems to have worked as Texas Instruments has ordered an |
| undirected, weighted graph. Their R&D department has come up with a list of |
| requirements that will ensure the graph is a hit with Gen Z schoolchildren. |
|
|
| To start with, the graph must have **N** nodes numbered 1 to **N**. It must |
| have no self-loops and at most one edge connecting each unordered pair of |
| nodes. The weight of each edge must be an integer between 1 and 1,000,000, |
| inclusive. The graph does not need to be connected. |
|
|
| The graph must also satisfy **M** customer requirements, the _i_th of which |
| states that the shortest distance between two different nodes **Xi** and |
| **Yi** must be equal to **Zi**. No two requirements pertain to the same |
| unordered pair of nodes. |
|
|
| Carlos's goal is to find any valid graph consistent with all of these |
| requirements if possible, or to determine that no such graph exists. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of graphs that Texas |
| Instruments has commissioned. For each graph, 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 graph, print a line containing "Case #_i_: " followed by either |
| an integer **E** and then a description of a valid graph if possible, or the |
| string "Impossible" if no valid graph exists. |
|
|
| A graph description contains **E** lines, where **E** is the number of edges |
| in your graph. The _i_th line contains the space-separated integers **Ai**, |
| **Bi**, and **Wi** indicating that there is an edge between nodes **Ai** and |
| **Bi** with weight **Wi**. Please keep in mind that your graph must satisfy |
| all of the requirements stated above (both the fundamental requirements |
| dictated by Texas Instruments, and the **M** customer ones). |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 350 |
| 2 ≤ **N** ≤ 50 |
| 1 ≤ **M** ≤ 1,000 |
| 1 ≤ **Xi**, **Yi** ≤ **N** |
| **Xi** ≠ **Yi** |
| 1 ≤ **Zi** ≤ 1,000,000 |
|
|
| ### Explanation of Sample |
|
|
| In the graph described by the first sample case's sample output, the shortest |
| distance between nodes 3 and 1 is 5 (along the path 3 -> 2 -> 1), as required. |
| **Multiple other outputs would also be accepted for this case.** |
|
|
| **Multiple other outputs would also be accepted for the third and fourth cases.** |
|
|
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|
