CF1552F Telepanting

An ant moves on the real line with constant speed of $ 1 $ unit per second. It starts at $ 0 $ and always moves to the right (so its position increases by $ 1 $ each second). There are $ n $ portals, the $ i $ -th of which is located at position $ x_i $ and teleports to position $ y_i < x_i $ . Each portal can be either active or inactive. The initial state of the $ i $ -th portal is determined by $ s_i $ : if $ s_i=0 $ then the $ i $ -th portal is initially inactive, if $ s_i=1 $ then the $ i $ -th portal is initially active. When the ant travels through a portal (i.e., when its position coincides with the position of a portal): - if the portal is inactive, it becomes active (in this case the path of the ant is not affected); - if the portal is active, it becomes inactive and the ant is instantly teleported to the position $ y_i $ , where it keeps on moving as normal. How long (from the instant it starts moving) does it take for the ant to reach the position $ x_n + 1 $ ? It can be shown that this happens in a finite amount of time. Since the answer may be very large, compute it modulo $ 998\,244\,353 $ .

The first line contains the integer $ n $ ( $ 1\le n\le 2\cdot 10^5 $ ) — the number of portals. The $ i $ -th of the next $ n $ lines contains three integers $ x_i $ , $ y_i $ and $ s_i $ ( $ 1\le y_i < x_i\le 10^9 $ , $ s_i\in\{0,1\} $ ) — the position of the $ i $ -th portal, the position where the ant is teleported when it travels through the $ i $ -th portal (if it is active), and the initial state of the $ i $ -th portal. The positions of the portals are strictly increasing, that is $ x_1

Output the amount of time elapsed, in seconds, from the instant the ant starts moving to the instant it reaches the position $ x_n+1 $ . Since the answer may be very large, output it modulo $ 998\,244\,353 $ .

Explanation of the first sample: The ant moves as follows (a curvy arrow denotes a teleporting, a straight arrow denotes normal movement with speed $ 1 $ and the time spent during the movement is written above the arrow). $ $$$ 0 \stackrel{6}{\longrightarrow} 6 \leadsto 5 \stackrel{3}{\longrightarrow} 8 \leadsto 1 \stackrel{2}{\longrightarrow} 3 \leadsto 2 \stackrel{4}{\longrightarrow} 6 \leadsto 5 \stackrel{2}{\longrightarrow} 7 \leadsto 4 \stackrel{2}{\longrightarrow} 6 \leadsto 5 \stackrel{4}{\longrightarrow} 9 $ $ Notice that the total time is $ 6+3+2+4+2+2+4=23 $ .

Explanation of the second sample: The ant moves as follows (a curvy arrow denotes a teleporting, a straight arrow denotes normal movement with speed $ 1 $ and the time spent during the movement is written above the arrow). $ $ 0 \stackrel{454971987}{\longrightarrow} 454971987 \leadsto 406874902 \stackrel{48097086}{\longrightarrow} 454971988 $ $ Notice that the total time is $ 454971987+48097086=503069073 $ .

Explanation of the third sample: Since all portals are initially off, the ant will not be teleported and will go straight from $ 0 $ to $ x\_n+1=899754846+1=899754847$$$.