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## 문제

There are many cities in the Kingdom of JOI. The road network satisfies the following conditions:

• (Condition 1) The cities are numbered from 0 to N − 1. Here, N is the number of cities in the Kingdom of JOI.
• (Condition 2) The cities are connected by N − 1 roads. We can pass each road in both directions. We can travel from every city to any other cities if we pass through several roads.
• (Condition 3) We can travel from the city 0 to every city by passing at most 18 roads.

Every day, in the Kingdom of JOI, many people depart from the city 0 to other cities. Because many people have two destinations, they sometimes ask the following queries: for two different cities X, Y, which one of (0), (1), (2) is satisfied?

• (0) We necessarily pass the city Y when we travel from the city 0 to the city X.
• (1) We necessarily pass the city X when we travel from the city 0 to the city Y.
• (2) Neither (0) nor (1).

Note that, in the above situation, exactly one of (0), (1), (2) is satisfied. When X = 0, we consider (1) is satisfied regardless of the value of Y. Similarly, when Y = 0, we consider (0) is satisfied regardless of the value of X.

It is known that, just like the road network in the Kingdom of JOI, the above Conditions 1–3 are satisfied in other countries as well. In the Kingdom of JOI, people plan to develop the following two machines so that they can be used also in other countries.

• (Machine 1) Given the number of cities N and information of the road network, this machine assigns a code to each city. A code is an integer between 0 and 260 − 1, inclusive.
• (Machine 2) Given the codes of two different cities X, Y assigned by Machine 1, this machine answers the query.

If large integers are assigned as codes, it is difficult to treat them. We want to develop machines so that smaller values are assigned as codes.

Note that, when Machine 2 is used, neither the number of cities N nor information of the road network is directly given to the machine.

In order to develop two machines as above, write the following two programs:

• Given the number of cities N in the Kingdom of JOI and information of the road network, the first program assigns a code to each city.
• Given the codes of two different cities assigned by the first program, the second program answers the query for these cities.

## 구현

You need to submit two files .

The first file is Encoder.cpp. This file implements Machine 1. It should implement the following function. The program should include Encoder.h.

• void Encode(int N, int A[], int B[])
For each test case, this function is called once.
• The parameter N is the number of cities N.
• The parameters A[], B[] are integer sequences of length N − 1 describing the road network. The elements A[i], B[i] (0 ≤ i ≤ N − 2) mean there is a road connecting the city A[i] and the city B[i] directly.

Your program should call the following function.

• void Code(int city, long long code)
This function assigns codes to the cities.
• The parameter city is the city to which a code is assigned. The parameter city should be an integer between 0 and N − 1, inclusive. If the call to this function has parameters outside this range, your program is considered as Wrong Answer . It should not be called twice with the same parameters. If it is called twice with the same parameters, your program is considered as Wrong Answer .
• The parameter code is the code assigned to the city whose number is city. The parameter code should be an integer between 0 and 260 − 1, inclusive. If the call to this function has parameters outside this range, your program is considered as Wrong Answer .

Your program should call the function Code exactly N times. If the number of calls to Code is not N when the function Encode is terminated, your program is considered as Wrong Answer .

If the call to the function Encode is considered Wrong Answer, your program is terminated immediately.

The second file is either Device.cpp. This file implements Machine 2. It should implement the following function. The program should include Device.h.

• void InitDevice()
This function initializes Machine 2. The function InitDevice is called once before calls to the following function Answer.
• int Answer(long long S, long long T)
This function answers each query. For each query, the function Answer is called once.
• The parameters S, T are the codes assigned to two different cities X, Y by the function Encode.
• To answer the query, the return value of the function Answer should satisfy the following conditions:
• The return value should be 0 if we necessarily pass the city Y when we travel from the city 0 to the city X.
• The return value should be 1 if we necessarily pass the city X when we travel from the city 0 to the city Y.
• Otherwise, the return value should be 2.

Hence the return value of the function Answer should be an integer between 0 and 2, inclusive. If the return value is outside this range, your program is considered as Wrong Answer . If the return value is inside this range but it does not satisfy the above conditions, your program is considered as Wrong Answer .

## 입력

The sample grader reads the following data from the standard input.

• The first line contains two space separated integers N, Q. This means there are N cities, and Q queries are given.
• The (i + 1)-th line (0 ≤ i ≤ N − 2) of the following N − 1 lines contains two space separated integers Ai, Bi. This means there is a road connecting the city Ai and the city Bi directly.
• The j-th line (1 ≤ j ≤ Q) of the following Q lines contains three space separated integers Xj, Yj, Ej. This means X = Xj and Y = Yj in the j-th query, and your program will be considered Wrong Answer by the sample grader if the answer to this query is different from Ej.

## 출력

When the program terminates successfully, the sample grader writes the following information to the standard output. (The quotation mark is not written actually.)

• If your program is considered as correct, the sample grader writes the maximum of the codes assigned to the cities in the following form “Accepted : max_code=123456.
• If your program is considered as Wrong Answer, the sample grader writes its type in the following form “Wrong Answer .

If your program is considered as several types of Wrong Answer, the sample grader reports only one of them.

## 제한

• 2 ≤ N ≤ 250 000.
• 1 ≤ Q ≤ 250 000.
• 0 ≤ Ai ≤ N − 1 (0 ≤ i ≤ N − 2).
• 0 ≤ Bi ≤ N − 1 (0 ≤ i ≤ N − 2).
• Ai ≠ Bi (0 ≤ i ≤ N − 2).
• We can travel from the city 0 to every city by passing at most 18 roads.
• 0 ≤ Xj ≤ N − 1 (1 ≤ j ≤ Q).
• 0 ≤ Yj ≤ N − 1 (1 ≤ j ≤ Q).
• Xj ≠ Yj (1 ≤ j ≤ Q).

• N ≤ 10.

## 서브태스크 2 (92점)

There are no additional constraints. In this subtask, the score is calculated as follows:

• Let L be the maximum of the codes assigned to the cities in all test cases for this subtask.
• The score for this subtask is:
• 0 point if 238 ≤ L.
• 10 points if 236 ≤ L ≤ 238 − 1.
• 14 points if 235 ≤ L ≤ 236 − 1.
• 22 points if 234 ≤ L ≤ 235 − 1.
• ⌊372 − 10log2(L + 1)⌋ points if 228 ≤ L ≤ 234 − 1. Here, ⌊x⌋ is the largest integer not exceeding x.
• 92 points if L ≤ 228 − 1.

## 예제

Here is a sample input for sample grader and corresponding function calls.

Sample Input Sample Calls
Machine 1 Machine 2
6 5
4 1
0 3
4 5
3 2
3 4
2 4 2
1 0 0
5 1 2
5 3 0
4 1 1
Encode(6, {4, 0, 4, 3, 3}, {1, 3, 5, 2, 4})
Code(0,0)
Code(2,4)
Code(4,16)
Code(1,1)
Code(3,9)
Code(5,25)
InitDevice()
Answer(4,16)
Answer(1,0)
Answer(25,1)
Answer(25,9)
Answer(16,1)

## 제출할 수 있는 언어

C++17, C++14, C++20, C++14 (Clang), C++17 (Clang), C++20 (Clang)

## 채점 및 기타 정보

• 예제는 채점하지 않는다.