Problem:
Score : 1700 points

Problem Statement

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.

Sample Input 1

3
4 4
6 10
12 11

Sample Output 1

2 4
4 1
4 7

In the first query, (2,3), (3,2), (3,4) and (4,3) have a Euclidean step count of 2. No pairs have a step count of more than 2.

In the second query, (5,8) has a Euclidean step count of 4.

In the third query, (5,8), (8,5), (7,11), (8,11), (11,7), (11,8), (12,7) have a Euclidean step count of 4.

Sample Input 2

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

Sample Output 2

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

Solution in Python (3.8.2) (judge verdict: Wrong Answer)

from math import gcd
q = int(input())
mod = 10**9+7
for _ in range(q):
    x, y = sorted(map(int, input().split()))

    ma = 0
    i, j = 1, 1
    while j <= x and i+j <= y:
        i, j = j, i+j
        ma += 1
    if x == 1:
        print(ma, x*y % mod)
    else:
        ans = 0
        if x >= i:
            ans += max(0, (y-j) // i+1)
        if y >= i:
            ans += max(0, (x-j) // i+1)
        i, j = 1, 1
        for c in range(ma-1):
            k, l = j, i+j*2
            if gcd(k, l) == 1:
                for _ in range(ma-c-1):
                    k, l = l, k+l
                if x >= k:
                    ans += max(0, (y-l) // k+1)
                if y >= k:
                    ans += max(0, (x-l) // k+1)
            i, j = j, i+j
        print(ma, ans % mod)