⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀⠀⠀⢀⣠⣴⣄⣀⡤⣖⡥⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣽⣾⠿⣹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣟⣯⣷⣿⣻⢾⣻⣽⣟⡾⣻⣿⢡⢣⡝⣆⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠒⡲⣤⣀⣀⣠⣼⣿⢯⣟⡾⣽⣯⣟⡾⣹⢫⢏⡿⣙⠖⣯⡓⣬⢧⡻⣽⣔⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣰⠢⡜⠱⢭⠻⡝⢫⠙⠈⣯⢎⡽⣻⠾⣝⠮⡵⣉⡖⡸⣄⢫⠷⣱⠚⣦⢻⡵⣾⡴⡴⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣼⡱⢐⡉⢌⡛⣜⢣⠒⣀⠨⣗⢾⣽⣻⣍⠳⣎⠡⢂⡕⢠⠏⣑⢢⡙⡴⢫⣽⢳⢯⠳⠎⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⢳⡣⢆⠣⡜⡜⣢⠙⡤⢃⢯⡿⣾⡱⣎⡝⣞⠠⠃⣼⠓⡸⣄⠡⡘⣜⢳⢮⡟⣯⣛⠖⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⣝⡘⢣⢳⣙⢦⢻⡴⣛⢮⡽⣿⣽⣿⣾⢯⣳⢨⡷⢉⢧⢸⢰⡱⢮⣹⠲⣝⡷⢯⣓⠄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⣄⡳⣞⣬⢳⣝⡹⢖⡣⢿⣿⡿⣯⣛⠦⡽⣠⢉⢦⣹⢮⣝⡷⣎⡳⣙⡞⣯⣟⣮⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣷⢯⣞⣳⣎⡷⣹⢞⣿⣿⣟⣧⢿⡹⡕⠢⣜⣧⣟⡞⣮⢳⣭⢳⣍⠾⣱⣻⠾⣽⢧⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣌⣳⢿⣶⣿⣿⣿⣿⡿⣜⡧⣟⣾⣽⡿⣟⣯⣟⢾⣫⢳⢩⡑⣾⣱⡞⣽⢣⠻⣜⢧⡸⢫⢵⣩⢟⣭⣟⣾⣻⣴⣲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⣭⣻⣟⡿⣿⣾⣿⣽⣿⣯⣿⡿⣟⣯⣟⣿⡻⣞⡽⢂⡏⢦⡑⢧⢇⡛⣬⢃⡟⣜⠣⣝⢣⠞⣦⠻⣼⢎⡷⣝⡾⣝⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢳⣽⣳⢿⣾⢿⣻⣷⣯⣿⢾⣽⡷⣿⣻⣳⢽⡲⣹⠱⣊⠱⣘⠣⢌⡓⠎⡴⢡⠎⡔⢪⠱⣎⠼⣙⢦⡻⡜⣯⠽⣎⢷⣫⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣳⣾⡿⣿⣻⣿⢿⣾⣿⢾⣟⣯⡟⡷⣣⢟⢢⡑⢆⠓⡌⠱⣈⠑⠢⡘⠰⣀⠣⢎⠜⣥⡓⢮⡹⣍⠾⣱⢫⡜⣣⢟⡮⣗⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡾⣟⣷⣿⡿⣟⣿⣿⣻⣾⣟⡾⣣⢟⡱⢣⠎⢆⠱⣈⠒⠌⡡⠐⡌⡑⢄⠣⢤⠙⣢⡙⢦⡙⢆⡳⢌⢳⠡⢲⢍⡷⣫⢾⡽⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣵⣿⢿⣻⣷⣿⡿⣟⣷⣟⣷⢫⣞⡵⢫⡜⣡⠚⡌⠒⠤⡁⢎⡰⠱⢠⡑⢎⡱⢊⠦⡡⢏⢲⡉⢦⠑⡊⠦⡙⢬⡚⣜⡳⢯⣻⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⢻⡾⣿⣟⣯⣷⡿⣿⡽⡾⣭⠷⣎⣳⢣⠞⡤⢋⠬⡑⢢⡑⢦⠡⢋⠴⡉⢆⠰⣁⠲⢑⢪⠱⣘⠢⠑⡌⠱⡘⢢⢕⡪⣝⢧⡳⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠞⣯⣟⣯⣿⢯⣷⣟⣷⣻⣽⢳⣻⢵⣣⢏⡞⣡⢋⠖⡩⢆⠱⡈⢆⡉⠦⡑⠌⡐⠠⢂⠡⢎⡁⠆⡐⠡⢌⠱⢨⠑⣎⠱⡎⢧⡳⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡺⣵⣻⣽⣾⣟⣷⣻⣞⡷⣽⣫⢗⡯⡞⡼⡸⢥⡚⣜⠱⣊⠱⢈⠄⡒⠤⢑⠠⠄⡁⠂⠔⠂⠌⡐⠠⠑⡨⠐⣁⠣⡌⡒⢍⣣⠳⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⡳⣯⣷⢿⣞⣯⣷⢯⣟⡷⣽⢫⡞⣵⢳⣙⢦⡹⢌⡓⡄⠣⢌⠒⢠⠒⡈⠐⢠⠀⠅⡈⠐⢠⠠⢁⠊⠔⣁⠢⢁⡒⢍⠢⡐⢭⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢽⣳⢯⡿⣯⣿⢾⣻⣽⣻⢼⡳⣝⢮⢧⣋⠶⡱⢣⠜⡰⢡⠊⡔⠡⠐⣀⠃⠄⡈⠔⠠⢁⠂⡐⠠⢈⠂⠄⠢⢁⠒⡨⠑⡌⠠⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡵⣻⢿⣽⢯⣟⣷⣻⢮⣏⡷⣫⢞⡧⢞⣱⡙⢦⡙⠴⡁⠎⡄⠡⠒⠠⢈⠐⠄⡈⠐⡀⠂⡀⠁⠂⠌⠠⢁⠂⡘⢄⠡⠐⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣻⣯⣿⣻⢾⣽⣻⡼⣳⠽⣮⢝⡺⡴⣙⢦⡙⠦⣉⠒⡌⠡⠌⡁⠂⠌⡐⠠⠁⡐⠀⠠⠁⡈⠄⡁⠂⠄⠡⠌⡐⠠⠐⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡷⣟⣾⣽⣻⡞⣷⡽⣫⡟⣼⢫⡵⢳⡍⢦⡙⢦⠡⢎⠰⡁⢒⠠⢁⠂⠄⡁⠂⠄⡈⠐⢀⠐⠠⠠⠁⠌⡐⠁⠄⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣻⢞⡷⣯⣽⣳⢿⣱⡻⣜⢧⣏⡳⣜⢣⡙⢦⡉⠦⡑⢌⡐⢂⠂⠌⡐⠠⢁⠐⠠⢈⠀⠌⠠⠁⠌⡐⠠⠁⡐⠀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⣽⣳⢯⢷⣫⢷⣹⡝⣞⡲⡵⣊⢧⡙⢦⠱⡑⡌⢢⠐⠌⡐⠂⠄⡁⠂⠌⡐⠠⠈⠄⠡⢈⠐⠠⢁⠂⠄⠂⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠾⣽⣫⢯⡷⣯⡳⣝⢮⡳⢵⣋⠶⣙⠦⢣⠱⡘⠤⡉⠔⠠⢁⠂⠄⡁⠂⢄⠡⠈⠄⡁⠂⠌⡐⠠⢈⠀⠌⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⣋⣷⣻⡵⣻⡝⣮⡝⢶⡩⢞⡥⣋⢧⢣⡙⡔⢡⠊⣁⠂⠌⡐⠠⢁⠂⠄⡁⠂⠄⡁⠂⠄⠡⠀⠌⢀⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠳⡝⣧⣛⠶⣭⢣⡝⣎⠶⣍⡖⣣⠜⡘⡄⢃⠤⢈⠐⠠⢁⠂⠌⡐⠠⢁⠂⠄⠡⠈⠄⠁⠂⠄⡈⠄