1552F · Telepanting

Problem: 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 $998\,244\,353$.

Input Format: 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 $x_1<x_2<\cdots<x_n$. It is guaranteed that the $2n$ integers $x_1, \, x_2, \, \dots, \, x_n, \, y_1, \, y_2, \, \dots, \, y_n$ are all distinct.

Output Format: 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 $998\,244\,353$.

Note: 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 $x_n+1=899754846+1=899754847$.