[SOLUTION] Fancy Arrays SOLUTION CODEFORCES

Let’s call an array $a$ of $n$ non-negative integers fancy if the following conditions hold:

at least one from the numbers $x$ , $x + 1$ , …, $x + k - 1$ appears in the array;
consecutive elements of the array differ by at most $k$ (i.e. $| a_{i} - a_{i - 1} | \leq k$ for each $i \in [2, n]$ ).

You are given $n$ , $x$ and $k$ . Your task is to calculate the number of fancy arrays of length $n$ . Since the answer can be large, print it modulo $10^{9} + 7$ .

Input

The first line contains a single integer $t$ ( $1 \leq t \leq 50$ ) — the number of test cases.

The only line of each test case contains three integers $n$ , $x$ and $k$ ( $1 \leq n, k \leq 10^{9}$ ; $0 \leq x \leq 40$ ).

Output

For each test case, print a single integer — the number of fancy arrays of length $n$ , taken modulo $10^{9} + 7$ .

Example

input

Copy

3 0 1

1 4 25

4 7 2

1000000000 40 1000000000

output

Copy

Note

In the first test case of the example, the following arrays are fancy:

$[0, 0, 0]$ ;
$[0, 0, 1]$ ;
$[0, 1, 0]$ ;
$[0, 1, 1]$ ;
$[0, 1, 2]$ ;
$[1, 0, 0]$ ;
$[1, 0, 1]$ ;
$[1, 1, 0]$ ;
$[2, 1, 0]$ .

SOLUTION

# Function to calculate the number of Monocarp’s starting moves that result in a win, draw, and loss
def calculate_moves_outcome(n, ax, ay, m, bx, by):
# Create a list of tuples (attack, defense)
player1 = [(ax[i], ay[i]) for i in range(n)]
player2 = [(bx[i], by[i]) for i in range(m)]

# Sort the lists by attack values in descending order
player1.sort(key=lambda x: -x[0])
player2.sort(key=lambda x: -x[0])

# Initialize counters for wins, draws, and losses
wins1, draws, wins2 = 0, 0, 0
j = 0 # Index for player2

for i in range(n):
while j < m and player1[i][0] > player2[j][1]:
j += 1
if j == m:
wins1 += (n – i) # Player1 wins all remaining moves
break
elif player1[i][0] < player2[j][1]:
draws += 1 # Player2 wins this move
else:
j += 1 # Player1 and Player2 draw this move

for i in range(j, m):
wins2 += 1 # Player2 wins all remaining moves

return wins1, draws, wins2

# Read the number of test cases
t = int(input())

# Iterate through each test case
for _ in range(t):
n = int(input())
ax = list(map(int, input().split()))
ay = list(map(int, input().split()))
m = int(input())
bx = list(map(int, input().split()))
by = list(map(int, input().split()))

# Calculate the number of wins, draws, and losses for Monocarp
wins1, draws, wins2 = calculate_moves_outcome(n, ax, ay, m, bx, by)

# Print the results for this test case
print(wins1, draws, wins2)

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