B - Smartphone Addiction

Score : $200$ points

Problem Statement

The battery of Takahashi's smartphone has $N$ mAh capacity. At time $0.5$, $1.5$, $2.5$, and so on (that is, at time $n + 0.5$ for every integer $n$), the battery charge decreases by $1$ mAh.
Takahashi will leave his house with his phone fully charged at time $0$, visit a cafe $M$ times, and return home at time $T$.
He will stay at the $i$-th cafe from time $A_i$ to time $B_i$. During this stay, he charges his phone, so the battery charge does not decrease. Instead, at time $n + 0.5$ for every integer $n$, it increases by $1$. However, if it is already equal to the battery capacity, it does not increase nor decrease.
Determine whether he can return home without the battery charge dropping to $0$ on the way.

Constraints

• $1 \le N \le 10^9$
• $1 \le M \le 1000$
• $1 \le T \le 10^9$
• $0 \lt A_1 \lt B_1 \lt A_2 \lt B_2 \lt A_3 \lt B_3 \lt \dots \lt A_M \lt B_M \lt T$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $T$

$A_1$ $B_1$

$A_2$ $B_2$

$A_3$ $B_3$

$\hspace{15pt} \vdots$

$A_M$ $B_M$

Output

If Takahashi can return home without the battery charge dropping to $0$ on the way, print Yes; otherwise, print No.

10 2 20
9 11
13 17

Yes

The battery charge changes as follows:

• Time $0$ (leaving home): $10$ mAh
• Time $9$ (the beginning of the stay at the first cafe): $1$ mAh
• Time $11$ (the end of the stay at the first cafe): $3$ mAh (He charges his phone in a cafe.)
• Time $13$ (the beginning of the stay at the second cafe): $1$ mAh
• Time $17$ (the end of the stay at the second cafe): $5$ mAh
• Time $20$ (getting home): $2$ mAh

During this process, the battery charge never drops to $0$, so we print Yes.

10 2 20
9 11
13 16

No

This case is the same as Sample Input/Output 1 until he starts his stay at the second cafe with $1$ mAh charge.
When he ends his stay there at time $16$, the battery charge is $4$ mAh.
Then at time $19.5$, it drops to $0$, so we print No.

15 3 30
5 8
15 17
24 27

Yes

The battery charge drops to $1$ mAh when he gets home, but it never drops to $0$ on the way.

20 1 30
20 29

No

The battery charge drops to $0$ at time $19.5$.

20 1 30
1 10

No

Note that when the battery charge is equal to the battery capacity, staying at a cafe does not increase the battery charge.