Score : $400$ points

Problem Statement

$N$ cards, numbered $1$ through $N$, are arranged in a line. For each $i\ (1\leq i < N)$, card $i$ and card $(i+1)$ are adjacent to each other. Card $i$ has $A_i$ written on its front, and $B_i$ written on its back. Initially, all cards are face up.

Consider flipping zero or more cards chosen from the $N$ cards. Among the $2^N$ ways to choose the cards to flip, find the number, modulo $998244353$, of such ways that:

• when the chosen cards are flipped, for every pair of adjacent cards, the integers written on their face-up sides are different.

Constraints

• $1\leq N \leq 2\times 10^5$
• $1\leq A_i,B_i \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $B_1$

$A_2$ $B_2$

$\vdots$

$A_N$ $B_N$

Output

Print the answer as an integer.

3
1 2
4 2
3 4

4

Let $S$ be the set of card numbers to flip.

For example, when $S=\{2,3\}$ is chosen, the integers written on their visible sides are $1,2$, and $4$, from card $1$ to card $3$, so it satisfies the condition.

On the other hand, when $S=\{3\}$ is chosen, the integers written on their visible sides are $1,4$, and $4$, from card $1$ to card $3$, where the integers on card $2$ and card $3$ are the same, violating the condition.

Four $S$ satisfy the conditions: $\{\},\{1\},\{2\}$, and $\{2,3\}$.

4
1 5
2 6
3 7
4 8

16

8
877914575 602436426
861648772 623690081
476190629 262703497
971407775 628894325
822804784 450968417
161735902 822804784
161735902 822804784
822804784 161735902

48