P5269 Owen-O Learns to Drive Again

Please imagine a picture of Owen-O learning to drive.

When Owen-O practices driving, he often uses an “oak car” for training. This oak car has a total of $N$ gears. Each second, Owen-O can increase or decrease the gear by $1$. Initially (at time $0$), the gear is $1$. The car’s RPM range is $[L,R]$, and initially the RPM is $L$. Each time he upshifts, the RPM becomes $L$; each time he downshifts, the RPM becomes $R$. Each second, Owen-O can also step on the accelerator to increase the RPM by $X$, then take $\text{min}$ with $R$. If the RPM stays continuously at $=R$ for $K$ seconds, then the engine will stop working and the car will stop at the instant those $K$ seconds end (even if downshifts happen during those $K$ seconds, it is still counted as this situation). We consider all operations to happen instantly at the beginning of each second, and the gear-change operation happens before the acceleration operation. The distance the car travels during that second is RPM $\times$ gear. Now you are given Owen-O’s sequence of operations during practice. You need to compute the total distance traveled.

The first line contains six integers $T,N,L,R,X,K$, where $T$ is the total time. The next $T$ lines each contain two integers $x,y$, describing the operation in this second. Here, $x=0$ means upshift, $x=1$ means downshift, $x=2$ means the gear stays the same; $y=0$ means do not step on the accelerator, $y=1$ means step on the accelerator. (Do not ask why there is no brake.)

Output one integer in one line, the total distance traveled under the given operation sequence. If Owen-O upshifts while the gear is $N$, or downshifts while the gear is $1$, then the given sequence is invalid; output $-1$.

For Sample 1: In the first second, the gear is $2$ and the RPM is $6$; in the second second, the gear is $3$ and the RPM is $1$; in the third second, the gear is $3$ and the RPM is $6$; in the fourth second, the gear is $3$ and the RPM is $10$; in the fifth second, the gear is $2$ and the RPM is $10$. For Sample 2, the engine stops working after moving for two seconds. For $30\%$ of the testdata, there are no gear operations (i.e. guaranteed $x=2$); for another $30\%$ of the testdata, there are no acceleration operations (i.e. guaranteed $y=0$); for all testdata, it is guaranteed that $1\le T,N,L,R,X,K\le 10^6$ and $L\le R$. Translated by ChatGPT 5