⡿⣼⢳⢾⣹⡻⣼⢏⡷⣏⡷⣛⣮⢟⣧⠏⠹⡆⣰⡏⠀⠀⢀⢀⡈⠽⠞⠋⠓⠒⠳⢯⣛⡼⢣⣏⢯⣇⣂⠀⠀⠈⣞⣽⣯⢿⡽⣞⡿⣯⣷⢿
⡿⣜⡯⣗⢯⣳⢯⠾⣝⡾⣹⡽⣎⣟⠎⢀⡀⠹⠶⠵⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⢳⣎⠷⣭⢻⣧⠀⠀⢸⣿⣞⣯⣟⣯⢿⣳⣯⣿
⣿⡱⣟⠾⣭⢷⣫⢿⣹⢞⣳⡽⠚⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡳⣎⠿⣜⣧⢻⣅⠀⣼⣻⣞⣷⣻⣞⡿⣽⣳⢿
⡷⡽⣭⢿⣹⡞⣵⡻⣎⣿⡱⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷⣭⢻⡜⣾⢯⣥⡾⢯⡷⣞⣷⣻⢾⡽⣯⣟⣿
⣿⣱⢟⣮⢷⣹⣳⢽⡳⢾⡍⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣜⢧⡻⣜⢯⡞⣽⡻⣼⣻⡼⢯⡿⣽⣳⣟⣾
⣷⡹⣞⢧⣯⣳⢽⡺⡽⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢯⡳⣝⢮⡽⣞⡽⣣⢿⣹⢯⣟⣷⣻⢾⣽
⣷⢫⡽⣞⣵⣫⠾⣝⣳⢯⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠛⠮⣷⣹⢞⣽⣫⢯⣟⡾⣵⢯⣟⡾
⣯⢏⡷⣏⡶⣏⡿⣭⠷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣏⡷⢯⡾⣽⣻⢾⡽
⣟⢾⣹⡞⣽⢞⡵⣏⡿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⢿⣽⣯⣽⢳⣯⢯⣟
⣟⡮⢷⣹⣏⡾⣽⢺⣵⢻⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣤⣵⡿⣞⡿⣼⣻⢞
⡿⡼⣏⡷⣞⣽⢺⣝⡮⣟⣽⡀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣯⣗⢯⣟
⣿⣱⠿⣼⢳⡞⣯⡞⣵⡻⣖⢷⡀⠀⠀⠀⠀⠀⢀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣻⢞
⡷⣭⢟⣧⢿⣹⢶⣛⣧⢟⡽⣞⣳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⣤⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣜⣯⣻
⡿⣼⢫⣞⢧⡟⣾⡹⣎⣟⠾⣵⣫⢷⣀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⡀⢀⡤⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⣷⣛⡶⢯
⡿⣜⡿⣼⢫⣞⠷⣽⣹⢞⣯⣳⠽⣞⡽⣆⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⡰⠛⣺⣷⣾⣶⣶⣶⣦⣤⣤⣀⣉⠙⠻⢿⣿⡹⣞⡵⢯⡽⣖⡯⣽⢯
⣿⡱⣟⣼⣛⣮⢟⡧⣟⢾⣱⣏⢿⣹⢞⣽⣻⡤⡇⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⣿⣟⣷⣻⣿⢧⣽⣻⣿⣷⣾⣳⠽⣮⢽⡻⣼⢳⡽⣣⣟
⡷⣽⡹⣶⢫⡾⣝⡾⣹⣞⣳⢞⡯⣞⢯⡶⣏⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡽⣞⣷⣿⣿⣻⣿⣿⣿⣯⢳⣏⢷⣫⢗⣯⢻⣼⢳⡽
⣟⢶⡻⡵⣯⢳⣻⢼⣳⠽⣎⣟⠾⣭⢷⡻⣜⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⣷⢾⣷⣟⣿⣿⣿⣚⠷⣮⢻⣜⢯⡞⡽⣎⡿⣼
⣟⢾⡹⢷⣹⣏⢷⡻⣼⢻⣝⡞⣯⡽⣞⡽⣭⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⢯⣟⣿⣾⡽⣿⣷⣭⢻⡜⣯⢞⣧⢻⡝⣾⡱⢯
⣟⢾⣹⣏⠷⣞⢯⣳⢏⡿⣼⣹⢧⡟⡮⠽⠾⠧⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣾⣿⡽⣿⣽⣿⣮⢳⡻⣜⢯⡞⣧⢟⡶⣻⠽
⣟⡮⢷⣺⢻⣝⡾⣹⢾⣹⢶⣛⠎⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⣷⣻⣿⣿⢣⡟⣼⣣⢟⡼⣫⢞⡵⣻
⡿⡼⣏⡷⣛⣮⡽⣳⢏⣷⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⡟⣼⢣⡟⣮⢳⣭⢻⣜⣳
⣿⣱⢯⣳⢟⡶⣻⢵⡻⣎⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣽⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣽⣿⡜⣧⢻⣜⡳⣎⠷⣎⠷
⣷⡹⣞⢧⣟⣳⣭⢷⡻⢝⡢⠽⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡿⣿⠟⠿⠿⠶⣄⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡜⣧⢻⣜⡳⣭⢻⣜⡻
⣷⢫⡽⣞⢮⣷⡏⠟⠸⠣⠙⠉⠌⢳⣤⣤⣤⣤⣶⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠙⣿⣷⢄⠀⠀⠀⠈⢶⠛⣿⡾⣱⢏⣮⢳⣭⢳⣎⠷
⣯⢏⣷⠟⡃⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠈⣃⠈⢿⡱⣏⣞⢳⡎⡷⢎⡟
⣟⡾⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⡁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠐⣆⢸⣟⡼⢎⡷⣹⢭⡻⣜
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⡜⣯⡜⢧⡳⣝⢮
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸⣤⢹⢇⡿⣸⡌
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠉⢚⣧⡛⣼⡱⢞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⡨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⡀⠀⢘⡇⠀⢿⣦⠀⠀⣶⠻⣔⢯⡹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡩⢄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣬⠧⠀⠀⡜⠀⠀⣽⠀⠀⣻⣞⣧⠀⣯⢟⡼⢎⡳
⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⡓⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣠⢳⡾⠃⠀⡸⠁⠀⣸⣹⠃⠀⣼⡱⣞⢧⡝⣎⡞⣭⢳
⡇⣀⠂⠄⡀⢀⠀⡀⢀⠀⢾⣱⡝⡰⢀⠂⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢂⢡⢒⡭⣖⡟⠃⢀⠄⢀⡠⠞⡱⢾⠀⣐⠢⡕⢮⡻⣝⠶⣙⠶⣭
⣇⠰⡈⠔⣀⠂⡐⡀⠆⣌⢹⡷⣎⡕⢂⠌⠠⢀⠂⠐⢈⠠⢈⠐⠠⡁⣜⢢⣏⣾⣽⡿⠖⠙⠒⠂⠁⠀⣜⣱⠇⠐⠂⡰⢩⠳⣝⣯⢏⣝⡺⢴
⢆⢣⡙⠲⣄⢳⡰⣱⢚⡴⣫⣿⡞⣜⠢⡌⣁⠂⠌⡐⠄⡒⠤⡉⢖⡱⣮⣟⡾⠟⠉⠀⠀⠀⠀⠀⠠⣘⠴⣻⠊⠁⠀⠥⣋⡝⢮⣟⢼⢲⣙⢮
⡈⢆⠩⢓⢬⢣⡳⣵⢫⣞⣵⣻⣿⣭⠳⡔⢢⢉⠔⡈⢆⡑⢢⠙⡌⠓⠁⠁⠰⠘⠀⠀⠀⠀⠀⣀⠳⣌⠿⠁⠀⠀⡘⢰⢣⠞⣧⡟⣎⢳⢎⡞
⡿⡍⠒⠉⠎⠃⠇⠩⠛⡘⠃⠛⢻⢿⣯⡝⢦⣉⠦⣑⠢⡈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠛⠈⠀⠀⢀⠂⢥⠣⣏⡻⡴⣟⡬⢳⢎⡽
⢳⡌⡐⠀⠀⠀⠀⠀⠀⠀⠀⢄⡘⢮⠽⣿⣳⢎⡳⣌⢳⡘⣤⢃⡌⡰⣀⠂⡄⢠⡀⢤⠐⠈⠁⠀⠀⠀⢀⢀⠢⣘⢦⡛⣶⡹⡵⣿⠸⣍⡞⡼
⢧⡒⢥⢊⠔⡀⡀⢠⠀⢆⡉⢦⣙⢮⡻⣽⣏⠍⠓⠫⠓⠛⠆⠛⠜⠱⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⢠⠐⡄⢎⣲⡱⢮⡝⣶⡹⣵⡿⣙⠦⣝⠼
⡴⣙⣎⠮⡜⣥⢣⢣⡙⢦⣙⠶⣭⢞⣵⣣⣿⠬⡑⡄⢂⠄⠠⢀⠠⠀⠄⣀⠂⡐⢠⠠⣀⠠⢄⠣⣆⠽⣘⢧⢶⣹⢧⡻⢶⡹⢾⡵⣩⢞⡬⣛
⣷⣙⠬⣻⠵⣎⢷⣫⡝⣧⢏⡿⣼⢫⣶⢳⣻⣇⠳⣜⠢⣌⠱⣂⠥⣉⠲⣀⠧⡘⢦⠱⣌⡓⣎⡳⣜⢯⡝⣮⡟⣼⡳⣏⢷⡹⣟⢧⡓⢮⠖⣭
⣷⢫⣟⡁⠐⠛⠒⠓⠚⠓⠛⠈⠉⢿⣉⣧⣉⣾⡹⣬⢳⡬⢳⡱⢎⡵⢣⣓⢮⣙⡎⢷⡸⣵⢫⡵⣫⣞⡽⣣⡟⣧⢟⡼⣣⢻⡟⢦⣋⢧⡛⣖
⣯⣛⣾⡟⣍⢫⡙⣙⢋⡛⣹⢛⡿⢯⡿⣽⡻⣷⡇⢯⡳⣝⢧⢯⡝⣮⢳⡭⢶⡹⣞⢧⡻⣜⣧⢻⡵⣺⠵⣯⡝⣮⣛⠼⡱⣿⡙⢶⡩⢶⡹⢬
⣷⡹⣞⣟⣾⡱⣜⡜⣮⢵⣫⢯⡟⣯⣽⢳⡟⣵⣟⢣⡟⣼⢫⡞⡽⣎⣷⣹⡳⡽⣎⡷⣫⢗⡾⣹⢞⡵⣛⠶⣙⢦⡙⠮⣽⢧⡹⢖⣹⢲⡙⢮
⣷⢫⣗⢯⣧⣻⡜⡽⢪⠷⣹⠮⣝⡳⣎⠿⣼⢳⣻⡥⢞⡭⣗⢯⡳⣏⣶⢳⣝⣳⡝⣾⡹⣎⢷⡹⣎⡳⣍⢻⡘⢦⣙⣼⡏⢶⡙⣮⢱⢣⢏⡳