20643번 - Cowmistry 다국어

Bessie has been procrastinating on her cow-mistry homework and now needs your help! She needs to create a mixture of three different cow-michals. As all good cows know though, some cow-michals cannot be mixed with each other or else they will cause an explosion. In particular, two cow-michals with labels $a$ and $b$ can only be present in the same mixture if $a \oplus b \le K$ ($1 \le K \le 10^9$).

NOTE: Here, $a\oplus b$ denotes the "bitwise exclusive or" of non-negative integers $a$ and $b$. This operation is equivalent to adding each corresponding pair of bits in base 2 and discarding the carry. For example,

Bessie has $N$ ($1\le N\le 2\cdot 10^4$) boxes of cow-michals and the $i$-th box contains cow-michals labeled $l_i$ through $r_i$ inclusive $(0\le l_i \le r_i \le 10^9)$. No two boxes have any cow-michals in common. She wants to know how many unique mixtures of three different cow-michals she can create. Two mixtures are considered different if there is at least one cow-michal present in one but not the other. Since the answer may be very large, report it modulo $10^9 + 7$.

The first line contains two integers $N$ and $K$.

Each of the next $N$ lines contains two space-separated integers $l_i$ and $r_i$. It is guaranteed that the boxes of cow-michals are provided in increasing order of their contents; namely, $r_i<l_{i+1}$ for each $1\le i<N$.

The number of mixtures of three different cow-michals Bessie can create, modulo $10^9 + 7$.