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Ex - K-th beautiful Necklace

Score : $600$ points

Problem Statement

We have $N$ gemstones. The color and beauty of the $i$-th gemstone are $D_i$ and $V_i$, respectively.
Here, the color of each gemstone is one of $1, 2, \ldots, C$, and there is at least one gemstone of each color.

Out of the $N$ gemstones, we will choose $C$ with distinct colors and use them to make a necklace. (The order does not matter.) The beautifulness of the necklace will be the bitwise $\rm XOR$ of the chosen gemstones.

Among all possible ways to make a necklace, find the beautifulness of the necklace made in the way with the $K$-th greatest beautifulness. (If there are multiple ways with the same beautifulness, we count all of them.)

Constraints

• $1 \leq C \leq N \leq 70$
• $1 \leq D_i \leq C$
• $0 \leq V_i < 2^{60}$
• $1 \leq K \leq 10^{18}$
• There are at least $K$ ways to make a necklace.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $C$ $K$

$D_1$ $V_1$

$\vdots$

$D_N$ $V_N$

Output

Print the answer.

4 2 3
2 4
2 6
1 2
1 3

5

There are four ways to make a necklace, as follows.

• Choose the $1$-st and $3$-rd gemstones to make a necklace with the beautifulness of $4\ {\rm XOR}\ 2 =6$.
• Choose the $1$-st and $4$-th gemstones to make a necklace with the beautifulness of $4\ {\rm XOR}\ 3 =7$.
• Choose the $2$-nd and $3$-rd gemstones to make a necklace with the beautifulness of $6\ {\rm XOR}\ 2 =4$.
• Choose the $2$-nd and $4$-th gemstones to make a necklace with the beautifulness of $6\ {\rm XOR}\ 3 =5$.

Thus, the necklace with the $3$-rd greatest beautifulness has the beautifulness of $5$.

3 1 2
1 0
1 0
1 0

0

There are three ways to make a necklace, all of which result in the beautifulness of $0$.

10 3 11
1 414213562373095048
1 732050807568877293
2 236067977499789696
2 449489742783178098
2 645751311064590590
2 828427124746190097
3 162277660168379331
3 316624790355399849
3 464101615137754587
3 605551275463989293

766842905529259824