| Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. | |
| An alphabet tree is an unrooted tree with \(N\) nodes (numbered from \(1\) to \(N\)) and \(N - 1\) edges. Initially, the \(i\)th edge connects nodes \(A_i\) and \(B_i\) in both directions, and is labeled with a uppercase letter \(C_i\). Two edges incident to a common node are always labeled with different letters. | |
| Alphonse has \(Q\) events to process, the \(i\)th of which is one of two types: | |
| * `1` \(U_i\) \(L_i\): Add a new node to the tree by connecting it to node \(U_i\) with a new edge labeled with uppercase letter \(L_i\). Newly added nodes are numbered with integers starting from \(N + 1\) in the order they're added. | |
| * `2` \(U_i\) \(K_i\) \(S_i\): Print the final node Alphonse will end up at if he: | |
| - Starts a journey at node \(U_i\). | |
| - Repeatedly traverses a previously untraversed edge (on this journey). If his current node has multiple untraversed edges, he picks the edge labeled with the letter that comes earliest in the string \(S_i\). | |
| - Ends the journey once there are no more untraversed edges at the current node, or \(K_i\) edges have been traversed on the journey. | |
| Please help Alphonse determine where each journey will take him. | |
| # Constraints | |
| \(1 \le T \le 20\) | |
| \(2 \le N \le 300{,}000\) | |
| \(1 \le Q \le 300{,}000\) | |
| \(1 \le A_i, B_i \le N\) | |
| \(A_i \ne B_i\) | |
| \(C_i, L_i, S_{i,j} \in \{\)`'A'`, \(\ldots\), `'Z'`\(\}\) | |
| \(1 \le U_i, K_i \le 600{,}000\) | |
| The sum of \(N\) over all test cases is at most \(1{,}100{,}000\). | |
| The sum of \(Q\) over all test cases is at most \(1{,}100{,}000\). | |
| For each event, it is guaranteed that: | |
| - \(U_i\) is a valid node in the tree at the time of the event. | |
| - \(L_i\) is different from all existing labels of edges incident to \(U_i\) at the time of the event. | |
| - \(S_i\)'s letters are distinct, and are a superset of all edge labels in the tree at the time of the event. | |
| # Input Format | |
| Input begins with a single integer \(T\), the number of test cases. For each test case, there is first a line containing a single integer \(N\). Then, \(N - 1\) lines follow, the \(i\)th of which contains space-separated integers \(A_i\) and \(B_i\), followed by a space, followed by \(C_i\). Then, there is a line containing the single integer \(Q\). Then, \(Q\) lines follow, the \(i\)th of which is either `1` \(U_i\) \(L_i\) or `2` \(U_i\) \(K_i\) \(S_i\). | |
| # Output Format | |
| For the \(i\)th test case, print a single line containing `"Case #i: "`, followed by space-separated integers, the answers to all type-2 events. | |
| # Sample Explanation | |
| {{PHOTO_ID:1175696903384597|WIDTH:550}} | |
| The first sample case's alphabet tree is depicted above, and yields the following journeys: | |
| - Event \(1\) traverses \(1 \stackrel{\texttt{M}}{\longrightarrow} 2 \stackrel{\texttt{E}}{\longrightarrow} 6\) (ended after no more edges from node \(6\)) | |
| - Event \(2\) traverses \(3 \stackrel{\texttt{E}}{\longrightarrow} 1 \stackrel{\texttt{T}}{\longrightarrow} 4 \stackrel{\texttt{A}}{\longrightarrow} 9\) (ended after \(K_2 = 3\) steps) | |
| - Event \(3\) traverses \(9 \stackrel{\texttt{A}}{\longrightarrow} 4 \stackrel{\texttt{T}}{\longrightarrow} 1 \stackrel{\texttt{M}}{\longrightarrow} 2\) (ended after \(K_3 = 3\) steps) | |
| - Event \(4\) traverses \(8 \stackrel{\texttt{M}}{\longrightarrow} 3 \stackrel{\texttt{E}}{\longrightarrow} 1 \stackrel{\texttt{T}}{\longrightarrow} 4 \stackrel{\texttt{A}}{\longrightarrow} 9\) (ended after no more edges from node \(9\)) | |
| - Event \(6\) traverses \(8 \stackrel{\texttt{T}}{\longrightarrow} 10\) (ended after no more edges from node \(10\)) | |
| The second and third sample cases are depicted below. | |
| {{PHOTO_ID:1142476319774543|WIDTH:650}} | |
| In the second case, the journeys are: | |
| - Event \(1\) traverses \(1 \stackrel{\texttt{P}}{\longrightarrow} 5 \stackrel{\texttt{C}}{\longrightarrow} 2\) (ended after \(K_1 = 2\) steps) | |
| - Event \(2\) traverses \(4 \stackrel{\texttt{P}}{\longrightarrow} 2 \stackrel{\texttt{C}}{\longrightarrow} 5 \stackrel{\texttt{P}}{\longrightarrow} 1\) (ended after \(K_2 = 3\) steps) | |
| - Event \(4\) traverses \(4 \stackrel{\texttt{P}}{\longrightarrow} 2 \stackrel{\texttt{C}}{\longrightarrow} 5 \stackrel{\texttt{U}}{\longrightarrow} 6\) (ended after \(K_4 = 3\) steps) | |
| - Event \(5\) traverses \(3 \stackrel{\texttt{U}}{\longrightarrow} 2 \stackrel{\texttt{P}}{\longrightarrow} 4\) (ended after no more edges from node \(4\)) | |
| In the third case, the journeys are: | |
| - Event \(1\) traverses \(3 \stackrel{\texttt{C}}{\longrightarrow} 2 \stackrel{\texttt{A}}{\longrightarrow} 1\) (ended after \(K_1 = 2\) steps) | |
| - Event \(2\) traverses \(4 \stackrel{\texttt{E}}{\longrightarrow} 2 \stackrel{\texttt{A}}{\longrightarrow} 1\) (ended after \(K_2 = 2\) steps) | |
| - Event \(3\) traverses \(1 \stackrel{\texttt{A}}{\longrightarrow} 2\) (ended after \(K_3 = 1\) steps) | |