Kenus the Ancient Greek

Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.

You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.

Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:

• (a,b) and (b,a) have the same Euclidean step count.
• (0,a) has a Euclidean step count of 0.
• If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.

Constraints

• 1 ≤ Q ≤ 3 × 10^5
• 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)

Input

The input is given from Standard Input in the following format:

Q X_1 Y_1 : X_Q Y_Q

Output

For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.

Examples

Input

3 4 4 6 10 12 11

Output

2 4 4 1 4 7

Input

10 1 1 2 2 5 1000000000000000000 7 3 1 334334334334334334 23847657 23458792534 111111111 111111111 7 7 4 19 9 10

Output

1 1 1 4 4 600000013 3 1 1 993994017 35 37447 38 2 3 6 3 9 4 2

inputFormat

Input

The input is given from Standard Input in the following format:

Q X_1 Y_1 : X_Q Y_Q

outputFormat

Output

For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.

Examples

Input

3 4 4 6 10 12 11

Output

2 4 4 1 4 7

Input

10 1 1 2 2 5 1000000000000000000 7 3 1 334334334334334334 23847657 23458792534 111111111 111111111 7 7 4 19 9 10

Output

1 1 1 4 4 600000013 3 1 1 993994017 35 37447 38 2 3 6 3 9 4 2

样例

10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10

1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2

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