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



Sample input:
3
4 4
6 10
12 11


Sample output:
2 4
4 1
4 7

Submission status: Wrong Answer

Solution code:
m=10**9+7;f=lambda a:max(0,(a-l)//k+1);r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=k=l=1
	while l<=x and k+l<=y:k,l,s,a=l,k+l,s+1,f(y)+f(x)
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a,i,j=a+(y>=k)*f(x)+(x>=k)*f(y),j,i+j
	if x<2 or y<3:print(1,x*y%m)
	else:print(s+1,a%m)

Submission status: Wrong Answer

Solution code:
m=10**9+7;f=max;r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x+y<4:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Wrong Answer

Solution code:
m=10**9+7;f=max;r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
		i,j=j,i+j
	print(s+1,a)

Submission status: Wrong Answer

Solution code:
m=1e9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Wrong Answer

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
	i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		if x>=k:a+=f(0,(y-l)//k+1)
#		if y>=k:a+=f(0,(x-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Wrong Answer

Solution code:
q=int(input())
m=10**9+7
for _ in range(q):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+max(0,(x-j)//i+1)
		i=j=1
		for c in range(s):
			k,l=j,i+j*2
			for _ in range(s-c):k,l=l,k+l
			if x>=k:a+=max(0,(y-l)//k+1)
			if y>=k:a+=max(0,(x-l)//k+1)
			i,j=j,i+j
		print(s,a%m)

Submission status: Wrong Answer

Solution code:
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(1, 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)

Submission status: Wrong Answer

Solution code:
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)

Submission status: Time Limit Exceeded

Solution code:
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 or y<=2:
        print(1, 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)

Submission status: Time Limit Exceeded

Solution code:
F=[0,1,1]
for i in range(100):
  F.append(F[-1]+F[-2])
P=0
X,Y=0,0
def f(a,b,c):
  if (a*F[c]+b*F[c+1]>X or a*F[c+2]+b*F[c+1]>Y)and(a*F[c]+b*F[c+1]>Y or a*F[c+2]+b*F[c+1]>X):
    return True
  global P
  if c:
    d=a+b
    while d<=X:
      if f(b,d,c-1):
        break
      d+=b
  else:
    d=a+b
    if b<=X and d<=Y:
      P+=(Y-d)//b+1
    if d<=X and b<=Y:
      P+=(Y-b)//d+1
  return False

mod=10**9+7
for i in range(int(input())):
  P=0
  X,Y=map(int,input().split())
  if X==1 and Y==1:
    print('1 1')
    continue
  if X>Y:
    X,Y=Y,X
  for j in range(100):
    if F[j+1]>Y:
      M=j-2
      print(M,end=' ')
      break
    if F[j]>X:
      M=j-2
      print(M,end=' ')
      break
  f(0,1,M)
  if M==1:
    print(X*Y%mod)
  else:
    print(P%mod)

Submission status: Accepted

Solution code:
m=10**9+7;f=lambda a:max(0,(a-l)//k+1);r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=k=l=1
	while l<=x and k+l<=y:k,l,s=l,k+l,s+1;a=f(y)+f(x)
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a,i,j=a+(y>=k)*f(x)+(x>=k)*f(y),j,i+j
	if x<2 or y<3:print(1,x*y%m)
	else:print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7;f=max;r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7;f=max;r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()));s=-1;i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		if x>=k:a+=f(0,(y-l)//k+1)
		if y>=k:a+=f(0,(x-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
	i=j=1
	for c in r(s):
		k,l=j,i+j*2
		for _ in r(s-c):k,l=l,k+l
		if x>=k:a+=f(0,(y-l)//k+1)
		if y>=k:a+=f(0,(x-l)//k+1)
		i,j=j,i+j
	print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x<2 or y<3:print(1,x*y%m);continue
	else:
		a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=lambda x:max(x,0)
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=f((y-j)//i+1)+f((x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f((y-l)//k+1)
			if y>=k:a+=f((x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:i,j,s=j,i+j,s+1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j,s=j,i+j,s+1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
p=print
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:p(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		p(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
r=range
for _ in r(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in r(s):
			k,l=j,i+j*2
			for _ in r(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
m=10**9+7
f=max
for _ in range(int(input())):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+f(0,(x-j)//i+1)
		i=j=1
		for c in range(s):
			k,l=j,i+j*2
			for _ in range(s-c):k,l=l,k+l
			if x>=k:a+=f(0,(y-l)//k+1)
			if y>=k:a+=f(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
q=int(input())
m=10**9+7
for _ in range(q):
	x,y=sorted(map(int,input().split()))
	s=-1
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:print(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+max(0,(x-j)//i+1)
		i=j=1
		for c in range(s):
			k,l=j,i+j*2
			for _ in range(s-c):k,l=l,k+l
			if x>=k:a+=max(0,(y-l)//k+1)
			if y>=k:a+=max(0,(x-l)//k+1)
			i,j=j,i+j
		print(s+1,a%m)

Submission status: Accepted

Solution code:
q=int(input())
m=10**9+7
for _ in range(q):
	x,y=sorted(map(int,input().split()))
	s=0
	i=j=1
	while j<=x and i+j<=y:
		i,j=j,i+j
		s+=1
	if x==1 or y<3:
		print(1,x*y%m)
	else:
		a=max(0,(y-j)//i+1)+max(0,(x-j)//i+1)
		i=j=1
		for c in range(s-1):
			k,l=j,i+j*2
			for _ in range(s-c-1):
				k,l=l,k+l
			if x>=k:
				a+=max(0,(y-l)//k+1)
			if y>=k:
				a+=max(0,(x-l)//k+1)
			i,j=j,i+j
		print(s,a % m)

Submission status: Accepted

Solution code:
q = int(input())
m = 10**9+7
for _ in range(q):
    x, y = sorted(map(int, input().split()))
    s = 0
    i = j = 1
    while j <= x and i+j <= y:
        i, j = j, i+j
        s += 1
    if x == 1 or y <= 2:
        print(1, x*y % m)
    else:
        a = 0
        a += max(0, (y-j) // i+1)
        a += max(0, (x-j) // i+1)
        i = j = 1
        for c in range(s-1):
            k, l = j, i+j*2
            for _ in range(s-c-1):
                k, l = l, k+l
            if x >= k:
                a += max(0, (y-l) // k+1)
            if y >= k:
                a += max(0, (x-l) // k+1)
            i, j = j, i+j
        print(s, a % m)

Submission status: Accepted

Solution code:
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 or y <= 2:
        print(1, 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
            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)

Submission status: Accepted

Solution code:
from math import gcd
q = int(input())
mod = 10**9+7
tan = 100
tab = [[[0]*2 for _ in range(tan)] for _ in range(tan)]

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 or y <= 2:
        print(1, 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
            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)

Submission status: Accepted

Solution code:
from math import gcd
import random,time

def gcdcount(n,m):
    x,y=min(n,m),max(n,m)
    if x==0:
        return 0
    else:
        return 1+gcdcount(x,y%x)

fibo=[0,1,2]
for i in range(100):
    fibo.append(fibo[-1]+fibo[-2])

gcdfibo=[[],[(1,2),(1,3)]]

for i in range(2,101):
    temp=[]
    for a,b in gcdfibo[-1]:
        temp.append((b,a+b))
    a,b=fibo[i],fibo[i+2]
    temp.append((a,b))
    temp.sort()
    gcdfibo.append(temp)

def solve(x,y):
    if x>y:
        x,y=y,x
    mod=10**9+7
    if x==1:
        return  (1,y%mod)
    if (x,y)==(2,2):
        return (1,3)
    id=1
    t=-1
    while True:
        if fibo[id+1]<=x and fibo[id+2]<=y:
            id+=1
        else:
            if fibo[id+1]>x:
                if fibo[id+2]>y:
                    t=0
                else:
                    t=1
            else:
                t=2
            break

    #print(t)
    if t==0:
        res=0
        for a,b in gcdfibo[id]:
            if a<=x and b<=y:
                res+=1
        return (id,res)
    elif t==1:
        res=0
        for a,b in gcdfibo[id]:
            if a<=x and (a,b)!=(fibo[id],fibo[id+2]) and b<=y:
                res+=(y-b)//a+1
                res%=mod
        return (id,res)
    else:
        res=0
        for a,b in gcdfibo[id+1]:
            u,v=b-a,a
            if u<=x and v<=y:
                res+=1
                res%=mod
            if u<=y and v<=x:
                res+=1
                res%=mod
        return (id,res)

mod=10**9+7
Q=int(input())
start=time.time()
for _ in range(Q):
    x,y=map(int,input().split())
    id,res=solve(x,y)
    if (x,y)==(2,2):
        res+=1
    print(id,res)

#print(time.time()-start)

Submission status: Accepted

Solution code:
from math import gcd
import random,time

def gcdcount(n,m):
    x,y=min(n,m),max(n,m)
    if x==0:
        return 0
    else:
        return 1+gcdcount(x,y%x)

fibo=[0,1,2]
for i in range(100):
    fibo.append(fibo[-1]+fibo[-2])

gcdfibo=[[],[(1,2)],[(2,3)],[(3,5),(4,7)]]
for i in range(4,101):
    temp=[]
    for a,b in gcdfibo[-1]:
        temp.append((b,a+b))
    b=gcdfibo[-1][1][1]
    a=fibo[i]
    temp.append((b,a+b))
    temp.sort()
    gcdfibo.append(temp)

def solve(x,y):
    if x>y:
        x,y=y,x
    mod=10**9+7
    if x==1:
        return  (1,y%mod)
    if (x,y)==(2,2):
        return (1,3)
    id=1
    t=-1
    while True:
        if fibo[id+1]<=x and fibo[id+2]<=y:
            id+=1
        else:
            if fibo[id+1]>x:
                if fibo[id+2]>y:
                    t=0
                else:
                    t=1
            else:
                t=2
            break

    #print(t)
    if t==0:
        res=0
        for a,b in gcdfibo[id]:
            if a<=x and b<=y:
                res+=1
        return (id,res)
    elif t==1:
        res=0
        for a,b in gcdfibo[id]:
            if a<=x:
                res+=(y-b)//a+1
                res%=mod
        return (id,res)
    else:
        res=0
        for a,b in gcdfibo[id+1]:
            u,v=b-a,a
            if u<=x and v<=y:
                res+=1
                res%=mod
            if u<=y and v<=x:
                res+=1
                res%=mod
        return (id,res)

mod=10**9+7
Q=int(input())
start=time.time()
for _ in range(Q):
    x,y=map(int,input().split())
    id,res=solve(x,y)
    if (x,y)==(2,2):
        res+=1
    print(id,res)

#print(time.time()-start)