Housewife Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠟⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠍⠀⠀⣫⣭⡀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢠⣀⢠⠀⠀⠀⢀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠁⠀⠀⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⠁⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠞⡼⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢠⣿⣿⠋⠀⠀⠀⠀⣸⣤⠿⣆⠀⠀⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣶⣿⡟⠙⡆⠀⠀⠀⢴⠇⢸⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠋⢚⠃⡀⢙⡦⠒⢡⡞⠀⡇⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠙⢻⡟⠛⠛⠛⠛⠃⠠⡇⠀⡆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢀⠱⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠈⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⡀⠀⠁⠀⢠⡀⠀⠆⠀⠈⠄⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⠄⣀⠤⢀⡀⠤⠐⣢⢄⡸⠯⢄⣐⣄⣠⠤⢄⠀⢠⠀⠙⢶⡞⠸⠇⣸⣇⣧⠀⠐⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣥⣽⣆⣛⢱⣶⣛⣡⣶⣏⣉⣶⠚⠓⢻⣤⢤⡽⣷⣾⢃⡘⠂⢀⡠⠀⣲⡾⠓⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠀⠉⠋⠀⠹⠿⠟⠙⠟⠛⠀⠙⠳⠝⠏⠩⠇⠙⠛⡁⠉⢉⡀⠌⠤⠎⠉⠉⠀⠈⠓⢦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⡀⠀⠀⠀⠂⠐⣀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠤⢴⡶⣶⠟⠐⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠋⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣿⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⢿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⡗⡲⠿⠟⠉⠉⠉⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⣀⡀⠀⢀⣄⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⡿⠀⢫⣥⠦⠀⠈⠋⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠶⠀⠀⠀⠐⠢⠴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠈⠒⠲⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⢳⡀⠀⠀⠀⣿⠛⠢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠹⠀⠀⠙⢄⠀⠀⠇⠀⢴⣼⡛⠛⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⣧⣤⣤⣤⠀⠰⡂⠄⢀⡾⠀⠁⠀⢰⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⠘⠋⠉⠉⠀⠈⠐⢌⣠⠋⠀⠀⠀⠀⢸⢀⡼⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢌⡄⠀⠀⠀⠀⢸⢸⠇⠀⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡜⠈⠀⠀⠀⠀⢀⣠⠂⠤⢀⣣⠀⠀⠀⠀⠘⣸⠀⢀⣰⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⢠⣾⣿⠇⠀⠔⢋⣁⠀⠩⠖⠀⠈⠉⠂⣤⠀⣰⡿⡶⢛⣁⡁⠉⢣⡀⢂
⠀⠀⠀⠀⠀⠀⠀⣶⠾⠭⠤⠄⣠⡟⠉⠻⣶⡤⢀⣀⣘⣆⣀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠙⢶⣭⣥⣤⡄⠁⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⣀⡿⠁⠀⢶⠋⠁⢀⣼⣿⣿⣿⣿⣿⣶⣦⣟⣿⣿⡃⣠⢀⡀⠿⠿⠿⠛⠛⠛⡄
⠀⠀⠀⠀⠘⢇⣬⠴⠊⠁⠀⠀⠀⠀⠀⣬⡲⡾⣻⣿⣿⣿⣿⣿⣿⡟⠉⠭⠥⠤⠭⠝⣻⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⡤⠖⠋⠁⢈⡉⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡤⠞⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠂⠒⠒⠛⠉⠁⠀⠀⠀⢀⠎⠛⠢⠤⠀⢐⢒⠀⠀⠸⣀⡀⠀⠀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢠⣤⢾⣿⣶⡟⣻⣟⠛⢟⢫⣠⡉⢭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣵⠿⡿⣹⢁⡿⠃⢸⠈⣮⣿⢷⢸⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡠⠚⠣⠶⠄⣧⢈⡇⣦⠄⠀⠐⠛⠘⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣼⡄⠀⠈⠁⠀⢸⢸⠇⣍⡀⠘⠛⠛⠛⠻⠿⠿⠿⠿⠶⠒⢁⡀⠀⣠⠞⣷⡐⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣓⠀⠀⠀⠀⣿⣿⢰⣿⡙⠛⠛⠲⠶⠶⠶⠦⠤⠶⠶⠖⠛⠁⠀⣧⣤⣤⡷⠶⢶⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣧⣭⠄⠀⠀⢸⣿⣇⣾⣃⡄⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⣁⣀⣀⡇⠀⠀⡇⠀⢸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⢀⡾⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠳⠶⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠙⠿⠏⢁⠀⠀⠀⠀⠀⠙⠓⠶⠤⠤⢤⠤⠤⠴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣶⣶⣤⣶⣾⣥⡄⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⠏⠛⢿⣿⣿⣿⡿⣻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣾⠐⠃⠈⠉⠙⠻⠷⠛⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠉⣁⣀⡞⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣯⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⡭⠶⠞⠋⠉⠹⡄⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⣉⠉⠛⠦⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢻⠛⠉⣁⣤⣶⡶⠞⠠⠄⣳⡼⢩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⢟⣻⠉⠀⠀⠀⠘⠻⠟⠇⠀⠀⣠⣌⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠉⣉⡩⡤⣶⣾⣏⢯⣗⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡿⠳⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢸⣟⣽⣷⢺⡛⢏⡀⢺⡜⡞⡾⡼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡛⡄⠀⠀⠀⠀⣀⣀⡀⠂⠼⢇⡾⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠆⢉⡼⡽⣌⣛⠻⢿⣾⣿⣼⣽⣷⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⢀⠀⠀⠀⠙⣶⣯⣽⢣⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠁⢣⢻⡫⢈⡟⠋⢀⣻⣧⢣⢿⢧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⠀⠀⠀⠙⢦⣀⠀⠀⠉⣱⠋⣄⡠⠔⠢⡄⠀⠀⠀⠀⠀⠀⡏⠀⢃⡈⣇⢿⣿⣿⣷⢶⣿⣿⣏⣏⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⢧⡀⠀⠀⠀⠀⠈⢻⡗⢚⣱⣶⣆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⡜⡜⣿⣿⣻⣹⣷⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⢠⡇⠀⠀⠙⠢⣄⠀⠀⠀⢸⣈⢢⠙⠌⠛⣆⠀⠈⠣⠤⠤⠀⠀⠀⠀⠀⢳⠀⠰⠃⢹⡿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠔⠋⠁⠀⢹⡄⣀⣴⡾⣟⢯⣢⡀⠀⠀⠀⠈⠀⢠⣞⣹⣛⡟⠛⠓⠒⠒⢲⡆⠀⠀⢀⡜⠀⠀⠀⠀⣼⠁⠀⢠⠃
⠀⠀⠀⢀⡈⢀⣴⣤⣤⡾⠛⢿⣳⣝⢮⢳⡽⣷⠀⠀⠀⣠⡶⢰⠶⣩⣭⠩⠭⠭⣭⠭⡏⠃⠀⣴⡏⠀⠀⠀⠀⢰⠇⠀⠀⡞
⢀⠔⢋⣡⣶⣿⣿⣿⣿⠃⠀⠀⠛⠻⢿⠳⣽⣎⠀⣠⣾⡋⢀⠴⢊⠴⢳⣶⣶⣶⡆⠀⣷⠀⠘⣹⡇⠀⠀⠀⢀⡏⠀⠀⠀⡇
⡇⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⣴⣶⣶⣶⡟⢯⣤⣾⡻⠁⠔⣁⠴⠥⠤⢌⣽⣭⡛⡇⢠⣿⢠⡀⣹⣿⠀⠀⢀⡞⠀⠀⠀⢸
⠙⢆⠸⡔⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⠻⣿⣵⠠⡏⣁⠔⠊⣡⠤⠖⣯⣶⡇⢸⢻⠈⣷⣿⣿⠇⢀⠞⠀⠀⠀⠀⡌
⠀⠀⠀⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⢟⢿⣧⡉⠀⢰⣫⠔⠒⠀⠲⣧⣀⢸⣬⠀⠘⣿⠊⣱⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢧⣀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣮⣳⣵⠀⠀⠰⡎⠑⠒⠒⠒⠒⠚⠁⢀⣼⡿⣰⠃⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠱⣼⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⡀⢉⡵⠋⠀⢀⡴⠃⠀⢀⣴⣿⡡⢊⣴⣿⠋⠀⠃⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢻⣿⠟⢉⡤⠖⠋⠉⠻⣿⣿⣿⡿⠁⠀⠀⢠⠎⠀⢀⠴⣫⡾⠋⣰⠟⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠐⡏⣠⢖⠉⠀⠀⠀⠀⠀⣻⠟⠋⠀⠀⠀⢠⣿⣿⠟⢁⡴⠋⠀⢠⡏⠀⠀⠀⠀⠑⢤⡀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠘⠓⠚⢇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡰⠛⠛⠁⠀⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣾⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣜⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠆⠿⠛⠉⠀⢷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⠀⡀⠀⠀⠀⢀⡈⢧⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠻⡦⢴⠀⠀⠀⠁⠀⢸⡁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⣷⣄⠀⠀⠀⣄⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠛⠹⠃⢀⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠶⢃⣴⠛⡢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠀⢠⣶⡇⢀⡴⢿⣋⣱⣟⡺⡊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢠⣿⣿⠞⣉⣐⣸⣿⣿⣯⣾⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⣀⣿⡿⡟⣯⣽⠟⢻⡆
⠀⠀⠀⠀⠀⣰⠓⢛⠛⠓⣬⡿⣈⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣾⣫⡘⢣⣿⠆
⠀⠀⠀⠀⣀⠇⠄⠨⣄⣹⣭⢼⣿⢿⣖⢲⣤⠀⢤⣤⣄⣤⣀⣀⣤⡀⠀⠀⠀⠀⠀⣠⣿⡿⣉⡼⠿⣿⢿⣿
⠚⠷⠶⣿⣋⣿⣭⣽⣿⣻⣏⡀⠡⢴⠤⠶⣿⢟⣻⣾⣿⣿⣿⣿⣿⣷⣄⣀⣀⣤⣴⣿⣾⣟⣩⣗⠾⣵⢟⡃
⠀⠀⠀⠀⠈⠉⠙⠉⠉⠙⠛⠛⠓⣲⣂⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⢿⣿⢿⣿⣽⡿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣴⡿⠛⠁⢽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠿⢿⣿⣿⣿⣟⢿⣿⣷⠏⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠰⢶⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣾⡾⡇

⠀⠀⠀⠀⠀⠀⣠⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⢀⣰⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠿⣿⣿
⠀⠀⠀⣠⡜⠟⠠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠈⠀⣀⢸⣿
⠀⢀⣼⢜⣁⠀⠶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣠⡉⠉⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠳⠂⠘
⠀⠸⠄⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢀⡄
⠀⠀⠀⠄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠐⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠓⠲⢄⠀⠀⠀⠀⠀⠀⠠⠶⣶⣄
⠀⠀⠀⠀⢀⣤⣠⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠚⠀⠀⠀⡄⠀⠀⠀⠀⠘⣻⣇
⠀⠀⠂⠀⢸⣽⠈⠳⠎⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣷⠀⠀⠐⠀⠀⢹⣿
⠀⠀⠉⣱⡦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⣭⡿⠏⠀⠀⣰⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠃
⠀⠀⠉⠉⠆⠀⠐⠋⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣴⣤⣦⡀
⠀⠀⠀⢠⡆⣿⠦⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣿⡇
⠀⠀⢠⣞⢷⡁⠀⠀⢰⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⡙⠆⠹⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡄⢸⣿⡇
⠀⣀⠻⠛⠈⠁⣤⣿⣪⠉⠄⢢⠀⠀⠀⠀⠀⠀⠾⡏⠉⣈⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢀⡟⠁
⠀⣿⠰⠶⠛⢫⣿⣳⣿⡆⠀⢰⠀⠀⠀⠀⠀⠛⠀⠙⢺⠻⠁⠀⠀⠀⠀⠀⢤⢰⣶⣖⣒⣻⣿⡄⠄
⠀⡿⠀⠀⠤⢿⣿⠩⠙⣿⠁⠀⠀⠀⠀⠀⠻⠆⢲⣶⣿⠀⠀⠀⠀⠀⠀⠀⡼⠟⣿⣿⠟⣿⣾⣿⢇⢄⡀
⠀⠁⠀⠛⠛⠋⠙⠛⠐⠛⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠛⠛⠛⠛⠛⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣲⣥⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣶⣴⣿⠟⠛⣉⣼⣿⣿⣷⣶⣿⣿⣿⡟⢳⣄
⠀⠀⠀⠀⠀⢀⣤⣸⣷⣿⠿⠋⢀⣤⣾⢋⡟⢿⣿⣿⡏⢻⣿⣿⣏⡸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⣠⣿⣿⡟⠯⠟⠀⠉⠉⠁⢩⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠁⠀⠀⠀⠞⠓⠒⣒⡒⠼⠛
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠰⣿⣿⣿⣿⢠⠶⠶⣤⡀⠀⠈⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠞⢻⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢠⢿⣿⣯⠁⢀⣠⣤⡀⠁⠀⠐⠛⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠒⠤⣀⣀⣤⡤⠤⠬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠆⢿⡿⠇⠀⠈⠉⠛⠁⠈⠀⡀⢾⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣦⠁⠀⠀⠀⠀⠀⠀⠐⠀⢠⠄⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡁⠀⠀⠀⠆⡤⠤⠶⠤⣤⡠⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁⠀⠀⠀⠘⢷⣦⣴⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠀⣰⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠲⠴⢾⡛⣦⣀⡤⠊⠁⢄⠈⠳⠴⢦⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣁⡈⠙⢶⣦⣀⠀⢻⡄⠀⠀⣩⠇⠁⠀⠈⣹⠖⠒⠒⠂⠀⠈⠃⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣏⠛⢷⣶⣿⣿⣿⣦⣷⡆⣰⡏⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣦⠀⠀⠀⠀⢺⡇⠂⠀⠀⠀⠙⠭⡻⣿⣿⣿⣿⣿⡏⠀⠀⡆⣼⣿⡀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣷⡄⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠙⢮⣽⣿⣿⣿⣷⣤⣤⣇⣿⡟⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⣿⣿⣿⣷⣶⡤⠠⠀⠀⠀⣀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣄⡙⠋
⠀⠀⠀⠀⠀⠀⢀⡔⠻⣄⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⢠⣼⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⠳⡄⠀⠀⢿⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡹⢦⣤⠴⠞⠛⠙⡆⠀⢻⣽⣿⣿⣿⣿⣿⠀⢠⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢰⣧⣤⠤⠶⠶⢒⣠⣄⠀⡀⢿⣻⣿⣿⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⡬⠾⡄⠀⠀⣴⡾⠟⢋⣥⣾⣛⠈⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢨⠀⠀⠀⠐⠠⠀⣀⣴⡟⠁⠈⠻⣷⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠋⠒⠠⡀⠀⣠⣬⣽⡆⠀⠀⠀⠈⠻⢟⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠉⣶⣿⠍⠉⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⡟⠂⣾⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠁⢀⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⡎⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠎⠀⠸⠋⣴⠟⣋⣴⣿⣿⣿⣿⣿⠟⣿⣷⣤⣄⣀⣀
⢰⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠟⣋⣀⣴⠟⠙⣦⣽⡋⠙⢷
⡌⠀⠀⠀⠀⣾⣃⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⣟⣛⣉⣩⡤⠖⠛⠁⠈⠉⠀⠀⢹⡇⠀⠐⢾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣾⣿⣿⣿⣷⣶⣶⣶⣶⣾⡷⠀⠀⠘⠁⠀⠀⠘
⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠶⠶⠛⠻⠿⣷⣦⣍⡉⠉⠉⠉
⣈⣉⣙⣛⣓⣶⣶⣶⣶⣶⣶⣿⣿⣟⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠶⠚⣿⣟⠛⠛⠲⢤⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⡞⠁⢀⣼⡿⠉⠻⢦⣄⡀⠈⢷⡀⠀⠀⢸⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡉⠉⠁
⠀⠀⢠⡟⠀⣠⢞⡿⠃⠀⣰⠋⠉⠛⡿⢷⣷⣄⡀⠀⠀⠀⢸⡷⢦⡄⠀⠀⢀⡤⠖⠒⠚⠁
⠀⣀⣸⡁⣾⡵⢻⠀⣠⠄⢹⡀⠚⠁⡰⠀⣿⢧⢻⠀⠀⠀⠺⠇⢠⣿⠀⠀⠀⠉⠙⠒⢢
⡜⢩⢭⡻⣿⠴⠚⢃⡀⠀⡠⢩⡒⠚⠁⢀⣿⡋⣸⠀⡐⠉⠦⠠⡄⠈⠀⠒⠛⠉⠉⠉⠁
⡇⣷⠆⣷⣿⣇⠀⠀⠀⠀⣄⠈⢀⠞⠀⣼⠃⠉⠁⠀⣇⡀⢀⢚⡇⠐⡼
⢷⠙⠛⠋⣼⣙⣦⣀⠀⠀⠈⠉⠁⣠⣾⠃⠰⣒⠦⠤⢤⡍⢉⣍⠀⠀⢀⣠⣿⣄
⠈⣑⡾⠟⠉⠁⢀⠉⠳⡖⠒⠒⠋⠉⢫⡙⢦⣸⠆⢸⣿⠂⠈⣯⠳⡴⠟⠓⠒⠺⢧⣴⣶⣶⡄
⢚⣿⠦⣤⣠⡞⣯⠀⠀⠘⣆⣀⣀⣠⠤⠽⢺⡉⠀⢸⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠈⠉⣯⠇⡷
⢾⡁⠀⠀⠙⠦⣼⡤⡄⠀⠀⠀⠀⠀⠀⠀⣸⠧⣀⣼⣀⡤⢾⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⡞⠋⢙⣿⣿⠂
⠀⠉⠙⠓⣒⣦⢄⣀⡟⠀⠀⠀⠀⣀⣤⠾⠋⠀⣾⣍⣀⣀⣀⣀⣉⣉⣉⣛⣛⣛⣛⣛⣛⣵⣾⣿⣿⡿
⠀⠀⢠⣾⣿⠉⠉⠋⠀⠀⠈⠁⠙⣿⣿⣦⠀⠀⢻⠀⠠⣶⡄⠀⢴⣦⠀⠠⣶⡆⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣻⣿⣿⡆⠀⢸⡆⠀⣬⠤⠤⠤⠴⠶⠶⠶⠶⣦⠀⢹⣿⣿⣿⣿⡇
⠀⢰⢿⣿⣿⣷⣷⣦⣤⢤⣖⣻⣾⣿⣿⣿⣧⠀⠘⡇⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⡅⠈⠙⣿⡏⠉⠉⣿⠀⠀⠀⠀⠀⣿⠀⠈⠛⠛⠒⠒⠶⠶⠶⠾⠧⠀⢸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢳⣀⣸⠇⢹⣆⣴⡃⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡿⠁
⠀⠀⠀⠀⠀⠀⠐⠿⠛⠳⠌⠛⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡒⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣠⠀⠀⠀⠀⠀⠀⢠⢞⣻⣿⣻⣧⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢸⠀⠀⠀⠀⠀⠀⠻⡟⢉⠀⡉⠟⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣖⢰⡋⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡰⠹⡀⠈⠢⡀⠀⢀⣀⠀⣈⢀⢂⡀⣠⠔⠂⠂⠒⠒⠈⠉⠁⣀⠤⠋⢸⠀⢠⠀⢸⠀⠀⠀⠐⣲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠉⠀⣀⡤⠖⠃⠀⠋⠶⠶⣤⣖⠒⠛⠛⠉⠁⠀⠀⠀⡖⠠⠋⡆⢸⠀⠀⠀⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⣀⠀⠈⠉⠉⣽⣅⣠⠴⢄⡀⠀⠀⢦⡀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠈⠂⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⡌⠀⡛⠁⠤⠀⠑⡆⠀⢨⠑⠦⣀⠀⠉⠒⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⢹⡆⠤⠀⣀⡿⠀⣸⠀⠀⠈⠑⠲⠶⠴⠏⠸⠞⠲⢞⣛⡛⠛⢛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠋⢀⠠⠂⠀⢘⠂⢀⣅⡀⣀⣴⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⡄⠒⠁⠀⠀⠀⠸⠀⠀⠀⠁⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠢⠀⠀⠀⣇⠀⠀⠀⠙⠀⠀⢁⠀⠀⠀⠀⠨⠀⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡈⠀⠀⠡⡀⠀⠈⠒⠒⠂⠀⠠⡀⢈⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠑⠤⠄⠀⣀⠀⠀⠀⣸⡟⠧⣄⠀⠀⠀⢠⢡⣾⣿⣷⡌⡆
⠀⠀⠀⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠉⠀⠀⠀⠈⢄⠻⠿⠿⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣟⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣀⣀⣉⣁⣁⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣫⡅⠀⠀⠐⣟⠁⠀⠈⠛⠀⠀⢠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠀⠸⢤⣤⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠂⠀⠀⢶⡄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡽⠛⡋⠀⠀⠀⠀⣶⣧⡀⠀⠀⠀⠀⠀⠀⣰⠅⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠈⠓⠀⠀⠠⠤⠏⠀⠀⠀⠀⠀⠄⠀⠄⢀⠀⠁⠀⠀⠸⠥⠜⠁⠀⠀⠀⠀⠂⢿⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢈⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠟⠉⠿⣿⣿⣟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢢⡄⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⢀⢺⣿⣿⠟⠀⣀⠠⠀⠀⠀⢈⣠⡄⠈⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⣤⣴⣷⡀⠀⠀⢀⣾⡟⠁⠀⠀⣤⣾⣷⡄
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠘⠻⠿⣿⣷⠖⠀⣼⣿⡇⢠⣠⣶⡿⠛⠛⠠
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⣿⣷⣷⣟⢋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠂
⠀⠀⠀⠐⠶⢶⠿⠆⠰⠿⠥⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣶⡀
⠀⠀⠠⣭⣷⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷
⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢻⣿⡇⠙⡇
⠀⠀⠀⠀⡇⢶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠘⠃⠃
⠀⠀⠀⠀⠙⠺⠤⠤⠤⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠠⣭⣛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⣼⣏⣙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠉⠙⠛⠋⠩⠿⠶⣄⡘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡓⠆⠀⠀⢀⠀⠀⠀⠀⢼⠁⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣾⡟⠙⠂⠀⠀⢨⣍⠀⠀⠀⢾⠄⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⣾⠇⢨⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⢰⣦⣭⣿⡇⡖⠦⢄⡀⠀⠀⠀⢀⣀⡿⣡⡴⠿⢟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡻⢿⣿⣇⠙⠉⠋⠀⠀⠀⠀⢹⠿⠷⣿⠗⠐⢺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠛⣿⣦⡄⠀⠀⠀⠀⠀⠘⣟⡻⠶⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠋⠁⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠈⣿⣉⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢋⠙⠛⠻⡻⣦⣔⢀⠀⠀⠉⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠄⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⣀⡀⠀⠀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠇
⠀⠐⠒⠒⠒⠒⠒⠀⠒⠒⣿⣒⣐⠒⠀⣂⣋⣿⠐⢛⣷⡄⠒⠀⠂⠀⠘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⠿⣿⡟⠁⢰⡿⣿⡷⢄⡀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣳⣾⣶⣶⣗⣳⣄⠘⠧⡈⢛⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣞⡿⢻⠿⣾⣝⣿⢷⡄⢉⣬⠾⢷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣻⡾⠁⠀⠘⣇⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣛⣛⠛⡛⠛⠛⠛⢻⣿⡇⠀⠀⠀⢻⡀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠈⠛⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠂⡙⢂⠓⣀⠂⠘⣀⠛⣘⠓⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⢁⣉⣴⣿⣿⣿⡿⠿⠯⣬⣅⡉⢀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⣾⣿⣭⠤⣄⠈⠛⢷⣟⢥⡄⢀⠀⠀⠀⠀⢀⡴⠎⠁⠠
⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢈⢻⡀⠲⠄⡶⠶⠖⠋⠉
⠀⠀⠩⠙⠲⠦⠤⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠋⠉⠉⠀⠀⠀⢹⣧⠀⡌⢇⣷⡄⠲⣿⡄⠀⠀⠠
⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠉⠒⠶⢤⣤⠴⠖⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⢴⣌⣾⣿⣔⣿⡇⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠦⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⠋⡝⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣤⠃⢀⠀⠀⠀⢸⡄⠀⣀⣀⠀⠀⠀⠀⢠⣶⣛⣀⠀⠀⠀⠉⠀⣤⡃
⠀⠐⠐⡐⠐⡀⠀⠀⠀⡐⢂⠀⠀⠀⢂⢲⡉⣱⠀⠀⠀⢃⠀⡨⢿⣿⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⢘⣠⣿⠂⢂⠀⡐⡐⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣂⣀⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠠⣬⣤⣴⣶⡾⠀⠀⠀⢠⣿⣿⣉⣀⣀⣾⡟⡀
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⣀⠠⠀⠘⠛⠿⣦⡀⠈⠛⠛⠋⠀⠀⢀⣴⠟⣿⣿⡂⠀⠾⣿⡟⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠖⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⠴⠋⠁⠀⢸⣿⠆⠤⠟⠉⠀⠀⠀⡤⢄
⠀⠀⠄⠄⠀⠀⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⢦⡀⠀⣄⠀⠀⠀⠀⢀⣠⣴⠿⠃⠀⠀⢀⣼⡃⠈⠁
⠀⠤⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢹⡧⢀⣼⣆⣽⣶⠿⠟⠛⠉⠀⠀⠀⠀⠾⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠂⠂⠉⠉⠀⠀⠀⠀⠀⠀⠠⠟⠠⢠⣾⣿⣾⣿⡿⠗⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣠⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡄⠀⠀⠛⠁⣤⣴⣆⠀⠁
⠀⠀⠠⠤⠤⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠤⠤⠤⡀⠉⠀⠀⠐⠟⠀⠀⠘⠛⠀⠀⠈⠁⠀⣀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀⠀⠀⠠⣦⠀⠀⠈⠉⠀⣸⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⢀⡀⠈⠉⠀⠀⢠⡄⢠⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⡀⠀⢀⡨⠁⠸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⣀⠤⠀⣌⢙⡀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣰⠆⣄⢲⡀⠀⠀⠀⠀⠀⢠⡿⠾⣧⠠⠉⠀⠀⠀⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⢠⡶⣤⠐
⠀⠀⠀⠀⢹⣿⣛⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⠀⠉⠁⠨⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀
⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠆⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠺⠛⠻⠶⠗
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⣉⡣⠀⠀⠀⠀⠀⠀⠀⢠⣰⠦⠤⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠠⣦⡤⣦⣰⠄⠀⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠆⣀⢤⡀⠀⠀⠀⠀⠀⠀⠐⠷⠾⠆
⠀⠀⠀⠹⠜⠠⠀⠀⠀⠀⠀⠀⠀⢠⡉⠹⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⣟⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⢀⡀⠀⡀⠠⠄⠀⠀⠀⠀⠀⡀⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠴⠉⠁⠙⠄⠀⠀⠀⠀⠀⠀⢠⣥⣔⠀⠀⠀⠀⠀⠰⠿⠓⠒⠷⠇⠀⠀⠀⠀⠀⠀⠟⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢁⣾⣿⣯⣭⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠈⠛⠛⠉⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣨⣿⡿⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣹⡏⠈⠁⠀⣄⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⡤⠀⠐⢧⣬⣿⣷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠦⠤⣌⣿⣿⣅⠀⠀⣠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠿⢷⡄⡉⣩⣏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣷⣾⣷⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡿⢿⣿⣿⠏⠀⠸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⢩⣿⣿⣿⣆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣆⠀⢻⣿⣿⣿⡄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣼⢿⣿⣿⣧⡀⠙⠻⢿⣷⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣵⣾⡿⣸⡟⣧⠛⢤⣬⣥⣶⣧⣆
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠧⠿⠧⡿⢆⣍⠀⠀⣺⣧⠹⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠉⠛⠻⠿⠟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠘⠻⠇⠰⠄⠀⠀⡆⠀⠀⢀⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠀⠀⠀⠀⠀⠀⡀⠃⠀⡀⠀⠀⠀⠉⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⡈⠒⠂⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡒⠰⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠁⢰⡣
⠀⠀⠀⠀⠀⠀⣿⣙⣿⢸⣿⣿⢈⡍⣿⢿⣿⢸⣿⣻⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡣⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⢠⣝⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣷⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠠⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠔⠊
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡆⠉⢈⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠢⢾⡆⠀⢀⣀⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⣀⣤⡖⢀⣾⡿⠀⠀⠀⠀⠀⠀⢹⠇⠀⢀⣸⠷⣄⣠⠾⣿⣿⠿⣿⡏⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣭⣥⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣟⣒⣻⣗⣒⣿⣟⣒⣻⠃⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⣻⣗⣒⣺⣗⣒⣿⣷⣒⣺⠀⠀⠀⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢾⣿⠀⣾⣷⠀⣿⡗⠀⣿⣤⣄⣀⣠⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣽⣯⣭⣽⣯⣬⣽⣯⣬⠟⠿⠿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠰⠒⣒⣶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣲⠖⡶⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠛⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⢿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣋⣭⠶⠶⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣭⠖⣫⣥⡶⠿⠟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡷⣟⣭⣶⠾⢛⣛⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡙⣌⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣫⡽⠖⣛⣯⡽⠿⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⡳⡽⣎⢳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣝⣿⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣄⣄⣸⣿⣿⣿⣿⣿⣿⣟⣿⣇⠀⠀⠀⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⣴⠿⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⡟⢻⣿⣿⣿⣿⡿⣿⢸⣿⠀⠀⠀⣿⣿⡽⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣻⡿⠟⠀⢸⣿⣿⣿⣿⢷⡸⣆⢿⣷⣤⣴⣿⣿⣹⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⢰⣾⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡘⣧⢹⣦⡻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⠙⠾⠿⠛⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⢷⡘⣧⣙⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠧⣙⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠛⢿⡿⣧⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⣶⠀⠀⠀⠀⠀⣿⣿⣟⠳⠦⠽⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⣷⣼⣷⡄⠀⠀⠀⠐⢿⣯⣤⣾⡿⠀⠀⠀⠀⣰⣿⣿⣿⣟⡲⠞⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⠾⣫⢿⣿⣿⣦⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣠⣾⣿⣿⣿⡿⠻⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⣴⣚⣿⡿⠋⠙⠳⠦⣤⣄⣀⣀⣀⣤⣾⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠀⠈⠙⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣷⡏⢽⠑⢀⣀⣀⣀⣀⣤⠤⣦⣶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠜⠉⠉⠉⠛⠋⠉⠉⠁⠀⡀⠴⠊⠁⠀⠐⠒⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠞⠻⡙⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠠
⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⣠⣀⡀⣀⣘⠶⠶⠶⠖⠒⠒⢒⡚⣋⣉⣉⣉⣉⠉⠁⢠⡤⠠⣤⣤⡤⢤⣄⠀⠀⢠⡄⢀⢤⠉⠒⠒
⠀⠀⠠⠤⠄⠀⠀⠘⠿⠇⠀⣿⣟⣿⣿⠀⠀⢹⡏⠀⠀⢈⠁⣻⣿⣗⣂⢸⠀⠀⢸⣙⣛⣿⣿⣯⣹⡇⠀⠀⢨⡆⢿⣨
⠀⠠⠐⠒⠂⠰⠂⢀⣀⣀⣀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠀⠛⠛⠛⠘⠘⠀⠀⠸⠟⠐⠛⠛⠁⠈⢁⣠⣤⣬⣥⣄⣀⣀
⠀⢨⠀⠀⠀⠀⠂⠀⠉⠹⣟⣭⡽⣛⡻⢤⡐⠀⠀⠀⠀⡄⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⢆⠩⠍⠀⠈⠱⠭
⠀⢒⣀⣀⡤⠦⠀⠄⠐⠂⠀⠁⠉⠉⠉⠉⠙⠛⠛⢶⣷⡄⠀⠂⠀⠀⢠⣄⡀⠀⠀⠛⠙⠉⠶⠆⠄⠤⠂⠀⠐⠶⠄
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠀⠀⠴⣇⠀⢹⣄⡀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⣀⠀⣀⡐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠾⠈⠈⣳⣤⣤⣤⣤⠥⠒⠁

⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⢀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⢀⣼⣿⣦⠀⠀⠀⠀⠀⢿⣇⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣋⡀
⠀⠋⣿⣿⣯⠑⠀⠀⠀⠀⠀⠈⢹⣿⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢧⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⢹⢿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢤⣤⣤⠀⠀⠀⢀⣀⣀⢼⣿⡆⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠈⠳⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠸⠸⠀⣶⡆⠀⠀⠀⠀⠀⠸⠏⠇⢸⣿⡟⠀⠀⠀⢸⣿⡿⠸⠿⠀⠀⠀⣈⣶⡆⠀⠀⠀⣶⣶⣶⠞⠈⣿⡟⠀⠀⠘⢆⠀⠀⠀⠀⠀⠛⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠋⠉⠀⠀⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠇⠛
⠀⠀⢀⣀
⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⣿⡟⢧⣀⣀⡀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣮⠁⠀⠀⠀⠀⠀⠀⢸⣷⣆
⣀⣠⣿⣧⠋⠀⠈⠀⠀⠀⠀⢸⣿⡀⠑⢢⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⠓⠀⠀⠠⠾⠶⠀⠀⠀⠀⣼⣿⠉⠙⣷⠀⠀⠀⠀⠀⠀⣼⣟⠈⢦
⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⠒⠒⠒⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⡷⡆⠀⠀⠀⢠⣿⣿⠀⠀⣣⣀⣤⣄
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢸⠖⠒⢲⡄⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠈⠀⡇⠀⠀⠀⠈⣿⡟⠀⠀⢻⣻⠿⠷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢩⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢾⠷⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡃⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⣤⣠⠄⣺⣛
⠀⠀⣸⣿⢃⣴⣆⣰⣶⡄⠀⠀⠀⠀⢠⡏⣿⡏⢳⣀⠀⠀⠀⠀⠀⠀⣯⡴⣷⣿⣄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡓⢬⡄⠀⠀⠀⠀⠀⠰⠿⠿⣿⣸⣿⣿
⠀⢠⣿⣿⡀⢸⣿⣿⡏⠀⠀⠀⠀⠰⠶⢶⡞⠳⢺⡿⠳⠂⠀⠀⠀⠀⠀⢸⣿⣿⠙⠆⠀⠀⠀⠀⠀⢡⣿⣿⣿⡇⢸⣿⠀⠀⠀⡖⠒⠒⠒⢲⠀⢩⣿⣿
⠀⠸⣿⣿⣷⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢙⡶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣷⠀⠀⠀⠀⠀⠈⠙⡿⣿⠁⠀⠋⠀⠀⠀⡇⠀⠀⠀⢸⠀⠸⡿⡿
⠀⠀⠿⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠃⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⣰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⣿⣿⠿⠛⠉⠙⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣼⡇⠀⠀⠀⠀⠀⠸⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡆⠀⠀⠀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⡁⠀⠀⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣅⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠉⠙⣦⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⣀⣿⣿⣦⣿⡷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣧⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⠿⠿⠿⠿⠟⠛⠁⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣶⣶⣬⣿⣿⣿⠏⠀⠘⣿⣿⣿⡆⠀⢀⣀⠠⠶⠚⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢈⠿⠿⢿⣿⠏⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⢠⡾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡞⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠻⣿⡆⠀⠀⠀⣼⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⡾⠀⠀⠈⠉⠓⢦⣤⣤⠀⠀⠀⠀⠹⡷⠒⠖⠚⠹⣷⡀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢻⡶⠀⠀⠀⠀⣾⠿⠁⠈⠉⠑⠶⠶⣶⡶⠖⠀⠀⠈⠻⠶⣄⠀⠸⣇
⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⡦⢹⡟⠓⢶⣶⣶⣦⡀⠀⠻⡄⠀⠀⠀⠀⠀⠙⣷⠀⠻⣄
⠀⠀⠀⢠⡿⠀⠀⠀⠀⣼⠇⢸⡇⠀⣼⣿⣿⣿⡇⠀⣀⣹⣆⠀⠀⠀⠀⠀⢹⣧⠀⢹⡄
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠛⠀⢸⡇⢰⣿⣿⣿⠉⠓⠈⠉⠁⢀⠀⠀⠀⠀⢀⠀⢻⣇⠈⢧⡀
⠀⠀⠀⠁⠀⠀⠀⠀⢰⡆⠀⣾⡇⣸⣿⣿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠈⣿⡄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⡇⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⢸⡇⠀⠸⣿⠀⢻⣄⡀
⠀⠀⠀⠀⠶⢦⣀⠀⣿⠃⠀⢼⣿⣿⡏⠀⠀⠀⠀⠀⣀⣠⡾⠀⠀⠹⠀⢸⡇⠀⠀⢻⣇⠀⠈⠈⢻⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠠⣬⣉⡁⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡇⢸⡇⠀⠀⠘⣿⡄⠚⠟⠛⠁⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡇⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⣼⣿⠃⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣿⡆⠀⠉⠉⠀⠉⠉⠑⣾
⠀⠀⠀⠀⣰⣿⢇⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠹⣿⣤⣀⣤⣤⣤⡶⠾⢋
⠀⠀⠀⢰⣿⠏⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣏⠉⠉⣉⣉⠀⠀⣸
⠀⠀⢠⣿⠏⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠉⠀⢠⡿
⠀⢀⣿⡏⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣼⠇
⠀⣸⡟⢰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⠀⠀⢠⠟
⠰⡏⢠⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣷⡆⠀⢤⣤⡶⣞
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⡿⠛⠾⠷⠟⠛
⠒⠦⣤⣌⣙⠛⠛⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠉⠙⠻⠶⣤⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣟⣛⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⢉⡽⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠶⠶⣤⣤⣤⣌⣉⣉⣉⣁⣀⣤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⡟⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡇⢸⡁⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠻⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⣽⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠉⠀⠀⠻⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠤⠄⠀⢶⣽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣠⡅⢰⣶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡦⠀⠀⢰⣶⣿⡇⣺⣿⣙⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢃⣤⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢸⣟⡇⠢⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⣴⣧⠀⠀⠀⠔⠀⠀⠀⢀⡄⢸⢯⣇⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡀⠀⠀⣴⢧⣼⡄⢀⣡⣴⡞⠛⣷⠿⠂⡻⠿⠫⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⠿⠇⠀⠠⠧⣼⣿⣷⣿⣿⠟⢛⠞⢁⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡷⠀⠀⠰⣴⣇⣸⠃⡽⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠆⠠⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⣧⡆⠀⢠⣏⣠⡋⣹⠟⠡⠈⠠⠀⠀⣠⣶⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠀⠀⠀⣠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⢸⣇⡻⠹⠃⠀⢀⣀⢠⣼⣿⣿⣿⡇⠀⢸⡄
⠀⠀⠀⠀⣠⡀⠀⠀⠄⠔⠀⠀⠀⣴⠾⠿⠻⠈⠰⠶⠾⠿⠿⠛⠛⠀⠁⠀⠀⠀⠐⠤⡄⡘⢋⡟⠁⠀⠀⠙⣿⣶⣿⡿⢛⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡄⠀⣠⢠⣶⣠⣤⡄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠴⠆⠀⠱⠇⡝⠀⣠⣴⣶⣿⣿⡿⠋⠀⢰⡏⠀⠀⠀⡇
⠀⠀⣠⠀⠀⠀⠱⣾⠀⠀⠈⠉⠛⣷⡶⠄⡀⠐⠷⠀⠤⠀⠀⠀⢤⡾⠟⠋⠀⠀⠀⣠⡄⠀⠀⠀⠉⠉⠙⠛⠋⠀⠀⠀⢸⡅⠀⠀⢀⡇
⠀⠀⠁⠀⠀⠀⠀⠡⠤⠴⠶⣣⡴⣿⣵⣿⡀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⡠⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠘⢿⣶⣶⡿⠁
⠀⠀⠀⣿⠁⠐⠒⠛⠛⢿⣿⣟⡇⠛⠳⣾⡉⠛⠒⠀⠐⣿⣿⡏⠀⠀⠀⠐⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⢄⠀⠀⠉⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⢸⣧⡀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢺⣿⣿⠀⠀⠀⣿⡿⠗⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡟⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⢠⡄⠀⠠⣮⠍⠀⢊⣷⠀⠠⠀⣠⠀⠀⠀⠀⠀⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣾⣿⡾⢥⡤⠀⢸⢿⣇⣈⣠⠞⠀⣿⣯⡇⠀⠀⠀⠀⢼⠇⢈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠽⠋⢉⡄⠸⣼⠹⣿⡻⢧⢴⡟⠁⠙⢦⣠⠀⣤⠀⠀⠀⡿⣿⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢿⡆⢀⠬⠋⢰⢛⣀⣾⣿⣶⡞⠁⠀⠀⣼⡗⠀⠀⢂⠀⠈⠑⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⡆⠀⠀⠀⡿⠿⣡⣄⣭⣽⠄⢠⡎⠀⠀⠀⢀⣿⣷⣶⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠠⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣾⣄⠀⢸⡷⢿⣼⡁⠵⠾⡀⠀⠀⢀⣠⡴⠺⣿⣿⣿⣷⡰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣽⡀⠌⠁⠀⠀⢀⣴⡞⠋⣹⣶⡴⠿⣿⢁⣸⣿⡍⠳⣦⡀⠀⠀⠀⠀⠀⢻⣡⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣷⠀⠀⠀⠀⣿⣿⣿⣾⡿⢿⣧⣤⣿⣿⡟⠻⣷⡀⠠⠉⢢⠀⠀⠀⠀⣴⢗⣰⣷
⠀⠀⣠⣿⣿⡏⣿⣿⣿⣛⣿⣿⣿⣿⣿⠿⣏⣽⠉⠆⠀⢀⣼⣿⣿⡟⣻⣦⣾⣿⣿⡏⣻⣷⣿⣿⣿⣠⣶⠀⠇⠀⠀⣼⡟⠸⠋⠈⠓
⠀⢀⣿⣿⣿⣷⣿⣿⣟⡸⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⣰⣿⣿⡟⣿⣿⣿⣿⣟⣿⡇⣄⠀⠀⣿⠗
⢀⣾⣿⣿⣿⠟⠻⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣼⣿⡟⠁⢿⠀⠀⠻⣦⡀
⡜⠻⠛⠒⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⠀⢸⣿⣿⣧⣾⣿⣿⣿⡟⣻⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠄⣸⠁⣤⣦⣬⣿⣷⡀⠀⠘⠃
⣽⣶⣄⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢻⡿⣿⣿⣟⣉⣿⣿⣿⡿⣿⣶⡿⣿⣿⣿⣿⣿⠟⠀⣿⠑⠀⢙⡱⢿⠿⢧⡄⠀⣠
⠈⢿⣿⣟⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⢻⣿⣿⢿⡄⠀⢛⢻⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣶⣿⣿⣿⣹⣿⡗⢀⠇⣠⢶⣿⢳⣄⢠⡘⠀⣦⡉
⠀⠀⠙⢿⣿⣽⣤⣬⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⡄⠀⠀⠀⣻⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣼⣿⡿⣿⣉⣿⡟⠀⢻⣾⣿⣧⡈⢀⠀⢸⣿⡟
⠀⠀⠀⠀⠙⠿⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣷⠁⠋⠁⣄⡿⢻⣿⣱⣽⣿⣷⣾⣿⣿⣽⣿⠀⢀⣿⡇⠉⠀⠀⠀⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡃⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⡟⢻⣿⣶⡿⢿⣧⣠⣿⣿⣿⣭⣿⣧⣽⡿⣥⣿⠀⣸⣿⡆⠀⠀⠀⠀⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⣿⣿⣿⠋⡇⠀⠀⠈⠛⠻⢿⣿⣄⣿⣿⣿⡿⢋⣡⡜⢻⣿⣿⣟⣫⣽⢰⠋⠁⣷⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⡐⣿⡷⢶⠟⠛⠋⠉⠀⡄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⣬⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⣠⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⡇⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⢴⣦⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣿⣿⣿⣿⠂⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠠⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣿⣿⣿⣿⣆⠀⠀⠇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡿⣿⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⣿⣿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠉⠻⣿⠛⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣶⡶⠂⠀⠀⠀⠀⠀⠀⠐⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣈⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠘⠛⢿⣶⣔⠀⠠⡀⠀⠀⠀⠀⠑⠢⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣭⣽⣿⣧⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡤⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠙⠿⠿⠛⠛⠉⠉⠙⠛⠃⠠⠿⢿⡆⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⢻⡄⢠⡟
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⣿⠀⠘⢆⡀⣿⢀⡇⢠⡆⠀⠀⠀⠀⠈⢻⣄⠀⣠⠖⢦
⠀⠀⠀⠀⢠⡇⠀⠀⠐⠒⠛⠀⠀⡠⠴⠦⣤⡀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢩⡟⢸⡏⠘⣇⠀⠀⠀⣼⠋⠀⢻⡆⣧⠀⠙
⠀⠀⠀⠀⢠⠻⣶⣶⣿⡿⠀⠀⠈⠠⠶⣿⠂⠀⠀⠀⣴⣶⢶⡀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣀⠼⠃⠈⠁
⠀⠀⠀⠀⣿⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠖⠲⣤⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢻⡆⢿⣆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⢠⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⢸⡇⠀⠀⢤⣤⣧⣤⡤⠄⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠘⣷⡈⠙⢣⠠⡄⠀⠀⠀⠀⢠⡶⠶⠦⠤⡴⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣠⠟⠀⢠⠄⡄⢸⡇⠾⢴⠆⢀⣴⠲⡄⢸⡇
⠀⠀⠀⠀⠀⠈⠛⠷⢿⠀⠘⢦⡀⠀⠀⠀⠙⠷⢶⠾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⣅⠀⠀⣿⠐⢁⢼⡇⠁⢸⡀⣿⡀⠰⠁⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠳⢦⣈⠉⠁⠀⠓⠀⠀⠁⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⣀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⢖⣁⣀⣠⡤⠤⠤⣀⡂⠒⠤⠿⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠉⠁⣠⢴⣿⠁⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡠⢾⣧⡟⠁⠀⠐⠛⠲⠦⠀⠀⢀⣀⡀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠂⠙⠁⠀⠀⠀⠉⠛⠀⠀⣴⣯⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠠⠄⠀⣀⡀⣈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠳⡀⠀⠀⠀⠠⠤⠈⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠎⠑⣄⠀⠙⠢⣤⣄⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠋⠀⠠⠀⠀⠑⠄⡀⠀⠈⠿⠏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⡠⠖⠚⠋⠉⠁⠉⠘⠒⢆⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠉⠁⠀⠀⠀⠁⡀⠀⠀⢈⣶⣾⣦⡀⠀⠉⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠔⢈⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⢀
⠀⢀⣠⡤⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣄⠔⠉⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⠔⢡⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡊⡄
⢀⠃⢀⠀⠐⢄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢂⠀⠀⣿⣿⣿⡆⠀⠀⢀⠄⠀⠀⠀⠃⠀⡄⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣏⢰⠰⡀
⡘⠀⠀⠱⠀⠀⠱⣄⠀⠀⠀⠀⢦⣀⡀⠀⠀⠀⠀⠢⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⢀⡴⣿⠁⠘⡆
⡇⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠱⣄⠈⠁⠀⠀⠀⠀⠀⠑⣼⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⣰⢃⢠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⢰⠃⠀⠀⠏⠐⣉⡠⢴
⡇⠈⠲⠤⠄⠀⠀⠀⠀⢲⡀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⡴⠁⠈⠙⠢⡀⠀⠀⢠⡞⠉⠀⠀⠈⠳⣄⣠⠆⠀⠀⠀⢀⠈⡁⠄⠚
⢧⠀⠀⠀⢤⣀⠒⠄⡁⠢⣷⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠆⣰⣋⣀⡀⠀⠀⠀⠀⢈⡷⢶⣚⣛⣣⣤⡄⡟
⢸⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⡀⠀⡶⠐⠐⠒⠀⠘⣋⣉⣭⣭⣭⣭⠄⠀⢠⠞⢁⣀⣮⣽⠫⢇⡜
⠀⡆⠀⠀⠀⠀⠀⠲⠤⣤⡼⡇⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⠇⡀⠀⡇⢀⡴⠒⣶⡖⠚⠛⠛⠛⠻⢿⣶⡁⠀⠠⣧⣴⡏⠀⣽⠋⣀⡈⣵⠇
⠀⢿⡄⠀⠀⠀⣀⡤⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠰⠊⠠⠼⠗⠊⣸⢀⠾⣇⠀⠀⠀⠀⠀⠀⠉⠛⠷⠂⠛⣛⡻⣵⣞⠛⠉⠉⠉⡳⣤⡀
⠀⠈⣷⠀⢠⡞⡫⠒⠈⠉⠙⣶⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠠⠂⣤⠋⠀⠹⣄⠀⠀⠀⠀⠀⣀⠤⢲⡿⠛⢁⠀⠈⡹⢿⣷⣄⠀⠀⠀⠝⠶⣄
⠀⠀⢹⡀⠏⠀⢀⠌⠀⠒⣫⠥⠒⠓⠦⠤⠤⠤⠤⠤⠤⡤⡤⠚⠁⠀⠀⠀⠀⠀⠄⣼⠀⠀⢀⠘⠳⣷⣦⣞⣉⣤⠚⠩⠚⠋⠁⢀⠔⣡⠞⠉⠙⢿⣦⣄⠀⠀⠂
⠀⠀⠘⣿⠀⢰⠞⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞⠁⠀⠀⠀⠀⠈⠻⢿⣄⡀
⠀⠀⠀⢻⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⣀⣀⡴⠒⠒⠒⠚⠻⣎⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢊⡵⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣
⠀⠀⠀⠈⢧⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠿⠋⠀⠀⠀⠀⠀⠀⠈⠓⢾⣁⡄⠀⠀⠐⠂⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠙⠂⠈⠓⠻⠶⢤⣄⣀⡀⠀⠀⡤⠄⡄⠀⢧⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣷⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⣠⣶⣴⣶⠖⠃⠀⠉⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣠⡾⢽⠋⠘⠁⠀⠀⠀⠀⠙⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠔⠁⠀⠀⣰
⠀⠀⠀⠀⠀⠀⢸⣯⠀⡿⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣌⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡄⢁⣤⠀⠀⣰⡿⠋⠀⢠⣄⠀⠀⠄⠀⠙⣀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢨⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⣿⡄⠀⠛⠁⠰⢄⡀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣮⣿⠀⠀⠀⠐⠁⠀⠀⡘⠻⠆⠀⠀⠒⠲⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠶⣿⣆⠀⠀⠀⠀⣠⡖⠀⠀⢀⠐⠂⠙⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡈⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠿⠹⣧⡀⠠⠘⠋⠤⠒⣈⣠⣴⣶⠏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠙⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠟⢿⣄⠑⠄⠀⠀⠉⢙⣛⡉⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠁⠀⠀⣀⠞⠀⣀⣀⣀⠀⠀⢀⡀⠚⢉⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠛⠛⢿⡒⠒⢲⠞⠁⠀⢀⢋⠉⠉⠉⠁⠉⢹⠀⣿⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠐⠠⡀⠹⣄⠙⢦⡆⠸⠃⠀⠀⢸⢀⡴⠂⠀⢠⠂⣸⡏⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣠⠶⠚⠛⠓⠘⢦⠹⡆⢸⡇⠀⠀⠰⠃⣼⡟⠀⠀⠀⢁⣀⣿⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠸⠁⠀⠀⠀⢄⠀⠈⠁⠐⢀⣿⢤⠀⠀⢀⠏⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⣄⠀⠀⠀⠀⠀⠑⢄⠈⠆⢻⡟⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠙⣦⡀⠀⠀⠀⠀⠀⠁⠘⠘⢁⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠐⠒⠒⠋⠉⢷⣄⠀⠀⠙⠒⠲⠚⠛⢢⠀⡼⠁
⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⢀⡿⣷⣄⠀⠀⠀⠀⠀⠈⢻⡁
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣴⢯⣴⠛⠻⣷⣄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⡸⠁⠀⠁⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣬⠈⠻⣧⡀⠀⠀⠀⠘⢆
⠀⠀⠀⢸⣿⣄⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠘⢿⣤⠤⠀⠒⠚
⠀⠀⠀⢸⡿⠀⠉⠑⠒⢰⡎⠤⣤⣤⡴⠖⠒⠉⠁⢹⠀⠀⠀⠈⢷⣤⣴⡆⠀⠀⢦
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⡴⢣⡿⣄⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⢳⡀
⠀⠀⢀⣿⠃⠀⠀⠀⠀⣠⠟⢠⣿⣄⠈⢷⡀⠀⠀⠀⢶⠀⠀⠀⠀⠀⢠⠙⢀⠀⠀⠀⠑⢄
⠀⠀⢸⣿⠀⠀⠀⢀⡞⠁⢠⣿⡉⢿⣆⠀⠙⢦⡀⠀⠸⡆⠀⠀⠀⠀⠈⠛⣁⡄⢁⡐⠛⠄⠑⢤⡀
⠀⠀⣼⡏⠀⠀⣰⠋⠀⣰⠃⠈⢿⣄⠻⣧⡀⠀⠙⢦⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠙⠲⠞⠁
⠀⠀⣿⠇⠀⠜⠁⠀⣼⠃⠀⠀⠈⢻⣦⡙⢷⣄⠀⠀⠉⠀⢡
⠀⢰⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠻⣷⡌⠛⠇⠀⠀⠀⠘⡄
⠀⢸⡇⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⢹⡀
⠀⡾⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⢷⡀
⢠⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠶⢶⣶⣶⣾⣧⣀
⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠉⠻⠿⣿⡶⠄
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠀⣛⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠇⠀⣆⡀
⠀⠀⠀⠐⡇⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠡⡀⣏⠰⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢸⠀⠀⣿⠧
⠀⠀⠀⠀⡇⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣟⡼⢿⣿⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠯
⠀⠀⠀⢠⡁⠉⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⠟⡃⢀⣀⢀⠀⠄⠂⠀⠀⠀⠀⠀⢸⣀⢀⣷⡶
⠀⠀⠀⢺⡁⣷⣶⠐⢸⣧⣄⠀⠀⢀⣀⣀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠐⢒⠸⣷⣶⣾⣿⣦⠀⣀⠀⠀⠈⠀⠀⠉⠉⠉⠁
⠀⠀⠀⢸⠀⣿⣿⣿⢿⡟⠃⠀⠀⠐⢋⣛⠛⠻⠝⠃⠀⠀⠀⠀⠀⠀⡐⢀⣴⣷⡤⠈⠙⠿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⠀⣿⣏⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠒⠀⠀⢦⢉⠉⠉⠳⢾⣀⣀⠈⢿⣿⣿⣿⡄⠀⠁⠀⠀⠠⠼⠂⠐
⠀⠀⠀⠀⠀⢹⡿⡾⣿⣽⣱⠆⣞⣻⠋⠀⠉⠀⠘⡎⠀⡌⡇⠈⠀⠀⠀⠀⠣⡘⠛⣛⣻⣿⣴⠄⠙⢿⣿⡇⠀⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢘⢧⡥⠋⠀⠤⠖⠦⠀⢻⠀⠀⠀⠀⠈⠀⠇⠁⡀⠀⠄⠀⠀⡀⢱⡈⠁⠚⠋⠙⣤⣄⠀⠙⣿⠀⠐⠀⢀⠀⡀⠁⢠⠆
⠀⠀⠀⠀⠀⠈⣨⣷⠀⠀⠀⠂⠀⠀⠀⢳⡄⠀⠀⠀⠰⢶⠖⠀⠀⠀⠀⣴⡀⠸⢧⢀⣤⠶⠶⢿⣷⣴⠆⠀⠀⠀⠐⢀⠀⠀⠠⠋⠐⠤⠄
⠀⠀⠀⠀⠀⢁⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⣛⣲⠀⠀⠐⠀⠀⠚⠃⠀⠀⠈⠁⠀⠀⠸⠟⠉⠒⢔⣳⣼⣧⡍⠀⣀⣰⣤⡀
⠀⠀⠐⠿⣿⣿⣯⣿⣿⣆⢀⠄⢈⣉⡿⠷⠗⠒⠉⠉⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠀⡤⠈⣧⣼⢿⣿⠉⠃
⠀⠀⠀⠈⠉⠛⠛⠉⠉⠙⠛⠋⢩⣴⡄⠀⠀⠀⠀⢤⠘⡦⠀⠀⠈⠉⣉⠉⠉⠉⠉⠉⠀⠀⡀⠸⠀⢸⣿⣿⠘⣿⣧⣰⡆⠀⢀⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣰⣶⠀⠉⠂⢸⣶⡆⠀⠀⠀⠈⠉⠉⢛⣛⣛⡟⠀⠀⠀⢸⣿⣿⠀⢻⣿⣿⣿⢹⡿⢟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠀⢸⣹⣁⡀⠀⠀⠀⠘⠛⠿⠿⣻⠁⠀⠀⠀⢸⣿⣿⠀⣸⣿⣿⣿⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⣤⣴⣶⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠿⠿⢻⢻⣿⠟⠀⠀⠀⠀⠀⣿⡿⠛⠉⠛⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠘⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⠿⢯⣥⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣠⡍⢁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠉⣑⣶⣶⣾⠒⠓⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠉⣿⣿⣟⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠟⠛⠛⠿⢿⡿⠀⠀⠀⠐⣿⣿⣿⠀⡀⢀⣸⡿⠟⠛⠉⠙⠿⢿⣿⣿⣿⣿⠁⠀⢀
⠀⠀⠀⣠⣤⠀⠘⣿⣿⡟⢯⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣖⠛⢿⣿⡟⠀⠀⡀
⠀⠀⢀⠈⠁⠀⢀⢹⣿⠳⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⢀⣼⣷⣄⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⡟⠁⣸⡿⠁⠀⠘⠋⠆
⠀⠀⢀⣤⣄⡠⣿⠇⠻⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⡿⠱⣷⡀
⠀⠀⠙⠛⠛⠁⠀⠀⠀⠙⢦⣿⣿⡿⣿⢻⣿⣿⣿⣿⣿⣤⣼⣿⣧⣴⣿⣿⣟⣋⣻⣷⣿⣾⣿⣿⠟⠀⠀⠙⠑⠾⠿⠖
⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣠⣴⣀
⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡀⣿⣿⣷⢸⣿⣿⠿⠿⣿⣿⣿⣿⡿⠛⡿⢻⣿⠉⢋
⠀⠀⠀⠀⠀⠀⠐⠿⠿⠶⠀⠻⠿⢿⣿⡿⠀⠀⠀⠹⣿⣯⣿⣛⣁⣼⣿⠋⠒⠚⠉⢿⠿⠿⢳⡘⠗⠀⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⡶⠂⠀⠐⠀⠀⠤⠶⠒⢿⣟⣷⠶⢶⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠈⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠅⠀⣿⠁⣦⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⢈⡆⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⡀⠀⠀⠀⠁⢀⠲⠀⣾⣿⡞⢃⠔⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣩⣭⡆⠀⠀⠰⠶⣭⣍⣛⠀⠀⠀⠀⣠⣼⡀⡇⢸⣿⡞⠋⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠁⠀⠀⠀⠀⡟⣿⣧⠸⠘⡻⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠶⠶⠦⠀⠠⠴⠶⠤⠀⠀⠀⠀⠀⠀⠄⠀⣽⣷⣤⣵⢷⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢰⣿⣷⣷⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣀
⠀⠀⠀⣸⣯⠀⠀⠀⠀⢠⣤⠀⣻⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆
⠀⠀⠔⣿⣿⡳⠀⠀⠀⠈⠙⢺⣿⡿⠆⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠐⣷⣾⣿⡄
⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠈⣿⡏⣤⣤⠀⠀⠀⣀⣀⢼⣿⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠙⢆⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠘⠉⢰⡶⠀⠀⠀⠀⠀⠋⠃⠿⠿⠀⠀⠀⢿⡿⠸⠇⠀⠀⣨⣶⠆⠀⠀⣶⣾⡖⠁⢿⠏⠀⠀⢡⣀⠀⠀⠀⠉⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣿⡷⡀⢀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⣾⣯
⠀⢀⣠⣿⡔⠉⠈⠁⠀⠀⢀⣿⡉⠓⡄⠀⠀⠀⠀⠀⠀⣸⣟⠙⠦⠀⠴⠷⠆⠀⠀⠀⣾⡏⠙⣶⠀⠀⠀⠀⠀⣿⡋⢳
⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠸⣿⠇⠀⡟⠚⠒⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⡘⠿⢿⠀⠀⠀⢸⣿⡇⠀⣵⣤⣶⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠉⠙⠂⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠇⠀⠸⠀⠀⠀⠀⠛⠀⠀⠸⠟⠉⠫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⢚⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⢺⣗⠀⠀⠀⠀⠀⠀⠀⡄⠰⡶⠀⠀⠀⠀⠀⠀⠀⣺⣗⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣾⠆
⠀⠀⠀⣾⡟⢴⣦⣰⣦⠀⠀⠀⠀⢰⢹⣿⢱⣄⠀⠀⠀⠀⠰⡷⣾⣿⣄⠀⠀⠀⠀⢀⣾⣿⣿⡗⣭⡀⠀⠀⠀⠀⠾⠿⢿⣼⣿⡇
⠀⠀⢠⣿⣷⠀⣿⣿⡇⠀⠀⠀⠘⠙⠻⡉⣹⠛⠋⠀⠀⠀⠀⠀⣿⣿⢈⡀⠀⠀⠀⠀⢼⣿⣿⡇⢻⡇⠀⠀⠋⠉⠉⠙⠀⢸⣿⡇
⠀⠀⠈⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠣⡀⠀⠀⠀⠀⠀⠀⣿⡟⠘⠋⠀⠀⠀⠀⠀⢻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⠚⣋⣩⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣮⣭⣯⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠓⠮⠛⠛⣋⣙⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢶⡆⠀⢀⣩⡀⠁⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⠉⢸⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠰⣦⢥⡀⠀⠠⣦⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠈⠆⠉⠉⣀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⢿⣏⠧⣀⠰⠍⠉⠉⠿⣷⡀⠀⠀⣀⠴⢒⠟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠏⡇⠉⠲⣼⢄⠀⠙⠛⢶⣤⡴⠚⠉⠛⣻⢛⡑⠦⢧⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⣇⡉⠑⠻⠈⠙⡢⢄⣀⠀⠙⠦⣤⠸⠃⢀⢀⡴⢿⠝⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⢘⣋⢀⢸⡄⢄⢰⠀⠈⠉⠳⡊⠀⠑⣞⠀⣠⠟⠂⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⣘⠃⠀⠀⠇⠈⡼⠐⠒⠒⠂⡟⢦⡘⠋⠳⣌⠟⣠⠞⠁⠸⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡠⡴⠛⠁⢀⢀⡹⠔⠋⠧⢀⡏⡑⢲⡖⠖⠾⢁⣀⣷⡀⠀⣸⢦⣁⡞⠡⡀
⠀⠀⠀⠀⠰⡀⢀⠘⣔⠡⢷⠼⢴⠚⠛⣏⠀⢀⡤⠖⣍⢀⡃⢸⣀⣀⠞⠁⠂⣀⢳⠀⠁⠠⠋⣳⣾⢄⡀
⠀⢤⠀⠲⡀⠱⡄⢻⡹⣄⠈⠇⠘⠇⣀⣩⣤⠬⢗⠒⠋⠝⣒⣀⠂⣀⠉⠩⠉⠂⣀⡀⠒⠠⠤⠈⣀⢊⠁
⠀⣀⠑⢤⠙⣂⣉⢥⡡⠤⢒⡚⢋⠍⢑⣂⣉⣠⣶⡶⢾⠿⣿⠿⢛⣭⣞⣡⣖⣴⡶⠠⣤⣐⣦⠶⠗⡏⡄
⠀⠀⠉⣀⣯⣖⣋⣒⡾⠭⠥⠶⢶⣛⣿⠿⠟⢛⣩⡈⠉⣩⡙⠛⠛⢉⣐⣬⡭⠖⠚⠉⣉⣡⣤⠶⣸⡀
⠀⠉⠉⠁⠘⡄⢈⣉⣙⣻⣶⣶⣶⣶⣶⣶⣶⣭⣭⣭⢽⣭⠴⠖⣛⣉⢁⢠⠦⣶⡾⠿⢛⣋⣥⣴⠿⠃
⠀⠀⠀⠀⠀⢵⣌⣙⣛⠛⠵⠖⠚⠻⢭⣡⡴⢖⢋⣡⣤⣖⣫⣭⣶⡾⢟⣋⣭⣵⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⠶⠶⢤⣥⣤⣙⣉⡒⠛⣛⣻⣭⣷⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠟⠁⣴⡛⡿⣰⣿⣷⣷⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣧⢠⡛⢛⣛⣯⣽⣶⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⢿⣷⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⡇⢽⣿⣿⡛⠁⢸⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢰⣄⣿⣿⣿⣿⣿⣿⣿⣉⣿⡟⡋⠄⢀⣁⣠⡄⠀⠎⠑⣻⣏⣏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡙⣤⣷⠞⢛⠛⠿⢿⣷⣶⣿⡷⢶⣦⣉⣻⣆
⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⡟⣰⣾⢯⣠⣿⣿⣜⢷⣿⣿⠟⣛⡛⢿⣿⣷⣦⣜⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⣴⣿⣯⡼⢋⣺⡿⠃⠀⠉⠃⣾⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣽⡟⠑⡪⠟⠋⠁⠀⠀⠀⠀⠀⢿⣥⣢⣾⡿⠑⣿⣿⡿⠽⢻⣿⡟⠁⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⣋⠀⠀⠀⠀⡀⠤⢀⡙⠦⠀⠉⠛⠋⣤⣾⣿⠏⢁⡠⠞⠋⢀⡄⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣧⣌⠀⠀⠀⠈⠩⢿⡶⠜⠀⠀⠀⠀⠀⠀⣈⣰⠀⠀⣀⣀⡤⠞⢄⡀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⡀⢄⡼⡗⠋⠉⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣼⢿⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢻⠟⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⡆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠎⠰⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⡄⠀⠀⢰⡚⠳⣤⡄⢀⡠⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⣀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠂⠀⡀⠈⢦⠀⠀⠛⠓⣡⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁⠀⠈⠙⠲⠤
⠀⠀⠀⠀⠀⠀⠀⣠⠮⠶⠒⠒⠲⠤⣀⠀⢳⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠂⠤⣀⣀⣀⣀⣀⣀⣠⠤⠴⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠋⠁
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣶⣄⠀⠈⠉⠙⠁⠒⠒⠀⠤⣤⣠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠁
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠻⣿⣦⠀⠐⠠⢄⠀⠀⢸⣦⠼⣛⢧⡀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⠲⣞⣸⣧⢶⣾⣇⠀⠀⠙⢧⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠀⠀⠀⠀⢠⢼⡡⠚⠉⠀⠀⠀⠀⠀⢙⣆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣁⣀⠀⠀⠠⡀⠀⣀⢧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠞⣇⠀⠀⠈⠀⠀⠀⢱⡤⣷⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣣⠀⠴⡇
⠀⠀⠀⠀⠀⠰⠋⠀⠹⠀⠀⠀⠀⠀⠴⣧⠟⠁⠀⠠⠷⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠒⠻⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⢘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠀⣈⣳⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠟⠛⠛⠛⠋⠉⠉⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣇⣥⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⣿⣿⣻⣄⠀⠀⠀⠀⠀⢀⠄⠀⣠⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣠⠴⢼⡡⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⡄⣧⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⣿⣿⣿⣿⣿⣼⡄⠀⠀⡼⠁⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠧⠷⠒⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡻⣿⣿⣿⣿⡟⣷⠀⠠⡇⠀⠀⣸⠀⢠⣇⣀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠏⣠⣍⠙⠋⠀⠀⠳⠤⠴⠉⠛⠉⠉⠉⠙⡄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⢈⡄⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠇⠀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⢣⠀⠘⣧⠀⠀⠀⠀⠘⡟⠛⠋⠡⣖⣚⣧⣶⣒⣢⠤⠔⠒⠢⠤⢤⠀⠰⡗⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⡆⠀⠀⠀⠀⠘⡄⠀⠀⡇⢀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠳⠖⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣿⡛⢦⣄⣀⣀⣀⣸⠀⢐⠁⠀⡄⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⡷⡄⠀⢀⡽⠁⠀⠀⢸⠀⠴⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡴⠋⠀⠀⠀⠀⢾⠭⠥⣶⣒⣒⣺⣇⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⣎⡀⠀⠀⠀⠀⠀⣧⡄⠘⠂⠠⠄⠀⡸⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠉⠛⡁⡳⣄⠀⠀⠀⢀⣧⢤⣀⣀⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢺⣦⡵⣄⡴⠋⠀⠀⠘⢾⠧⢌⣁⣇⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢰⡀⢀⣤⡟⠛⠉⣠⣆⡀⡈⢢⣈⡗⠒⠒⠪⠭⠭⠭⢍⣉⣉⣉⣉⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠳⠋⠙⠶⠖⢺⡁⠀⣇⡽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⣹⠖⠒⠒⠒⢺⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠈⠀⠀⠀⢰⠀⡌⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⠶⣴⣏⣉⠀⠀⠀⠀⠀⠙⠻⡞⠷⠀⠛⢲⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⣀⠀⡞⠙⠾⠾⠛⠛⠓⢒⣢⣤⣤⣼⣧⢤⣔⣚⣿⡭⠛⡷⠈⠉⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⠭⠭⠭⠭⠭⠭⠽⠛⠛⠋⠉⠉⠉⠉⠉⠙⠒⠻⠬⠭⠭⠿⠭⠭⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⣀⠔⠀⠀⢀⡴⢀⢳⠹⣿⡇⢀⣀⡀
⠀⠀⠀⢀⣴⠁⠠⠀⠀⣿⡂⠀⢸⣿⡿⠃⣼⡇⣿⣦⡀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢻⡿⠟⠉⠤⠚⠉⠸⠟⠛⠛⣶⣄⢸⡆
⠀⠀⠀⠰⠢⠀⠀⠀⡠⠐⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠲⠿⠻⣿⣷⡀
⠀⠀⠀⠀⢣⣄⢀⣠⠀⠀⠀⠀⠀⢠⠟⠀⢀⣠⡀⠀⠀⠀⠀⠹⣿⣷⠄
⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢠⣼⣯⡑⠠⠤⢀⣈⠳⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠖⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡙⠂⢐⠳⢿⡇⢨⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣴⠃⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⠿⣯⣙⣿⠻⠄⢠⢹⣿⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡴⢋⣭⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣀⣸⣃⣤⣤⣤⠤⢂⢠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣄⡀⠙⠿⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⢰⠛⠻⠮⠛⠿⢿⠿⠿⠿⠿⠷⠾⣿⣿⠿⠿⠿⠿⠟⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⢻⡿⣶⣄⣀⣀⣴⠂⠀⠀⠀⠀⠀⠀⣼⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡆⡾⢀⣿⡿⠟⠉⠣⢄⡀⠀⠀⠀⠀⠀⢸⠁⠈⠙⠛⠋⠉⠛⠟⠋⠉⠉⠛⠛⠿⠋⠉⠙⠋⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⡤⢴⣄⠀⠀⠀⠀⢰⠒⠂⠁⠀⠀⢿⠷⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠾⡟⠀⠀⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⡏⠀⠀⠹⣗⢦⢀⢸⡈⠀⠐⠶⠲⠀⠺⠂⠀⠀⠄⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠛⣀⣴⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠄⠀⠀⠙⠆⠀⣸⠃⠀⠀⠀⠀⠀⠈⠍⠑⠒⢆⣤⣦⣤⣤⣤⡤⠤⢤⣤⣤⣴⣦⠰⠦⡀⠻⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠐⢈⡄⡞⠀⠀⠀⠀⠀⠀⣁⠔⠚⠓⠋⠀⠁⠀⠉⠉⠀⠀⠀⠉⠂⠚⢿⠀⠀⠀⠀⢠⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⢀⠀⢹⠀⠀⠀⢲⡖⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⡀⠸⣇⠀⠀⠀⡿⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣿⢿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⡂⠀⡉⢀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⣞⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠅⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡟⣇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡼⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠶⠚⠛⠃⠀⠀⠀⠀⠀⠇⣼⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢈⡀⢀⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠐⠛⠀⠀⠀⠀⠀⠀⠐⠛⠉⠇

⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⡶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⢷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠻⢿⣿⣿⠤⢺⡇⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠋⠉⠙⢟⣻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠄⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠉⠽⠂⠀⠐⠇⠃⠀⠀⠀⠀⠀⠀⣰⣿⠖⠀⠀⠀⠓⠀⣨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣿⡀⢀⠀⢀⠀⠈⣷⣶⣍⢻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠡⠀⠀⠐⠄⠀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠡⠘⠒⠊⠀⠀⠈⣿⣿⣦⣿⠏
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⢣⠐⢄⠠⢀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠶⠋⠀⠀⠙⠛⠛⠉
⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠑⢄⠛⢡⠑⢦⠀⠀⠀⠀⠀⠀⢀⠒⠙⢨⢫⠃⠀⠈⠀⠀⠀⠑⢄
⠀⠀⢠⣏⣰⣶⣶⣤⡀⢀⣈⢷⣤⠀⠀⢂⡔⢁⢑⠀⢃⠈⠁⠀⠀⠀⠀⠀⠃⡠⠀⠀⠈⠳⠋⠀⠄⠀⠀⣀⣀⣀⣄
⠀⠀⢸⣷⣽⣷⣝⢿⣿⡉⠁⠀⠁⠀⠀⠀⠀⠸⠫⠀⠀⣧⠃⠀⠀⢒⠚⠁⣰⠁⠀⠠⠃⡀⠀⠀⠀⠀⠸⣧⡀⠀⠈⠱⡄
⠀⠀⢸⣿⣿⣿⣿⣷⣬⣙⢠⢰⡀⢷⡄⠀⠀⣧⠀⠀⠀⠀⢰⢰⠀⠀⠀⢼⠇⠀⠀⡤⠊⠀⠀⠀⠀⠠⠀⠛⢮⡀⠀⠀⡇
⠀⠀⢸⣿⣿⣿⡿⢿⠻⢿⣦⣿⣿⡏⠹⠆⡀⢸⡆⠀⠀⠀⣶⠄⡱⠀⣀⣠⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⢸⣿⣿⣟⠿⣳⢮⣔⣧⢾⡷⠤⣄⣴⣿⣌⠁⢀⠂⢠⣿⡰⣥⡚⠁⠘⡀⠀⡆⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠄⠀⠀⠀⡃
⠀⠀⢸⣿⣿⣿⣿⣮⣕⣭⡻⠏⣇⣠⠀⠈⠋⠈⢦⡄⠀⣾⡿⣱⡏⠐⡀⠀⠐⠄⠰⠀⠀⠀⠀⠒⠛⠿⣄⠀⠀⠈⠢⡀⠀⡅
⠀⠀⢸⣿⣿⣿⣿⣿⡛⠯⣗⡠⢿⣧⣠⠀⠀⠀⢸⣇⣼⣿⣟⣩⣷⡦⠶⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣘⣦⠀⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣯⣥⣤⣿⠰⢿⣇⣴⠀⠀⠀⣻⣿⣿⣿⣿⣿⡟⠀⠈⠂⢀⣄⣀⡀⠀⢀⡀⠀⠀⢰⣿⡑⠄⢀⠀⡠
⠀⠀⠈⠛⠉⢩⣵⣿⣿⣿⣿⣿⡿⠂⠿⢿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⢀⣴⣿⣿⣶⡏⠀⠀⡇⠀⠀⠈⠈⡇⠈⠀⠈
⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣵⣾⠃⣨⠂⢤⣸⣿⣿⣿⣿⡟⢿⣿⣷⣾⠿⠿⢿⣦⠀⠀⠀⢡
⠀⠀⠀⠀⠿⠿⠛⠛⠋⣹⣿⣿⣿⣿⣿⣿⢂⣬⣽⡾⠻⢿⣿⡇⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣿⣿⡏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠿⠋⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⢿⣦⡀⣷⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣿⣿⣿⡿⣿⢧⡀
⠀⠀⠀⠀⠀⠿⣷⣿⣿⣿⣿⣿⣷⠙⢿⡷⠁⡀
⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⢿⣯⣿⣦⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣿⣷⣿⡽⣟⢻⣷⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⢹⣿⡷⠊⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠤⣦
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠑⠂⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠛⢓⢂⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⢙⡛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡾⣿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠰⣷⢿⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⣾⢿⣾⡿⠁⠻⡄⠀⠀⠀⠀⠀⠀⠴⠞⠛⠋⠀⡗⠒⠖⠞⠿⠿⠻⠖
⠀⠀⠀⠀⢀⣻⣿⣻⣿⠀⠀⠀⠀⠀⣤⠂
⠀⠀⠀⠀⣼⣯⣿⠿⣷⡄⠀⠘⠀⠤⠚⠃
⠀⠀⠀⠀⢿⣻⡿⣷⣿⢃
⠀⠀⠀⠀⠈⠻⠷⠿⠿⠿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡰⠚⠉⠉⠙⠲⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⠙⠓⠖⠋⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠊⠀⠑⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢀⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠴⠛⠛⠀⠀⠀⢀⣠⡥⠴⠮⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠬⠿⠒⠛⢳⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡤⠖⠒⢦⣠⡾⠋⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣀⣀⣀⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡭⠭⣤⡀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡧⣶⠀⠀⠀⠀⣸⣿⣿⠆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣩⣛⢳⡆⠀⠀⢹⡞⠁⠀⠀⠀⠀⢀⡤⠖⠒⠀⠀⠀⠀⠀⠘⠛⠛⠶⣄⣀⡴⣇⠈⠁⠀⠀⠀⢀⡴⢃⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠉⠀⠀⣠⠿⣄⠀⠀⠀⠀⠀⣿⠀⠐⠀⠀⠀⠀⠀⠀⠠⠤⡤⣄⠈⠻⠀⠈⠳⠶⠤⠶⠚⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡒⠋⠉⠛⠛⠓⠶⡄⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⢀⠶⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⡀
⠀⠀⠀⠀⠀⢸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠙⡄⠀⠙⢦⣀
⠀⠀⢀⡴⠒⣿⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⠠⠴⡖⠋⠀⠀⠀⠈⢦⡴⠋⢉⡶⠂
⠀⠀⠘⠷⢦⣬⣙⣛⡒⠒⠒⠛⠛⠛⢛⣉⣉⣥⡤⠾⡄⣰⣧⠤⠀⠀⠀⠀⠀⠀⠀⡜⣡
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠊⠉⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠣⠘⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠚⠉⠓⠀⢀⡀⠀⠀⠀⠀⠀⡏⠀⠀⠘⢾⣠⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⠬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠉⠉⠐⣟⠓⠒⠲⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠎⠛⠛⠉⠁⠀⠀⣿⢵⣿⣿⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠄⠀⢰⣿⠀⢷⣿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⣿⣷⠞⠗⠀⢸⣿⢟⣾⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⡿⠉⠈⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢶⠦⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⡄⠃⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠈⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠄⠀⠀⠀⢀⣠⣘⣛⡳⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣄⠀⠀⠀⠀⣨⣿⠛⣡⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣶⣾⣿⣿⣿⣿⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⢹⠏⠉⠁⠀⢾⣷⢿⡄⠀⠀⠀⠀⠀⢀⢴⣶⣆⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠤⠄⣀⣿⡧⠀⠛⢹⣿⣿⡮⣿⣷⢶⣿⣿⣿⢻⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⠀⠀⠋⡏⠀⣁⠘⠿⢿⣅⣿⣟⣸⣿⣿⣿⣜⡿⠿⠿⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠉⠀⠐⠲⡏⠉⠀⠀⠀⠀⠉⢻⠛⠛⢿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣒⠂⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⣸⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣠⡄
⠀⠀⠀⠀⠀⠀⠰⡀⢸⠀⠀⢣⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⣠⣀⣀⠀⣤⠀⠀⠀⠋⢘⡀
⠀⠀⠀⠀⠀⠀⢀⡡⠃⠠⣀⣰⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠢⡀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⡎⢠⠲⡄⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠇⢀⠀⢀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⣄⠀⢀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⢇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣧⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣁⡈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡲⠂⠀⢀⠞⠌⠁⠜⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢚⠑⠊⠑⠄⡷⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣠⢊⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⡖⠋⠢⣀⡸⠀⢀⣴⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠊⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⣃⡀⠤⢐⡲⡪⡟⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢹⠚⡟⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠡⠤⠒⠈⠀⠀⠀⠈⠑⠬⡑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣅⠀⡀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣎⣿⠶⢂⠀⠀⠀⠀⠀⠈⠀⠰⠦⠤⠀⠀⠀⠀⠄⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⡵⠛⣻⠀⠀⠀⠧⠒⠀⠀⠀⠀⡀⡑⠀⠀⠀⠀⠀⠲⡄⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣡⣶⡿⠿⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣤⣤⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠹⢅⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢋⡉⠉⣥⠄⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡼⠿⠋⠀⠐⠲⡖⠂⠐⠈⠉⠉⠀⠠⠀⢸⠂⠉⢄⠀⠀⠀⠀⠀⢀⡼⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⡶⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢉⠝⠢⡀⠀⠀⠘⢌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⡀⠠⠤⠴⠶⢦⠀⠀⠀⠀⠀⠀⠀⢠⢹⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢘⣼⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⠉⠑⠛⠲⠫⠥⢐⣾⣿⣶⣦⣬⣿⡶⠒⠊⠉⠀⢤⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠰⠶⠦⢄⡀
⠀⠀⠀⠀⠀⢀⠤⠒⠋⠉⠉⠉⠉⠙⠻⠿⣶⣄⡈⠐⠄⣀⡀
⠀⠀⠀⠀⠠⠁⠀⣀⣠⣤⣤⣦⠀⠀⣠⣴⣶⠯⣽⣶⣄⠀⣙⠳⣮⡐⣄
⠀⠀⣠⣾⣶⡷⠋⠉⠀⠠⠛⠋⣀⡄⠀⠁⠀⠀⢀⣈⣻⣼⣿⣿⣿⣿⣮⣧⡀
⠀⠜⡹⠃⣿⡇⠀⠀⠀⠀⠀⠠⠏⠀⠀⢠⣦⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠠⣰⠃⠀⢻⡗⠀⠀⢀⣴⡆⠀⢧⡀⣠⣤⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⡇⠀⣀⠀⣲⣶⣾⣯⡹⢾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀
⠘⢻⣷⣰⣿⣿⣿⣿⣿⣿⡿⠿⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠙⡿⢿⣷⣾⣷⠿⣿⣿⠀⣀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⢸⠀⠙⠟⠿⢿⡿⠟⣿⢿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⣀⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠺⣦⣿⡧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣈⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠚⠿⠿⠷⠆⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⣠⣤⣴⣶⠶⠾⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠟⠋⠛⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠈⠉⠉⠙⠛⠻⠟⣿⠟⠉⠀⠀⠁⠀⠀⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠹⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣠⣔⣋⣀⡀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⢹⠀⣿⢰⡇⢠⡇⢰⠏⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠈⠈⠀⠀⠈⠃⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⠀⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣘⡀⠀⠀⠀⠠⣶⣶⡆⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠈⠉⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢀⣴⣿⣟⣿⣿⣦⡀⢀⡇⠀⠀⢻⠀⢀⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣿⣿⣿⣿⣯⡿⣿⣿⡾⣅⣀⣀⣼⣷⣿⣿⣿⣿⣟⣽⡄⠀⠀⠐⠠⠄⠀⠀⠀⣀⡀⠔⢫⡍⡉⢻⠒⠚⣉⣍⢉⠐⠀⡀⣀
⠀⠀⣿⣾⣿⣿⣿⣷⣿⣿⣻⢿⡀⠀⠁⠀⠈⢿⡿⣟⣿⣾⣿⠃⠀⠀⠀⠀⠀⠠⠀⠈⠉⠐⡶⠤⠄⢁⣉⡭⠿⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠻⠿⣷⣽⣻⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣦⡀⠀⠀⠀⣀⣤⠐⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡏⠀⢀⠀⠉⠻⣿⣿⣿⣿⣾⣿⠉⠉⠑⣦⣤⠾⠋⡡⠞⠋⠁⢀⣠⠔⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⡏⠀⠀⠀⠀⠉⠛⠛⠛⠳⠶⠖⠛⠉⠁⣠⠾⠦⠶⠶⠚⠋⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣳⣦⣄⣀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡋⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢻⣿⣿⣻⣿⡛⠛⠛⠛⠉⠁⠀⠉⠙⠛⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣄⣀⡿⠀⠀⠹⣻⣿⣿⢻⣿⡏
⠀⠀⠀⠀⠀⠀⢘⠀⠉⠛⠀⢲⡀⢹⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠳⣷⡾⢿⣿⢸⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠛⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⡿⠿⢛⣉⣈⣉⣛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣔⠠⢄⢠⣴⠾⣛⣭⠉⢉⢏⣾⣿⣿⣀⠀⠉⠉⣭⣽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⢄⣲⣶⣦⣧⣿⡄⢀⣯⣶⣿⣯⣤⣤⣽⣸⣿⣿⣿⣿⣿⡄⠀⠲⣮⣽⣿⣿⣷
⠀⠀⠀⠀⣠⣶⢿⣿⣾⣿⣿⣿⠛⢿⡿⣾⣉⣿⣿⣛⠋⠉⣻⣿⡏⠉⠀⢬⣿⣿⣧⠰⣿⠿⣿⣿⣻⣿⡇
⠀⠀⠀⣼⡿⢡⣿⣿⠟⠉⣉⣹⣧⣬⣿⣿⣟⣛⣛⠿⠷⢾⣿⡟⠀⠐⠶⣤⣼⣿⡿⣷⣼⣧⠈⢁⡼⣿⡁
⠀⠀⣸⣿⢁⣿⣿⢏⣴⣿⡳⠯⠛⢸⣿⣿⣿⣿⣿⡻⣶⣼⣿⣇⡄⢹⣿⣿⣿⡟⠇⠀⠙⡿⠀⢉⢤⡞⢣⡀
⠀⠀⣿⣿⠘⠋⢻⣾⣿⣿⢷⡆⢲⣿⣧⣝⠋⣁⣠⣥⣼⣯⣿⣿⠟⠉⠉⢻⡿⠁⢀⡀⠀⠁⠀⠈⠙⠿⠁⠈⠑⡄
⠀⠀⣿⣿⡇⠀⢸⣿⣿⠇⢸⠀⢸⡿⣿⠟⠓⡟⣵⡿⠋⢙⣿⡙⠦⠀⠀⠈⠁⠚⡉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⣿⣿⡇⢰⠈⣿⣿⢀⢸⠀⠸⣿⣜⢦⣼⢸⡟⠀⣤⣘⣏⢣⡀⠀⠀⠀⠀⢠⣴⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢻⣿⣷⠸⣇⠸⣿⣿⡜⣇⠀⠻⣿⣿⣿⣾⢿⣦⣭⣿⣿⢿⣿⣏⠁⠀⠐⠛⠁⠝⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠈⢿⣿⣷⣿⡄⠹⣿⣷⠹⣦⡀⠙⢿⣿⣿⣿⣿⣿⡝⢿⣷⣬⣛⣷⣦⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⣠⡦
⠀⠀⠀⠈⢻⣿⣿⣿⣄⠹⣿⣷⡹⣿⣦⣀⠙⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⢉⣄⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠙⢿⡿⣿⣷⣼⣿⣷⣜⢿⣯⡛⠶⣌⡙⠛⠿⢿⣿⣿⣿⣿⣿⣷⣵⣆⠀⠠⣀⠀⢀⣀⡀⠠⠄⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣮⡻⣿⣿⣿⣿⢷⣍⠛⠷⢦⣽⣷⣤⡀⠀⠀⠀⠉⠙⣿⣿⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣎⠻⣿⣏⢣⡻⣷⣄⠀⡈⠙⢿⣿⣄⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠱⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⣎⢧⣿⣾⣿⣷⡈⠳⡄⠻⣿⣆⠀⠀⠀⠋⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣟⣛⣛⠛⠻⢿⣿⣿⣆⠘⡆⢹⣿⡄⠀⠀⠀⣠⠀⠁⠀⠀⠀⠈⠀⠀⠈⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⣤⣤⣬⡙⣿⣆⠀⣿⣿⡙⣆⢸⡀⣿⡇⣠⢾⡿⢃⠄⠀⠀⠀⠀⠀⢀⡴⢚⠅⠶⢂⣙⣦
⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⣋⠉⠻⣿⡿⣼⣿⢀⣿⣿⡇⠘⢠⡇⣿⢟⡅⢊⢧⡂⠀⠀⠀⢀⣠⠴⣫⢾⣧⣂⠼⣫⢭⡶⠶⠶⢤⣀
⠀⠀⠀⠀⠀⢸⣷⣿⡇⣾⣿⣷⣧⣿⣿⣟⣥⣾⣿⣿⣷⣾⣿⣷⢿⣾⣿⡆⠀⠉⠓⠒⠚⠉⠡⢊⣧⢟⡴⢣⠊⠀⠜⠀⠀⠐⣶⡌⠳⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡉⢉⣿⣿⣯⣉⣉⣽⣿⣿⣿⡟⣿⣿⣜⣿⡏⣢⠀⠄⠀⠀⠀⡉⠠⢢⡵⠊⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⡀⢻⡄
⠀⠀⠀⠀⠀⠀⢿⣿⡻⣿⣿⣾⣿⣿⣿⣿⡿⠿⠛⣱⣿⣧⣾⡇⠙⠳⢬⣟⣦⡔⠀⣀⣀⣤⠾⠋⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡙⠻⢿⣿⣿⣿⣋⣤⣾⡿⢃⡿⠋⡇⠀⠀⢠⡄⣿⠉⠉⠉⠁⠀⣴⡴⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣿⣿⣿⣭⣵⠾⠋⠀⠀⣧⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⢇⡾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⣸⡃⠛⣯⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⠷⢸⡇⠀⠀⠀⢰⡟⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠠⢹⣷⣤⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢠⡈⣷⠀⠀⠀⣸⡇⣀⠀⡀⠀⣸⠀⢀⠀⠀⠀⠀⠀⢀⠞⠀⠋⣹⡞⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠄⠠⠆⠠⠀⠠⠁⠸⠇⠀⠀⠁⠹⡇⠀⠀⢻⣧⠸⠀⠷⠀⠸⠀⠸⠀⠀⠀⠀⠰⠛⠆⠀⠀⠀⠈⢿⠿⢿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⡀⣀⣀⣀⠀⠀⠀⣸⡽⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠉⠁⠀⠀⠀⠀⠠⢉⡀⢱⠀⠂⠰⠄⢺⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠻⠀⠀⠀⠀⡩⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢠⠤⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡶⠉⠉⠒⠂⠤⠤⠤⠊⠁⠀⠀⠀⠀⡔⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡾⠁⠀⣴⣦⠀⠀⠀⣀⣀⠀⠀⢠⠟⣧⠀⠀⢧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢀⠁⠀⠘⠃⠈⠀⠀⢈⡀⣈⠁⠀⠀⠀⠁⠀⠀⢈⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢉⡀⠀⠈⠉⠒⠠⢤⣤⢤⡤⣤⣄⠒⠈⠁⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠀⢀⡄⢠⣤⠀⠀⠑⢆⠀⡠⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⠀⠀⠀⠀⢰⠟⢧⣼⢿⠂⠀⠀⠀⣦⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢉⠍⢉⠠⣥⠀⡐⠀⠁⠀⠀⠀⠘⠀⠀⠀⢀⡰⠛⡄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡠⠖⠙⠓⡄⠀⠀⠀⠀⣠⠟⠀⠒⠊⠀⠋⠛⢇⣶⠀⠤⠀⠴⣤⣀⠀⠀⠀⠱⡜⠀⣠⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡞⢧⠀⠀⠀⠆⠀⠀⠀⠀⣇⡠⠆⠀⠀⠐⠢⠄⢛⠀⠀⠀⠀⠀⢀⠉⠉⠙⠢⣄⠉⠲⣥⣤⠜
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠳⠎⠀⢀⠜⠀⠀⠀⠀⢀⣮⠀⠀⠀⠀⠀⠀⠀⠒⢦⡀⡠⠄⠒⠉⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣠⠊⠀⠀⠀⠀⠀⠏⠰⠦⠄⠠⠆⠰⠶⠚⠛⠉⠁⣠⡔⠙⠃⠀⠀⠀⠠⡀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡴⠐⠲⠆⠀⠀⣄⢤⣀⠄⠆⠀⠀⠀⢀⣠⣄⠀⠀⠘⢄
⠀⣀⣀⡀⠀⣀⡀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠚⠣⡤⠶⠄⢠⠛⠀⠀⢈⡀⠀⠀⠀⢀⠟⠀⢈⣆⠀⠀⠀⠣⡀
⡞⠁⠈⠉⣿⠉⠙⡆⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⡴⠊⠐⠂⢤⡴⢄⣀⠀⠀⠙⠳⣴⠊⠁⠙⡀⠀⡄⠛⠻⡉⠀⠆⠀⠀⠀⠘⠢
⠰⠀⠀⢸⣸⠀⠀⡇⠀⢸⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⢠⡀⣀⡉⠀⠀⠀⠀⠈⠛⡤⠚⠃⠀⣍⡀⠀⡹⠒⢻⣄⠀⢀⡤⠤
⠀⢃⠀⠀⠁⠀⡔⠀⠀⠀⡀⠀⠀⠀⠀⠙⠒⢦⠀⠀⠀⠀⢀⡉⢽⡁⠀⠀⠀⠀⠀⠀⣀⠐⠈⠓⠀⠈⠈⠧⠐⠁⠀⠁⠉
⠀⠀⠣⡀⢠⠎⠀⠀⠀⠀⠁⠀⡰⠒⠢⣀⠀⣀⠉⠒⠂⠈⠁⠀⠀⠈⠲⡆⠶⠀⠶⠊⢹
⠀⠀⠀⠙⡏⠀⠀⠀⠀⢰⠀⠀⡇⠀⠐⠃⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠰⠾⠀⠀⢡⠀⠘⠃⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡆⢸
⠀⠀⠀⠀⢧⣤⠀⠀⠀⠀⡇⠀⠀⠣⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡇⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡄⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⡇⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡈⣧⠀⢀⣇⠼⡄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⣘⢇⣴⠖⣫⡝⠴⠦⠤⠤⠄⢀
⠀⠀⠀⣀⡀⠼⠦⠤⠤⠼⠦⠀⠒⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⠀⢻⣦⣸⣆⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣀⢘⡿⣿⣿⠟⠛⠻⣦⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣥⣾⣿⣿⣿⣶⣤⣿⣿⣎⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠋⠻⣿⠏⠉⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢏⡀⠀⠀⠀⠀⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠴⢤⣄⣨⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠀⠀⠀⠀⠉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣠⠀⠀⠀⠀⠀⠀⢰⣿⣤⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡶⠂⢀⣤⠤⠀⢾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⠁⠀⠀⣀⢸⣿⣿⣿⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢂⣤⡶⠛⠙⣿⣿⣿⠋⠉⠉⠉⠁⠀⠀⠉⠛⠛⠦⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣻⡿⠛⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⣿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⡿⠟⢁⣾⣇⢼⣟⣿⣿⣿⣿⣷⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠁⠂⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠦⢤⣤⣤⣤⣤⠆⢄⡀⢀⢀⣠⣄⣀⣀⣐⣿⣿⣄⠀⠈⠣⢣⠾⠿⡿⣯⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠘⣿⣿⣿⣿⣿⡅⠀⠲⣶⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⡧⠤⣾⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⠀⢻⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠈⠀⠀⠈⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣤⣤⣤⣤⣀⠀⢰⣶⣶⣶⣄⠀⢰⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣿⣶⠀⠀⢀⣠⣶⣤⣤⣤⣀⣠⣤⡄
⠀⠀⠀⠉⠙⠻⠿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣷⡀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣯⡀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢠⣤⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⣿⣿⠃⣀⠀⠛⠟⢃⡀⠀⠀⠀⠘⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⣠⠒⠢⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠁⠊⡸⠑⠒⠒⠙⢄⠑⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠳⣅⢠⠀⠘⣀⡨⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠒

⠀⠀⠀⠀⠀⣀⠴⠒⠉⠉⠉⠓⠦⡀
⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠲⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠐⠀⣀⠀⠑⠀⠀⠰⠀⠀⠀⢀⣀⡥⣶⣷⡦⢶
⠀⠀⢀⡇⢀⣠⠄⠁⠉⡉⠉⠁⠀⠊⠱⡀⠔⡀⡔⣰⣤⣠⠀⠀⠀⠀⡆⠀⠈⠀⠱⠀⠀⠒⢸⠀⠅⠀⠀⠀⠆⠀⠹⢻⣬⡭⣴⠐
⠀⠐⣯⠔⢢⡎⠀⠀⢁⡀⠀⠀⡤⣶⠂⢹⢈⡀⣃⣾⣿⡍⡀⡀⠀⣤⢧⠀⣈⢀⠶⡄⠀⠰⠘⡄⠀⠀⠁⠀⠀⠀⠀⠠⡟⣮⣝⡇
⠀⢀⠟⠉⠁⡁⠀⠐⠛⠉⠀⠀⠹⠀⠀⠀⠀⣐⣯⣿⣿⡟⠀⠑⠠⠈⠁⡀⠹⠀⠀⢠⠖⡄⠀⢀⢐⠺⠒⠀⢀⠀⠈⢢⠚⣿⣝⣇⡄
⠀⠈⡀⠙⠿⡃⠀⠀⠤⢀⣀⠤⠐⡶⠂⠀⣠⣿⣿⡟⣿⡟⠀⠣⠀⠀⠀⡄⠈⠁⠁⠀⠙⡁⠀⠓⠈⣦⣤⣬⣷⠧⠶⠋⢸⣿⣿⣿⠅
⠀⠀⠑⠝⣒⠛⠃⠐⡄⠀⠉⠛⠉⠀⣠⣿⣿⣿⣯⡥⠘⠐⢠⡀⠀⠀⢀⠀⠛⠒⠀⠒⠒⠃⠠⠤⠀⠀⠓⠀⠀⠀⠂⠀⠨⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢰⢼⣿⣿⣿⣿⠿⠄⣄⣀⣤⣤⣀⡰⠃⢦⡀⠀⢀⠞⠂⢠⡀⠀⠀⠰⠀⠀⠀⠀⠇⠀⠛⠟⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣆⡒⢓⠛⢻⠯⠀⢙⠒⠲⠶⠬⠟⠉⠒⠒⠀⠠⣄⣠⠄⢉⠭⣤⠀⠀⠑⠚⠒⠃⢀⣀⣀⣈⣿⣿⣿⡅
⠀⠀⠀⢄⠀⠀⠄⢀⠌⠓⠦⢠⣴⡁⠉⠒⠲⠄⠛⠛⠐⡚⠀⠠⢄⡀⠀⠀⠡⡤⠍⠙⠛⠉⠉⠉⠉⠈⣠⣶⣾⣶⣷⣿⣋⢻⢿⣿⣿⣿⠇
⠀⢀⠀⠀⠑⠄⢂⡁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠂⠀⠀⠈⠀⠀⠠⢀⢌⡉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⣱⣿⣿⣿⣿⠟⠝⠂⠀⠀⠂⠻⠏
⠀⡄⠀⠀⠠⠐⠁⠘⢶⡴⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠔⢋⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣻⠏⠄
⠀⠁⠀⠀⠋⣠⠔⠈⠀⠀⠐⠄⠀⠀⠉⠀⠀⠀⠀⠀⡰⠀⠀⠐⢋⠀⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠏⢫⡟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢂⣴⣦⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠉⠁⠈⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡁⠈⣀⠀⠴⠉⠰⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠈⠁⢀⣀⣤⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⡿⣿⣿⣷⠦⣤⡉⣁⣼⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣿⣿⠶⢿⣿⣄⡀⣨⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠏⠀⣸⣿⣿⡛⢉⣀⣾⣿⣽⠋⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⠠⣿⣿⣿⣵⣿⣿⣿⣇⡿⠁⠀⠀⠀⠀⡎⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢨⡟⡷⣮⠙⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠁⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣸⢇⢿⡀⠈⠛⢁⣿⣿⣿⣿⠿⢦⣤⣀⣀⠀⠀⡸⠀⢻
⠀⠀⠀⠀⠀⠀⡴⡂⢾⠱⣄⣿⢸⡞⣷⡈⠉⢹⣿⣿⣯⣅⣀⣀⣀⣀⣀⣉⣉⣀⢠⠞
⠀⠀⠀⠀⠀⠀⠣⡙⠚⠛⠺⢿⣿⣧⠈⠁⠀⣼⠟⠁⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡔⢨⡇⠺⣌⢻⡿⣦⣤⠿
⠀⠀⢀⣠⠴⢂⡿⠋⣰⣿⢰⡇⠈⠁⢠⠀⠀⠀⡛⣦
⠀⠀⠣⣀⡴⠋⠀⣰⣿⡏⠀⣷⣀⠀⠈⠀⠀⠀⠃⠈⢷⡄
⢤⡄⠀⠉⠀⠀⢠⣿⣿⠃⠀⠘⠏⢷⡄⢇⠀⠀⢸⠀⠈⠻⣄⠐⢤⡀
⠀⠙⢷⣄⢀⡠⣾⣿⣿⠀⠀⠀⠀⠀⠉⠘⡄⠀⠈⡀⠀⠀⠹⣧⡀⠉⠳⣄⡀
⠀⠀⠀⠱⢛⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢁⠀⠀⠀⠘⢷⣄⢀⣼⢻⣦⡄
⠀⠀⠀⣠⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡀⠀⠀⠀⠈⠻⣿⣵⠟⡀⠛⠆
⠀⠀⣰⡿⣿⢃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⡄⠀⠀⠀⠀⡽⢿⣆
⠀⣰⡿⣱⠏⣼⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⢶⣶⣾⣧⡄⠛⡙⠁
⠀⠟⢱⡟⢸⣿⣿⣝⡻⢶⣦⠴⠶⠶⠦⣤⣤⣤⣄⣠⡧⠀⢠⣟⣛⣉⣥⡄⠑⣄
⠀⢠⡿⢀⣿⣿⣥⣉⠛⠻⠶⠒⠒⠲⠶⠶⠶⠦⣭⡤⠶⢑⣬⡛⢿⣿⣿⣷⠀⣸
⠀⠈⠁⠈⠉⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⡟⢦⠻⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⠿⡿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠙⢧⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠋⢴⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠰⠀⠀⠀⠘⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⡆⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔
⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠈⣀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡢⡀
⠀⠀⠀⠀⠀⠀⢰⠲⡄⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⢄
⠀⠀⠀⠀⠀⠀⠈⣷⢾⣦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠻⣿⣿⠆⠀⠀⠀⠀⡻⢄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⠀⠀⢀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣷⣄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢀⣠⡶⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⢻⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡒⠀⠀⢀⣠⠾⠛⢉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣈⣉⣙⡛⢋⣡⣤⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣛⠿⣿⣷⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠉⣀⣹⣿⣿⡏⠩⣽⡳⣤⠀⢰⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣅⢸⣿⣟⠛⠿⣿⡿⠿⢿⣿⠦⣸⣿⢿⣿⣮⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢯⡋⣹⣧⢿⣿⣥⠴⡀⢻⣷⠶⣿⣿⣿⣿⣧⣼⣋⡛⢿⣿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⡦⠁⠹⠀⠘⢹⣿⠿⢅⣾⣿⡞⠻⠻⢿⢿⣧⡉⣻⣿⣮⡿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢀⡀⠓⠂⠋⠀⠠⢟⣿⠻⢿⣿⠶⢿⣿⡇⢿⢻⣿⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⣄⠀⢀⣀⣬⣿⡴⢈⣿⣷⣮⣿⠁⡎⣼⣿⢡⢆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠳⠄⠀⠀⠀⠀⠀⠀⣀⣀⣴⣟⣹⡿⣿⣿⣿⣿⡿⠃⣰⣳⣿⠃⣾⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣬⣹⣿⣿⣿⣾⣿⣿⠿⣋⣤⣾⣳⣿⢃⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⠛⠋⣩⣔⣫⡽⢛⣿⣿⣷⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠁⠸⣿⣿⡁⠀⠀⣠⣾⠟⡉⢀⣴⡻⣻⡿⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⢰⡿⢡⢊⣴⣿⣿⣿⣽⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⠭⣓⢤⡀⠀⠀⠀⠀⠳⣦⣄⣿⢃⢃⣾⣿⡏⣠⡖⣫⣭⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠒⠲⡎⢝⢴⢷⣝⠢⢄⣀⣀⡴⠋⣸⣿⣼⣌⣸⣿⣇⢿⣿⣿⣋⣭⡻⣿⣧
⠀⠀⠀⠀⠀⠀⢀⡞⣰⠏⠀⠀⠁⠀⠨⠳⣟⠄⠂⠀⠀⠀⡈⢿⡿⣿⣿⣿⣿⣟⡛⣿⣿⡿⠛⣷⣿⣿
⠀⠀⠀⠀⠀⢀⣸⠃⡏⠀⠀⠀⠀⠀⠐⡄⠈⠓⠦⣤⣄⣤⡾⠷⠋⢹⣷⢿⣍⠛⢿⣿⣿⣿⣾⠿⣿⡟
⠀⠀⠀⢀⡴⣿⢻⡇⡇⠀⠀⠀⠀⠀⠀⠘⣎⣿⡀⠀⠀⢸⡇⠂⠀⢸⠙⠷⣝⣷⣾⣿⣟⣫⣵⠞⠋
⠀⠀⣰⣫⣶⡏⣨⢧⠀⠀⠀⠀⠀⠀⡄⠀⠈⠸⡇⠀⠀⣸⠰⠀⠀⡏⠀⠀⠀⠉⢹⡉⠉⠁
⠀⠀⢹⣿⣿⠙⠈⠈⣣⡀⠀⠀⢠⠀⡇⢠⢀⢄⣷⠀⢀⡏⠆⠀⡸⠁⠀⠀⠀⠀⣾⠇
⣀⣀⣈⣁⣁⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣈⣈⣁⣀⣈⣁⣀⣀⣁⣀⣀⣀⣈⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⢍⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⣤⡀⡻⢙⣓⢆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠻⣴⠛⡀⢀⢚⡇⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠂⠀⢳⠀⠈⣧⠘⠓⡗⢻⠀⠀⠀⢀⣶⢿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠑⡀⡏⣧⠀⠈⠂⠀⡇⣧⠀⠀⠀⣲⣒⣚⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤⣤⣦⡀⣤⣄⣐⣠⣤⣤⣧⣠⡤⠀⠐⠀⡐⠀⢰⣖⣿⢿⣿⣆⠀⠀⢠⢷⣿⠀⢀⣸⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢛⣛⣛⣧⣬⣿⣿⣿⣷⡿⡿⡿⠃⠉⠁⠀⠁⢀⣿⡟⠃⠚⠛⠛⠲⠤⠊⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⢋⠉⢫⠟⡯⠞⠼⢸⠚⠃⠃⠀⠀⠀⠀⠰⢧⠛⠻⠓⠀⠀⣴⣤⣦⣴⣶⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠂⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⡿⠀⠄⢀⡀⣠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠉⠩⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠝⢻⡇
⠀⠀⠀⡇⠀⠀⠀⢀⠇⠀⠠⢢⠆⠀⢠⠢⣤⠀⣄⠞⠀⠀⠀⠀⠀⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢸⠃
⠀⠀⠀⣿⣤⣤⣄⡞⠀⡠⠁⡼⠀⢠⠃⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡀⠀⠈⠉⠁⠂⢀⡀⠀⠒⠒⠨⠭⢀⣐⣢⡤⠴⣿
⠀⠀⠀⠘⣫⡇⡈⠀⣰⣧⣄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠀⠀⠀⠀⢒⣒⡠⠀⠀⣀⣀⣠⡤⢴⣶⣿
⠀⠀⠀⠀⢿⡿⠀⠰⠁⠈⠁⠀⡼⠓⠻⢷⠆⢴⣤⣄⡀⢀⣀⣀⣀⣤⣄⣀⣠⣄⠀⢀⣀⣀⣀⡀⣀⣭⣽⡗⠲⠾⠟⠋⠉⠀⠀⢹⣧
⠀⠀⠀⠀⡾⠁⣠⣾⠆⡸⠀⢰⡁⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡿⠛⠛⠉⠁⠙⠛⠉⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⢀⣠⣄⣤⠟⠛⣅⠈⢀⡄⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠉⠉⠁⢀⣤⣿⣾⠿⠋⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠀⣀⣀⣀⣀⣀⣀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⢲⣾⣿⡒⡄⠀⣒⣀⡤⠤⠴⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⡄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⠻⠷⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢧
⠀⠀⠀⠀⠀⢀⣤⡾⣫⣰⡆⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇
⠀⠀⠀⠀⣰⠋⠸⡑⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇
⠀⠀⠀⠰⡇⠀⠀⠈⠁⠁⠀⠀⠑⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣸
⠀⠀⠀⠀⣧⣤⣶⣾⣆⠁⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡏
⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⣼⣿⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜⠁
⠀⠀⣇⠀⠀⠀⣼⣿⡟⠀⣸⣿⣿⣿⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡸⠁
⠀⠀⢸⠀⠀⠀⣿⠋⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡰⠁
⠀⠀⠀⢠⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡴⠁
⠀⠀⠀⠀⣄⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠈⠀⡟⠉⠉⠉⠉⠉⠉⠉⠣⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠈
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠠⠿⠯⠍⠉⠉⠐⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠟⡄⠀⠀⡜⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⢆
⠀⠀⠀⠀⠀⠀⠸⡀⣸⠀⠀⠱⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⡤⠤⠄⢠⡖⠀⠀⠀⠛⢸⡀
⠀⠀⠀⠀⠀⠀⢠⠜⠓⠒⢄⡶⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠉⠣⢀⣀⠘⠋⠠⡀
⠀⠀⠀⠀⠀⢐⠁⡴⠳⡄⠈⡤⠶⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠑⠄⢣⠀⠱⠀⠀⠀⢠⠁⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠈⠉⠁⣴⠇⣠⠃⢰⠀⠘⡄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⠚⠓⢄⠀⠀⠀⣼⠀⠀⠱⣬⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠾⠔⠦⠰⣧⡠⠟⠀⡀⠀⡇⠀⠀⠀⠀⡠⠔⠀⠉⠉⠉⠉⠉⠉⠉⠁⠙⠒⠔⢳⡀⠀⢸⠇⠀⠀⡿⢠⠖⣶⡁
⠀⠀⠀⠀⢀⡀⠤⣇⠀⠠⠔⠊⠀⠀⠐⠂⠁⠀⠀⠱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⢠⣧⠞⡁⣀⠈⠆
⠀⠀⠀⢸⠁⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢣⠀⠀⠀⠉⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠊⠁⢈⡴⠁
⠀⠀⠀⠀⠱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢢⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠒⠀⠀⡆⠀⠀⢀⠏⠏⠂⡔⠁⠈⠉⠉
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠤⠒⠤⢴⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡠⠊⡜⠀⠈
⠀⠀⠀⠀⠀⠈⠆⢠⢄⠀⠀⠀⡤⠞⢢⠀⢀⠇⠀⢀⣰⣺⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠴⠊⣷⡟
⠀⠀⠀⠀⠀⠀⢘⠘⡈⠢⠚⢄⠇⠀⢀⡉⠭⣐⣾⠿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⡦⢾⡟
⠀⠀⠀⠀⠀⢀⠘⠒⠓⠒⠒⣒⣊⣉⠥⠒⠉⠈⠙⠢⣌⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠣⠊
⠀⠀⠀⠀⠀⠘⠓⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣉⠢⣄⣀⣄⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠎⠁⠀⠑⢴⠶⠒⠁⠀⠀⠀⠀⢀⡞⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠩⣿⣟⣖⠒⠲⠶⠖⠊⣁⣏⣀⣠⡀⠀⠀⠀⠀⠀⠀⡅⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢠⢄⣴⣷⠟⠋⠈⡆⢀⡠⠔⠈⠀⠀⢰⣭⢭⡍⠉⠉⣐⡒⠋⠀⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⣵⣯⣤⡿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⢃⠁⠀⠀⠀⠀⠀⠈⢇⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠵⠞⠛⠛⠛⠛⡳⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣶⣿⣿⠿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠠⢤⣀⡶⠒⠒⠀⠀⠀⠀⠀⣀⣀⣤⠴⠊⠉⢆⣠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠟⢻⠁⠈⠙⡙⣏⠉⠉⠉⠀⠀⠀⠑⠂⠀⢸⠀⠊⠢⠀⠀⠀⠀⠀⠣⠼⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣿⠀⠀⠀⠐⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⢁⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⡡⠋⠳⡀⠀⠀⠈⠢⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠢⡀⠀⠤⠴⠖⠶⡖⠂⠀⠀⠀⠀⠀⠀⢵⠛⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⢸⣼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠂⠒⠛⠲⠭⠔⠖⢶⣿⣷⣶⣠⣝⣷⠶⠐⠊⠁⠀⢆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠐⠒⠂⠃⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠺⣶⡤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡏⠀⠀⢨⡂⠀⠀⠀⠀⠀⠇⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⣜⡀⠀⢸⡁⣤⠄⢸⠇⠀⠀⠀⠀⠀⣸⣆⠻⣿⡆⠀⠀⢀⣀
⠀⢠⣤⣤⠀⢀⣀⣤⡌⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⠛⠉⠀⠀⢠⠇⠀⡀⠋⠀⠀⠂⠀⠀⠀⢹⣿⣿⣾⠿⠀⠀⠀⠙⡿
⠀⢸⣿⡋⠀⢰⡿⠿⡿⣽⣽⠇⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣰⣄⣤⡁⠀⠀⠀⠀⠀⣹⡿⣿⡀⠀⠀⠀⢠⣶⣤⣤
⠀⠈⠛⠁⠀⣿⢄⠀⠳⠗⠃⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⣯⢟⣷⣿⡆⠀⠀⠀⢀⣿⣶⣼⣇⢰⠏⠀⢸⣿⣿⣿⡇
⠀⠀⢀⠀⠀⠉⠙⠃⠀⠀⠀⠀⢉⣠⣴⡖⠂⠀⢠⣶⡀⠀⠀⠀⠀⠀⢻⣯⡏⣟⣿⣿⡇⠀⠀⠀⠸⠿⠿⠿⢿⠿⡀⢀⣬⣶⣿⣿⡇
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣟⣶⣷⠀⠀⠀⠐⠀⠀⠀⠀⠚⠛⠃⠉⠉⠉⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠀⠙⠛⠿⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠈⠐⠃⠐
⠀⢸⣿⡇⢸⡇⠀⡇⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢈⠀⣴⡾⠛⠻⣦⠀⠀⠀⠂⠀⠀⠂⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤
⠀⢸⡿⠧⠸⢧⣤⣇⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⢈⣿⣷⠦⣚⢹⣷⣄⠀⣁⢈⠀⡁⢈⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢛⠻⣿⣧
⠀⢠⣤⣴⣶⣿⠟⣋⠀⠀⠒⠿⢿⣠⣤⡈⣛⠂⢘⣠⣿⣿⣆⡛⣹⣿⡁⢄⠄⡡⠈⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⢈⠁
⠀⠀⠈⣿⣿⣤⣴⣿⡆⠀⠀⠀⠀⠀⠉⠓⠛⠀⢪⢿⣿⠿⠻⢭⣭⣿⡔⠀⢀⣀⣀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢊
⠀⢠⣤⡟⢏⠀⠀⢘⣥⣤⡄⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⣷⣩⢂⡙⣿⣷⣿⣿⣽⣶⣶⣾⠀⠀⠀⢸⠿⣼⡟⠇⠀⠐⠇⠀⠀⠀⠀⠀⠰⠄
⠀⢘⣿⢧⣾⠀⠀⢻⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⢾⣿⣏⠻⣦⡡⢬⣾⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠈⠉⠰⠆⠀⠀⠀⠀⠂⢄⠀⢠⡅
⠀⢸⣷⡄⠁⢠⡀⠀⠈⠙⢻⡟⣷⢤⣄⠀⠀⠀⣻⣿⣿⣵⣿⣿⣷⣟⢝⡫⣿⣿⡧⢀⠌
⠀⢸⣿⠇⠀⠀⢶⣶⠖⣥⠀⠠⠙⠩⣿⢹⣶⡆⣿⣿⣿⣿⣿⣿⣿⣯⢾⡵⢻⠿⠕⠬⠢⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡀
⠀⠀⠀⡞⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋⠉⠉⠹⡧
⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡤⠀⠀⠃⣿
⠀⠀⢀⣤⠀⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠴⠂⡎
⠀⣾⡿⣷⠀⠛⢹⣦⠀⠀⠀⠀⠀⢄⡐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⡀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠒⣢⡞
⠀⣿⠗⣸⠀⣤⣼⣿⡇⠀⠀⠀⠀⠈⠀⠀⠆⠀⠁⠀⡃⠀⠀⠀⠀⠀⠀⢉⠁⡶⠙⠈⢤⡀⠀⠀⠀⠉⣉⣩⡝⠛⠊⢀⠀⠀⣾⡿⣊⣩⣷
⠀⠀⠺⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⢀⣀⣠⡀⠛⠀⠾⠗⠒⠒⠛⠋⠀⢀⠀⢠⣾⠿⠀⠀⠀⣾⠃⠀⠀⣲⣿⣿⣿
⠀⣴⣾⣯⣭⢹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠋⣩⣿⠣⠯⠁⠀⠀⠀⠀⠀⠀⠀⠂⠐⠓⠒⠶⠀⠠⡀⠹⡀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠂⠘⠗⠀⠈⠈⠻⠛⠛⠛⠛

⣾⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣠⣉⣵⣀⣴⣿⡇⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠁⠀⠀⢃⠀⠀⢠⡤⢀⢀⡀⠀⠀⠀⠞⢿⣉⣍⡉⠛⠛⠃⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠁⢀⠂⠀⠀⠀⢿⣶⣿⠛⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣿⠇⢠⣴⡀⢮⣶⣴⠏⠀⣾⣇⣾⡽⡇
⠀⠨⡇⠀⠀⠀⠀⠀⠉⠐⠊⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⢰⡦⡁⡄⣢⡩⣠⣿⡻⠏⠀⠀⠉⠙⠛⠀⠔
⠐⠐⠓⠐⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠸⡟⠻⠷⠿⠇⠀⠘⠪⡧⠀⠀⢙⠚⠁⠀⠘⡀⠀⠀⢀⠤⠴⠶
⠀⠀⠀⣠⠒⠛⣛⣠⠀⢠⣾⡄⠀⢠⡤⣴⠃⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⢈⡋⠀⢐⠀⠀⠀⠀⡾⠆⠀⠘
⠀⠀⠀⡟⠃⠀⢈⠛⠀⠈⠛⢧⠶⠂⠀⠈⠀⠀⠀⢠⠀⠀⠀⠐⡄⠀⠠⡜⠁⠀⢱⠂⠀⢀⣴⣇
⠀⠠⣾⡧⠀⠀⠀⠀⠀⠀⠀⢠⣆⣾⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠛⠂⠂⠚⠂⠀⠀⠙⠃
⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⣼⠛⠃⠚⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠈
⢦⠐⠤⠀⠀⠀⠀⠀⠀⢠⠨⠿⣄⠂⠸⢿⣄⠀⠀⠀⠰⡇⠀⠀⠀⠄⣻⠛⠿⠚⠟⠀⠀⠀⣠⢲⡆⣄
⠁⠁⠐⠀⡀⠀⠰⡀⠀⠜⣠⡬⡤⣿⣤⣿⣿⡇⠀⠀⣾⡧⠀⢠⣦⢶⡏⠀⠠⠉⠄⠀⠀⢀⡟⠛⠒⢿⡆
⠁⢶⡿⢘⣿⣿⡿⠁⠀⠀⠉⠿⠤⠤⢌⢉⠉⠁⠀⠀⠙⠻⠶⠊⠀⠀⠁⠀⠀⠠⠄⠀⣀⡻⣧⡀⠀⢿⣏
⡠⠢⡂⣼⠿⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣍⣽⣆⣴⣿
⠀⠀⣩⡁⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠧⠤⢅⡙⡛
⠀⠀⢤⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠀⣀⣆⢠⣴⣦⣶⠃
⠀⢰⡈⠀⠀⣹⠀⠀⠀⣰⣟⣱⣦⣦⠀⠀⠀⣴⢄⢄⡀⣍⠍⣸⡟⡿⠃⠀⠀⢀⣔⠋⢙⣥⡆
⠁⠀⠉⠉⠉⠉⠀⠀⠀⠙⠻⠿⠇⣈⠀⠀⠀⠷⢵⠌⠁⠈⠪⠋⠀⠀⢖⠀⠀⢸⠋⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣙⠃⠀⡀⠈⠀⢺⣽
⢠⡄⠀⠀⠀⠀⢿⠷⣷⢼⣷⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢤⠙⠀⢈⠖⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⠂⡀⢀
⣤⣇⠀⠀⣀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠐⠀⠶⠠⠀⠀⠀⠀⠀⠀⠀⠸⢿⣾⡿⠙

⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⠀⠀⠀⠉⠉⢭⣛⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠓⠒⢂⣰⣶⣶⣦⣤
⠀⠀⠀⠀⠠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⣀⣤⣤⣴⣶⡾⣿⡛⠛⣿⣿⣙⢻⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠒⠀⠈⠉⠛⠻⠿⣿⠟⠛⠛⠛⠻⠃⠀⠈⠛⠋⠀⠀⠁⢀⣿⣯⡙⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⣼⣿⡟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢸⡃⢀⣠⡩⠀⠀⠀⢀⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⢷⠀⢿⢯⠀⠀⠀⠀⠀⠞⠉⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⣽⣟⣛⡒⠢⠀⠀⠀⠈⢀⡆⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⡾⠉⠉⠀⠀⠀⠀⠀⣆⡴⠋⢁⠇⣿⢬⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣄⡀⣠⣀⣀⣠⣾⠟⠀⡠⠊⣰⠧⠔⠢⠭⡿⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡟⣿⣿⣿⣿⣿⠟⠁⡠⠊⠀⢠⡿⠄⠠⢒⣶⣿⡇⡏⡟⠛⢻⢿⢶⡆
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠙⢿⣟⠋⠐⠊⠀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣷⣷⢺⣾⢺⣾⡗
⠀⠀⠀⣿⠟⢏⠁⢀⣼⣿⡿⣿⣿⣿⠿⡻⢟⣽⣷⡀⢰⣿⣶⡶⠆⠀⠀⢸⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣸⣿⡇
⠀⠀⠐⣧⢂⢸⣇⣿⠏⣜⡔⡸⣹⠟⠔⣉⡭⢹⣿⣿⣼⠟⠣⠙⣿⡆⣰⡿⢻⡿⠿⢿⣿⡿⣿⣿⢻⢸⠸⣻⡟⣿⡇
⠀⠀⠀⠇⠸⠘⣿⡇⣰⢿⢰⣡⣿⣴⣽⣦⣠⣾⣿⣿⣷⠀⣀⣤⡌⢿⣿⡵⣩⣏⢽⣛⣿⣾⣏⣹⡐⢸⠀⡏⡇⡿⡇
⠀⠀⠀⠀⢰⠀⣿⣷⣏⠻⣯⠣⡱⡜⣍⢏⢏⢧⣿⣿⣿⡿⠛⠉⠁⣾⣏⡞⡵⣫⢛⣶⣻⣭⢽⣿⡟⣻⣿⡇⣇⣧⡇
⠀⠀⢀⣧⣬⣆⣿⣿⡸⣸⣿⣷⡱⡜⣜⢎⢫⢢⡻⣿⣿⣷⠀⠀⢠⣿⢫⢞⡟⣵⡿⡽⣱⡳⣫⠏⣼⣿⢷⣇⣿⠹⡃
⠀⠀⠀⠿⢉⣛⢻⣿⣷⣿⣿⣿⣷⡜⣜⢮⢧⡻⣷⡙⣿⣿⣷⣀⣼⣱⢫⢎⡞⣟⣼⡽⡿⣷⣧⣾⢻⢸⡸⣿⢹⢸⡇
⠀⠀⠀⢃⢈⡟⣿⣿⣿⣿⣿⢻⣿⡿⣌⢮⢣⡱⡝⣟⣮⣿⣿⣿⣿⣧⢏⣾⠞⣿⣟⣞⣿⣵⣿⣼⠸⡆⣧⣿⣾⣾⡇
⠀⠀⠀⣾⣼⣏⢸⣿⣿⣿⣿⣿⣽⣿⣿⣎⢣⡳⡘⢾⣿⣿⣿⡟⣻⣣⢻⢏⢞⣟⣟⣾⣿⡏⣿⣿⡇⣇⢻⢛⣿⣿⡇
⠀⠀⠀⢹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡳⣽⣿⣿⣿⡿⠹⣱⣳⣣⢯⢞⣾⡾⣿⣿⣧⢻⡧⡇⢿⢸⣆⢹⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡴⠚⠉⠉⠙⠢⣄⠀⣀⠠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠦⠎⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠺⡁
⠀⠀⠀⠀⠀⠀⠀⡞⣇⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠒⠙⣒⡒⠚⠉⠈⠹⠿⠏⠉⠣⡀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠘⣧⣿⣠⣀⠀⣹⠄⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢋⣁⣀⠀⣠⠴⠒⠛⠛⠳⣄⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⢰⠋⠁⠀⢀⣸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⡖⠉⠀⠈⢻⠁⠀⠀⠀⠀⠀⢸⠀⠘⣆⠸⣅
⠀⠀⠀⠀⠀⣾⠀⠾⠭⢭⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⣴⠄⠀⠀⢸⣿⣷⠀⠀⠀⢀⡞⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⡬⢭⡷⠀⠀⣽⠋⠀⠀⠀⢀⡴⠒⣛⣓⠒⠏⠈⠉⠉⠓⢦⡴⠻⣄⠀⠀⢀⣠⠞⡰⠒⢆⢸⡤⠞
⠀⠀⠀⣀⣬⢧⣀⣀⡤⠾⠽⣦⣀⡀⠀⠘⢧⡈⠉⢉⣉⣠⠀⠘⡟⠒⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠤⢋⡞
⠀⠀⢸⢀⡤⡀⢠⠖⢦⠀⢀⣄⠀⡷⠀⠀⠀⠈⠉⢩⡖⣺⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⣏⠀⠙⠃⠈⢂⣙⣉⣸⠀⡷⠤⢤⣀⡀⠀⢫⠀⠱⡀⠀⠀⠑⠀⠀⣀⣀⡤⠤⠴⠒⠋⠁
⠀⠀⣀⡞⠋⡉⠉⠉⠉⠉⡍⢹⢀⣷⠀⠀⢀⣹⠖⠉⠡⡀⢈⡦⣶⡞⠉⠉
⠀⣎⣁⠧⠤⣬⣤⣤⡤⠤⠤⠾⠞⣉⣿⠉⡟⠀⠀⠀⠀⠘⠉⠀⢿⠁
⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠖⠚⠛⠉⠈⢉⡟⠁⠀⠀⠀⠀⠀⣰⡊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⠇⠱⡌⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⠀⠀⠀⠀⠀⠀⡼⠀⠀⣇⢀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⠀⠈⠙⠒⠒⠢⠤⠒⢧⡀⠀⠉⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣤⣤⣄⣀⣀⣀⣀⣀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠉⠁⢿⡉⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠨⠭⠅⠯⠬⠛⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⢟⣿⣯⣥⣄⡉⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⠯⠛⠉⠈⠁⠈⠉⠛⣿⠿⠻⣿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣀⠀⠀⠀⣀⣀⠀⠀⠘⣿⣿⣿⣟⡛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⣿⣿⣯⡭⣄⠀⠀⠡⣤⡀⠀⠀⠙⣿⣿⣿⣷⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⣏⣿⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠙⠨⣙⣛⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡀⠦⢀⣀⠀⠀⠀⠀⠀⠀⣤⣴⣯⡿⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡸⣭⣭⡽⠃⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⡈⠁⠀⢀⡠⠄⠀⢰⣿⣟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠉⠉⠑⢤⡀⢠⠃⠀⠘⢯⡉⠁⠀⠀⠀⢸⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢢⠙⣏⠀⠀⠀⠈⢻⡀⠀⠀⠀⢸⡇⠈⠢⡤⢀⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⡤⠤⠤⠔⣲⠆⢀⠂⠇⢌⡂⠀⢀⣀⣤⣵⣄⡀⠀⠀⡇⠀⠢⣬⣤⠈⠱⣄⠀⠒⡄
⠀⠀⠀⠀⢰⠁⠀⠀⡸⠀⠀⢀⡴⠛⢷⠇⠀⠈⢾⠙⠛⠉⠉⠉⠀⠑⢍⠢⣀⡇⠀⠄⠉⣁⠀⠀⠈⠀⠉⠙⠢⡀
⠀⠀⠀⢠⠃⠀⠀⢀⠁⢀⡴⠋⠀⠀⠀⢣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⡆⠀⠀⠀⠂⢠⠄⠈⠀⠀⠀⠈⠳⢄
⠀⠀⢠⠃⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⡆⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠃⠀⠀⠀⠀⠀⢹⣲⠀⢄⣀⠀⠀⠀⠙⢆
⠀⠀⠂⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢱⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢻⣃⠁⢉⢻⣶⠄⠀⢸⡧⣶⣶⡾⣶
⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣴⣷⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⠀⢈⣩⡉⣉⣩⠁⠀⠀⣾⢷⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡴⣏⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⣇⢠⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡘⠀⢹⠏⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡌⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⡘⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡴⠃

⠀⠀⠀⠀⠀⣠⣶⠶⢻⡿⢶⣄
⠀⠀⠀⠀⣰⡟⠻⢧⣸⣃⣤⣿⣧
⠀⠀⠀⠀⢹⡷⠿⢚⡟⢻⣅⣀⣿
⠀⠀⠀⠀⠈⢻⣤⣿⠀⠀⠘⠻⣇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠘⠳⣆⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀
⠀⠀⠀⠀⠀⣸⡇⠀⠂⠀⠀⠀⠀⠀⢀⡀⠈⠿⣦⡀
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀
⠀⠀⠀⠀⣼⠏⠀⠀⠀⡀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⡈⠿⣦⡀
⠀⠀⠀⢠⡿⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠈⠿⣦⡀
⠀⠀⠀⣼⠃⣀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⡻⣦⡀
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⢿⡆⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠄⢒⡈⢧⠈⠇⠈⠛⡶⠟⠋⠉⠀⠀⠀⠈⠙⠻⣆
⠀⢰⡟⠀⠄⠀⠂⠀⠉⠀⡀⠀⡄⠠⠒⢶⡈⠃⠀⠀⠀⢀⡈⣣⠄⠓⠈⠁⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⣾⠃⣀⠀⠀⢴⡀⢳⠀⠁⠀⠀⠀⠳⣀⠳⠈⠙⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹
⢸⡟⠀⠇⠀⠳⡀⠱⠄⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⢸⣧⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣿⠰⠃⠀⠀⠐⠂⢲⠒⠒⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠠⠄⠈⠄⠀⠀⠀⢀⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠
⢿⡰⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢷⣦⠀⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡏⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢿⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢸⣇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⡀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⡤⣀⣠⣆⠀⡴⣴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠦⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠿⢿⠃⢸⠟⠐⢷⡿⣦⣤⣀⡀
⠀⠀⠀⠀⣴⠟⠛⠛⢛⠉⠁⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠳⣦
⠀⠀⠀⣸⡇⠀⠀⠀⢈⠀⠀⠀⠤⠀⠦⠆⠰⠮⠀⠄⠀⠀⢀⠀⠀⠀⠀⢻⡆
⠀⠀⢠⣟⠀⠀⠀⠀⠸⠀⣀⠀⠀⠘⠀⠈⠀⠀⠈⠀⠀⠀⡀⠒⠀⠀⠀⠈⣿
⠀⠀⢸⡏⠀⠀⠀⣀⠳⠀⠀⠄⠈⠙⣉⠀⠡⡀⠌⢨⠀⠠⠀⢵⣀⠀⠐⠈⢻⡄
⠀⠀⣾⠃⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢘⢀⠀⠀⠀⢸⡇
⠀⢀⣿⠀⠀⠀⠀⠋⠰⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠺⠃⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠐⠇⠀⠀⠀⢹⡆
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣇⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⠀⠀⠀⣼⠇
⠀⠀⠙⠶⠶⠟⠉⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠙⠻⠶⠾⠋
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⡻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣄⣠⡍⠛⡟⡟⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⢿⣿⣿⣿⣇⠐⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⣿⣿⡋⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⡏⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠋⠙⠛⠛⠉⠀⢀⠀⠀⠀⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠉⢘⠛⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⡀⠺⣿⠃⢃⣀⣿⡀⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠀⣴⢿⣷⡶⠘⠛⠿⠛⠛⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠳⣤⣤⣶⠉⠀⠀⠀⠀⠀⠀⠠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣄⠀⢀⢀⣤⠀⠀⣶⠂⠀⠀⢰⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⠈⢻⠏⠀⠘⠛⠁⠀⠀⠀⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⠛⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣹⣿⣦⠀⠀⠀⠀⢾⠛⡻⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⡇⠀⣤⡀⠀⠈⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⢀⣠⣤⣶⠊⣡⠀⠀⠀⠀⠀⢿⡇⠀⣴⠞⠿⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠉⣡⠀⠀⠛⠋⠀⠀⠀⢿⡀⡀⠀⠀⠀⠈⠉⢻⣤⡄⠀⠀⣶⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⣶⣿⡿⠀⠀⠙⣦⠀⠙⢡⣶⡦⢤⣀⡀⢠⣿⣷⠆⠀⠈⠀⠀⠉⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣴⣿⠋⢋⣀⡀⠀⠀⠀⣷⡀⠈⢩⣀⠀⠈⠛⠁⣀⣷⣶⠓⠀⠀⣄⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⣿⣿⢿⣿⡟⢻⣦⣤⠀⠙⣷⠀⠀⠛⢻⣿⣂⣿⣿⢙⣿⣦⡄⠀⠿⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣧⡀⠿⢿⣿⡿⣅⠘⠀⠉⣦⠀⠀⠉⠀⠛⠛⠛⠛⠋⠀⠁⠀⠀⠛⠀⢠⡟
⠀⠀⠀⠀⣤⣿⣿⣤⣿⣧⣬⡟⢱⣶⣾⣿⣤⣿⣦⡀⠀⣿⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣂⠀⠿
⠀⠀⠀⠀⣬⠿⣿⣿⡛⢻⣿⣇⣠⣿⣧⣼⣿⠛⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡀
⠀⠀⢀⡿⠁⠀⠈⢿⣿⣿⣿⣏⠛⣿⣿⣿⣿⣿⡿⠀⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠛
⠀⣤⠛⠀⠀⠀⠀⠀⣿⣧⣼⣿⡿⠀⠹⣤⣽⣏⠀⣾⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣴⣿⢯⣤⠌
⣄⠀⠀⠀⠀⠀⢠⣴⡏⠀⠈⠉⠀⠳⠀⢿⣿⣿⣄⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠐⠒⠶⠀⠀⠀⢤⡀⠀⠀⣠
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⢼⣿⣁⣿⠀⠙⠛⠀⠀⠀⣀⣀⣤⣤⡆⢀⣶⠀⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠉
⠀⠀⠀⠛⠀⣤⡀⠀⠀⠀⠀⠀⠛⠣⣤⡀⢈⣿⠟⠛⠛⠛⠀⠀⠀⠀⠿⠉⠍⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠚⠛⠃⠀⠀⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⣤⡄⠀⠀⠀⠘⠛⠻⣶⡶⠿⣶⣦⣄⣤⣤⣀⣤⣤⡀⠻⠿⠛⠣⣤⣄⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⢠⣥⡄⢸⣷⠀⠻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⢀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⠀⠀⢻⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠀⠛⣿⠁⠀⣴⠁⠀⠚⡀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⢿⣅⠘⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠛⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠃⠀⡿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣾⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣿⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠈⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⣀⣠⣤⣶⣿⠃⠂⢤⣤⣤⠄⠀⢀⣠⣤⣤⡄⠀⠘⠛⣤⣤⣀⠀⠀⠀⢀⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢿⣿⣿⠛⣿⣿⣦⡀⠀⠀⢀⣴⣿⣷⣤⡀⠀⢀⣤⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣼⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⢿⣿⣤⣿⣿⣿⣿⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣸⣿⣿⣟⡛⢱⣷⣾⣿⣿⣧⣼⣿⣿⣿⣿⣀⣿⣿⣿⠀⣿⣿⣿⣅⣽⣿⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣿⣿⣿⣿⡀⣸⣿⠛⢻⣿⣿⣿⣅⣽⣿⣿⠟⢿⣿⠀⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⣏⣸⣿⠟⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⠟⢿⣿⣶⣿⣿⣯⣤⣿⣿⠙⢛⣧⢸⣿⡟⢹⣿⣿⣿⣧⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣼⣿⣏⣉⣿⣿⢾⣿⣷⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣏⣹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⡿⢿⣿⣿⣿⣿⣤⣼⣿⣏⣹⣿⣿⣶⣿⣿⠿⣿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⣠⣿⣿⣿⣿⣿⣷⣶⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣤⣿⣿⣤⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⣿⡏⢻⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣶⣿⣿⠟⣿⣿⣿⣿⣿⣿⡋⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⠉⣿⣿⣿⠛⣿⣿⣷⣿⣿⣧⣼⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠉⣭⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⠛⠉⠀⠀⠈⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣷⣄⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣿⣦⡀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⣿⣦⣀⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢿⣿⠟⢻⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣧⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣦⣤⣤⣶⣶⣶⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⢉⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠁⠐⠠⣦⡀⢂⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⠃⠀⠀⢀⠀⠀⠀⣼⡟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣿⡄⠀⠀⠀⢀⣽⠀⠈⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣴⣿⠀⠀⠀⠀⣼⡿⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⠀⠀⢸⣿⣿⠄⢳⠀⣴⣿⠃⣠⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⡴⣿⣿⣧⣄⣿⣷⣿⣧⣾⣿⣟⠉⠄⠀⠀⠀⡴⠀⣾⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⠁⠀⠀⠉⠛⢿⣿⣿⠻⠥⣸⣿⣿⣾⣧⣄⣸⣾⣧⠄⠻⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⠀⠀⠀⠀⠜⠉⠀⡀⠙⠛⡿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠙⠻⢿⣶⣤⣷⣶⣀⣤⡄⠒⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⢠⣿⣟⣿⣽⣆⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⣦⣠⡿⠛⢉⣿⣿⣿⡏⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡏⠓⠄⠀⢿⣿⣯⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⢿⣿⣿⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠈⣿⣿⣿⣇⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⣿⣿⣿⣿⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡉⠁⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠁⠀⠀⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣸⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⠀⣿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣄⣀⢈⣵⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣉⣉⡄⢠⣏⣻⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠶⠛⠛⠛⠛⠛⠷⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠒⠶⢤⣀⠤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡡⠄⠲⠶⣆⠀⠀⠈⢿⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠸⣷⣌⢿⣼⠿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠼⢉⣤⡄⡀⠀⠀⡹⠄⠀⠿⠁⠁⢙⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡁⠉⣟⠛⠉⣅⠱⠀⠀⠀⠐⢂⠄⠀⢠⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢠⠀⠈⠉⠉⢆⣤⠀⠀⠀⠨⠅⡤⠊⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠜⠈⠅⠆⢄⠀⢘⡀⢀⣀⣀⣀⠇⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⡅⠀⠀⠒⠁⠈⠊⢉⠍⠃⢀⡾⠁⠈⠁⣸⠋⠁⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣈⠑⠃⠀⠒⠶⢎⠁⣠⠟⠈⠀⠉⠀⠐⣣⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⠒⡍⠰⠒⠀⢦⠀⠀⠀⠀⢠⢁⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠁⢀⠷⡀⠑⢄⠀⠰⠃⠀⡈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡔⠁⠀⡟⢄⠀⠱⠁⢀⣾⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⡗⠀⠴⢓⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⠀⠀⠀⠀⠀⡿⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⡆⠀⠀⠀⢰⡇⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡇⢡⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠖⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⡆⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠨⠤⠭⠿⣶⢏⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡉⢡⡤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⣿⠀⡠⠂⠀⣀⠄⠀⠉⠀⠀⠀⠀⠀⠀⢀⡤⢸⡁⠀⠀⠈⠣⣴⠀⢰⣶⣭⣝⣩⠍⠐⢄
⠀⠀⠀⠀⣸⣿⣇⠁⠐⠾⠋⠀⠠⡆⠀⠀⠀⠀⠀⠀⠈⡧⠳⡌⠀⣐⠀⢠⣰⠀⠈⠉⠙⠛⢷⢂⣵⠀⡀
⠀⠀⢀⡤⠿⣿⣧⣶⡅⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠈⡇⠬⢄⠀⠈⠀⣤⣬⠀⠸⠰⡦⠀⠈⢻⠿⠰⡇
⠀⠀⠸⣾⣷⣾⣿⡏⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠀⠳⡈⢉⣀⠀⣾⡿⠁⠀⢠⣀⠀⠀⠀⣥⡄⠀⢷
⠀⠀⠀⣠⣾⢮⠍⠀⠀⡠⠓⠢⠂⠀⠀⠀⢀⣤⣤⡀⢀⣀⢇⠀⠀⢠⠉⠁⠀⢀⣬⣡⣤⠔⠃⠉⣷⣿⡟
⠀⢀⢞⠕⡑⡡⢊⠗⣮⣤⠆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⠈⢀⠔⠁⣾⣶⣤⣿⣿⡟⠋⢲⠀⣪⡇⠄⠢⢄
⠀⠛⢯⣮⢎⠔⡡⢊⣦⡝⢲⣲⢢⠀⠀⣿⣿⣿⣿⣿⣿⣿⡔⠁⣨⣷⣿⣿⣿⣿⣿⣿⡢⣠⣾⣿⣷⣿⣦⠆⠈
⠀⠀⠀⠈⠋⡰⡱⢻⠣⡘⣠⣿⢿⠀⠀⠿⢿⣿⣿⣿⣿⣿⡅⢰⣿⣿⣿⣿⡏⠈⣿⣿⣴⣿⣿⣿⣿⣿⢃⣤⠆
⠀⠀⠀⠀⠀⠱⢧⢿⣴⢻⢩⣷⠌⠀⠀⠀⠺⠛⠋⠃⠙⠿⢳⠿⠛⠉⠉⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⢤⡴⠖⠶⠾⠁⠀⠀⠀⠀⠀⠈⠷⣤⡤⠖⠚⠉⠉⠉⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⢀⣀⠴⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢱⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠂⠉⠀⣀⣩⡍⠁⠀⠀⠀⠐⠖⠂⠀⠀⠈⢢⡀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣼⣀⣀⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠊⠁⣀⡀⠀⠀⣠⡴⠞⠛⠛⠙⠒⢆⠀⠑⡄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠛⠉⠁⠈⣿⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⠀⠀⣠⠞⠋⠉⠉⢳⡞⠉⠀⠀⠀⠀⠀⠀⠈⡆⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣤⣤⠤⠴⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣼⠁⠀⠀⠀⠀⠈⣧⣠⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⠳⠒⠒⢦⡀⣸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣺⡇⠀⠀⠀⢠⡿⢾⡿⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠬⣍⣿⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠁⠒⠂⠀⠈⠉⠉⠙⠳⣤⡴⠿⣆⠀⠀⠀⠀⢀⣠⠞⢠⡖⠒⢄⠀⣇⡠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡼⢻⣄⠀⠀⠀⠀⠀⢿⡀⠘⠙⠁⠛⠒⠀⠀⠀⠖⡶⠦⠄⠘⠃⠀⠈⠛⠓⠖⠛⠋⠀⠀⠈⠀⢀⠌⢠⠃
⠀⠀⠀⠀⡔⠊⠉⠉⠉⠘⣛⡋⠉⠉⠉⠉⠛⢲⡄⠀⠀⠈⠛⠂⠀⠀⠐⠒⠘⠄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⣇⣠⠋⢆⠀⡜⢀⠙⢄⠀⣰⢶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⢸⡇⠀⠈⠉⠀⠀⠀⠀⠉⠀⢠⠀⢸⠣⠤⢤⣀⣀⠀⠀⠈⡀⠀⠙⣄⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡀⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⡿⠛⠓⠆⠀⠀⠀⠐⠂⠒⠘⠀⠨⢄⠀⠀⠀⠀⠓⣤⠖⠙⢆⠀⠈⠳⣄⡠⡄⠒⠈⠉
⠀⢀⡠⠖⢻⡅⠀⠃⠌⠀⠀⠀⠠⠈⠀⢸⣀⣼⠂⣤⠤⢴⡖⠋⠁⠀⠀⠈⠳⡤⠞⠉⣷⠗
⠀⠘⠳⢦⣬⣛⡓⠒⠒⠒⠒⠒⠚⠛⠛⣋⣉⠡⣤⢾⣀⣰⣇⡠⠀⠀⠀⠀⠀⠀⠀⢰⢛⡄
⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠻⠇⠈⠛⠃⠐⠋⠉⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠳⡀⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢧⠀⢀⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠛⠦⣀⡀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢮⡄⣸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣁⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠚⠻⣄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠭⠭⣿⢟⣛⣛⣻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠧⠤⣀⣀⢀⣹⠦⣄⣀⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠭⠤⠶⠛⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢖⣫⣽⣲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢯⣶⣿⣿⣿⠿⡶⠟⢿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡮⠯⠚⠉⠁⠀⠀⠀⠀⠀⠀⠇⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢾⣿⠀⢀⡤⠤⠤⠀⠀⢀⡘⣫⣤⡉⣿⡗⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠜⣼⠁⢁⣴⣶⡶⠀⠀⠘⠈⠛⠚⠁⢹⡃⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⣟⣠⣿⠆⠀⠈⠁⠀⠀⠠⡀⢰⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀⣨⣤⣀⠀⢠⡏⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡧⠀⢦⠀⠀⠀⢴⣯⡥⠶⣶⡏⠀⢸⣁⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⠀⢸⣆⠀⠀⠀⠈⠙⠛⠉⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠾⢿⡍⠲⠤⣀⣀⣀⣠⣴⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠔⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠠⠚⠒⠒⠂⠄⢢⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡷⣿⣽⡣⣀⡀⣀⣀⡠⣴⣿⣿⣦⡀⠀⠀⠀⠀⢐⣉⣉⠁⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⢒⠖⠛⠙⢿⣾⣿⠁⠁⠑⠉⠉⣡⢾⣿⣻⣿⠛⡦⢀⠀⠀⠨⣀⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠼⠠⠀⠁⠀⠀⠀⠀⢳⡇⠀⠀⠀⢐⣽⣿⡿⠿⠛⠋⠀⣀⣀⣉⣲⣤⡙⠏⡆⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⢔⠔⠁⠀⣀⠤⠒⠒⠚⠋⠉⠉⢀⠄⠂⠛⣦⠘⢄⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⠔⠀⠐⣄⠚⠀⠀⠀⠀⠀⠀⠠⡠⠊⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠱⣼⣷⠈⠳⠀⠀⠀⠀⣧⢀⡀
⠀⠀⠀⢠⠃⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢠⠗⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡀⠀⠀⠈⢻⣇⢸⣦⡆⡆⣤⢻⢀⣝⣄
⠀⠀⡴⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢠⠋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠋⠉⠉⠉⠁⠲⢄⠙⣾⣿⡇⡇⡿⠙⢸⣿⡕
⢀⠞⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢓⣿⣿⢧⡇⡷⣀⣼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠋⠙⢿⣿⣿⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠛⠋⠃⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠵⠂⠀⠀⠀⠰⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⢋⠃⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⢿⣿⡷⠂⠀⠀⠀⡠⠀⠀⠀⠀⠋⠀⡀⠀⠉⠛⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⣿⣿⠏⠀⠀⢰⡞⠉⠀⠀⠀⠀⢀⣴⣾⡏⠀⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⠃⢸⣷⣿⣿⣿⡄⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⣴⣾⣿⣿⡧
⠀⠀⠀⠀⠀⣎⠆⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⠘⢿⡟⣿⣿⣿⡶⠃⠀⣼⣿⣿⢁⠀⡀⢌⣷⡽⠿⠛⠫⠄⣤
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⣿⣿⣿⣎⣳⣝⣿⡟⠁⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡿⠑⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡏⠀⠀⣀⠉
⠀⠀⠀⢀⠀⣠⣤⣴⣶⣦⠀⠀⠴⠶⠆⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⣀⣉⣤⡄
⠀⠀⢀⡀⣀⣈⣉⣙⣛⡙⠒⠒⠛⠉⠉⠁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⠤⠴⠾⢯⡭⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡇⣿⣿⡇⠈⠻⣧
⠀⠀⠀⠀⠀⣷⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢻⡟⢿⠀⠀⠻
⠀⠀⠀⠀⠀⠹⡌⡫⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⣤⡤⠤⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣟⣿⣩⣿⣬⣏⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣯⣭⣭⣿⣯⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣄⣀⣀
⠀⠀⠀⠀⣠⣶⠀⠀⠀⠠⣀⠙⠳⣄⠀⣠⡴⠾⣛⡃⢤⣤⣀⡀⠀⠀⠀⠀⠐⣿⣿⠋⢹⣿⠟⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⣄⠁⣠⣷⠄⣿⣿⣿⡆
⠀⠀⠀⣤⡴⢸⣤⣾⣀⡤⢟⡶⣢⣿⣾⣿⠵⠞⠻⠿⣛⣿⣿⣿⣷⣄⠀⠀⠐⠏⠙⠻⠦⠙⠦⣴⠟⣻⣷⠀⠀⠀⠀⢠⡾⢟⡋⣬⡉⠛⠋⠻⠛⠿⠃
⠀⠀⣰⢏⡀⠃⠈⠛⠻⠷⠿⠾⣏⣸⡏⣇⡀⠀⠘⣿⣿⣿⡛⢻⣿⣿⡆⠀⠀⡄⠀⠐⣂⡀⠀⠘⢦⡿⢻⡇⠀⠀⠀⠸⣷⣾⣷⠟⠀⠴⠎⠀⣠⡿
⠀⠀⢹⣦⡧⠀⠴⡄⠀⠀⡤⠢⢈⣽⠃⠻⣬⠀⠀⢈⣽⣿⣷⣤⣿⣿⡇⠀⠀⠛⠃⠀⠉⠁⠀⠰⠦⢶⣿⠃⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⢀⠘⠉
⠀⠀⠀⠹⣧⠘⠋⠓⠂⠀⠙⠟⠸⢏⡞⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡇⠀⠀⠁⢀⡀⠀⠀⠀⠀⣴⠿⠃⠀⣠⣤⣤⣤⠂⡎⠀⠀⠀⢿⣽⠄⠁
⠀⠀⠀⠀⠈⣦⠀⠠⠆⠀⠀⠀⠋⠘⢹⢶⡶⠂⠀⠀⢀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠸⠟⠋⠀⠀⠈⠰⡄⠀⠀⠙⣿⣿⣿⡀⣣⠆⠰⣁⠀⠈
⠀⠀⠀⠀⣀⢱⡀⠰⣶⡶⠀⡄⣤⠇⠘⢸⠁⢀⣠⡔⠃⢠⠔⠉⠉⠉⠀⠀⠀⠀⠀⠒⡒⠀⠀⠀⠀⠁⠀⠀⠀⠻⣿⣿⣿⡑⠌⠁⢿⣷⡄
⠀⠀⠀⢸⣿⡄⢹⡒⠠⠀⠊⠀⣿⠀⢀⠀⠛⢉⡉⠉⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠆⠀⡐⠁⠀⠀⠀⠀⠀⠈⠛⢿⣷⠀⠹⠾⠿⠇
⠀⠀⠀⠈⣛⠃⠀⠙⠀⠀⠁⠀⠛⠀⠀⠀⠀⠛⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣰⣾⡿⡋⣉⣩⣟⣶⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠦⠄⠀⠀⠀⠀⠀⠀⠀⣞⣛⣓⡣⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠛⠛⣆
⠀⠀⢸⡟⠛⠛⠋⠻⣿⣄⣠⣦⠀⠀⠀⠀⡴⢋⣄⣁⣀⣀⣠⣶⣦⣴⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢈⡿
⠀⠀⢈⡇⠀⠀⡀⠀⣼⡿⣿⣿⠄⠀⢀⡞⠀⠛⠻⣿⡿⣿⣿⠟⠉⣿⣧⠀⠀⠀⠀⠛⡆⠘⠃⠀⠉⠀⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢃⡀⢯⡄
⠀⠀⠈⠛⠀⠠⠬⠄⠈⠻⣿⣿⣦⠀⢈⣿⡟⠀⠀⠀⠀⠉⠀⠒⠀⢻⣿⠁⠀⠀⠀⡉⢿⡀⢠⣀⡄⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣶⣷⣾⡿
⠀⠀⠲⣀⣀⣀⠀⠀⠀⣴⣿⡿⠋⠀⠸⣿⡇⠰⠟⠃⠀⠠⠈⠛⠋⠸⡿⠀⠀⠀⠀⠀⠀⠓⠀⡉⠰⠆⠈⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⡀
⠀⠀⠀⡙⠋⠁⣀⠤⠊⠙⠋⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢀⣀⠄⠂⠀⠀⠠⠂⠀⠐⠄⠘⠆
⠀⠀⢿⣿⣿⡇⠀⠀⠀⣨⣦⡀⠀⠀⠀⠀⠉⢳⠀⠰⣦⣦⣾⠆⢠⠋⠀⠀⠀⠳⣿⣿⠀⠀⣡⣀⡀⠀⢸⣦⠀⢠⡀
⠀⠀⠀⢿⣿⣇⣀⣤⣾⣿⣿⡿⠇⠀⠀⠀⠀⠀⠁⠄⠈⠋⠁⡰⠃⠀⠀⠀⠀⠀⠙⢿⣤⣾⣿⣯⣹⣆⣸⡗⠀⢸⣿⠀⠀⠀⠀⢠⣤⣦⣤⣤⣶
⠀⠀⠀⠀⠉⠛⠿⠟⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠯⠿⠗⠀⠚⠁⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⡰⢟⡰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢸⡃⠀⠀⣿⠂⠀⠀⠀⠀⢀⠄⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠁⠀⣀⡇⣀⡀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣠⣔⣀
⠀⠀⠀⠀⣤⠉⠀⣼⡁⠀⠀⠀⠀⠰⠏⡄⠂⠘⠓⠘⠒⠀⠀⠀⠀⠀⠰⠗⠀⣿⣶⣾⠿⣿⣿⣿⠟⠐⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣙⠿⢷⣦⣀
⠀⠀⠀⢾⣧⣤⣦⣼⣷⠀⠀⠀⣴⣶⣿⠀⠀⠤⠀⢀⣀⡃⠀⠀⠀⠀⢹⣷⡄⠈⠀⠀⠀⠀⠀⠴⡄⠈⢧⠀⠀⠀⠀⠀⣰⠏⢈⠻⢿⣿⣿⣿⣍⣭
⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⢿⣿⣅⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠈⢥⡀⠀⠞⠃⠀⠀⠀⢀⣤⠀⠸⣧⠀⠀⠀⣼⡏⠠⢤⠀⠀⠀⢙⠹⠟⣽⠁
⠀⠀⠀⠀⠀⠉⠁⠠⠀⠀⠀⠀⠈⠻⠟⢀⡀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠰⠛⠀⠀⡄⠀⠈⠀⠀⣥⢀⣶⣿⡟⠀⠀⠀⠀⠀⠰⡄⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡂⠀⠙⠶⠀⠀⠀⠀⠀⠘⠒⠆⠀⠈⠉⠂⠀⠀⢀⠀⠀⢀⠴⠂⠀⠘⠈⠻⠟⠃⠀⢓⡠⢀⡀⠀⠀⣸⡟
⠀⠀⢠⣤⣄⣀⣀⣤⠀⠀⠀⠀⠻⣿⣿⠄⠀⠈⠁⠀⠀⠀⠀⠀⠂⠥⠀⠀⠀⠀⠀⠀⠀⠉⠡⡤⠂⠀⠀⠀⠀⠀⠀⠀⠢⡈⠈⠉⠁⢀⣴⣟⢿⠆
⠀⠀⠾⠿⢿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠠⣤⣬⡦⠀⠀⠠⠄⠀⠀⠀⠐⡇⠀⠈⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠘⠉⢭⠱⠖⠀⠀⠀⠀⠀⠀⠀⠀⣄⢁⣶⢀⣀⣰⣉⣾⣛⢒⡆
⠀⠀⠈⢈⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⠀⣠⡀⠀⠀⠀⠀⠀⠈⣄⠀⠀⡀⡇⠠⡄⠀⠠⠀⢷⡀⠀⠀⠀⠀⠀⣹⣿⠛⠉⠁⠈⢹⡟⠫⡿
⠀⠀⠀⠀⢱⠀⡈⠀⠀⠀⠀⠀⠀⠀⢦⢑⠂⠀⠍⠀⢂⠀⠀⠘⡄⠠⢁⣷⠀⠁⠀⠀⠁⣾⡅⠀⠀⠀⠠⣇⣾⢻⣶⠆⠀⠀⠈⠀⢤⠀⡄
⠀⠀⠀⢘⡛⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⢫⣤⡀⢀⠀⢛⡄⠀⠀⡇⠀⠠⠙⣆⠐⠷⠆⣀⣿⠆⠀⠀⠀⠀⠙⣻⠀⠉⠀⠀⠀⠀⠀⠀⢠⡸⣦
⠀⠀⠀⡸⠃⢨⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠒⠉⠀⡀⠀⠠⠀⠀⠀⢻⢠⡄⠠⣄⠀⠀⠈⠃⠀⣧⠸⣷
⠀⠀⠀⠃⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⢶⣿⣷⣈⠀⠀⢀⣤⡴⠀⠀⡀⠀⡰⠀⢀⡀⠑⠀⠉⠀⠀⢸⠘⢧⠀⠈⠀⠀⠀⠀⠀⣿⣧⣿⠃
⠀⠀⢠⠀⠀⠾⠃⠐⠛⣆⠀⠀⠀⠒⠀⠃⢈⠉⠘⢿⢻⡏⠙⠂⠘⢿⣿⠀⢀⣴⡴⣄⠀⢸⣇⠀⢰⣦⠀⢀⣿⡀⣾⣦⡀⠀⢴⣶⠆⢀⣿⣿⣿
⠀⠀⠨⢀⣀⡈⠀⠀⠀⡈⠆⠀⠀⠀⢀⠀⠁⠀⣶⡎⠰⡟⠀⠀⠀⠀⠉⢶⣧⣿⣦⣙⡷⢿⣦⠀⢸⡿⠁⠘⣿⣿⣿⣿⠿⣦⡄⠀⠀⠈⠿⠛⠥
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠁⠈⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣾⣿⡿⠿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣀⡤⢶⡻⠿⣿⣿⣿⠋⠩⠶⢟⣷⣾⣷⡀
⠀⠀⠀⠀⠀⢀⣠⣄⣤⣤⣦⣵⠀⣸⣯⣾⣯⣥⣼⣹⣿⣿⣿⣶⡀⠨⣭⣿⣿⣷
⠀⠀⠀⣠⡾⣿⣿⣿⣿⡟⢻⣿⣯⣽⣿⣏⡉⣹⣿⠏⠀⠰⢽⣿⣆⢿⡿⢿⡿⣿⡇
⠀⠀⣼⡟⣼⣿⢋⣠⣶⣿⢗⣿⣷⣾⣿⣭⡉⣿⡏⠀⣬⣳⣿⣿⠛⠷⣷⠀⠔⣹⡄
⠀⢰⣿⠰⠟⣯⣾⣿⣧⠤⣼⣟⠻⠛⢛⣊⣻⣿⣿⠥⠚⢿⡿⠋⠀⠀⠃⠀⠸⢿⠎⠂
⠀⢸⣿⡇⠀⢿⣿⡇⣿⠰⣿⣿⠗⢾⣻⡿⠋⣻⡟⠦⠀⠀⠃⠒⡉⢀⡀⠀⠀⠀⠀⠀⠈⠂
⠀⢸⣿⡇⢷⠸⣿⣀⢿⠀⢿⣾⣦⣷⣟⠀⣦⣿⣱⣀⠀⠀⠀⣠⢾⣧
⠀⠀⣿⣿⡼⡆⢻⣿⡜⣆⠈⢿⣿⣿⣿⣿⣾⢿⣟⠿⣧⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⣀
⠀⠀⠈⢿⣿⣿⡄⢻⣿⡼⣷⣄⠙⢿⣿⣿⣿⣮⣿⣿⣿⣿⠟⡁⠀⠀⠀⠀⠀⠀⠖⠒⠚⠋
⠀⠀⠀⠀⠻⣿⣿⣦⣽⣿⣞⢿⣟⠶⢌⡛⠿⠿⣿⣿⣿⣿⣿⣵⡆⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⡷⣍⠙⠲⢽⣷⣄⠀⠀⠀⠉⢿⣿⣿⠀⠠⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⣦⢹⡷⡝⣝⣷⣄⠢⡉⢻⣷⡀⠀⠀⡸⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡿⠾⠿⣿⣿⣧⡘⢆⢻⣷⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣭⣭⣭⡙⣶⡀⢻⣿⢳⠈⡄⣿⣀⣴⡶⠃⠀⠀⠀⠀⠀⡠⢞⠭⢶⣙⡆
⠀⠀⠀⠀⢠⣟⣿⢋⣭⢉⢻⣟⣿⢇⣼⣿⡆⣅⣇⣿⣻⡀⠳⣄⣀⣀⣠⠴⣫⣾⢷⡴⡫⢶⠖⠒⠲⢤⡀
⠀⠀⠀⠀⢸⣿⣇⣿⠿⢿⣿⡿⠷⠿⣿⣿⣿⣿⣿⢿⣿⠇⠀⠀⠉⠁⠠⠀⠾⣣⢋⠊⠀⠀⠀⠀⠹⣧⠙⣆
⠀⠀⠀⠀⠸⣿⡿⣿⣦⣾⣿⣿⣿⣿⠿⢫⣿⣸⡟⠳⢍⣾⣅⡀⠀⢀⣀⡴⠛⠁⣠⠀⠀⠀⠀⠀⠀⠸⡆⢻
⠀⠀⠀⠀⠀⠙⢿⣮⡛⠿⣿⣿⣿⣠⣴⠿⣹⠟⡇⠀⠀⡌⢻⠋⠋⠉⠁⣤⡆⡴⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠿⣿⠿⠷⠛⠁⠀⢳⠀⠀⢀⢸⡄⠀⠀⠀⡿⠸⠃⠀⠠⠀⠀⠀⠀⠀⠀⠁⡯⠈⣿⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠘⡆⠀⠈⠸⡇⠀⠀⢸⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼⢻⠦⣼⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⡀⠀⠀⠀⢹⡀⠀⠃⢿⠀⠀⢾⡆⡇⢰⠀⠇⠀⠀⠀⠀⢀⡼⡄⠀⠀⠁⣿⣿⡟

⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠠⠤⠀⣀⣀⣀⣩⣭⣥⣤⡤⠤⠤⠴⠶⠶⢤⣤⣤⣄⡉⠙⠲⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⠉⣉⣟⣛⣿⣟⣱⠄⣀⣋⣭⡥⠴⠖⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢁⡿⣿⡶⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠘⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠏⣬⡔⠀⠠⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡄⠠
⠀⠐⠀⠀⠀⠀⢠⡶⢻⣾⠡⣿⣶⠇⠀⠀⠂⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣰⠏⠀⠘⢆⣹⡿⡿⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠓⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡿
⠀⠀⠀⠀⠀⣿⢀⣀⣤⣤⣦⡐⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡇⠀⢀
⠀⠀⠀⢀⠐⠉⠙⢻⣿⣿⣿⣇⠀⠀⠀⣾⣿⣷⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡾
⠀⠀⠀⡄⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠁
⠀⠀⠀⡇⠀⠀⠀⢀⣾⣿⣿⠃⠀⣰⣿⣿⣿⣿⣿⣿⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣰⠃
⠀⠀⠀⢷⠀⠀⠀⠸⣿⡿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣰⠏
⠀⠀⠀⠘⡀⠀⠀⠀⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⠏⢀
⠀⠀⠀⠀⠅⢀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⠃
⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⣰⠃
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠘⠦⢀⣴⠿⠿⠿⠿⠿⠿⠿⠛⠛⢦⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠈⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠄⠠⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠁⠁
⠀⠐⢀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢿⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣀⡀⠈⠒⢶⣤⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣛⠛⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠰⡿⠟⠁⠀⣒⠲⠀⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⡀⠀⠀⠑⠂⠀⠂⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣔⣂⠙⢶⡆⢸⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠁⠒⠀⠀⠀⢀⡔⢁⡠⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡴⣇⣈⠁⢦⣀⠀⠘⠟⣰⡇⠀⠀⠀⢠⠀⠀⣤⣀⣀⠉⠀⣭⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⡟⠋⠀⠀⢈⣷⣶⣾⣿⣿⠆⠌⠠⠂⢀⣼⣿⣿⣿⣟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠙⢄⠀⠀⠀⠘⠛⠿⣿⠿⢛⠀⠠⠁⣰⣼⣿⣿⣿⡟⣾⣿⣿⠃
⠀⠀⠀⠀⡖⠺⠗⠚⡉⠘⡑⣄⠀⠑⠢⢤⣀⠀⢠⠈⢀⠄⠀⠄⢰⣿⣿⣿⡿⢏⣼⣿⣿⠃
⠀⠀⠀⢸⡿⣻⣦⠛⢀⡀⠀⡋⠲⣤⠖⠀⠸⠟⠃⢸⣋⡀⠀⢐⣾⣿⣿⡏⠀⠄⣿⠟⠁
⠀⠀⠀⢸⣿⡏⠻⣦⡈⠛⢄⠘⡷⢀⡴⠓⠂⠘⠛⣊⠀⣙⢦⠿⠿⡿⠿⢢⠑⠏⠀⠀⠀⢀⡀
⠀⠀⠀⠸⣮⡁⣶⠈⢑⣶⠘⠆⠘⠉⠀⠀⢀⡴⠁⠹⠆⣻⣀⠛⢴⠃⡀⠀⠌⠀⠀⠀⣴⡯⠢⠄
⠀⠀⠀⠀⠘⣯⢀⣐⢶⣁⠰⠇⠀⢀⠠⠐⠉⢀⡀⢌⠀⢀⡝⠒⢚⡀⠁⡐⠀⠀⠀⠀⠉⠵
⠀⠀⠀⠀⠀⢿⠀⠀⢹⣿⠥⠚⠈⠁⢀⠔⠂⠉⠀⠈⡴⣘⡇⠠⠊⠀⢰⠃⠀⠀⠀⠀⠀⠈⡖⠈⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠀⣠⠖⠁⠀⠀⠀⢠⠊⡅⢃⠹⠒⠟⢀⠂⠀⠀⠀⡄⡄⠀⠈⠑⠊
⠀⠀⠀⠀⠀⠘⣇⠀⢀⣀⡴⣞⠁⣰⠀⠀⠀⠀⢢⡀⢈⡁⢀⡆⡠⠃⠀⠀⠀⢀⠏⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢛⡩⠀⠀⢻⡋⠉⠉⠀⠀⠀⠈⠟⢒⣷⠀⠀⠀⠀⠀⠀⣠⣞⠄⠘⠄
⠀⠀⠀⠀⠀⠀⠠⣇⡤⠞⠁⡼⠃⠀⠀⠐⠒⠂⠀⠁⢸⣷⣂⠀⠀⠀⠀⠀⣿⠀⣤⡄⢘
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡼⠀⠀⠀⠀⠀⠻⢦⣠⠤⢄⣀⠤⠀⠠⠄⠀⠀⣠⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠘⣧⠀⣄⠀⠀⠀⠀⠀⠀⡀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⠀⠀⠀⠀⠹⣦⡙⠃⠀⠀⠀⢀⣠⠃⣿⠀⣿⡇⠀⠀⢀
⠀⠀⠀⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠾⠶⠶⠶⠦⠤⠼⠷⠷⠦⠤⠤⠽⠤⠤⠿⠤⠿⠧⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣠⣤⣤⣶⣶⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠪⢭⡛⠛⢉⠥⠔⠊⠉⠛⢿⡋⢔⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⣠⣇⡀⢧⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢋⣡⣿⢿⣿⣦⣌⠑⠒⠤⠤⠤⠶⡿⣿⣇⠙⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⠾⠿⠛⠛⠛⢹⡿⢙⣿⠿⢿⣿⣇⢄⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⠃⠀⠀⠀⠀⠋⠉⠙⠻⣿⣧⣤⣾⣿⠿⣎⡾⣿⣿⡏⡿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⣷⠀⢘⠀⠀⠀⣾⣶⠶⡄⠀⣿⣿⡀⣸⣿⣽⣿⣿⡥⢀⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠁⢀⠂⠀⠀⠀⠉⠉⠁⠀⢀⡿⠉⢳⣿⡟⢿⣿⣿⣵⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⡷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⡫⣱⡀⢙⣋⣁⡴⠂⠀⠀⠀⠀⠀⠀⣾⣿⢷⡉⠛⠂⠈⢭⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣞⣾⢿⣧⠈⠉⠁⠀⢀⠠⠀⠁⠀⠀⠆⢸⣿⡄⠁⠀⢴⣶⣴⣶⣾⡿⢇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢾⣿⣅⣠⣴⣶⡅⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣷⣶⣾⣿⡿⢋⠀⠨⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠹⣟⢿⣿⣯⡻⣿⣛⢛⣽⠇⠀⠀⠀⠀⠀⠀⠀⠁⡘⠉⣿⣿⡏⠠⢗⢹⡆⠀⡟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢧⠈⢈⡻⣿⣿⣶⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡟⠙⣷⠀⠋⢸⡇⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡈⠛⢷⣦⣤⣽⣿⣿⣿⡿⣱⡙⠶⣤⣀⣀⣀⣀⠀⢰⢏⠀⠀⠃⠀⢻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠸⣧⢷⣷⣿⣿⣿⣿⢳⠿⣎⠉⠛⠒⠄⣀⠀⠀⠸⣏⣶⣾⣿⣶⣦⣤⣘⣧⠀⢸⣷⣴⣦⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⢿⡿⢁⠏⡄⠌⠳⢦⣀⡀⠀⠀⠀⠀⢿⣿⠿⣿⣿⣿⣿⣿⣿⠀⣼⢉⣸⠻⣧⡀⠀⠀⠀⠀⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠁⠚⠒⠒⢒⡶⢂⠈⠙⠓⠶⢤⣤⣸⣿⡇⢿⣿⣿⣿⣷⣿⡆⡟⠉⠉⠀⢉⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣶⣋⣖⣁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⣿⡟⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡄⠉⠉⠉⠙⠓⠒⠒⠦⠤⠤⣭⣛⣿⣿⣿⣁⣴⡃⠀⠀⣽⠃⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢠⣤⡠⠀⣸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠶⣾⡿⣿⠘⠋⠀⣀⡴⠃⠀⠀⠀⠀⠀⠑⢄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⣄⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠉⠉⠉⠉⠉⠉⠙⣷⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⠀⠀⢀⣀⡀⣼⣧⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣻⡇⢈⣭⠯⠀⠀⠨⡭⡁⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠀⡹⠄⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⣛⠀⠀⠀⠀⠀⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠟⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⢸⡏⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⢀⣤⣠⣦⣴⡓⣤⠶⣲⢓⡤⠤⡗⠒⢪⠃⠰⡗⢐⡃⢰⣿⡦⣵⣿⠾⣦⡤⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⣿⣼⣿⣿⠚⠿⠷⠼⢿⣦⡇⠰⢛⡽⡿⠁⢈⣷⠟⠃⡸⢻⠇⢼⡿⣿⣿⠁⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠓⠀⠀⠀⢹⣁⠒⠄⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠈⠢⠼⡞⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡤⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⡼⢲⠀⣤⠄⠀⠤⠀⣶⢐⡄⠀⡀⠀⣄⠄⠠⠐⠀⠀⠠⠰⠆⠀⠄⠺⢛⣟⣤⣦⣀⣀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⢠⡾⠓⠒⡂⠴⠒⢠⠀⢷⡆⠒⠁⠀⠙⠀⠀⠈⠀⠉⠐⠀⠀⠈⠐⠀⠁⠈⠀⠀⠂⠀⠀⠈⠁⠻⣡⠖⠈⠱⠦⠤⠄⢱⠄
⠀⠀⠀⠀⠀⢠⠋⡴⠀⠀⢠⠤⠄⣠⢷⣀⡅⢀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠼⡇⢰⡆⠉⢦⣤⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⣾⡀⠐⠷⢀⣠⣉⣹⣁⣠⣀⡀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣬⣤⡜⡶⠚⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠸⡁⠈⠷⠤⡠⠤⢄⡳⠟⠉⠉⠉⠛⠓⠒⠚⠀⠀⠀⠀⠉⠉⠁⠈⠉⠋⠑⠋⠉⠛⠘⠉⠉⠉⠉⠈⠹⣧⠽⢃⣲⢶⣛⡽⠁
⠀⠀⠀⠀⠀⠀⠓⡀⠘⠶⢉⣡⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠅⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢋⣵⣶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⢰⣿⣿⣷⣿⣷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠈⠛⠛⠉⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⡿⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣃⣤⡌⣿⡿⠉⠝⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⡷⠀⢼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⣿⣿⣿⣧⣴⠖⠀⠀⠲⣶⣾⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠦⢦⡈⣿⣿⣧⡀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣿⠻⢿⣿⣆⠀⠉⠋⢹⡏⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⣿⣧⣨⣤⣉⣶⣶⣾⡿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢻⣿⣿⣟⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⡀⠀⢿⣿⣿⣿⣿⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢹⣿⣿⣿⣷⡄⠈⢻⣿⣿⣿⣷⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡛⣠⣿⠃⣿⣿⣿⣿⠆⠀⣠⠉⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣴⢿⣿⡟⣸⣿⢸⡇⠉⢶⣦⣤⣶⣶⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣿⣿⣧⣿⣏⣼⣷⠄⣍⠀⠀⠀⣻⣷⡘⢧⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⡛⣓⠀⢀⣿⣿⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠋⠛⢱⣄
⠀⢫⡡⢷⢮⡀
⠀⠀⠀⢸⠠⠛⢦⡀
⠀⠀⠀⢘⠁⠀⢀⠓⢄
⠀⠀⠀⠈⣇⠔⠁⠀⡸⠳⣄
⠀⠀⠀⠀⡇⠀⡠⠊⠀⢀⠜⠳⡄
⠀⠀⠀⠀⢸⠀⠀⢀⠔⠁⢀⡠⠜⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⣸⠦⠖⠀⠒⠉⠁⠀⠀⠀⠀⢀⣀⠀⠙⢦⡀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⣀⠀⠀⢱⠖⠋⠡⢤⡀⠀⠀⠀⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹
⠀⠀⡼⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⡌⠀⠀⠀⠈⠳⡀⡠⠐⠐⠀⠒⢅⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣀⣀⣀⠠⠤⠼⠼⠤⠤⠤⢤⣀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣇⠑⠊⢢⡴⠂⠘⣀⣀⠒⠒⢲⠁
⠀⠀⡇⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢆⠀⠈⠁⠀⢰⣌⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⢧⠀⠀⠀⢀⠤⠘⢆⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠉⢽⣉⡙⠫⠍⠉⠉⠙⠋⠙⣿⠉⠛
⠀⠀⠈⢦⡀⢰⠁⠀⠀⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⠲⣤⡴⠊⠉⠳⠞⠁
⠀⢀⣀⡀⠵⢼⣄⠀⠀⠀⡘⠀⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠒⠁⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⣠⠞⠁
⢰⡁⠀⠀⠀⠈⠳⣍⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠒⠒⠶⠶⠒⠋⠁
⠀⢳⢤⣀⣀⡴⠒⣌⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏
⢠⠏⠀⠀⠀⡴⠒⠁⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⡸⠂⣀⠠⣄⠈⠓⡄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠧⡴⠹⡀⢹⠑⠒⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⢦⡤⢤⡤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡌
⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⣧⣁⣠⣤⣶⣶⣶⣤⣤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣄⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⣿⣿⡇⠀⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣿⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠈⠀⠀⠁⠈⡞
⠀⠀⠀⢺⣿⣿⣿⣿⡟⢻⣿⠃⠀⠀⠀⠀⣀⣈⣈⠉⠁⠀⠀⠀⠀⠀⠠⠤⣤⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡉⢷⡀
⠀⠀⠀⠘⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠂⠀⠀⠀⣴⢿⡟⢲⢦⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡠⠀⠳⠾⢧
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⢀⣠⣤⣶⣶⣶⡙⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠻⠋⠁⢀⠠⡖⠛⠿⣉⣻⣦⣸⣿⣿⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣤⣤⣄⣀⣉⠉⠀⡴⠁⢀⠀⠀⠁⠀⠀⠀⠀⠈⠻⡟⢿⣿⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠾⠛⠁⠀⠈⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢹⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⣀⡄⠈⣃⣾⠁⠀⢀⣀⣤⣄⡀⠀⠀⠀⠠⠶⠶⠶⠶⠤⠀⠈⢋⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠀⢶⡌⢻⡆⠀⣉⣤⣤⠄⣄⠀⠀⢠⣿⠖⠶⠶⠾⠖⠀⠀⢨⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠊⠉⠉⣀⣠⠑⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⠖⠒⠒⠛⢩⡍⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠋⠀⠀⠀⢠⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⣄⡀⠀⠀⠀⣀⣤⠀⠀⢘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠛⠁⠀⢀⡟⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁


⣿⢿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠒⠒⠒⢒⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣀⣀⣠⣮⣟⣿⣿⣿⣿⣿⣶⣶⡤⡖⠒⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢞⣠⣤⣴⣾⣿⣿⣿⣷⡀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⢹⣿⣾⣿⡿⠿⠛⠹⣿⣿⣖⡴⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠈⠉⠛⠻⠿⠟⠉⢙⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣉⣉⣉⣩⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇⣤⣀⠀⠀⢀⣤⣤⡀⠀⢻⣿⣟⣿⢸
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠸⠇⣚⣲⠀⠀⠀⢖⣒⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠻⣿⣿⣿⣿⡥⠴⠮⠀⠀⠀⠲⠶⠄⠀⠀⠏⠗⢿⢺
⠀⠀⠀⣠⣴⣖⣻⣿⣿⣶⣄⠉⠉⠁⠀⠀⠉⠉⠸⣿⣿⡟⠛⠛⠛⠛⠋⠁⢿⣿⣿⣾⣷⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢸
⠀⢀⣾⣿⠟⢋⡁⢴⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⣸⡉⣻⣧⠤⠀⠀⠀⠤⠄⠈⢿⣿⣿⣄⠀⠀⠀⢻⡄⢈⡋⢈⣀⡀⠀⠀⠀⠀⢲⣿⡟⢸
⢰⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⡎⠻⡜⠯⠟⠁⣰⣿⣿⣿⡧⠶⠖⠀⠀⠐⠟⠂⢸⢿⣿⡿⠀⠀⠀⠀⢣⡀⠒⠒⠀⠀⠀⢀⡠⠀⣸⢦⡽⢿
⢹⣿⣿⣿⣿⣿⡝⠛⠋⠥⠛⢀⣰⣷⣤⣤⡞⠁⢻⣏⣹⡇⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⢐⡶⣲⣶⣶⣶⣿⣄⣀⣀⡤⠒⠁⣠⠞⠁⡿⣻⢇
⣿⣿⣿⣾⣿⡷⢟⣛⡀⠀⠀⣤⢼⣿⣿⡟⠃⠀⠀⠻⣿⣿⣤⡈⠫⠉⠁⢀⢴⣿⣿⣫⣿⣿⣻⠁⣴⣿⡇⢩⣍⠀⠀⢀⡼⠃⢀⣴⡿⢃⣾
⠹⣿⣿⣿⠻⠀⠉⠋⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⡀⡀⠀⣠⡴⣿⡶⠶⠚⠁⠘⢯⡟⢿⠿⢿⢿⣾⣿⣿⣧⣾⡟⣳⣴⠟⠀⣴⢿⡟⣡⣾⣿
⠀⠀⠙⠻⠿⢦⡀⠀⠠⣤⢤⠆⣰⣗⡫⠶⠶⢱⡺⠆⣠⠳⠘⡿⠓⠀⠀⠀⠀⢸⠃⠀⠀⢰⡀⠀⡎⣿⢇⣿⣧⣿⠏⢠⠞⠡⣫⣾⣿⠟⠁
⠀⠀⠀⠀⠀⣌⠙⢶⣤⣀⣀⠔⠁⠀⠀⡜⢊⠉⢀⠀⡟⠢⡔⢺⡄⠀⠀⠀⢀⢯⡀⠀⠀⢸⠁⠀⣿⣿⢸⣿⣿⠃⠀⣁⡀⣀⣤⣾⡇
⠀⠀⢀⣴⠞⠅⠑⠠⣀⠁⡯⠲⢤⡀⢠⡿⡀⠑⢄⠱⡁⡌⠈⢱⢌⢢⡀⢀⠆⠀⠙⢄⡜⢩⡦⠞⡁⣿⡿⠻⢃⣴⣿⣡⣾⣿⠿⠋
⠀⠴⢿⣄⡈⠑⢮⠂⢌⠑⢄⠘⠀⠙⠊⠁⠀⠀⠴⢀⣄⠀⠀⡎⠀⢈⢾⡇⠤⠆⢀⡨⠔⢩⠀⠀⢀⡫⠂⣀⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠈⠉⠛⠲⠶⠤⠷⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠈⠁⠉⠁⠀⠀⠀⠈⠁⠀⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣷
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⣠⣾⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀
⠀⠀⣀⣀⣀⢸⢻⣿⣿⢻⡆⠀⠀⠀⠀⠀⠺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠑⢶⡟⢹⣿⣿⢈⡗⠀⠀⠀⠀⠀⠰⡇⣸⣿⣿⠀⡟
⠀⠀⣿⣿⣿⣀⣹⣿⣿⣏⣀⣀⣀⠀⠀⠀⢀⣿⣿⣿⣦⡀⣀⠀⢀⢀⠀⠀⠀⢸⣇⣾⣿⣿⡏⠀⠀⢀⣀⣀⣀⣀⣵⣿⣿⣿⣶⣧⣤⡄
⠀⢸⣿⣿⣿⠛⢿⣿⣿⡿⠛⠛⢿⠀⠀⠀⠘⣿⣿⣿⠉⠰⣿⣷⣾⣿⡤⠀⠀⠈⣹⣿⣿⣿⣷⠀⠀⠈⠉⠙⠉⠉⢹⣿⣿⣿⢿⡏⠉⠋
⠀⢸⣿⣿⣿⠀⢸⣿⣿⠃⠀⠀⢸⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⠈⠃
⠀⢸⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⠏⠀⠀⠹⠿⠿⠿⠀⠀⠀⠀⠙⢻⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⣹⣏
⠀⠀⣿⣿⣿⠀⠸⡟⣿⠀⠀⠀⣿⠀⠀⠀⢀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⡟⡿
⠀⠀⣿⣿⣿⠀⢀⡇⣸⠀⠀⢨⣿⠀⠀⠀⢿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠏⣇
⠀⠀⢿⣿⡿⠀⠈⠁⠈⠁⠀⠀⠸⠀⠀⠀⠀⠀⠘⠶⠤⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠸⠃⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢠⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠀⠀⠀⣾⣿⡗⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠘⣿⣿⣿⢿⣿⣿⣿⡇⣾⡿⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⣿⣿⡇⢈⣽⣿⣿⣶⣿⡇⠀⠀⠀⠀⢀⣴⣿⣯⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠐⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣼⣿⣷⡄
⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣻⣿⣿⡄⠀⠀⠀⢰⠛⣿⣿⠙⠦⠖⠉⠰⡇⣿⣿⡇⣳⠀⠀⠀⠀⠀⠀⣿⣿⣏⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢈⣿⣿⣿⠁⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠙⠉⢹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣴⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⡿⠿⣿⣿⣿⠿⡗⠀⠀⠀⠀⠐⢿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⡏⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⡇⡀⣿⣿⡇⢸⡇⠀⠀⢠⣤⣤⢸⡏⡇
⠀⠀⠀⠀⠀⠀⣸⠃⠀⡿⢠⣶⣶⣦⡄⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⣇⠀⠀⠀⡧⡇⢹⢸⠃⠘⡇⠀⠀⠸⣿⡏⢸⠁⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⣇⠘⠛⠛⠛⠋⠀⠀⠀⡇⠈⣇⡀⠀⠀⠀⣄⣀⠀⣇⠸⠀⠀⠀⡻⠇⠾⠸⠆⠤⡇⠀⣠⣶⡧⠈⠉⠔⠃
⠀⠀⣀⠀⠀⠈⡁⢀⠀⠉⠀⣀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠈⠈⠁
⣿⣏⢭⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⠁⢸⡇⠀⠀⠉⠙⠛⠓⠚⠛⠛⠋⣸⠏⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠛⠷⢶⣤⣤⣤⡴⠶⠛⠁⠀⠀⠈⣠⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⠀⠀⠀⣘⢷⡒⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢹⣧⣤⢀⡈⢉⣙⡛⠀⠐⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠂⠀⠀⠀⠀⠴⠀⠈⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡂⠀⠀⠀⠀⠀⠀⠀⠄⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⣤⣤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡝⠂⠀⠤⠍⠻⠀⢿⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢛⡛⠀⠀⠀⠀⠀⠀⠄⢨⣄⣠⣩⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡄⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⠆⠠⠀⢤⠄⠀⠀⠀⣉⣸⡁⠀⠀⠐⠀⠒⡗⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⠁⡆⠀⢸⠆⠦⠀⢠⡌⣿⠀⠀⠀⢐⣟⠀⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⡏⣤⠀⠀⣿⡖⠂⠀⠰⢷⡿⠀⠀⠀⢸⣿⡉⣿⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢣⡌⠀⢀⣙⣛⠀⠀⣳⣾⠇⠂⠀⠓⠚⣿⠗⣿⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⡈⠁⠀⢀⣉⣭⡀⢀⢻⡿⠀⠀⠀⠀⠀⣿⣿⢷⠀⠸⣧
⠀⠀⠀⠀⠀⠀⡙⢘⣿⡇⠛⠀⠀⠠⠤⠤⠀⠀⣾⠇⠀⠀⠀⠀⠀⠸⣿⢉⠀⢠⣿⡀
⠀⠀⠀⠀⠀⡀⠁⢼⣽⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠁⠀⠈⡁⠀⢀⣤⡴⠶⣦⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠁⣀⣼⠋⠠⣴⡿⠛⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡆
⠀⠀⠀⡈⠀⠀⠀⠀⠀⢻⣿⣶⣶⣯⠀⠄⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣧
⠀⠀⢀⡁⠀⠀⠀⠀⡀⡀⠉⠉⠉⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠈⠉⠉⠛⣇⠀⢀⣠⣶⣶⣶⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠙⢃⠀⠙⠛⠿⠛⠋⠀⢀⣦⣨⣀⣀⣀⡀⠀⠀⠀⣀⣠⠛⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢠⣀⠀⠀⠀⢠⣾⣿⣷⡄⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠉⠁⠀⢀⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⣾⣿⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣤⡘⠋⣁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠿⠁⠀⠀⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠈⢸⠇
⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⣸⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡟⠀⠇⠀⣿⣿⣧⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣅⣀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠂⢻⣿⣿⣿⣤⣤⣶⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⢿⡿
⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡖⣭⠛⣿⣷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡼⠇⠉⠀⢠⡟⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡅⠀⠀⠠⠤⠸⡶⠎⠹⣧⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠆⠀⠐⠻⠂⠑⢺⡿⣿⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣄⣄⠀⠀⠀⢈⢁⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢀⠀⠀⣬⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠶⠊⠁⠀⠀⡿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢰⠀⠀⠀⡴⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣴⡆⠀⡏⠀⠀⠈⠀⣤⣄⣀⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣻⡏⢳⡀⠀⠀⠀⣠⠞⠋⠉⡝⣿⠛⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡿⠀⠀⢷⡄⠀⠞⠁⠀⢀⠶⣨⠃⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡼⡇⠀⠀⠘⠟⠀⠀⠀⢀⢾⣻⡇⠀⠀⠀⠚⣹⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢟⣻⣟⣝⣋⡹⣙⢍⡫⣝⢻⢎⡽⠀⣠⣷⡀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢻⣿⡿⣷⣦⡆⠢⠸⠻⠷⢷⣼⣡⣽⣾⣿⣿⡟⢄⣠⡤⡟⢆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠁⠀⠀⠀⡩⠁⠈⣿⢯⢘⣈⡁⣃⣘⣉⡕⢾⣿⣿⣿⠀⠀⠹⡼⠀⠈⠳⢤⡀
⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⢀⠠⠄⠊⠀⠀⠀⢸⡆⡂⡆⠆⠰⠸⠰⠆⣻⣿⣿⠇⠀⠀⠀⠘⢆⠀⠀⠀⠈⠓⢄
⠀⡔⠁⢀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢧⠥⢭⢭⢨⠉⡩⡋⢽⣿⣟⣤⢞⡧⡤⣄⢀⡙⠦⡀⠀⠀⠀⠑⢤⡀
⠀⡄⡐⠻⠚⡆⢤⣀⣠⡤⠔⠀⠀⠀⠀⠂⠀⠀⠀⠓⠒⠒⠲⣶⣦⣍⣾⠿⣿⣿⣤⡶⣞⠛⣁⡏⠀⠀⠙⠒⠤⢄⠀⠉⠠
⠀⣿⢷⣾⣷⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⠖⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⢀⠆
⣼⡏⣿⣿⣿⠂⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⣠⣤⠖⢋⡁⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠈⢻⣿⡍⢃⡴⠂
⠙⣧⣌⢟⣛⠉⢀⡒⠀⠀⠀⠀⠀⠁⠀⢘⡉⢀⡔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣀⠄⠀⠀⠘⢻⢿⣿⠇
⠀⢩⣿⢻⣍⡑⣋⣉⠓⠒⠤⠠⢤⡀⡴⠼⠀⢉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠊⠀⢀⠁⣿⡏
⠀⠈⣿⣧⣿⡠⣡⢽⣖⣿⠻⠦⠰⠐⠒⣶⠷⣬⠾⣭⣾⣭⣯⣽⣏⣱⣉⣾⣐⣲⡞⢺⣂⣞⣳⣏⣲⣏⣿⣽⣿⣿⠃
⠀⠀⢹⣿⡾⣧⢹⠮⣿⠼⣧⠻⡀⣳⠀⣺⠄⣺⡍⣿⢛⡄⢸⠠⢿⡥⣾⢩⡟⢽⠍⢽⠏⣿⣭⣿⢽⡯⣿⣼⣿⡟
⠀⠀⠀⢻⣿⣽⣆⢫⠸⡧⢻⡇⣷⣸⡦⢼⣖⢺⡇⢻⡷⣷⢼⣎⢽⣏⣿⢰⡏⢹⠄⣽⢛⣿⢲⡿⣾⢯⣏⣿⡿
⠀⠀⠀⠀⠫⣷⣨⡍⠡⠯⠄⣿⡸⣷⢷⣚⣷⡐⣿⠘⣇⢸⢘⣀⢸⡇⣿⢸⡏⣽⡆⣿⢺⡿⣾⡇⡿⣾⣽⡿⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣾⣽⣌⣋⣹⠳⠾⠐⢻⡄⢿⠸⠎⣿⢸⡇⣿⣸⡇⣾⢰⡟⣻⠧⣿⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠷⠾⠷⠾⡶⣿⣾⣿⣿⡿⣷⡿⣾⡿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⠀⠀⠀⠈⠀⠠⡀
⠀⠀⠀⠀⣄⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠒⠀⠤⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣜⠀⠀⢰⣼⠀⠀⠀⣦⡀⠀⠀⠀⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡈⠓⠀⠨⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠸⢩⠀⠀⢸⠈⠀⠀⠀⠁⡇⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠈⠈⠀⠀⠧⢀⠤⡀
⠀⠀⣀⠠⠼⠦⠤⠼⠴⠦⠤⠤⠤⠧⠤⠄⠠⠄⠥⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⡀⠀⠀⠈⠩⢀⠃⠀⠀⡠⠁
⠀⠀⣇⣀⠀⠀⠀⢀⡀⢀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠃⠀⠀⠀⠀⣁⠤⠐⠈
⠀⠀⠙⡇⠉⠒⠊⠁⠀⠀⢄⡼⠀⠈⠑⠒⠉⠀⠑⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⡠⠔⠂⠀⣀
⠀⠀⠀⡇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠴⠖⠉⠀⠓⢄
⠀⠀⠀⡇⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠔⠊⠀⠀⠀⠀⡆
⠀⠀⢠⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡡⠊⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠒⠒⠲⠶⢶⠒⠒⠒⠲⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⠄⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣠⠀⠀⠈⠉⠘⠒⠒⠒⠠⢤⠔⠈⠀⠀⠀⠀⠀⢀⠤⠊⠉⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠠⢝⠢⠭⠤⢤⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠉⠉⠉⠒⠢⢀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣏⣙⠒⠦⢤⣀⡀⡠⠐⠁⠀⢀⣀⠀⣀⣀⣀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠉⠙⠒⢦⢔⠉⠀⣀⣤⣾⣏⡄⢩⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠑⡂⠀⠀⠀⠀⢸⢨⣥⡇⢠⠄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣪⡿⢁⣤⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣌⡑⠦⣅⡂⠸⣇⣘⠃⠸⡎⡟⣀⣤⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⠧⠴⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣳⣦⣍⡛⠷⣍⣦⠛⣼⢸⢱⢦⢀
⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢉⣠⣤⣶⣶⣶⣦⣤⣀⡉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣍⣓⢧⣈⣀⢀⢘⠠
⠀⠀⠀⠀⣼⣿⠟⣋⣅⣈⠿⢿⣿⡟⣋⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣛⣷⣶⣤⣌⣉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣻⢿⣘
⠀⠀⢀⣤⣿⣷⣿⣲⡗⠻⠀⠀⠀⠈⠉⠙⠛⠻⠿⡿⢿⠋⠉⠉⡟⠛⠛⠿⡿⢿⣋⣩⣤⣾⣿⣿⠶⠦⣬⣉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⡀
⠀⣾⣿⣿⣿⢿⡟⢽⡇⠀⣯⣿⣷⡆⠀⠀⠀⠀⠀⡧⢺⠉⠁⠉⠟⠀⠀⠀⡦⠀⠀⢽⣉⠙⢻⠶⠾⠿⣿⣿⡿⢿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣯⣟⡇
⠀⣿⠛⢿⡀⢹⡇⣺⡇⠀⣛⣻⣿⡇⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⢸⠀⠀⠸⡿⠶⢶⣿⣿⣗⠀⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⣷⡇
⠀⣿⠀⢾⣇⣸⣧⣾⡇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡿⡄⢰⣿⠛⣶⠀⠀⠀⡇⠀⠀⣿⣿⣿⢀⣤⠰⠦⡄⠀⠀⠀⢀⡀⠈⢹⡇
⠀⣿⠀⢸⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡃⣀⣀⣀⣀⣀⣿⣿⣿⣀⣿⣿⡃⢠⣿⡄⣷⣤⢴⣿⣿⣿⢸⣿⣾⣠⣾⡰⠀⣿⣯⢉⡄⢸⡇
⠀⣿⠁⠨⠿⠷⠖⠚⢻⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣷⢀⣴⣶⣶⣶⣶⣯⣭⣭⣭⣭⣭⣭⣿⠀⠀⣿⣿⣿⠸⡏⣿⠏⢿⠃⠀⡟⢹⠛⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡀⣿⣿⡀⣸⣀⣤⣿⢿⡀⣧⣼⣷⣤
⠾⠿⠿⠿⠾⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⢸⣷⣿⣏⠃⣿⢸⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣘⠛⢨⣭⣭⣭⣴⣶⠶⠶⠖⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿