| Carlos has been working in technology so long that he's starting to feel a bit | |
| burnt out. Hoping to rejuvenate himself, Carlos has been seeking out more | |
| artistic opportunities. | |
| Yamaha, the well-known creator of musical apparatus, has approached Carlos | |
| with a request that might be right up his alley: they'd like him to design a | |
| brand new instrument. Immediately, Carlos knows what to do. | |
| _ "You may have seen Pat Metheny's 42-string guitar, but that's nothing | |
| compared to what we're going to make together." _ | |
| Carlos presents his plan for a 1,000-string guitar, complete with programmatic | |
| tuning so that you don't need to turn 1,000 knobs by hand. Yamaha's market | |
| research suggests that these sorts of guitars would be great for playing | |
| palindromic chords, chords where the first string plays the same note as the | |
| last string, the second string plays the same note as the second-to-last | |
| string, and so on. Carlos is quickly tasked with developing default tunings | |
| for the strings so that the guitars are ready to play right out of the box. | |
| For various integers **K**, Carlos wants to find a set of at most 1,000 | |
| strings on which exactly **K** distinct palindromic chords can be played. The | |
| guitar's strings are arranged in a line, and each one must be tuned to a note | |
| from the set {A, B, C, D, E, F, G}. A chord is then played by strumming a | |
| contiguous subset of 1 or more strings. Two chords are considered to be | |
| distinct if there is at least one string that is used in one chord but not the | |
| other; chords involving the same notes but different strings are considered | |
| different. | |
| For example, if **K** = 9, a set of 7 strings could be tuned to the notes C, | |
| A, B, B, A, G, E in order from left to right. You can play 7 different | |
| palindromic chords by strumming single strings, the chord BB by strumming the | |
| 3rd and 4th strings, and the chord ABBA by strumming the 2nd, 3rd, 4th, and | |
| 5th strings. This is a total of 9 distinct palindromic chords. ** | |
| Output any non-empty string of valid musical notes, with length at most 1,000, | |
| representing the tunings of sequential strings. An aspiring musician must be | |
| able to play exactly **K** distinct palindromic chords on these strings. It's | |
| guaranteed that there is at least one valid output for each possible valid | |
| input. | |
| ### Input | |
| Input begins with an integer **T**, the number of tunings that Carlos needs to | |
| figure out. | |
| For each tuning, there is a single line containing the integer **K**. | |
| ### Output | |
| For the _i_th tuning, print a line containing "Case #_i_: " followed by a | |
| string of up to 1,000 characters representing a tuning of strings as described | |
| above on which exactly **K** distinct palindromic chords can be played. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 500 | |
| 1 ≤ **K** ≤ 100,000 | |
| ### Explanation of Sample | |
| In the first case, "ACE" is a valid output as it contains exactly 3 | |
| palindromes: "A", "C", and "E". On the other hand, "DAD" would not be valid as | |
| it contains 4 palindromes. | |
| In the second case, "GAGA" is a valid output as it contains exactly 6 | |
| palindromes: "G", "A", "G", "A", "GAG", and "AGA". | |
| **_Note that other outputs would also be accepted for each sample case._** | |