You are developing a game consisting of (n) levels, numbered from (1) to (n). Levels are connected by a transition network consisting of (m) transitions, numbered from (1) to (m). Transition (k) connects levels (a_k) and (b_k) bidirectionally ((1 \le k \le m)). A single run of the game starts at level (1). Each time a player enters a new level, the player must complete it and then move to another level that is directly connected by a transition and has not been completed in the run. The run ends successfully when level (n) is completed. A sequence of pairwise distinct levels starting from level (1) and ending at level (n) is called a successful path if a player can complete the levels in the order given by the sequence in a single run. A transition network is well-designed if it satisfies both of the following: For any level (i) ((2 \le i \le n-1)), there exists a successful path containing level (i). For any pair of levels (i) and (j) ((2 \le i \lt j \le n-1)), at most one of the following holds: There exists a successful path containing levels (i) and (j) with level (i) appearing before level (j). There exists a successful path containing levels (i) and (j) with level (j) appearing before level (i). There exists a successful path containing levels (i) and (j) with level (i) appearing before level (j). There exists a successful path containing levels (i) and (j) with level (j) appearing before level (i). Your first task is to determine whether the given transition network is well-designed or not. If the network is well-designed, you have a second task. Let (S) be the set of pairs of integers ((i, j)) ((1 \le i \lt j \le n)) such that levels (i) and (j) are not directly connected and adding a bidirectional extra transition between them keeps the network well-designed. You are interested in the s