Goblet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠶⠖⠒⠒⠛⠛⠛⠛⠉⠉⠉⠛⠛⢛⣓⣾⣶⣶⣶⣶⣶⡶⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠜⠿⠻⠿⠟⠟⠁⠀⠲⠶⠶⠳⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠛⣿⣿⣿⣿⣿⣏⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠠⣶⣿⣿⣿⡿⡿⠛⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡉⣿⣿⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣽⣿⣿⣿⣿⢹⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⠻⣿⣿⣧⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡛⠿⣿⣟⣋⡀⠀⢀⡀⠤⣤⡠⢒⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣋⣙⣃⣠⣰⣶⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣟⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣴⣶⡾⠿⠿⣿⡿⠿⣿⣿⠻⠇⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⡭⣏⣷⣖⠐⢠⡆⢀⡻⣭⠃⢬⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣤⣵⣦⣹⠃⠻⣾⣼⣿⣷⡋⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢳⣿⣿⣧⣿⣿⣿⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣷⣦⣿⣤⡱⡇⢶⣧
⠀⠀⠀⠀⠀⠠⢀⢻⣶⣶⣤⡀⡀⠂⣤⠀⢄⣲⣚⣂⡟⣽⠿⢯⡀⢓⣦⡂⣽⣧⠤⠷⣂⢔⣀⠀⠀⠀⠈⠀⠠⠊
⠀⠀⠀⢀⢤⣾⣟⣿⣿⣿⡿⣛⣁⣻⣿⡧⣿⣿⣿⢏⣴⣿⢵⠹⢹⢻⡏⢅⣷⡻⣶⡂⠼⠀⢇⣐⣒⡁⠲⣄⠀⠂⠔
⠀⠀⠀⠤⢾⣿⣯⣽⡿⢡⡞⣭⣾⣿⣿⣿⡿⠛⢿⣿⣏⠘⣣⠂⢃⡀⠳⣼⢰⣕⡊⠿⡔⠾⣟⣉⠉⠉⠳⡄⢳
⠀⠀⠀⠫⣽⣽⣿⣿⣇⢨⣛⢛⣻⠟⣫⣵⣶⣿⣻⣻⣿⢀⠀⣠⣾⣿⣦⡡⠄⣿⣿⣷⣶⣤⣌⢩⣛⣿⣓⣁⢸⡧⡤⠤
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡙⢧⣿⢸⣿⠏⣿⣼⢋⣿⡟⢠⣦⢿⣿⣈⣿⢷⡄⢿⡛⠦⠙⠟⢻⡇⢻⣩⠽⢣⣿⠃⣿
⠀⠀⠀⠀⠀⠴⠟⣿⣿⣿⣿⣦⣍⣈⠛⠶⠿⠿⢿⣛⡐⣎⢯⣛⣙⣷⡬⢞⣟⣲⣬⡛⠲⠶⠚⠀⡨⣥⣾⣿⠟⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠻⢿⣿⣿⣟⣳⣶⡶⢂⢉⣭⣭⢥⡄⢉⡀⣠⢬⡥⣍⢐⣀⣌⣷⣼⣷⠿⠟⢋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠬⣍⣛⠛⠻⠋⠿⠟⠷⠾⠶⠷⠼⠶⠿⠯⠿⠽⠗⠛⢋⣋⣩⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠀⠄⠰⠐⠶⠒⠒⠒⠛⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠴⡢⠴⣿⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣤⣤⣤⣄⣀⣀⠀⣀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠉⠉⠁⠶⠇
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣯⣽⣦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣿⣿⡿⡷⠓⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⡳⡖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢨⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣬⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣤⣶⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⣿⢦⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⡀⠀⡀⣀⠀⣀⠈⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡟⣻⡏⠉⠙⠋⣛⣛⢛⣳⣶⡦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣽⣿⣱⠟⠓⠀⠀⠀⢐⣈⠨⠥⢈⡋⣃⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠿⣿⣷⣿⣽⡀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣷⣿⣿⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⢬⣽⣿⣿⣷⣶⣶⠶⠶⣶⠶⢶⣶⣾⣿⣿⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠉⠛⠛⢻⣟⠿⣟⣟⣛⣛⡛⢻⠯⣉⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠐⠂⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⢸⠅⢻⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠄⠂⠀⢸⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠐⠀⠀⠭⠥⠤⠗⠀⡖⣀⣇⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠉⠀⠀⢸⣥⠠⠏⣣⡽⣉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣛⣂⣀⣺⣯⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠩⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡅⠀⣿⣿⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢉⡴⠂⢸⡃⠀⣼⣿⡴⠋⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠀⠈⠳⢤⢿⡿⠀⠀⠈⠐⠁⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠈⠀⠀⠀⢀⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⢻⡛⣻⢟⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣤⣷⣾⣷⣾⣤⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡅⠀⠀⠐⢾⣿⡁⢈⣿⡧⠄⠀⠀⢨⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⡿⢿⡿⡟⠛⠂⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠀⠀⠀⢁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢄⠄⠀⠀⠤⢄
⠀⠀⠀⠀⠊⠀⠀⢠⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⡀⠘⠄⠀⠁⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣒⣉⣀⣤⣤⠤⠶⢶⠶⢶⡶⠶⡶⠶⠤⣤⣬⣉⣉⡲⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡗⢭⠃⢟⣠⠾⠀⠀⣶⡞⢷⣶⠀⠀⠷⣄⡻⠘⡭⢻⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⢷⠈⣧⠘⠛⢢⠞⣿⢣⡄⣠⠞⣿⠳⡔⠛⠃⣼⠁⡿⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠹⣦⠄⠀⣧⡛⠈⣳⣀⡄⢠⣄⣞⠁⠓⣼⠀⠠⣴⠏⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠙⠦⣀⡈⠧⠀⢹⢿⣦⣴⡾⡍⠀⠼⢁⣀⠴⠉⠀⠀⢠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠄⠀⠀⠀⠀⠈⢻⠷⠤⠟⢯⣅⣬⡽⠻⠤⠾⡟⠁⠀⠀⠀⠀⠠⠀⡠⠂
⠀⠀⠀⠀⠀⢠⠀⠴⢃⠄⠉⠂⠀⢀⠀⠀⠀⠈⠐⢆⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⡀⠀⠐⠉⡄⠳⡄⠀⡆
⠀⠀⠀⠀⠀⠈⢂⠀⠀⠊⠀⠆⡆⢀⠄⡠⠀⡍⠓⠀⠉⢗⡐⢚⡿⠉⠀⠚⢩⢀⠀⠀⡄⡆⠈⠣⠈⠂⢀⡐⠁
⠀⠀⠀⠀⠀⠀⢌⡈⠂⠄⡀⠀⠁⠈⣄⠃⠀⠀⠑⠀⠀⢸⠏⢹⡇⠀⢀⠈⠀⠘⠂⠁⠐⠈⠀⢀⠠⠐⢁⡱
⠀⠀⠀⠀⠀⠀⠀⠉⢡⠀⠀⠀⢹⡤⠄⠀⢒⡆⠀⠃⣠⠟⠃⠘⠻⣄⢈⠀⠰⡒⠀⠠⢤⡏⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢀⢄⠀⠀⣷⣠⣠⠆⢈⠤⠿⣟⣱⡀⢀⣎⡻⢿⠤⣁⠱⣄⣄⣾⠀⠀⡠⡀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠈⠈⠀⢀⡼⠊⣏⢻⣴⣿⣦⣤⣁⣀⣈⣁⣀⣠⣤⣴⣿⣦⡟⣹⠁⢧⡀⠀⠁⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠧⠄⠀⠒⠂⠀⠦⠃⠰⠇⠱⠿⣽⣟⣻⡟⢿⡟⢻⣟⣻⣟⣻⡯⠿⠎⠸⠆⠘⠴⠀⠐⠒⠀⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢤⣤⣤⣤⣤⡤⠤⡤⣤⠤⡤⣤⢤⢤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⢀⢏⢣⡟⠀⡰⠡⠃⡀⠀⢀⠎⡜⣼⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠌⠄⡸⠀⠠⢁⠀⡀⠀⢀⠂⡜⣜⢎⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⠁⠠⠁⠀⠠⠡⠀⢀⠌⣼⢊⠎⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⠀⠃⢀⠂⠀⠠⠁⠀⢀⠌⢌⢂⠎⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠄⠀⠀⠀⠀⢀⠎⢌⢂⢂⠂⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⠀⢀⠎⣌⣄⣦⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⢉⡉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⠐⣢⣘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠞⠁⠈⠳⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣭⣭⣭⣍⣩⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠌⢍⠉⠉⠛⢛⡛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⣛⠛⢉⠉⡩⠁⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠡⢀⣺⠿⢮⡢⡀⠀⡀⢀⣠⣦⡄⢀⠀⢀⢔⣵⠻⣦⡀⢰⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣁⠔⡀⠹⣦⣴⠟⠁⠀⠢⠘⢿⡿⠃⡀⠀⠈⠻⣦⣴⠏⢀⠸⡈⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⢦⣤⣬⣥⣤⣤⣤⣤⣅⣶⣖⣈⣤⣤⣥⣤⣬⣥⡤⠴⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢰⡆⠀⠂⠀⠈⠉⠉⠉⠉⠉⠍⠉⠉⠀⠀⠀⠀⠀⡦⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠇⠀⢀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⠬⠄⡀⠀⠄⠀⢠⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⢦⣤⣤⣤⢤⣤⣶⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⣻⣏⣓⡀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡀⡀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡐⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢃⠁⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⠀⠀⠀⢀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠤⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢆⠀⢉⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⡔⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣶⣶⣤⣬⣤⣀⣀⣂⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣴⠒⠩⣝⠋⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠐⠒⠒⠤⢤⡍⠉⢐⠄
⠀⠀⠀⠀⠀⠀⠈⠙⠉⠚⠛⠛⠿⠷⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠖⠛⠋⠭⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢡⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢣⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠃⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣣⡀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⣀⣀⣀⢀⣼⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠡⣤⡜⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠂⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⠁⠀⠀⠡⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⠒⠒⠒⠂⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡆⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣧⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠀⠉⠉⠃⠀⢀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⢸⡿⠿⠿⠿⠿⠟⠇⠀⠀⠘⠋⢩⣿⣿⡟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠃⠀⠀⢀⣤⣿⣿⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣾⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡄⠀⡀⢀⣼⣿⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡷⢥⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⣿⣿⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⣼⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⣿⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠙⠛⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⢺⣿⣀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⣷⣼⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣸⣿⣿⣿⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠙⣿⢿⣿⣿⣿⣆⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⢋⠌⢀⣿⣿⣿⣿⣿⣿⡣⡀⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣯⣴⣤⣤⣤⣍⣄⣈⣁⣬⣭⣭⣤⣲⣤⣤⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣭⣿⣿⣿⣿⣿⣷⣿⣿⣦⡶⠶⠶⢿⣿⣿⣿⣾⣿⣿⣿⣭⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠶⠛⠻⠹⠿⠹⠿⠿⠻⠏⠙⠓⠙⠛⠁⠀⠀⠀⠘⠘⠛⠙⠓⠙⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⠏⣰⣆⠹⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠋⠈⠛⣿⣿⠋⣴⣿⣿⣦⠙⣿⣿⠟⠁⠙⢿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣦⣀⣴⣿⣿⣦⠙⣿⣿⠋⣴⣿⣿⣦⣀⣴⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧⠘⠃⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣾⣭⣽⣿⣻⣯⣭⣷⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⣿⣿⣿⣾⣷⠛⣿⣏⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣥⣝⣧⣽⣿⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡯⠘⣽⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠈⣿⣿⣿⣶⠖⠖⠒⠲⠶⠶⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⠒⠂⠲⠖⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⢶
⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢾⢫⣾⣿⣿⣿⣿⣟⣿⣿
⡰⣟⣻⣿⣷⣶⣒⣶⣶⠶⡶⠶⠶⠶⣶⡶⢶⣶⡶⢶⢶⢶⣶⣶⣶⣶⣦⠲⣶⣶⣶⣶⣶⣶⣶⠶⠷⣶⣿⣿⣿⡿⢻⣿⣿⠛⣻
⠻⣿⠿⢿⠿⠿⣿⣯⣿⣿⣿⣿⡶⣮⣮⡿⣮⡯⡵⠼⣿⣿⡿⠿⢿⠿⠿⣿⡿⣿⣿⣿⣿⣇⡑⣿⠿⠟⣿⣿⣿⣷⡿⡿⠿⢿⠿
⠀⠘⢧⠩⢸⡱⢡⡿⠟⣴⣿⡷⠾⠷⠛⠬⢿⣖⣥⢴⣄⡉⠇⠤⠀⠀⢈⡥⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠉⠹⠥⢸⠉
⠀⠀⠀⠫⢄⠑⢨⣿⣦⣾⣿⣷⡗⠀⠀⠀⠀⠈⠛⢾⣶⣿⣳⣄⣢⣾⣿⣦⣾⣿⡛⢿⣾⣿⣿⣿⣿⣟⣿⣷⣼⡇⢀⠖⠛⠁
⠀⠀⠀⠀⠀⠙⢺⣋⣭⣹⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠻⡿⢛⠛⠙⠛⢻⣿⡇⢺⣆⣾⣿⣿⣿⣿⣿⣿⣿⣼⣩⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣬⡻⢿⣿⣷⣤⡄⠀⠀⠀⠀⠸⣽⣟⡛⢛⣿⣿⡑⢭⣿⣿⣿⣿⣿⣿⣞⣵⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢽⡻⢿⢿⣦⣀⠀⢀⣀⣨⠉⠉⠡⣃⣀⣩⣿⣿⣿⡿⣟⠯⠚⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⣽⣿⣿⣶⣿⣶⣶⡶⣷⣾⣿⣫⣥⠚⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡙⢋⠙⡀⠀⢁⡈⢑⡁⢈⣉⣙⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢙⡿⡡⣕⠱⢯⠚⠄⠛⣾⠟⠛⡳⣿⢯⡿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡵⣶⡂⠿⣿⣼⢀⡀⡄⢠⡼⡀⣾⣿⣿⢺⢳⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⡏⣷⡐⢠⣍⢼⣷⣴⣢⣾⣯⣿⣥⣿⣆⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣀⡐⢀⣼⣿⣀⣘⠩⣻⣿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣙⡟⡉⣁⣟⣿⣁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⠭⠿⢭⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠤⣿⠄⠀⠼⣧⠼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣞⣿⡛⠒⠛⡦⣖⢺⠟⠾⢿⡋⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠞⠛⠀⠀⠀⢿⢛⡿⠀⠀⠐⠻⣿⣍⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⣶⣾⠋⢁⠀⠀⠀⠀⡴⠋⠽⢿⣦⠀⠈⠙⢌⡿⣷⣦⣞⠉⢶⣄⡀
⠀⠀⢠⣄⢠⣄⡤⠘⠂⣤⠟⡽⠶⢋⣹⣷⣄⡀⡀⠠⣴⣶⠄⠊⠉⠠⢌⠝⣶⣄⠀⣁⣀⠊⣹⣿⣿⣬⣇⣐⣦⣳⣀⣤⣰⣄⣆

⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⢻⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⡦⠀⠀⠀⠀⢴⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡟
⠀⠀⠀⠀⠘⣷⠶⠶⠶⠶⠶⠶⠶⢶⡦⠀⣠⠞⠋⠀⠀⠀⣠⣄⡀⠀⠈⠙⢦⡀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⣾⠇
⠀⠀⠀⠀⠀⢻⣤⠖⠒⠶⣄⣀⣀⡾⠁⣼⠋⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⢻⣆⠘⣧⣀⣀⡤⠖⠒⠦⣤⡟
⠀⠀⠀⠀⠀⠈⣷⣀⣀⠀⠈⠉⢹⠃⢸⡏⠀⢀⣴⣶⣦⣽⠿⠿⣿⣤⣶⣤⡀⠀⢿⡀⢹⡍⠉⠀⠀⣀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣯⠉⠓⠶⠶⢿⠀⢸⡇⠀⢸⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⡗⠀⢸⡇⢸⡗⠶⠶⠋⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⡀⠀⣸⡆⠘⣧⠀⠀⠉⠛⠋⣿⣿⣿⣯⠙⠛⠉⠀⢀⡿⠀⣸⡁⠀⢀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠋⠁⠙⣆⠙⣦⡀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢀⡾⢁⡼⠃⠉⠛⠉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⠛⠛⠉⠁⠈⠻⣦⣄⡀⠀⠉⠉⠀⣀⣠⡶⠋⠀⠈⠙⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠶⠶⠶⠶⠶⠶⠶⠶⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠦⠶⠶⠶⠶⠶⠶⠴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣤⣤⣤⣤⣤⣤⣤⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣻⠏⣾⠀⢸⡇⠘⣇⢹⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡟⢀⡿⠀⢸⡇⠀⣿⠀⢿⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⠃⢸⡇⠀⢸⡇⠀⢻⡄⢸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠘⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⢸⠃⠀⢸⡇⠀⢸⡇⠀⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠸⡇⠀⠘⣧⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣀⡼⠃⠀⢀⡿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠙⢦⡀⠘⢧⡀
⠀⠀⠀⠀⣀⣤⡤⣶⡶⠶⠞⢋⣴⠞⠉⠀⢀⣤⠞⠁⠀⠀⢸⡇⠀⠀⠘⢷⣄⡀⠀⠙⠳⣤⡙⠲⠶⢶⣶⢤⣤⣀
⠀⠀⠠⣞⡉⠀⣼⠋⠀⠀⢰⠏⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⢻⡄⠀⠀⠹⡆⠀⢉⣳⠄
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠉⠉⠀⢁⣀⣀⣀⣀⠀⣀⣄⢀⣀⣀⣀⣀⡉⠈⠉⠉⠒⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⡒⡖⠛⠛⠉⢈⡩⠋⠘⠒⠂⠘⢃⠀⠒⠣⠙⠍⡁⠩⠛⠛⢖⠒⠤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡓⣠⠃⠰⢀⠀⠂⡠⠀⢀⣦⡀⣶⣶⢀⣴⡀⠀⢄⠐⠀⡐⠆⠘⣄⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠐⠆⠂⠤⠄⠖⡠⢂⣯⣜⠿⣿⣿⠿⣡⣿⠑⢄⠲⠠⠄⠐⠸⠂⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡈⠹⠀⠸⡄⢰⡀⢈⣴⣿⣷⣼⣧⣾⣿⣦⡁⢀⡆⢠⠇⠈⠋⢁⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠃⠀⠀⠀⣸⠁⠀⣶⠛⣫⣾⣷⣝⠛⡶⠀⠈⡇⠀⠀⠀⠘⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠻⡄⢀⢉⠿⣯⣿⣟⣽⠾⡁⡀⣠⠟⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣷⣄⣸⠿⠝⠟⠋⠙⠻⠊⠻⣅⣡⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⡉⠉⣉⠉⠉⠙⠋⠉⠉⢉⠉⢉⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⡀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡃⠀⠀⢚⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠅⠰⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠀⠘⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢺⠅⠈⠆⠀⠀⠲⠁⠨⠗⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠛⠀⠈⠁⠀⠁⠀⠑⠎⠀⠈⠀⠊⠁⠀⠛⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⢥⠒⡡⡤⢀⡀⢀⣀⠀⢀⡀⠀⣀⡀⣀⡀⠦⢌⠒⡬⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠭⣜⣖⣊⣃⠐⠫⠀⠜⠃⠀⠉⠃⣘⣑⣒⣧⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢻⣿⣶⠖⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⠖⠶⠒⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⣼⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣨⣠⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣿⣿⣿⣭⣭⣜⣿⣿⣟⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡏⢿⣿⢿⣿⣿⣧⣿⣿⣷⡾⠇
⠀⠀⠀⠀⠀⠀⠘⢯⠻⣿⢻⡿⢯⣿⣿⠿⠿⠽⢿⣻⣷⣍⡛⠍⠉⢝⡭⣾⣿⣿⣿⣿⣿⣿⣾⡿⢿⡝⠿⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣻⣿⢾⣿⣿⣇⠀⠀⠀⠉⠻⣾⣿⣷⣿⣿⣷⡿⡿⢿⣿⣿⣿⣿⣿⡿⡿⢇⠝⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢷⣻⣿⣿⣶⣀⡀⠀⠀⠙⣧⣦⣤⣴⣿⢷⣿⣿⣿⣿⣿⣿⣿⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣄⠀⢀⣩⠛⠗⣿⣍⣹⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⢾⡷⢷⣿⣾⣿⣽⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣬⢥⣤⣠⣤⡤⣤⣤⢤⣽⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣧⡟⣾⣧⣏⡥⠄⣘⡇⣴⣿⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⡖⡽⡭⢷⣷⡴⣿⣿⣽⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢿⣥⡾⠿⣦⣵⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣶⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢠⡀⠀⣠⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠿⠾⣧⣦⡿⢿⣿⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠋⠀⠀⢹⢿⡇⠀⢙⣻⣯⡓⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣈⢿⣟⠀⠀⠀⢀⣘⣥⣼⠿⠄⣀⠙⠛⣻⡷⣥⣽⣢⡄⣀⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠿⡿⠿⠆⠛⠓⠋⠀⠚⠛⠛⠐⠈⠙⠋⠀⠁⠈⠉⠙⠙⠲⠻⠶⠿⠿⠿⠿⠿⠷⠿⠿⠿⠧

⠰⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠲⠶⠶⠶⠶⠶⠶⣶⣶⣶⣦⣶⠶⠶⠶⠶⡶⠶⠶⠶⠲
⠀⠇⠀⠀⠀⠀⠀⢀⣛⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⡇⠀⠰⠘⣧⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣈⣋⡁⠠⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢺⢿⣿⣿⣿⣿⣷⠀⢰⡷⠞⠃⣀⠀⢸
⠀⡄⣀⡀⢀⠀⣠⢏⣿⠁⠀⠀⣀⡀⠀⠸⣄⣴⡀⠔⠉⠉⠀⠀⠸⡃⠀⠀⣀⡀⠀⠀⡀⢀⣶⣤⣼⣿⣿⣿⣿⣿⢿⣄⡸⢳⡞⠛⠋⠀⠸
⠀⣿⣿⣇⣼⣆⢠⡄⠙⣊⠀⠁⠉⠁⠀⠁⠃⠀⢀⠼⠉⠁⣠⠶⠖⠁⢰⡎⠛⡇⠀⢰⡿⢻⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⠷⣈⡁⡾⢋⠁
⠀⢻⣆⠋⣀⢈⡧⠀⣴⣟⢞⡀⠠⠖⢠⠀⠀⠀⠀⠀⠀⠀⢽⣦⣶⠄⢸⣥⣿⠃⢸⣅⣉⣉⣻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠑⠈⠒⡇
⠀⠈⣮⠻⣯⠘⠀⣤⢋⡿⣮⣷⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⠉⢱⡄⠻⠊⣝⢢⡾⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣤
⠀⠀⠙⣄⠀⢀⣈⠆⢨⡿⡣⠈⣇⠀⠀⠸⠃⠀⠀⣀⠀⠀⠈⠃⠈⠁⠀⠀⠹⣿⠃⢠⣿⢏⣿⣿⣿⣿⣿⣿⣾⣿⣿⢻⣿⣿⣷⠒⠺⠏
⠀⠀⠀⠙⢦⡉⠁⠈⠶⠇⢦⡱⠌⠋⠀⠁⢀⠂⠀⠉⠀⠀⠀⠘⠛⠀⠶⠀⢸⢇⡐⣼⡿⣎⣻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⠻⠃⢠⠄
⠀⠀⠀⠀⠀⠙⢦⡀⡀⠈⠻⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⢻⢿⡿⣾⣿⣿⡿⡿⠿⢿⣿⣿⣿⡿⣫⣶⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⠙⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠇⠹⠬⠃⠈⠁⠁⠈⠀⠀⠀⠀⠀⠈⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢅⣀⡄⠀⠀⠀⢀⠀⣠⠖⠉⠉⠙⠳⢦⠀⣤⡶⠶⠶⣤⣴⣶⣟⠿⠟⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠄⠀⠈⢲⡄⠀⠀⠀⣀⠀⠀⣿⣯⣤⠄⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡀⠀⠸⡇⠀⠀⢠⣹⣦⢸⣿⣿⣿⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡁⠀⠀⠈⠘⠛⢸⣿⠿⢏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊⠛⢣⠘⠟⠃⠂⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣈⣀⢈⣉⣁⣀⣀⣀⣈⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠙⠉⠉⢩⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠠⠤⢼⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡜⠀⠀⠀⢻⡉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣾⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢠⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⣜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠆⠀⠀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⠛⠁⠀⠁⠀⠀⠘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⡿⠋⠙⠉⠉⠋⠀⠈⠉⠙⢻⣿⣿⣿⣟⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠙⠛⠛⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣄⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣤⣶⣾⣿⣿⡟⣿⣿⣟⣋⣉⣤⡤⠿⢬⡁⠀⠉⠐⠖⠛⠛⠙⠒⠛⠒⠒⠲⠶⠶⣶⣢⡤⢤⣤⠀⠀⠠⢄
⠀⢠⡆⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣤⣴⠶⠶⣤⣀⠀⠀⠀⣠⣴⣦⡀⠲⠋⠨⠷⠦⠘⠂⠀⢠⣾⠃
⠀⣿⣳⣦⣀⣈⣉⠉⢉⡉⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⢀⣰⣶⣶⣄⣻⣿⡷⠛⢷⡀
⢠⡏⣿⣯⡉⣿⠷⡦⠤⠤⠄⠀⣀⡀⣀⡀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⠖⠶⠿⣿⣿⣿⡿⣿⣿⣟⣛⠻⡿⢏⣴⡶⠄⠈⠃
⢸⡟⣿⣿⡇⢻⡀⢰⠐⠛⢛⣿⠿⣡⠌⠡⠀⠠⠄⡀⠀⠤⠠⠀⣤⠈⢹⡇⢠⡤⠠⡶⣟⡿⠟⠀⢤⣿⡟⠻⣿⣟⣿⡿⣻⠀⣳⡌⣹
⠈⠃⣿⣿⣷⣾⡇⠀⣠⡞⠻⠄⠂⣤⠒⠀⠀⠔⢢⣤⡀⠀⠄⠀⣿⠀⢸⡇⠀⠀⢠⡿⠋⡠⠀⠀⠀⢉⡹⠷⠿⣭⣿⣿⣿⣤⣿⣅⣽⣤
⠀⠀⣿⣿⣿⣿⡁⢀⣹⠎⣡⠖⠉⠀⠀⠀⠀⠀⠉⠀⠉⠷⣄⡈⠁⠀⠘⡏⢠⡆⠉⠰⠋⠀⠀⠀⠀⣼⣧⡖⠀⣹⣿⣿⣿⢻⣿⣿⣿⡏⠀⠀⢀
⠀⠐⠿⢿⣿⣿⠀⣼⠃⣼⠋⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⡇⢸⠄⠀⠀⠀⠀⠀⡠⠞⠉⠻⣿⠀⣿⣿⣿⣿⢸⣿⣿⣿⢁
⠀⠀⣴⣼⣿⣿⠀⡟⢸⣇⠀⠀⣤⠚⠀⠀⠙⣆⡀⠀⠀⠀⠀⠉⡆⠀⠀⡇⠘⠀⠄⠀⠀⠀⢰⠀⢠⡄⣀⡜⢿⣿⣿⣿⣿⢾⣿⣿⣿⢸
⠀⠃⠹⣿⠅⢹⣀⠷⢠⣿⢷⢸⡁⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡆⠀⠀⠀⢸⠀⢸⣷⢻⣄⣨⣼⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⢯⠀⢸⣿⣆⢸⣿⢀⣸⣇⡀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣿⠀⠀⡗⠀⠀⠀⠀⠀⠀⡌⠀⢸⣿⣿⣿⠿⢻⣿⣿⡿⣿⣿⣿⣋
⠀⠀⠀⢸⡆⢀⣷⠉⡐⢿⣿⣍⣷⠛⣶⡶⠤⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⡗⢠⠀⠀⠀⠀⠀⡇⠀⣇⣦⢸⣿⣆⣿⣿⣿⢇⣿⣿⣿⡟
⠀⠀⠀⠀⢵⠄⢙⠃⠀⢸⣿⣯⣿⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡃⠀⠀⣧⠸⠀⠀⠀⠀⢸⠃⢀⣿⣿⣿⡿⢫⣿⣿⣿⣼⣿⣿⣿⠃
⠀⠀⠀⡄⠀⡷⠸⠀⢸⡄⢉⠈⠉⢿⠀⠀⠀⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⢹⡟⣿⣿⣿⢣⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠆⣏⠈⣷⠈⡀⠀⠀⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⣼⣧⣿⣿⡷⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣹⡆⠸⣄⠃⠀⠀⢸⡀⠀⠄⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢁⣿⡗⠛⣽⣿⡿⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡄⠉⢰⠂⢡⠀⠂⠀⠈⢡⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆⠀⣦⣿⠃⢢⣿⡿⣡⣿⣿⣿⡏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⠀⠇⠰⠀⠀⠈⠀⠀⠀⠀⠸⡄⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⡇⠈⢨⡿⠀⣸⣿⢇⣼⡏⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣇⠀⠀⠰⣿⢠⣿⠀⠀⠀⠀⢀⡁⠐⢉⣤⣴⣿⠟⣼⡏⣴⠟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡶⠀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⢸⣼⣿⠀⠀⠀⠀⢸⣿⠿⠛⠋⠻⡉⠳⠋⣰⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢤⠐⡋⠠⠤⠂⠀⣠⣄⠀⢸⣿⣿⠀⠀⠀⠃⠁⣤⣄⣶⣯⡤⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠶⠄⣤⣤⣀⣀⣼⡛⣿⣤⣤⣤⣶⣾⣿⠏⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢣⣼⣿⣿⣿⣿⣿⢻⣿⣿⡿⣹⣿⡜⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⡵⢿⢁⣽⣯⢸⣿⡿⣽⣿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⣧⠀⢸⣿⡟⢸⣿⠿⢟⣉⣇⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣼⠿⣁⣴⠈⣷⢸⣿⠀⠛⣁⣩⣼⡶⡀⠰⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋⠀⠁⠀⢠⡾⠁⠈⠉⠹⠆⣯⣼⣿⣾⣿⣿⣿⣿⣯⢻⡄⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠣⠄⢀⡀⠰⣿⣇⠀⠀⠰⠖⢸⡏⢹⣯⠹⡟⣛⣿⣭⢿⣿⡇⠀⡀⢃⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⣤⢸⡖⠻⢏⣳⡀⠀⠀⠛⠃⠘⠿⠟⠋⠉⠀⠹⠛⠏⣠⣦⣄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠎⠀⠀⠈⠉⠁⠀⣴⢟⣠⡄⢳⡄⠀⠀⠀⠀⠸⠀⢁⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡦⠸⣷⣄⣄⠀⢉⡉⢾⣿⣿⡧⣌⠀⠀⠁⠀⣶⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠜⠿⠿⠿⣿⣧⣶⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡼⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣰⠇⠀⠈⢿⣿⣿⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣤⡴⠊⢁⠘⠁⠀⣰⠏⠀⠀⠀⠘⢿⣿⣧⠀⠁⠀⡀⠀⣤⡀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⢘⣻⡦⠀⠈⠀⠀⠀⠀⠛⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⢫⣿⣦⣴⡦⣴⣤⣤⢤⡤⠈⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢺⣆⠀⠀⠾⠗⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⠿⣵⣊⣜⠃⠀⣸⠏⠀⠀⢀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢠⣦⠀⠀⠈⢩⡿⠂⠈⠙⡇⠀⠀⠀⠀⠲⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣿⠿⠶⡿⠷⠰⣿⠯⣀⣠⣶⡇
⠀⠀⠀⠀⠀⢰⠷⠔⠒⠉⠀⠰⠂⠀⠀⠀⠀⢀⠀⠄⠀⠈⠙⠉⠉⠉⠋⠉⠉⠉⠉⠈⠉⠉⠁⠀⣠⠀⠿⠋⠰⠦⠀⠀⠠⠄⠀⠉⠛⢳⡄
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠛⠂⠀⠀⠀⠀⠀⠀⢰⡤⠄⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⢳⡄⠀⠀⠀⠀⠀⢀⡗⢠⡖⢲⠄⠀⠀⠀⠀⠀⠀⠠⡼⢟⣀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⢳⣤⡀⠀⠀⠀⢠⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⡀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⠛⠄⡙⠦⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡁⠈⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⢃⡿⠃⠀⠀⠀⠀⠀⣀⠀⠤⠀⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣄⠀⠒⣦⡔⠒⠀⠀⠐⣿⡅⠀⢨⡄⠀⠠⠄⠐⠒⣾⣥⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⠂⠀⠀⠉⠀⠀⠸⠆⠀⠀⠖⠂⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣷⣿⣀⠀⢀⣶⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡀⣸⣷⡀⢠⣼⣿⣿⣿⣄⣿⣿⠃⣼⠇⢀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣘⣿⡽⡷⣽⣻⢿⣿⡟⢹⣿⢹⣿⣿⡟⣼⡏⢀⣾⡿⠁⣠⡶
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣷⡙⠽⣿⡈⡿⡇⠀⣿⢸⣿⣿⣿⣟⣠⢿⣿⣤⣾⣿⡆⣠⣤
⠀⠀⠀⠀⠀⣤⣄⣈⣻⣿⣧⡙⠌⠳⡄⠙⠄⠐⢿⠀⠸⠀⡛⣼⢻⡿⢣⡿⢟⣿⣿⠿⢋⣜⣡⣤⡀
⠀⠀⠀⠀⠀⠀⠙⣯⣝⠻⣿⡛⢄⠀⠀⠀⠀⠀⣀⣄⣤⣄⣁⠀⠏⠀⢋⡴⠋⠈⢀⣴⡿⢟⣿⣛⡁
⠀⠀⠐⢶⣦⢤⣤⣿⣾⣝⣆⠙⠢⠀⠀⢀⡴⠋⢩⡴⠶⣦⠉⠻⣦⡀⠊⠀⠀⠀⠉⢀⣚⣯⡟⠋⠁
⠀⠀⠀⠀⢿⡿⠷⠶⣭⣍⡛⠓⠤⠀⢠⡟⢀⣀⣸⡇⠀⣿⣀⣀⡈⢷⠀⢀⣠⣔⣾⣽⣿⡍
⠀⠀⢰⣿⣿⣿⣿⣦⡤⠤⠀⢀⡀⠀⢸⡁⣏⠉⠉⠁⠀⠈⠉⣈⣿⢸⡇⠀⠼⠿⢿⣿⣿⣶⣶⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣛⣻⣿⠿⠶⠶⠦⠀⠀⠘⣧⠙⠋⠙⡇⠀⣿⠉⠉⢁⡾⠀⢠⣄⣀⣀⣒⠛⠻⠿⣿⣿⡛⠁
⠀⠀⠀⠈⠋⠹⢿⣿⣷⣄⣀⣠⣤⣤⣀⣈⣷⣄⣀⣷⣤⣿⣀⣠⣞⣁⣀⣀⣀⣀⣉⣉⣙⡛⢭⣍⣉⠁
⠀⠀⠀⢠⣴⠾⢛⣻⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣭⣛⣻⣿⣃
⠀⠀⠀⠀⠀⠀⢛⡭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⡉⠈⠉⠉
⠀⠀⠀⠀⢠⣶⠟⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠉⠛⠂
⠀⠀⠀⠀⠈⠀⠀⠰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠸⠋⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠈⠙⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠿⣄⠀⠀⢀⡼⠀⠀⠈⣷⢀⣠⡴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣰⡏⠀⠀⠀⢸⡷⠞⠿⢦⡀⣀⣠⣿⣯⠵⢶⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢸⣿⢦⣤⣤⣟⠀⠀⠀⡼⠋⠉⠉⢻⡄⠀⠀⣹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣤⣼⣇⠀⠀⠀⣹⠞⠛⠛⢷⡄⠀⠀⢠⣧⣴⣟⠉⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠋⠀⠀⠙⣶⠶⠲⣯⠀⠀⠀⠀⣿⠶⢾⣋⠀⠀⠈⢳⡄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠷⢿⣆⣠⣤⣼⡇⠀⠀⠘⢳⣤⠴⣾⠁⠀⠀⢹⡆⠀⠀⣸⣧⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢾⠁⠀⠀⢠⡟⠁⠀⠈⣳⡤⠤⣤⣾⡷⠛⠉⠛⢦⣠⠾⠷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣼⣷⣤⣠⣼⣧⣀⠀⣼⠁⠀⠀⠀⢻⡀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠻⣦⣀⣀⣠⠞⠛⠶⠴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡛⡃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠂⡄⠀⠀⠀⠀⠀⢀⠄⠀⠈⣵⡍⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⣄⣤⡤⠄⠀⡤⠀⠀⠀⠀⠀⠠⡄⠀⠤⣤⣄⡄⡠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠶⠴⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⢠⣀⠀⠀⠘⣇⠙⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠈⣇⠈⢻⡟⠀⢤⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣷⡄⠉⢦⠀⠛⢥⡀⢧⠈⣳⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⠻⣇⠀⠀⠀⠈⡑⢦⣉⡈⢷⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢀⣉⣆⠀⠀⠠⣄⠰⣿⣷⣶⡤⣴⣾⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣞⣂
⠀⠀⠀⠒⠯⣘⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠚⢛⡹⠖⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠀⠀⠀⠛⡉⣀⣀⣀⠀⠀⡰⠆⠀⠀⠀⠀⠶⡀⠀⢀⣀⣀⡈⡙⠃⠀⠀⠰⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠈⠁⢀⡈⠀⠀⣀⡀⠀⠈⣀⠀⠉⠁⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡠⡀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢀⣀⣠⡄⠀⣿⠀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣤⠶⠛⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠉⢁⠞⠧⠀⠀⠀⠀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠂⠈⠉⡏⠉⠙⠓⠶⣤⡀⠀⠀⣀⡴⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⣦⠚⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣼⠤⠤⠤⠤⠀⢰⡇⠀⢨⠀⠀⣸⣄⣀⣠⣤⠀⠰⠋⠀⠀⠀⣼⠠⡤⣤⠤⢤⣤⡄⠀⠀⠀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠐⠋⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠈⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡴⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠚⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣬⣿⣷⣶⣦⣤⣶⢤⣶⣿⣥⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠰⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠻⠿⠓⠒⠒⠃⠂⠐⢻⣶⣿⣿⣿⡿⢿⣿⡿⠛⣿⡟⠀⠀⠀⠀⢀⣻⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠍⠉⠁⠀⠀⣰⡟⠁⢸⡟⠀⠀⠀⠀⣠⡟⣻⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢠⡟⠀⢠⡟⠀⠀⠀⢀⡞⣁⢿⣽⠏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⢀⣨⡟⠀⣴⠟⠀⠀⠀⣰⢿⡷⣦⡾⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠚⠀⠀⠀⠀⢻⣦⡀⠁⠀⠀⠀⠀⠀⢍⣠⠤⣴⠟⣡⡾⠁⠀⠀⠐⣰⣻⠾⣷⡏⠀⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡐⣄⢀⣀⣴⣶⣶⠞⣣⡾⠋⠀⠀⠀⠀⣴⡿⠿⠛⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⣤⣀⣷⠈⢿⣿⠛⢿⡀⠠⣿⠀⠀⠀⣀⡴⢻⠋⢁⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⣠⡼⠋⠀⠚⠉⣹⡇⠀⠸⣿⠔⢾⡇⠀⣿⣤⣶⣋⣁⣀⣸⣦⣇⡈⢈⣷⣾⠋⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⣰⡟⠀⠀⢰⣧⣄⠐⣿⠀⠹⡏⠉⠁⠀⣼⢫⡟⣻⣿⠋⠙⢻⣷⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⢀⣤⣶⡶⢛⠁⠀⠀⣠⣿⠁⠀⠀⢸⡏⠀⠀⢽⡄⠀⢻⡆⠀⠀⠘⠿⣿⡿⠿⡿⣾⡿⣿⣦⣾⣤⡀
⠀⠀⠀⠀⠀⠶⠶⠶⠒⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⠀⠀⠀⣼⠃⠀⡀⢹⡇⠀⢘⣿⣦⣤⣤⣶⣿⣷⣴⣦⣿⣤⣬⣭⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⠛⠛⠛⠛⣛⣛⣻⣿⣤⣨⣽⣿⣤⣽⣟⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⣤⣿⣟⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠛⢿⣷⡶⠃



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⢦⡀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠧⠀⠂⠐⠚⠋⠙⠛⠀⠐⠀⠼⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠀⠀⠀⠠⠈⡀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣈⣠⡔⠀⠀⠉⣀⡁⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠂⡈⣂⠙⠀⠀⢲⣄⣁⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠘⠓⠃⢃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡙⠘⠚⠃
⠀⠄⣀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡈⠀⠀⠀⣀⠢
⠀⠀⠉⠃⡀⠂⠈⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⢀⠘⠉
⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡁
⠐⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠰⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙
⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡁
⠀⠄⣀⡄⠁⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠈⢰⣀⠠
⠀⠂⠉⠀⠀⠀⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠀⠀⠀⠉⠐
⠀⠀⠀⢠⠤⡄⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⢠⠤⡄

⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠁⢸⠙⠻⠀⠀⡄⠉⠁⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠄⠁⠍⢠⠀⠀⠞⠋⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⠐⢀⠁⠀⠀⠀⠀⠀⠀⠈⡀⠂⠀⠀⠀⣞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡞⠁⠄⠀⢤⣄⣠⡤⠄⠠⠀⢳⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠞⠁⠄⠀⠀⠀⠀⠐
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠒⠀⠀⠀⠐⣁⠒⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⣇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣩⣹⢏⢩⣉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡿⢸⣾⣿⣷⣦⠂⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣤⠖⠂⠔⢄⠀⠀⡀⠤⠤⢇⣿⣇⠸⣿⣿⣿⣿⣿⣦⠃⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⢠⠰⠀⠀⠁⢆⡀⢸⡻⣿⡄⢿⣿⣿⡏⠋⠉⠛⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠈⠊⣘⠒⣋⠵⢶⣶⠕⠊⠀⠀⢹⢿⣿⣮⣿⣿⡇⠀⠀⠀⠀⢀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⡋⠁⠀⢀⣰⠖⠓⢄⠙⢿⣿⣿⣧⡀⠀⢀⠴⠏⣐⠀⡤⠤⠤⠤⢀⡀⡠⠤⠤⠄⠀⣀⣄⣀
⠀⠀⠀⠰⠶⠀⠐⠈⢀⡔⠂⢽⣿⣦⡼⠿⣿⡀⢀⣸⠄⠠⢼⣿⡿⠇⢀⠇⠀⠀⠀⠑⣧⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⢻⡆⠉⠒⠤⡀
⠀⢀⠔⠀⠀⠀⠀⠀⢸⣄⣄⣸⠻⢿⡷⣄⣸⠛⠻⣯⡀⠀⣸⢻⣿⠂⡎⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠿⢀⢀⣐⣷⣷⣄
⠀⡈⠀⠀⠀⠀⢀⣖⡉⠙⣿⡡⡢⣨⡿⢿⣿⣷⣤⣿⣿⡛⠣⣼⣿⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣧
⠀⠡⡈⢉⠀⠀⠈⠿⣿⣷⡷⠻⣿⣿⡷⡆⣹⡏⠉⠙⣿⣷⣦⣿⣿⣿⣿⣍⠒⠶⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣛⣛⣛⣛⠛
⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠿⢿⡾⣽⣯⣵⣴⡟⡉⠉⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣥⡖⠉⠉⢉⣩⠭⢍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣾⣿⣦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡯

⠀⠀⠀⠀⣀⡤⠶⠆⠄⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⠶⠶⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡾⠋⠀⠀⠀⡀⠀⠉⢧⠀⠈⠉⠳⣄⠀⠀⠀⠀⠀⠁⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠁
⠀⠀⢸⠁⠀⠀⠀⢰⠀⠀⠀⣸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠈⠀⠀⠀⠀⠘⡧⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡋⠉⠉⣻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠢⣄⣀⣀⣀⣤⢾⣯⣀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⣾⣿⣿⡿⠀⠀⠀⠀⠖⢿⠏⠭⠶⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢰⣆⠀⠘⡋⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠉⠉⠉⠉⢻⡀⠀⠀⠀⠀⠀⢿⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢪⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⠁⠀⠈⢧⠀⠀⠀⠀⠀⠘⠗⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣘⡆⠈⢢⡄⣀⠙⢷⣄⠀⠀⠀⠀⠀⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡆⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣀⣀⣀
⠀⠀⠀⠀⣀⣴⠾⠛⠙⠆⠀⠀⢹⠆⠀⠙⢿⡄⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠟⠿⠛⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⣟⠉⠉⠛⠉⠉⢹⡆
⠀⠀⣠⡾⠋⠀⠀⠀⠀⠐⠁⣠⠊⠑⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⣠⠎
⠀⢐⡋⠤⠤⠖⠊⠀⢀⠤⠊⠀⠉⠒⠤⣄⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡦⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠞⠁
⠀⠈⠳⠤⠤⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⠿⢇⡀⠀⠀⠀⠀⣐⣀⣀⣤⣀⣀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠀⠀⡒⠶⢶⢶⡶⠶⠷⠖
⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠊⠀⢀⠠⠌⠁⡂⠀⢠⢠⢭⠥⠀⠀⠠⢆⠯⠠⢯⠴
⠀⠀⠀⠘⢿⡿⠃⠀⠀⡘⢿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠛⠺⠷⣷⡞⠚⠓⠛⠛⠀⠀⠘⢧⢀⡟
⠀⠀⢠⡟⠉⠉⢻⡇⣸⣿⣼⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣑⠒⠤⠤⠤⣉⣉⣁⡈⠀⠀⠀⠀⠉
⠀⠀⡾⠁⠀⠀⠀⢿⡟⢸⣿⣿⡇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠁⠀⠀⠀⠀⠈⠁⢸⣟⣿⡇⠈⠁⠀⠀⠀⠀⠀⢀⣀⡴⠃⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣏⣿⡇⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠿⠻⠇⠀⠀⠸⠏⠿⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡶⡶⢶⢶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⢶⣶

⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣿⠉⣿⠉⠉⠉⡏⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⣿⣀⣿⣀⣀⣠⣧⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⠀⠀⠄⠀⠀⠆⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠷⠿⠿⠿⠿⠿⠿⠿⠀⠀⠦⠤⠤⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠸⠉⣼⢃⡖⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣆⠂⣞⠠⠤⠤⠤⠤⠤⢀⡀⢸⠀⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢈⠋⡸⡟⡦⠤⠤⠤⠤⠤⠖⢻⣤⣷⣤⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⡠⠁⢃⠀⠀⠀⠀⠀⣄⠀⣼⣯⣯⢞⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠱⠀⠸⡀⠀⠀⠀⠈⠁⢀⠛⠻⣯⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⢵⡀⠀⠀⠀⠀⡼⠤⢄⣈⣛⣛⠟⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢨⠀⠘⣷⣄⠀⢀⣾⡥⠶⠒⠻⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡤⣀⠴⡿⡷⠀⠈⢧⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣞⣏⢀⠀⡤⠒⠉⠉⠉⠐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠁⠀⠈⠢⢄⡀⠀⠀⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣄⣀⣠⢐⠀⠀⠀⠀⠈⠐⠐⠋⠁

⠀⢐⡀⠀⣂⠀⠀⠀⠄⠀⠀⠀⢜⠀⠀⠆⠀⠀⠀⢤⢀⠠⡀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠠⠄⠄⣠⢀⠄⡀
⠠⠈⠂⠐⣈⠂⠀⡄⠂⠄⠀⠁⠉⠀⠂⡥⠆⠑⣈⢰⣇⡠⠋⠢⡀⣿⣰⣧⣝⣀⣇⣷⣅⣿⢀⠔⠀⠀⡤⠀⠠⡄⠀⢠⡔
⠀⡀⢀⠀⠄⠄⡀⢔⢊⠌⠀⡨⣀⠀⠀⠰⠀⠰⠢⢓⠀⢳⡄⠐⠀⠙⠛⠛⢻⡿⢛⡛⠛⠁⢨⠀⡳⠆⠀⣼⡄⢡⣶⡄⠁⢳⡇
⠀⠁⠁⠂⠀⠸⠅⠃⠀⡂⢀⠦⢀⠀⠁⢉⠀⠀⠾⠂⢠⠂⠀⡠⠊⠖⠀⡀⠈⠀⢠⡄⠀⠈⠀⠁⠠⠘⠠⣀⠔⠢⣀⠄⠁⠤
⠀⡠⢂⠅⠰⢈⢐⠀⡨⠇⣀⠪⠂⠁⢀⠰⢁⠀⠀⠠⠂⢰⡁⠄⠀⠀⠀⢧⢀⠀⡇⢩⠀⡞⢰
⠠⠠⠠⢐⠀⠂⠂⠂⠰⠃⠀⠄⠡⠂⠂⠀⠠⠃⠠⠄⠠⠊⠧⠤⠤⠀⠂⠈⠄⠀⠀⢸⠀⠇⢸⠀⡀⡤⠀⠠⡢⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡀⠐⢖⠂⠡⠀⠁⠀⠀⠔⠀⠈⠀⠀⠈⣹⡄⠀⠀⠐⠤⠘⠀⡄⢼⠀⠀⠈⢀⠀⠀⠀⢀⠅⠘⢂
⢀⠄⠀⢠⢀⠀⠀⣉⣂⠐⠻⠲⡶⠲⡶⠆⠀⠰⢆⠂⢒⠶⡀⠁⡁⠈⠁⡀⠀⡐⠈⠀⠁⢀⠀⠀⠀⣄⠀⢠⠀⠀⠌⠈⡀
⢀⣤⣤⣄⡏⢄⢾⣿⣿⠀⠀⠀⠹⠜⡀⠈⠤⠂⠈⠳⠎⠀⠈⠒⠀⠀⠂⠂⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠘⠈⠑⠠⠆⠀⠀⠠⠘
⠀⢿⣿⡟⠀⠀⠈⠋⠙⢆⠀⠀⠀⢀⢆⠀⠀⠠⠀⢀⠀⠲⠲⠒⠒⠒⠖⡤⢄⠀⠀⠀⠀⠞⠉⠀⠐⠄⠀⠀⠆⠀⠀⠀⣀⡀
⠀⠀⢖⠀⠀⠀⡠⠒⡀⠘⠣⢀⠠⠼⠸⠄⢀⠀⠀⡞⠀⢀⠀⢀⠐⠀⡆⠀⢘⠆⠘⠋⠁⠀⠀⠀⡀⠈⢆⠈⠀⠀⠃⡀⠉
⠀⠂⠛⠑⠀⠀⠅⠀⠀⢂⠀⠔⡀⠀⠀⠀⢰⡀⠀⡁⢈⠁⠄⠁⠈⠀⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⡀⡀⠀⠁⠀⠈⠂⠁⠀⠐
⠀⠀⠀⢰⡄⠸⡀⠀⠀⡘⢸⠠⠄⣁⠀⡀⠡⢸⠀⢃⢄⠀⠐⠐⠢⠀⠠⠄⢀⠇⠀⠀⠀⠀⠀⠤⠀⠸⠄⠀⠀⡀⠈⠁⠈⠂
⠈⡐⢠⠃⠱⡀⢡⠀⢀⠀⡸⣀⠀⠀⠀⠀⢐⠸⠀⠀⠘⠀⠤⠤⠤⠤⠄⠐⠁⠀⠰⠆⢰⣶⠞⠉⠁⠀⠙⢶⣶⠀⠀⠐⠈⠄⠁
⠀⠀⠎⠀⠀⠰⠀⠉⠈⠀⣈⠂⢩⠄⠀⠬⠀⠊⠀⣄⠀⠀⠀⢀⠀⠢⠉⠁⢐⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⢸⠁⢀⠠⠀⠒⢐⠢
⠀⢸⢀⡀⣀⠀⢇⠀⣸⠀⠉⠀⠙⠁⠀⠀⣠⠊⡀⠊⡀⣀⡂⠘⡬⢄⣲⠅⠈⡀⠓⠁⠀⠈⢀⡀⠀⠀⣠⠊⠀⢀⠘⠂⢒⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⢀⢐⠀⠀⡰⠁⠀⠀⢠⡅⠃⠀⠀⠀⠀⠀⡀⢀⠅⠀⠴⠢⠀⣤⣬⣷⣯⣤⣇⠀⠀⠑⠀⠐⢂⠁
⠀⢀⡀⠙⢋⢄⠀⠈⠢⠈⠈⠈⠀⠀⠐⠧⠐⠂⠉⢀⡀⡀⠀⠀⠀⠉⠐⠒⠁⠐⡿⠠⡤⠤⡄⡉⠉⣉⣉⣀⣀⢀⠀⠠⠄⠺⠃
⢀⢮⣈⠻⠧⣔⠢⡄⠐⠠⠀⠀⠀⢀⠆⢀⠀⢠⠞⠁⠀⠰⠞⠉⠁⠀⠀⠀⠂⡀⠀⢠⣀⣀⡃⠀⠀⠀⠀⣰⠀⠀⠀⡀
⠘⠉⠀⠤⠀⠐⠀⠅⠐⠀⠊⠀⠈⢂⣀⡈⠀⠐⠀⠁⠂⢐⠆⠼⠀⠷⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠈⠁⠈⠁⠉⠀⠁
⠈⠢⡀⠀⠀⣀⠤⠀⠚⠒⠀⠀⠂⠀⠙⠀⠀⠃⠒⠀⠀⠌⢸⠀⣇⠀⠍⠀⠀⠀⠘⠄⢱⠀⢉⠀⠀⠎⠂⠰⡰⢢⠚⠁
⠀⡠⠬⠉⠀⠀⢄⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠌⠀⠀⠲⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠂⠐⠃⠀⠊⠀⠐⠂⠀⠈
⠀⠀⠀⡖⠲⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣰⡌⢄⡸⠀⠀⡂⢀⠀⢀⠂⡂⠀⠚⠈⠇⢹⠈⠐⠄⠀⠡⠀⢡⠋
⠀⠒⠀⠀⠀⠀⠚⠀⠁⠂⠤⠁⠁⠀⠈⠉⠈⠒⠒⠒⠒⠒⠀⠂⠀⠁⠀⠑⠀⠈⠂⠘⠑⠃⠐⠂⠐⠁⠊⠐⠒⠀⠀⠀⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠙⣄⠀⠀⢽⠀⠀⢸⠀⠀⣠⠇⠀⠀⡘⠁⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠹⣆⠀⠘⣧⠀⣸⠇⢀⣿⠀⢀⡾⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠈⠳⣤⠀⠸⣧⡀⢿⣄⣹⣆⣸⡇⢠⡾⠁⢀⣴⠋⠀⠀⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠘⠷⣦⣽⡿⠛⠛⠛⠛⠛⠻⢿⣧⣼⠏⠁⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠴⢤⣀⣀⠀⠙⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠟⠉⠀⣀⣤⣤⠖⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠒⠚⠳⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠶⠶⠚⠒⠒⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⢀⣠⣴⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢶⣤⡀⠀⠀⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠁⣀⣴⠟⢻⣦⣄⡀⠀⠀⠀⠀⣀⣤⣾⠻⢷⣄⠈⠉⠛⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⡴⠋⠁⣠⡿⠉⣿⠛⣿⣿⣿⡟⢿⡍⠛⣷⠀⠙⠳⣆⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⣼⠋⠀⣼⠏⠀⣿⣿⣿⡇⠈⣿⡀⠈⠳⡄⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠞⠀⠀⢠⠟⠀⠈⡇⣿⡟⣧⠀⠈⣇⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡟⠀⠀⢸⡇⣿⡇⣧⠀⠀⠘⡆⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⣡⣿⣧⣜⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣶⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⢀⣶⠖
⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣶
⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠃⠀⠀⠀⣀⣀⣀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠿⠟⠋⠀⠀⠀⠀⢠⣖
⠀⠀⠀⣈⣿⣁⠀⠀⠰⣤⠉⠀⠰⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠿⠿⠿⣿⣇⠀⠀⠘⠀⣆⣀⠠⠀⣤⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠰⠟⠉⢹⡇⠀⠀⠀⠀⠀⣼⠁⢸⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠉⠃⡈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⢀
⠀⠀⠀⠀⢀⠀⠀⣴⣤⣤⡀⠀⣿⣠⣤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠂⠀⠀⠀⠀⢙⠇⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⣠⣷⣦⣄⡘⢿⠿⢷⠀⣿⣿⣿⠏⡀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⣿⡇⠀⠀⠀⠀⠀⠀⣬⣁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠸⣿⠄⣿
⠀⢠⣴⢛⡛⠛⠛⠁⠀⠀⢀⡀⠉⣶⣤⣶⣷⣤⡀⠙⢿⣧⡀⠀⠀⠿⠀⢸⣦⠀⣴⣦⣼⣃⠀⠀⠀⠀⠀⠀⠛⣙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠛
⠘⣿⣿⣾⡏⢀⠀⠀⠀⠀⠀⠃⠀⠉⠘⠛⠉⠁⠀⠀⠀⠈⠻⠞⢿⡗⠀⠀⠋⣀⣿⣿⠟⠛⠻⣦⠀⠀⠀⠀⠘⣿⡦⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰
⠠⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⡄⢷⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠛⢿⡗⠀⠀⠀⠿⣿⣆⣰⣜
⠀⠀⠉⠻⠀⠀⠀⠀⠀⠰⣾⣿⡇⠀⠀⠀⠀⠀⠀⡀⣧⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⠬⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠃⠀⠀⠶⠂⠀⣠⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⣶⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣾⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘
⢸⡁⠀⠀⠁⠀⣀⣠⣤⣤⣀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⠃⠀⠀⣤⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠛⠃⠀⢶⡄⠀⠀⠀⠀⠀⣰⣆
⢸⣷⡀⠀⠀⠀⣽⣿⣿⣗⠀⠀⠀⠀⠸⠛⣿⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠆⠈⠁⠀⠀⠀⠀⠸⣿⣿⡆
⠈⢹⣷⡆⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⢀⣾⡁⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒
⢸⣟⢁⣌⣅⠀⠀⠀⠈⠻⢷⣦⢴⣿⣸⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠋⠁
⢈⣉⠉⠻⢿⡇⠀⠀⠀⠀⠀⠀⠘⠋⣩⡄⠀⣀⡠⠘⠉⠀⣹⣿⠿⠿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠐⣷⠀⠘⢠⠄⠀⢈⣿⣂⠀⠀⠀⠀⠀⠀⠐
⠸⠿⣿⣦⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠉⠆⢩⡄⠀⠀⠀⠀⠙⠉⠛⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣿⢻⣧⠀⠀⠀⠀⠠⣤⣄⠀⢾⠁⠀⠀⠀⠀⠈⣥⡀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡇⢰⣶⣶⣷⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⢹⣷⣾⠇⠀⠀⠀⠀⠀⣿⣷⡈⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣍⣛⡛⣧⠟⠉⠛⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠹
⠠⠀⠀⢀⡄⠀⢀⡀⢀⣀⣀⣀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⠏⠉⠈⠉⠻⣿⡿⠛⠿⠃⠀⠀⠀⠀⢹⣏⣽⡆
⠀⠀⠀⣼⣣⣶⣾⣿⣿⣿⣿⣿⣷⡶⠈⡅⠀⠀⠀⠀⠀⠀⠀⠈⢁⣄⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣿⣠⣤⠀⠀⠀⠀⠀⠀⣁⣀⡴⣶
⠀⠠⢾⡿⠿⠿⠿⠿⡿⠿⠟⠋⠁⠙⠰⠀⠀⠀⠀⠀⠀⠀⢾⢿⠿⠿⠿⢶⡦⡤⠏⠀⠀⠀⠀⠀⠸⠹⡿⣿⠀⠀⠀⠀⠀⠈⠉⠻⣧⢷⠀⠀⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠡⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠃⠢⡀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢠⠃⠀⠀⠙⢦⡀⠀⠐⠧⣄⣄⡴⠋⠀⠀⢳⡀⠀⠀⠀⠀⠀⡼⠀⠀⠙⢄⠀⠀⢀⣸⡀⠀⠀⢀⡴⠋⠀⠹⡀
⠀⡞⠀⠀⠀⠀⣠⠀⠀⠐⠆⠀⠹⣧⠀⠀⠀⠀⢧⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠳⠄⡇⠀⠀⢀⣠⠏⠀⠀⠀⠀⢳
⢰⠁⠤⠤⠤⠴⣧⠀⢸⡤⠦⠢⡀⣿⠦⠤⠤⠀⠸⡆⠀⠀⢀⡏⠀⠀⠀⠀⡶⠒⠒⠃⠀⠀⠘⠁⠀⢠⠀⠀⠀⠈⣇
⣼⠀⠀⠀⡀⠀⠸⠢⠀⠀⠀⢀⠕⠇⠀⢀⠀⠀⠀⣧⠀⠀⢸⠁⠈⠉⠉⠙⣇⣀⣀⡀⠀⠀⢀⣀⣀⣸⠉⠉⠉⠀⢸
⡇⠀⠀⠀⡌⠉⠒⠒⠒⠲⠶⠒⠒⠒⠊⢁⠀⠀⠀⢸⠀⠀⣾⠀⠀⠀⠀⠀⠈⠉⠉⣯⠀⠀⣸⢦
⡇⠀⠀⠀⣱⠀⠒⠒⠲⠶⠖⠒⠒⠀⠀⣎⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⡠⠂⠀⡷⠀⠀⣽⡀⠳⣄
⡇⠀⢠⠞⠉⣇⠐⠒⠶⠶⠶⠒⠂⠀⣸⠋⠳⣄⠀⢸⡀⠀⡇⠀⠀⠀⣠⠞⠁⠀⡜⠀⣬⡄⠈⢷⠀⠈⠳⡄
⡇⠐⠁⠀⠀⢈⡆⠀⠀⠀⠀⠀⢀⡴⡇⠀⠀⠈⠢⢸⡇⠀⡇⠀⢀⠜⠁⠀⠀⡐⠀⡆⠀⠀⢢⠈⢧⠀⠀⠘⢦⡀
⡇⠀⠀⠀⠀⡜⠀⠀⠰⡶⠶⠖⠋⠀⠹⡄⠀⠀⠀⢸⡇⠀⡇⠐⠁⠀⠀⠀⣰⠃⠀⠉⣈⣁⡉⠀⠘⣇⠀⠀⠀⠙⠄
⡇⠀⠀⠀⡸⠁⠀⢀⠠⠷⠀⠠⢀⠀⠀⢳⡀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢠⠇⢄⠀⠀⠀⠀⠀⠀⡄⠘⣆
⡇⠀⠀⢰⠃⠀⠀⠘⠀⠠⠆⠄⠘⠀⠀⠀⢣⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⡏⠀⠈⠄⠈⠉⠁⠀⡸⠀⠀⠹⡄
⠀⠀⢠⠇⠀⠀⠀⠀⢀⠆⠀⣇⠀⠀⠀⠀⠈⢇⠀⢸⡇⠀⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠶⠒⠈⠀⠀⠀⠀⢹⡀
⠀⠀⠎⠀⠀⠀⠀⠀⡼⠀⠀⢹⡀⠀⠀⠀⠀⠈⠆⢸⡇⠀⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠈⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢤⠀⠙⠀⠀⣀⣀⣠⣤⠿⠂⡤⠤⠤⠤⠼⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡜⠀⠀⢣⡀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⢦⡀⠙⠒⠶⠤⠤⠤⠤⠴⠖⠋⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀⠀⠀⠒⣃⠄
⣀⣀⣀⣀⣉⣙⣒⣶⣶⣶⣶⣶⣶⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣢⣬⣭⣭⣤⣐⣊⣉⣉⣉⣉⣉⣉
⣿⣿⣯⣿⣯⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣿⣯⣿⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡏⢀⣶⣿⣿⣷⣶⣾⣿⣷⡄⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⠀⠈⠻⣿⣿⣿⣿⢿⣿⣄⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⠟⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠉⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⣿⣿⣿⣿⡿⠛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣿⣦⣿⣿⣿⡇
⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠁
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣈⣻⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⣹⣿⠂
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁

⠀⠴⣶⣦⡀
⠀⠀⠈⠛⠿⣆⠀⢠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠸⡄⢺⡟⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠸⡇⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢳⣸⡀⠈⣒⡁⣀⠀⢀⡀⠀⠀⠀⠉⢿⣿⣦⠀⠀⣴⣦⣼⣿⡄⠀⠀⠀⣀⣸⣷⣶⣷⣿⡁⠀⣇⣀⠠⠄⠚⠉
⠀⠀⠀⠀⢀⡠⠤⢿⣷⣤⣼⣿⣿⣷⣿⣧⠀⠀⠀⠰⣼⣿⡟⠀⠀⣿⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣣⣶⣾⠇⢀⡒⠄
⠀⠀⠀⢰⡁⢤⠀⠰⣾⣿⣿⣿⣿⡛⠉⠉⣷⣶⡄⠀⠈⠙⢧⠀⣿⡿⠋⢀⡀⠠⠶⢦⣭⣭⣽⣿⣼⣿⣿⣿⡇⠀⠀⡑⠤⠃
⠀⠀⠀⠀⣩⣼⣶⣾⣿⣿⣿⣿⣏⣿⡦⢀⣽⣿⣿⣦⡀⠀⣬⣿⣿⣤⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣼⣿⣧⢀
⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣿⣿⣟⠁⠀⠀⠉⠀⠘⠻⠿⢷⣿⡁⢙⣿⣿⣥⣀⡀⠀⠉⠀⠀⠉⠛⠛⠿⣤⣤⣼⣯⣿⣿⡏
⠀⠀⠀⠚⠻⣿⣿⣿⠿⠛⢻⣿⣿⣿⣶⡤⣾⣶⣶⡶⠂⠽⢿⣿⣿⡟⠉⠙⢿⣷⣶⡿⠂⠠⠀⡽⠀⠀⠉⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⢧⠾⠿⢿⣭⠀⠀⠀⢀⣾⣿⠁⢤⡀⠀⠀⠻⣿⣿⠆⠀⣘⣥⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡷⠻⠁⠀⠀⣠⣶⣿⣿⠉⠀⢸⣿⠢⣄⠀⠀⠈⠛⣝⠻⣅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠄⡞⣀⠤⡀⠀⠀⢻⣿⢿⣿⠄⠀⢿⣿⣆⠈⠀⠀⡀⢤⠈⡆⠈⠳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠉⠀⢰⡆⠳⣘⣦⡏⠀⠀⢸⢃⠜⠃⠀⠀⠀⠉⠻⠻⠀⠀⢇⣀⣠⢃⣀⠀⠈⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣽⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢠⡄⡠⠀⠀⠀⠀⠀⠀⣰⣶⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣟⣼
⠀⠀⠀⣨⡬⣀⢤⣅⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠛⣻⠟⠁⠀⠀⠉⣳⡤⠤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⢮⠿⠿⠿⣵⣄
⠀⢀⡞⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡞⢹⡇⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢻⡄
⠀⢸⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣸⣧⠞⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠳⣄⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⣀⣀⣠⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣤⠟
⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠘⠣⡀⠀⠀⠀⠀⠀⢠⠃⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⢐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢶⡶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠙⣳⣞⠋⠉⠙⢳⡄⠀⠀⠀⠀⢀⣀⡴⠛⠉⠙⠻⡅
⠀⠀⠀⠀⠀⠀⠈⠐⠲⠖⠒⠀⠀⠀⠀⣯⡀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢰⡏⢹⡆⠀⠀⠀⢻⠀⠀⢠⠞⠉⢹⡛⢳⡄⠀⠀⣻
⠀⠀⠠⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡀⠀⠈⠀⠀⠀⠀⢻⣄⠀⠀⠈⢷⡾⠁⠀⠀⣠⡟⠀⠀⣿⠀⠀⠈⢧⣀⣗⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠚⠋⠙⠓⠖⠛⠉⠀⠀⠀⠘⢦⣄⣀⣠⡾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠠⡿⠆⢀⣱⡖⠋⢻⠛⢦⡁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢄⣿⣆⠀⢰⡟⠀⠀⠌⠀⠀⣰⢟⣻⣿⣦⠀⠀⠁⠀⠀⠀⠀⠁⢀⣿⣓⣦⠀⠈⠂⠚⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⠀⠀⠟⠸⣆⠸⠁⠀⢀⠀⠀⠀⣵⣴⠿⠦⢽⠀⠀⠀⠀⠀⠀⢠⡀⡜⠀⢓⡛⠀⣀⡀⡴⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⡀⢀⠀⠀⠘⡆⠀⢀⠀⠄⠀⠀⠈⢢⣤⡔⠁⠀⠀⠀⣀⡀⠀⣠⠟⠀⠀⠀⢀⡀⡋
⠀⠀⠀⠀⢀⣀⣤⡾⠛⣛⠛⣛⠐⠇⡴⣈⠇⠸⡗⡐⢀⡀⣤⣬⣤⣥⣦⣀⡠⣀⠼⠌⠀⢋⠴⣶⠹⠛⠛⠛⠻⠻⣐⡤
⠀⠀⠀⠀⢎⢐⠡⠊⠁⢨⣤⠀⠈⠐⢜⢯⣦⡀⠙⠇⡤⠉⢑⡀⠀⢀⣚⣉⣤⡿⠁⠀⠀⠀⢉⠔⠀⠠⢼⡦⡄⠉⠈
⠀⠀⠀⠀⢀⠊⠀⢀⣀⣀⣁⠀⣄⠀⠀⢢⠉⠀⠀⠀⠀⠀⠺⠤⠔⠣⠞⠈⠁⠀⠀⠀⠀⠠⠁⢰⣄⠀⢸⡇⠀⠆⠐⠈⠆
⠀⠀⠀⠀⠈⠀⠀⠈⢉⢈⣉⣠⠋⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⡇⠙⢾⣇⠀⠢⠂⠀⠘
⠀⠀⠀⠀⢠⠀⠀⠀⢨⡈⢅⢨⡀⠀⠀⢠⠁⠀⣀⣤⡴⠶⠶⢞⣻⣿⡶⠶⢦⣤⣀⠀⠀⢃⠀⣀⣀⢌⢸⡏⠓⢦⡀⠀⡰
⠀⠀⠀⠀⠀⠣⡀⠀⠘⠁⠉⠈⠰⡀⡠⢂⣴⠞⠋⠁⠀⠀⠀⠈⣿⢹⡇⠀⠀⠈⠙⠳⣦⡈⠂⡀⠂⠀⢸⡇⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠀⠤⠔⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⠀⠀⠀⠈⠿⢦⠈⠂⠠⠸⠇⠄⢰⠁
⠀⠀⠀⠀⠀⠀⢳⠖⠀⠀⠀⢀⡤⣾⠿⠄⠀⠀⠀⢠⣤⣤⣤⣤⣿⣯⣥⣤⣤⡄⠀⠀⠀⠀⠧⣴⣀⠀⠀⠀⠀⢁⡤
⠀⠀⠀⠀⠀⠈⢹⡄⠀⠀⣴⡯⢛⠡⠂⠀⢹⡆⠀⠒⢄⠀⠀⠀⣿⠀⠀⠀⠠⠀⠀⠈⠻⠁⠐⠠⠉⢘⣧⠀⠰⢌⠁⣤
⠀⠀⠀⠀⠀⡀⢻⠇⠀⢜⡿⢠⠂⠀⢀⠀⠸⠇⠀⠀⠀⠑⠀⠀⣿⠀⢀⠀⣀⣀⣀⣀⣶⣀⣀⣀⡀⠌⢻⡇⠀⠈⡇⢃⡀
⠀⠀⠀⠀⠀⢻⠀⡆⠀⣄⠀⠄⠀⠀⠘⣏⡉⢉⡙⠀⠀⠀⠃⠀⣿⠀⠈⠀⢠⡶⠛⢻⣿⠛⠷⣄⠀⠘⢸⢶⠀⣄⣁⠜⠃
⠀⠀⠀⠀⢠⣴⡛⣧⡀⣴⡃⢁⠀⠀⠈⠉⠷⡏⠉⠀⠀⠀⡆⣰⠙⡆⢰⠀⢾⡅⠀⢿⣽⠂⢀⣿⠀⠀⢸⡾⠀⢌⣏⣀
⠀⠀⠀⠀⠈⠛⠂⢸⡇⠙⡧⠘⢄⠀⠠⠶⡔⢴⠦⠀⢀⡔⠀⢡⠀⡟⡄⠣⡀⠛⠿⠿⠛⠛⠛⢁⠴⠀⠲⠃⠀⣼⠋⣋
⠀⠀⠀⠀⠘⢦⡴⠈⢇⠳⠀⠀⠀⠑⠢⠾⠏⢻⠦⠐⠀⣰⠾⠶⠶⠷⠾⣆⠈⠒⠠⠤⠤⠄⠒⠁⠀⠀⠀⠰⡖⣺⠄⠁
⠀⠀⠀⠀⠀⠈⠓⠋⠻⣧⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⢀⡉⠀⠠⣤⡄⠀⠘⠀⠀⠀⠀⠀⠀⣲⡿⠁⠀⠀⠈⢱⣧⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠹⣦⣀⠀⠀⣀⣸⣇⣀⠠⠀⠄⠀⡈⢧⣀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⣼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣼⡃⢸⡃⠀⠀⠀⠀⡇⢸⣧⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⢙⠀⠐⠢⣀⠀⠉⠙⠛⠷⠶⠶⠶⠞⠛⠛⠉⠀⡠⠐⠀⣀⡀⠀⠂⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠙⠀⡶⠆⠈⠣⡀⢻⡔⣻⢇⡾⢿⢎⣹⡋⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠑⠈⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡀⠤⠀⡡⠀⠀⠀⢡⢀⢤⣤⠄⢀⣄⢠⠠⠀⠀⠀⠐⡆⣖⣀⣒⣂⠀⣀⡀⠁⢸⡛
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡜⠁⠀⠞⠋⠁⠀⠀⠀⡘⠈⠋⠁⠀⠈⠉⠘⠚⠃⠃⠀⠀⠀⣨⢦⢡⠉⠀⠙⠁⡀⢘⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢧⣛⠲⣦⡄⠀⠀⠀⡴⠁⠀⠀⠸⣀⡿⠏⢸⠁⠀⠈⢂⠀⠀⡈⠚⠈⠀⠀⠀⡀⢡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠂⢬⣍⣉⢶⡗⠴⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⣌⠛⠟⠛⠛⠻⠟⣉⠀⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡴⠖⡖⡴⢶⡼⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠸⢳⣶⢶⣶⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⣥⣤⡀⠀⠀⣄⡀⢠⢀⡀⣀⠤⢠⡤⣄⣀⢀⣠⡄⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠿⠁⠳⠆⠀⠀⠶⠒⠛⠛⠙⠛⠶⠂⠃⠐⠋⠀⠻⠶⠳⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣧⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣄⣀⣀⠀⣀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⡾⠀⠀⠀⠀⢀⡿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⡧⠀⠀⠀⠀⢸⡇⠀⠀⣾⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣠⠀⠀⠸⣇⠀⠀⢹⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠸⣇⠀⣿⠀⠀⠀⣿⠀⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠉⠀⡈⠂⠀⠀⠘⠷⠈⠀⠀⠈⠉⡿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣠⡿⠁⠀⢠⠂⠀⠀⠀⢰⡄⠀⠀⢀⠀⣠⡀⠀⣿⠀⣶⠀⢠⡄⢀⡀⠀⢰⠆⠡⡅⠀⠔⠰⠄⣤⠰⢹⡇
⠀⠀⠀⠀⣿⠁⠀⢀⡌⠀⠄⢾⠆⠸⠇⣽⠀⢸⡇⠸⠃⢰⡏⠀⢿⡀⢺⡇⠸⠏⢀⡿⠀⢸⡇⠠⣿⠀⠂⠿⠀⣼⠃
⠀⠀⠀⠀⠻⠄⠀⠈⠁⠀⠀⠈⠀⠀⠀⠓⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⠃⠀⠀⠘⠇⠀⠀⠀⠀⠣⠀⠀⠀⠀⢏⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡬⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⣁⡃⠀⠀⠀⠀⠀⢱⠚⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠺⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⡀⠀⠀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢀⠠⠀⠀⠐⠀⠑⢄⠈⠂
⠀⠀⠀⠀⠀⢀⠀⢤⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠔⢁⠄⠀⠈⠀⠀⠄⠀⠀⠠⠀⠔⠀⠀⢸⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡠⢂⡄⠀⢈⣨⡄⠉⠛⢄⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣐⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠈⠀⣀⡣⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐⠞⠉
⠀⠀⢀⠂⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠈⡆⠀⡠
⠀⠀⡀⠁⠈⠀⠐⠀⠀⢳⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⡯⡊⢀⠴⠁⠀⡀
⢸⡂⠧⠀⠀⠀⠀⠀⠀⢄⠁⠠⡄⠀⠀⠀⠗⢄⠀⢀⡀⠀⠀⠊⢀⡌⡀⠀⠀⠀⠀⢀⠇⠀⠁⠀⠈⠀⠀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠔⠑⠐⣦⣤⠀⠘⡀⠐⠁⠀⠀⠀⡠⠊⠉⠀⠐⠀⠀⠒⡤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⢑⠶⠀⡀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠶⠶⠶⠶⠾⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠈⠙⢶⣄⠀⠀⣴⠏⠁⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣤⣤⣤⣴⣶⣿⣷⣾⣷⣤⣤⣤⣤⣤⠤⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⠙⣿⣇⠀⠀⠀⠀⠀⢸⡏⠉⢻⡅⠀⠀⠀⠀⠀⣠⡿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠈⢿⣦⠀⠀⠀⠀⢸⡇⠀⢸⣷⠀⠀⠀⠀⣼⡿⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⢻⣇⠀⠀⠀⢸⣧⠀⠘⣿⠀⠀⠀⣰⡿⢁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢻⣧⡀⠀⠀⣿⡄⠀⣿⡇⠀⣸⣿⢃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠟⠛⠛⠳⠿⠷⠶⠿⠿⠶⠿⢷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⣶⣤⣄⣀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⢀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠈⠉⠻⣿⣶⣄⡀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣄⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠹⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠙⠻⣿⣿⣶⣦⣤⣤⣤⣤⣤⣄⣀⣠⣤⣶⣿⣿⠿⠋⠀⣠⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠞⣆⣨⣇⠉⢅⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠔⠂⠀⢳⡒⣶⣄⠀⣐⠇⢆⡄⠕⠉⢐⣾⣎⡦⠚⠘⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⣇⠆⠀⢸⣿⡇⠃⢘⣯⠦⣫⡴⠔⠀⠀⠙⢯⣟⠿⣾⣞⡇
⠀⠀⠀⠀⠀⠀⠀⢌⣀⣿⠀⡰⠈⡀⠀⣜⠙⣧⣄⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠉⠡⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠿⠷⠾⠷⠛⠑⠲⠿⠻⠿⣿⠀⠈⠀⠀⠀⠀⠀⠀⡈⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡦⠀⠀⠀⠀⠀⠀⢰⠄⠀⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣾⠉⠐⢢⠄⠀⢠⡴⢸⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⠀⠀⢀⠀⠀⠤⠌⢸⣿⡟⠀⡄⠙⡳⠴⡊⠋⢈⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠺⡀⠀⠎⣿⠃⢹⢀⠀⠀⠀⠀⢀⠈⡟⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡈⢸⡇⠀⠀⠀⢸⡀⠋⢻⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⢁⠉⠀⠄⠀⢠⠀⠁⠐⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀⣻⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⡀⠀⢠⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢇⠀⠀⠀⠀⠀⠀⢨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠀⠀⡂⠈⠈⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠧⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⡉⠉⠛⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠀⡷⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢀⣀⣀⠀⠀⠤⢾⠤⠤⠤⠤⠤⠄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⡀⠀⠃⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⠿⠿⠿⢿⡟⠛⣿⣿⣷⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⣿⣷⡖⠛⠛⠛⠓⠚⠛⠻⠿⠷⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠈⠉⠙⠛⠛⠛⠓⠒⠒⠛⠛⠛⠋⢉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣦⣄⣀⣀⡀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣭⣻⣏⣉⣤⣴⡿⠋
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠢⡄⠀⠀⠀⠀⢀⣤⣤⣄
⠀⢀⣤⡶⠟⠛⠋⠉⠉⣉⣷⡶⠶⠶⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⣿⣧⣴⢿⣶⣄⠀⣴⠟⠛⢦⠀⠀⠀⠓⠚⢠⡇⢀⣀⣠⡾⠁⢠⠦⠈
⣰⣿⠁⠀⠀⠀⠀⠰⠟⠋⠁⠀⠀⠀⠀⣀⣬⣽⣦⣤⣤⡞⠉⠻⣦⡤⣤⣞⠉⢹⡿⡿⠶⢾⣿⡁⠀⢈⣿⠿⠛⢷⣤⣾⣷⣴⡾⠳⣶⡟⢀⣽⣿⡧⢤⣄⠀⠓⠚
⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣨⣿⣷⣶⣧⣤⣀⣹⠇⣼⢃⠀⠀⠈⣿⣏⢛⡭⠿⢤⣤⣿⣏⠀⠀⣸⣦⣤⣿⣾⣿⣿⠿⢤⣀⠉⠛⣶⣄
⠀⠉⠛⢿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠁⠀⠀⠀⠉⠙⢿⣾⢃⡞⠀⠀⠀⠘⢿⣦⡀⠀⠀⡟⠁⠈⠉⣿⢻⣁⣀⣽⣿⡿⠃⠀⠀⠈⢳⡄⢈⣽
⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⣾⠁⠀⠀⠀⠀⠀⢻⣿⣶⠶⠳⣦⣤⡴⠛⠛⠻⣿⠟⠛⣷⣦⣼⣷⡀⠀⠹⣮⣿
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣛⣿⠿⢿⣶⣦⣄⣀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠼⠿⠟⠛⠓⠀⠀⠀⠀⠈⠙⠿⢷⣤⣿⠒⠳⣤⣤⣤⢿⣀⣀⣸⣇⡀⠬⢿⡷⠾⠿⣿
⠀⠀⠀⣠⡴⠞⢋⣡⡖⠛⠉⠀⠀⠀⠈⠙⠛⠿⣿⣶⣤⣀⣶⡿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⢛⣋⣹⣷⠖⢃⣀⡤⠤⠛⠛⣋⣥⣴⡾⠷⠚⠛⢷⠏
⠀⠀⠛⠛⠻⠶⠦⣤⣀⡉⠐⠒⠤⣤⣀⠀⠀⠀⠀⠈⠉⠙⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⣉⣉⣤⡤⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠦⣤⣀⡈⠑⠲⢤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⠉⠍⠙⠋⣉⣠⣤⠴⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣌⣉⣉⡒⠀⢄⡠⠄⠀⠀⢉⣁⣠⠤⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⠠⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠔⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠤⡐⠊⠐⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡠⠀⠀⠀⠐⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠆⠀⠸⠃⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠢⡀⠀⠀⠀⠀⠐⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣀⠈⠢⢀⠄⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢠⠔⠀⠀⡀⠀⡀⠀⠼⢠⠐⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠠⠀⠀⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⠄⠀⠄⠀⠀⠀⠴⠀⠢⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠬⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁⠘⠡⢀⠀⠀⠀⠠⠀⡆⠠⠀⠀⠀⢀⡠⠚⠀⠉⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠖⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠁⠀⠀⠠⣀⠀⠀⢀⠉⠀⠀⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠂⠀⠀⢀⠁⠢⡄⠁⠀⠂⠀⠀⠀⠈⠤⠆⢈⠀⠀⠀⠐⠀⠀⠈⠳⣱⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠠⠃⠀⣀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⢀⠀⠘⠠⠈⠀⠀⠀⠀⡐⠀⡀⠀⡻
⠀⠀⠀⠀⠂⡀⠀⠀⠀⢈⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠄⠁⠀⠁⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠘⢠⡄⠀⠀⠠⠂⠊⠀⠁⢀⠀⡀⠀⠐⡀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠂⠈⢀⠀⠄⠀⡠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⡆⠃
⠀⠀⠀⠀⠀⠀⡈⣁⠀⠀⠀⠀⠀⡱⠊⠂⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⡀⠀⠉⠀⠂⢀⡀⠀⠀⠁⠀⠐⠒⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⡀⠂⠀⠀⠀⠊⢛⣄⠘⠸⠀⣬⠛⠃⠀⠀⠀⠀⠠⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⢀⠔⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠈⠢⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠈⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠐⠀⡀⠀⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠀⠀⠈⠀⠈⠄⠁⠘⠀⠀⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡨⠉⠀⠔⠀⠎⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠔⢀⠐⠁⠀⠀⠀⠀⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠑⠀⠀⠐⠀⠐⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠒⠡⠄⢠⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣋⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠈⠡⡀⠍⠈⠅⠙⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠉⠉⠉⢉⣉⣉⣁⣀⣀⣈⣉⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠄⠀⠄⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠆⠀⣀⣀⡀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⣀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡁⣀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣤⠄⢀⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢠⣐⡂⠀⠀⠀⢸⠀⠠⠀⠀⠀⡠⠂⠀⠀⠉⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐
⠀⠀⡄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠘⠁⠈⠡⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆
⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠰⠆⠀⠀⠴⠂⠀⠀⠀⠀⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⠀⠄⠐⠖⠀⠀⠀⠃⠘⠶⡒⠛⠒
⠀⠀⠈⣀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢀⡀
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠂⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⠪⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠁⠀⠀⠀⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⠐⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⠀⠀⠀⡇⢀⡠⠄⠄⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢩⡉⠀⠀⠐⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⡀⠲⠤⠀⡀⠴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠱⠄⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⢠⡀⠉⠉⢰⡀⠠⠐⠚⠂⠄⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⢀⠀⡀⠢⣈⠂⠠⠁⠀⠰⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠐⠀⠈⢸⡇⠀⠀⠁⢂⠈⠙⠠⠄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠐⡠⠇⠀⠀⠀⠀⠦⠀⠀⠀⠈⠻⡀⢀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠁⠀⠢⣀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⢠⠀⠀⠀⠀⠄⠁⠀⠀⠀⠈⠁⢀⠠⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠤⠄⠒⠂⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠒⠀⢐⠓⠒⠒⠒⠒⠂⡏⠀⠀⠀⡀⠬⠓⠈⠉⠒⠒⠀⠠⠄⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠁⠀⢄⡠⣄⠤⠀⢀⠁⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠤⠠⠔⡡⢂⠄⠂⠉⠀⡀⠀⠀⠈⠁⠂⣅⠠⠙⢦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣆⠀⠀⠀⠀⡇
⠀⠀⢆⡀⠀⠀⠘⡀⠂⢈⠤⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⡄⠄⠀⠀⠀⠒⠂⡠⠀⡀⡀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡙⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠁⠀⠐⠠⠀⠀⠆⠀⠈⠀⠈⢀⡀⡁⢘⢀⠐⢣⣈⣁⣀⢈⡜⠀⢓⠀⠐⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠢⡀⠀⠰
⠀⠀⠀⠀⠀⠂⠀⠀⢰⡀⠀⠀⠀⠀⢰⠁⠁⡠⠀⢄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢁⠀⣄⡦⠟⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠸⠀⡀⢆⢋⡄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢈⠔⠂⠀⠐⠄⠠⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⠉⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⢅⡠⠂⠀⠁⠀⠀⠀⠀⠀⠑⢈⠂⢠⠌⡜⡀⠀⠀⠀⡐⢀⣹⣄⠀⠠⢣⠀⠀⠀⠀⠀⠀⡀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠃⠄⠀⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡁⢂⠜⠀⠀⠀⠀⠁⠀⢸⡀⠀⠀⠈⠀⢡⠤⠤⠄⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠂⠀⠀⠈⠀⠄⡀⠈⠐⠒⣒⠀⢓⡂⠠⢐⡠⠄⠁⠀⠀⠀⠀⠀⡆⠭⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠤⠶⢀⢀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡇⠀⠈⢰⠀⠀⠀⠀⡀⢀⡄⠀⢀⡀⣠⠆⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠢⠄⠈⠀⠂⠠⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄
⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⢆⣢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠲⣦⣿⣷⣯⣿⣻⡧⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣿⢹⣿⣿⣿⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠈⢻⢿⠵⢫⡿⢛⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠟⠛⠛⠛⠿⢿⣷⣦⣀⣴⣾⣿⡿⠿⠿⢿⣷⣶⣤⣀⠀⢸⡇⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣻⣍⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣼⣧
⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣃⡀⠀⢀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣘⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣤⣤⣤⣤⣶⣾⡿⠟⠋⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡠⠀⠀⠀⠀⢀⡦⢄
⠀⠀⠀⢸⠀⡸⠁⠙⢉⠇⠠⠀⣠⢐⠚⣓⠢⢀⠴⠂⠀⣴⢿⡭⠭⡻⣧⠀⡀⠀⠀⠇⢰⠙⠒⡉⡂
⠀⠀⠀⢸⣰⠁⠀⠀⣌⠀⠐⠄⡁⠬⣶⡟⠁⣇⡄⠀⠸⢟⡸⡀⠀⡀⣸⠀⠀⠒⠊⢊⠓⠘⠉⢏⡿
⠀⠀⠀⢀⠏⠀⠀⠀⠘⡄⠀⠀⠉⢄⡀⢀⡜⠁⡂⠁⡄⠑⢯⢮⣯⠾⠋⡀⡀⠀⠀⠈⠦⠀⠠⠼⠃
⠀⠀⠀⠈⠉⠉⠉⠉⡉⢁⠀⡀⢠⣤⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⡌⣳⡁⢸⠂⡒⠄⠀⠀⠀⠒⡱
⠀⠀⠀⠀⠀⡲⢠⡀⠂⠠⠃⢱⡟⣷⡎⡆⠐⠂⢰⠀⢰⠅⠀⠀⠀⠈⠁⠉⠁⢸⠀⡃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢵⠐⠃⠀⣀⢀⢀⢉⡒⠊⠀⠀⠀⠈⠀⠀⡀⠀⡀⢀⠀⡀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠱
⠀⠀⠀⠀⠀⠀⠤⣸⣆⠹⢯⣟⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠇⠑⠈⠠⠥⠀⠀⠀⠀⠀⢀⠰⠄⡀
⠀⠀⠀⠀⠀⢀⣀⡈⣁⡀⠀⠀⠀⡀⠀⠀⠙⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡍⣔⠁⠀⠀⠈⡴⡆
⠀⠀⠀⠀⡊⠁⠀⠀⠀⣈⠄⢀⠙⢋⡇⠀⠳⠀⣉⣀⣀⠈⢸⢓⣲⢀⢨⣿⣔⡁⠀⢿⠀⣴⣶⡀⠃⣤
⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠸⠤⠼⠆⠀⠀⠂⠡⣀⡠⠒⠉⠉⡀⠘⠿⠟⠉⠀⢁⠀⠈⠛⠞⠁⡀⠽
⠀⠀⠀⢸⡑⢒⠒⠠⢂⡄⠀⠰⢪⢍⢢⠀⠀⢶⠀⡀⠀⠀⠀⡼⣁⣥⠀⠀⠐⠉⠂⠉⢢⣀⣀⡜⠁
⠀⠀⠀⠘⠁⠀⠒⠒⠂⠝⠀⠰⢌⡈⠜⠐⠟⠀⠐⠘⠀⠀⠀⠀⠃⠔⠀⠀⠀⠉

⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⣤⣤⣄⣀⣀⣀⣀⣸⣿⣯⡿⠳⠿⠷⢿⣿⣄⣀⣀⣀⣠⣤⣤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣆⣀⣀⠀⠀⠄⣀⣀⠀⠀⣀⡠⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣸⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠛⠛⠻⠿⣶⣶⣒⣚⢻⡏⠉⠀⠀⣰⠟⣛⣛⣫⡭⠷⠛⠛⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣷⠀⢀⠀⣿⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣖⠀⣀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡟⡂⠩⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠈⠚⢡⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣿⠀⢸
⢸⠀⢸⡇⣴⣿⣟⣿⡿⠙⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⡀⠀⡾⣼⣿⡇⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⡄⠀⠀⣴⠛⠻⣿⣿⣷⡀⠀⣿⠀⢸
⢸⠀⢸⣿⣿⣾⣿⡿⠐⠓⣹⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠁⣽⠀⢺⡟⣿⠀⠀⠀⣴⠛⠻⣿⣿⣷⣄⢸⠀⠀⢸⣿⠙⠂⢹⣿⣿⡇⠀⣿⠀⢸
⢸⠀⢸⡏⢿⣿⣿⣿⣧⣼⣿⠃⠀⣿⢸⠀⣴⣾⣿⣿⣻⣦⡀⠀⠀⠀⢸⣿⠀⣿⠀⢸⡇⣿⠀⠀⢰⡏⠰⠄⣸⣻⣿⣿⣼⠟⠋⠉⠉⠉⠘⢿⣿⣿⠀⠀⣿⠀⢸
⢸⠀⢸⡇⣠⡿⢿⣿⣿⡏⢳⡄⠀⠺⠈⣼⣿⣿⣷⡟⢩⠍⣷⠀⠀⠀⢸⠸⡆⢻⠀⢸⡇⡿⠀⠀⠸⣿⣷⣾⣿⣿⡿⣿⠁⠀⢀⣀⣤⣄⣀⠀⠈⢿⢦⡄⣿⠀⢸
⢸⠀⢸⣷⠏⠉⠲⣬⠟⠛⠋⠙⠓⠻⣤⡸⣿⣿⣿⣿⣴⣶⡟⠀⠀⠀⢸⠀⣷⣨⣦⣾⠀⡇⠀⠀⣠⣽⣿⣿⠟⠉⣹⠷⣌⣰⠋⣭⣿⣿⣿⣷⠀⠈⡏⣇⣿⠀⢸
⢸⠀⢸⣿⠀⢠⡾⠁⠀⠀⠀⢀⠀⠀⠐⢝⢿⠟⠁⢿⣿⣿⡀⠀⠀⠀⢸⠂⢸⣷⣿⡿⢸⡇⢀⣾⠋⢻⣦⣄⣀⣤⢏⠞⠈⡟⠇⢻⢹⣿⣿⣿⡇⠀⡇⢹⣿⠀⢸
⢸⠀⢸⣿⠀⣾⠃⠀⢀⣴⣿⣟⡻⡿⢦⡀⣣⡻⣶⣶⣷⠟⢿⡄⠀⠀⠸⠦⣤⣭⣉⣡⡼⢃⣾⠇⠀⣆⠻⣿⣿⢫⢏⠀⠀⢿⣠⣶⠀⢻⣿⣟⡄⡰⢷⡄⣿⠀⢸
⢸⠀⢸⣿⢠⣿⡄⠀⢾⣿⣿⣿⢿⠟⠋⣷⣿⣷⡹⡿⠋⠀⠘⣷⡀⣠⠴⠶⠶⠶⠶⠶⠶⠾⣧⣄⠀⠈⠳⣽⣯⣿⣡⡀⠀⣨⠟⣯⣶⠿⠛⢿⣞⠁⠀⣿⣾⠀⢸
⢸⠀⢸⡏⢸⡇⢳⡀⠈⣿⣿⣿⠄⣀⣸⣿⢸⡟⣷⡹⣀⣤⡞⣿⣿⡇⠐⡆⠁⠀⠅⠠⠃⢀⡾⠘⣧⠀⠀⡏⣿⡿⠛⣷⣼⣯⣾⣛⣡⡤⠶⢤⡹⣧⠀⢻⣿⠀⢸
⢸⠀⢸⡇⣿⠀⠀⣹⢖⣩⠵⣭⡲⣜⢻⡏⢸⣧⢻⣷⠏⡍⣼⡽⠻⣧⣌⠻⠿⢿⣿⠷⠾⠿⢇⣴⠏⠀⠰⢰⢿⡷⣾⠙⢿⣿⣿⠟⢉⠀⠀⠰⣽⣿⡆⢸⣿⠀⢸
⢸⠀⢸⡇⣿⠀⣼⣿⠋⠀⠀⠈⠙⢷⣽⡇⢸⣿⣸⣥⠴⠷⠿⠦⢤⣌⡛⠻⣷⣶⣶⣶⡾⠟⢋⣡⡤⠴⠶⠿⢼⣧⡹⣇⣸⡿⣡⡴⠃⠀⠀⠀⢹⣽⣿⢸⣿⠀⢸
⢸⠀⢸⣇⣿⢠⣟⡟⠀⠀⠠⡀⠀⠙⣿⣧⣼⣻⣿⣅⡴⠒⠒⠶⠆⣀⣿⢿⡉⢉⣛⣉⣉⡷⣯⣀⠴⠒⠒⠒⣀⣼⡟⣿⣿⡿⠋⠀⣠⣎⠀⢰⡟⡯⣿⡟⣿⠀⢸
⢸⠀⢸⣿⡇⣸⣿⢱⠀⣗⡄⠙⢦⡀⣼⢻⡇⣯⠋⠻⢿⣿⣿⣿⣿⠿⠋⢸⢏⡟⣿⢯⢻⡀⠙⠻⢿⣿⣿⠿⠟⠿⣽⣿⠘⣧⣠⡼⣡⣿⡆⢸⠀⣿⢿⣿⣿⠀⢸
⢸⠀⢸⣿⡇⡏⣿⢸⡀⢿⣎⠢⣸⠟⠁⣾⢱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⢹⠘⡌⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⠁⣠⣿⣿⣷⡘⠀⢹⢸⣿⣿⠀⢸
⢸⠀⢸⣿⢸⡇⣿⠈⡇⢸⣿⣦⣼⡆⣰⢣⡏⣿⠶⢤⣄⣀⡀⠀⠀⠀⠀⢸⡾⡀⠸⡇⠀⣿⠇⠀⠀⠀⢀⣀⣤⠴⣾⣻⣯⢦⣰⣿⡇⣿⢿⣷⢦⣾⣌⣿⣿⠀⢸
⢸⠀⢸⡏⣼⠁⣿⢈⣷⠘⣇⣧⠙⢣⣯⡟⠀⣿⠀⠀⠀⠉⠉⠛⠛⠒⠶⢾⡇⣇⠀⠉⣸⡿⠶⠒⠛⠋⠉⠁⠀⠀⣿⢹⢻⣧⣿⢃⡇⣿⣾⢿⣆⣸⠟⣿⣿⠀⢸
⢸⠀⢸⡇⣿⢀⣿⠘⢿⠀⢿⠘⡆⠟⣿⡀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⢳⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⢠⡿⠁⢸⠁⡟⣿⢸⣿⣧⡀⣿⣿⠀⢸
⢸⠀⢸⣇⣿⣼⡟⠀⣸⡄⢸⣇⢻⡀⢹⣇⠀⢸⡷⢤⣤⣀⣀⠀⠀⠀⠀⠀⣿⠞⠛⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣼⡟⢸⣿⠃⠀⡿⠀⡇⣿⢸⣿⢸⡿⢿⣿⠀⢸
⢸⠀⢸⣿⠟⠋⡀⡀⣿⡇⠈⢿⡈⣧⠀⣿⡀⣸⡿⣶⢦⣬⣭⣉⣙⣛⣛⣛⡛⠒⢒⠒⣒⣚⣛⣛⣩⣭⢭⣶⡞⢻⡇⣿⡏⠀⢸⠇⢀⡇⣿⢸⣿⢸⣿⣽⣿⠀⢸
⢸⠀⢸⡇⣤⢰⢁⡇⠇⣧⠀⢸⣧⠸⣆⠘⢿⣿⡇⣿⢸⣿⡇⢹⢹⠍⣿⠿⢿⣿⡏⣿⡿⢭⡯⢹⣿⡇⢸⣿⠁⢸⣿⡿⠀⢀⣿⠀⢸⡇⡏⢸⣿⢸⡏⡟⣿⠀⢸
⢸⠀⠸⠷⠿⠾⠿⠷⠶⠿⠶⠶⠿⠶⠿⠶⠶⠿⠷⠿⠾⠿⠷⠿⠾⠿⠷⠶⠾⠿⠷⠿⠶⠶⠿⠾⠿⠷⠾⠿⠶⠾⠿⠶⠶⠾⠷⠶⠾⠷⠷⠾⠿⠾⠷⠷⢾⠀⢸
⢸⡀⢀⣄⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤⠤⠀⢸⠀⢸
⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣶⣿⣤⡾⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠙⠛⠋⠈⠙⢿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣧⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⠛⠛⠛⠋⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠏⠉⠁⠀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⠋⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣤⣤⣶⣿⣿⣿⣋⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⢠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿
⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢹⣿⣿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣷⣄⢠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣼⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠠⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⡄⢠⡄⠀⠀⠄⠀⠠⡄⢰⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠈⠷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⡆⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⡂⠄⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠙⢀⠀⡄⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠴⠂⠀⢀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠀⠀⠀⢃⠀⠀⠘⢇⠀⠀⠀⠀⢠⣀⢠⠒⣀⠀⢠⠀⠄⢀⡜⢦⠤⠄⠀⣀⡀⠀⠀⠂⠀⠀⢀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢦⠘⠀⠈⡀⠀⠀⠀⠀⢀⠂⢀⠀⠐⠀⠀⠀⠈⢀⠀⠒⠀⠈⠀⠙⠀⠀⠀⠐⠄⠁⡠⠂⠈⠁⢈⡀
⠀⠀⠀⠀⠀⠤⠴⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⠀⢀⢁⠐⠂⠀⠀⠀⠚⠋⠀⠒⠂⠀⠐⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠱⡀⢰⠆⠄⠁⠀⡀⠊⠁⠀⠀⠰⠄⡄
⠀⠀⠀⠀⠰⠀⠀⣤⠁⠀⠀⠀⠀⢰⠉⠐⠠⠀⣀⣺⡗⠈⠂⠁⠀⠁⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠘⣃⢀⠠⢀⣀⡀⠀⠰⣄⠤⠤⠆
⠀⠀⠀⠀⡁⠀⠀⠋⠀⢰⠀⠀⠀⠋⠀⠀⠠⠊⠁⠀⠹⣶⠀⠈⢀⠀⠀⠀⡠⠐⡴⣾⠰⠟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠙⡖⠂⢰
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡈⠀⠀⠋⠠⢀⡀⠀⠈⠘⣭⠓⠄⠀⠘⠉⠋⣄⠀⠀⠀⡀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠁⡄⠂⠊⠀⠀⠀⠈⠀⠀⠀⢠⠀⡀⠀⢀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⢨⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⢀⠔⠈⠁⠀⠀⢀⠀⠸⠂⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠒⠒⠘⠃⠀⠀⠀⠂⠁⠸⠀⠀⠀⠘⠈⠉⠉⠉⠁⠉⠀⠀⡤⠰⠆⠀⠀⠀⠀⢀⠐⠲⠁⠂⠀⠀⠀⠄⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⣰⠀⠀⠰⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠐⠒⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠃⠄⣀⠀⠀⠀⠀⠈⠀⠚⠠⠤⠀⠁⠀⣹⠉⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠔⠀⠀⢸⠀⡏⠈⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠉⠉⡋⠘⠉⠉⠉⠉⠂⢲⠀⠀⠀⢀⠄⠀⣠⡌⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠁⠀⠀⡄⠀⠰⠇⠀⢀⠀⠀⠀⡀⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠑⡌⠒⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠈⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢶⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡍
⠀⠀⠀⠸⠀⡆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆
⠀⠀⠀⠆⢸⠀⠀⠀⠀⡇⠀⡂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⢀⣀⠀⠂
⠀⠀⠀⠈⢾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠢⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠤⡇⠠⢤⡀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠉⠉⠉⠉⠀⣤⣴⢤⡀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡏⠉⠛⢻⡆⠀⣸⠃⠀⢘⡆⠀⣀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠻⣆⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣹⢦⡖⣊⣀⡀⢈⡇⣶⠛⠀⠀⣿⠛⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠁⠀⣿⣼⢳⢹⡏⠀⠘⡷⣟⣦⢻⡄⣰⠋⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⣿⣦⡀⡴⢛⡿⠀⠀⣿⠀⠀⠀⠀⣧⣀⡀⠛⡇⢸⢸⠃⠀⠀⣿⢻⣿⡏⠀⢻⡄⣤⠶⠏
⠀⠀⠀⠀⠀⠠⣴⠲⢾⠉⠻⣟⢛⡏⣻⠁⡏⠀⠀⠀⣿⠀⠀⢀⣀⡗⠪⠗⠀⠁⢺⣀⠀⠀⠀⠇⢸⣿⠀⠀⣸⣧⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠈⢳⠀⢹⢸⠀⡏⢠⡇⠀⠀⠀⣿⠂⠀⣾⠉⠙⢳⢶⣄⣀⣼⢿⠀⠀⠀⠀⢸⠉⣠⡴⣿⡻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠘⡇⢸⢸⡇⢁⣬⣧⡀⠀⣀⣤⣞⣋⠙⣻⣶⠟⠉⠉⠉⣧⢸⡛⠒⣶⠆⢸⣾⠁⠀⠀⢹⣿⡧⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⣧⠘⠈⠁⠈⣳⣴⣿⠟⢋⣛⣽⣯⣽⡿⠃⢀⡷⠂⢾⣻⡿⣿⣾⣧⢤⣸⢈⡇⡴⢶⡟⠈⢻⡇
⠀⠀⠀⠀⠀⠀⣾⣠⠊⠙⢧⡀⢀⣠⢴⣟⣙⣏⣿⡉⢹⣯⣿⣿⣷⠀⠉⡇⠀⠀⣠⢾⡏⢹⢿⣟⣿⡟⠀⠻⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣯⣀⠀⡴⠛⠉⠀⠀⠈⠉⢠⣿⡅⠸⣟⡟⣻⠋⠀⠀⠃⣤⣞⠁⠈⠳⣼⣶⢏⣾⡇⠀⠀⣸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢈⣽⠂⠳⣦⠀⢰⢫⣿⢦⡘⡇⣧⡴⣿⣳⠟⢦⡀⣴⢾⣟⣿⣹⡀⢰⠇⠈⢻⡟⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠸⡇⡿⢸⠈⣏⢻⠷⡟⠀⠛⠋⢀⡀⠹⡅⣼⣸⢸⣿⣧⠏⣠⠀⢸⡇⠀⠀⠀⣿⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⢻⡏⢸⡄⢹⡈⠿⣱⣧⠀⠀⢸⡟⢦⠹⡇⡿⠈⡟⠋⣰⠃⠀⢸⡇⠀⠀⠀⡟⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣦⡃⣸⡂⠈⣧⡰⠃⢿⠀⠀⢸⡇⠈⢳⣰⠇⠀⠷⠞⣿⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠉⠋⠀⠀⠈⠁⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢠⡞⠓⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⣸⡇⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠉⢻⡏⠉⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⣿⡇⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⣧⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣯⡀⠀⠀⢀⡷⠀⡏⣷⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⣿⠀⡇⠀⠀⠀⠀⠀⠀⠿⣄⣀⣠⡤⣾⠇⠀⠀⠀⠀⠀⠀⢹⣤⣷⢸⠀⢠⡧⠿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⡇⣸⠀⠀⠀⠀⠀⠀⠀⣸⡿⢹⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠸⡇⣼⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⣷⢸⠀⠀⠀⠀⠀⢀⣠⣿⡷⠾⠾
⠀⠀⠀⠀⠀⠀⠀⡇⢻⣀⡀⡇⢷⡀⠀⠀⠀⠀⠀⣀⣾⣿⠀⢹⣘⣳⡦⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠛⠚⠻⠂⠀⠀⠀⠈⠛⠉⠁⠀⠀⠉⠁

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢲⡲⠄⣠⡀⠀⠀⡀⠀⠀⡀⠀⣼⡗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡠⠴⠀⠈⠿⣶⣯⣇⠀⢀⣄⣫⣜⣁⣸⣙⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠄⢹⣿⠀⠛⠛⢋⠉⠉⠀⠈⡉⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢀⠈⠁⠂⠁⢸⣿⠀⠀⠀⠀⠈⡄⠀⠂⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠜⢸⠀⠀⢀⠸⣿⡆⠀⠀⠀⣸⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠧⠄⠠⠠⠠⠄⣿⢠⠀⠀⠀⡏⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⢹⡏⠀⠀⠀⡟⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢹⠀⢸⡆⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠁⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡾⣿⢣⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣇⣦⡀
⠀⠀⠀⢀⣠⣀⡤⣮⡾⣫⠞⢡⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣾⢏⣿⢠⡀
⠀⢀⣴⣿⣿⡗⣺⢫⡾⠋⣰⡿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡤⣄⣴⣾⣿⣿⣿⣷⣿⣼⣇⣾⡿⡇⡀
⣴⣿⣿⣿⣿⣗⡿⣸⡇⣾⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣧⢻⡆⣤
⣿⣿⣿⣿⣿⣿⡇⣿⢱⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⣝⠷⣝⡻⣿⣿⣿⣿⣿⣏⣾⣿⢻⣮⣻⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⡣⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⢷⣍⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠐⡤⡎⠘⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠓⠦⣧⣄⣀⣀⠀⢀⣀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡤⠤⠤⠴⠊⠁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⠒⠶⠖⠚⠋⠁⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠶⠶⠶⡏⠤⠤⠤⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢸⡇⠀⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡇⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢀⣀⠾⠅⠀⠈⣀⣀⠀⣠⣤⣤⡀⠀⠀⢱⠀⠀⠀⠁⠀⠀⢇
⠀⠀⠀⠀⠀⣀⣀⡀⢰⠋⡤⠀⠀⠁⢰⢏⡍⠑⠀⠀⡾⠫⠍⠙⠁⠒⠂⠱⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⡀
⠀⠀⠀⠀⢸⠋⠌⠁⠈⠀⠀⢀⡴⠒⠀⠀⠀⣰⠏⠉⠀⣠⠶⠲⠦⢀⣤⣤⡀⠀⢸⠀⠀⡌⠁⠀⠀⠀⢳⡀⠀⠀⢀⠆
⠀⠀⠀⠀⠈⠃⠀⣴⢋⠅⠀⢸⡄⠃⠀⣠⡄⢻⣌⠀⠀⠇⠘⠀⠀⢿⠑⠂⠈⠀⡼⠊⠁⠈⠁⠀⠀⠈⠑⢝⢦⡠⠃
⠀⠀⠀⢠⡶⢂⠀⠻⠄⠀⠀⠀⠛⠆⢸⡡⡀⠀⢨⡞⡃⠀⢀⡴⠛⠛⠓⣤⠖⠈⠠⠒⠀⠓⢆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠧⠀⠀⣠⠄⠀⢠⡶⢛⠀⠀⣑⠀⠀⠸⣇⠁⠀⠘⣇⠁⠀⠀⣿⡰⠄⣠⠀⠀⠰⡨
⠀⠀⠀⠀⠀⠀⠰⣏⠘⠀⠘⢧⣀⠀⠐⣏⠰⠄⠀⢀⡏⢁⠀⢠⡏⠀⡄⠈⡙⠛⢉⣉⣉⣉⡉⠉⠉⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⣤⠤⠀⠈⠙⠂⠀⠀⠳⣬⠤⠀⠙⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠂⠰⣭⣠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗

⠀⠀⠀⠀⣠⡄⠐⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⠀⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠂⢀⡄⠀⠀⠀⢠⠀⠀⠀⠿⠀⠀⠀⠀⠈⠓⠀⠀⠘⣿⣦⣾⡏⠀⠀⣠⣡⣄
⠀⠀⠀⠀⣠⣄⠀⠀⢡⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠙⠿⠁⠀⠀⡒⠄
⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠉⠀⣴⣶⣶⠀⠀⢸⣿⡇⠀⠀⠰⢿⣿⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠈⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢹⡇
⠀⠀⠀⠀⠀⢄⠀⠀⠤⠀⠀⢀⠀⢻⠀⢹⡇⠀⠻⠛⣇⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠟⠀⠸⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣿⣿⡯⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡇⢹⣿⣿⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⣿⣿⣿⣿⠿⠟⠋⣹⣟⠿⢿⣿⣿⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⡿⠁⠀⠀⠘⢿⡿⠃⠀⢹⣿⣷⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣿⣿⣿⠏⠀⠀⠀⣀⠀⢠⡄⠀⠀⢸⣿⣿⣇⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⣻⣿⠉⣿⠀⠀⣶⢠⣤⠁⠀⠈⣁⣤⣸⣿⣿⡿⡵⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⣷⣾⣿⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠀⢻⣧⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢹⣿⡿⡓⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣿⣿⠀⠈⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡿⠁⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠘⠃⠹⣾⣡⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠃⠀⠀⠰⠯⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⢂⣺⣿⡁⠀⠀⠈⣿⡃⡀⠙⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣣⡅⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠘⠂⣟⡹⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⠀⢠⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⣰⣿⠀⠇⢴
⠀⠀⠀⠀⠀⠀⠈⠋⠀⣰⣿⠀⠀⠀⠀⢶⡦⢼⠃⠀⠀⣠⣶⣿⣿⣿⡟⠋⠀⣀⣶⠤⠾⣿⣁⠘⠀⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡻⡆⠀⠀⣸⠇⠀⢾⠀⣠⣶⣿⣿⣿⡿⠟⠀⠀⣿⠇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠲⠆⠀⠀⠀⠾⠇⣴⡄⠀⣿⠂⠀⠙⠇⣿⣿⣿⣿⠋⠀⢀⣐⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣽⣄⣟⡀⠀⠀⠀⢹⠟⣿⠋⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢠⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠻⣤⡿⠃⠀⠀⠀⠀⠀⠉⠀⠠⣶⠃
⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠘⠇⠀⠀⣤⡄⠀⠀⠀⠀⣲⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣹⣦⠄⡀⠀⢀⠈⠄⠡⠌⠤⠀⡜⠀⠀⠰⠆⠀⠀⠀⠁
⠀⢰⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠙⠻⠾⣏⣷⣤⠴⢚⣚⡛⠒⢓⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠒⠈⢉⣭⠉⠉⠑⠂⠄⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠛⠀⠀⣹⣿⣷⡀⠀⠀⢀⠰⣄⠀⠀⡠⠊⠀⡀⠀⠀⠁⠀⠀⠀⠲⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠦⡡⠢⢈⡵⣤⠀⠀⠀⠋⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡷⠀⣀⣴⡿⠇⠸⠑⠆⢠⣄⣀⠀⠀⢠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⢸⣿⣿⣯⠀⠀⠳⡌⠀⡱⠀⠀⠀⠀⠀⢿⣮⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢨⣿⣿⣏⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣌⣀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⣀⠀⡠⢸⣆⠀⠀⠀⠀⠀⠀⠀⠱⠾⣿⠃⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠉⠐⠘⠧⠌⠀⠀⠀⠀⠀⠀⠀⢸⠟⠃⢾⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁⠈⠄
⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢿⣿⡆⠀⠀⠐⠺⣌⡃⠀⠀⠁⠄⢀⡀⠀⠀⣙⣡⠞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢾⣿⡄⠀⠀⠘⣂⡬⡇⠘⠃⠀⠀⢀⡄⠀⠀⢹⣉⢠⠇⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣿⡅⠰⠦⠀⣀⠻⣇⠀⠀⠀⠀⠈⠁⠀⠓⢀⡷⣎⢺⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠰⣿⣇⠀⠀⠀⠏⠱⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⡟⣡⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⠀⠀⠀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠶⢦⢩⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣿⣿⠀⠀⠀⢋⡀⠿⡀⢶⠀⠀⠛⠀⠀⠀⠁⠀⢛⠓⢯⠸⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣛⢻⠀⠛⠀⢠⣽⡶⡇⠀⠀⠀⠀⠀⠠⡆⠀⠀⠘⡟⢲⡄⢡⠶⠀⡦⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⡆⠀⠀⢀⠁⡹⢧⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⢹⣋⡾
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠃⠀⢠⡀⣷⢡⣻⡀⠀⠙⠀⠀⠀⠀⠈⠁⠀⠀⠽⠒⢤⡇⠀⠰⠄⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣀⣀⣸⡀⠀⠀⣘⢁⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⣸⡇
⣿⣿⣿⣿⢿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣤⣬⣥⣤⣯⣤⣽⣮⣤⣿⣡⣯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠖⠛⠋⠙⠛⠶⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣀⡀⢀⡀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣟⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⢀⠀⠙⠁⠈⠋⢀⡀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣻⣿⣿⡃
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢿⠀⠘⢷⣄⣀⣀⣤⡾⠃⠀⡿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢠⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢼⣿⢿⢿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠉⠉⠁⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⢿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠲⢤⣤⣤⡤⠖⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠠⠐⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠛⠛⠻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⠀⠀⠰⣿⣿⣿⠟⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⡄⠀⠀⣤⣾⠋⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⡄⠀⠐⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠉⠉⠙⢿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢼⡿⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⣿⣿⠋⠉⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡇⠀⠀⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣿⣿⣧⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⠀⠀⠀⠀⠸⠀⠈⠇⠀⠀⣿⣿⣿⠗⠶⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣶⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⢹⣿⣿
⠀⠀⠀⠇⠀⠐⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣿⡿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣾⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣦⡀
⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⣀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠁⠀⠉⠁⢀⡀⠈⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠁⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⡭⠈⠻⣿⠛⠁⠹⣧⠀⠀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⡟⠁⠻⢢⡌⠁⠀⠑⡄⠀⠁⠘⢦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣅⠀⠈⣹⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⡿⠇⠀⠃⢀⠿⢰⠀⠀⢠⡀⠀⢀⠘⣧⠀⠀⠌⠁⠀⠀⠀⠀⣰⡿⣥⠀⢀⠀⠀⢛⠀⠘⣦⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢈⢿⣿⣿⣿⡇⠃⠐⠀⠈⢠⡏⠀⠀⠀⠹⣷⡾⠀⠘⣦⠀⠀⠳⠂⠀⠀⣰⡟⠀⠈⢢⡾⠀⠀⠀⢣⡄⣿⠀⠀⠀⠀⠹⣿⣿⣿⣯⡟
⠀⠀⠘⠛⣿⣿⣿⣇⡀⡀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣽⣿⣿⡛⠁⠁⠀⠀⡴⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡀⠐⠀⠀⢸⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠘⣿⣿⣿⡀⣤⠀⠀⠀⣠⠇⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠉⠁⠀⢸⣿⣶⣀⣀⣠⣀⣴⣤⣦⣴⣶⣶⣷⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣄⣶⣶⣿⣦⣴⣤⣀⣠⣴⣦⣶⣶⣼⡇⠈⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣉⠀⠀⠁⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣏⣀⣀⣭⣤⣴⣶⣤⣭⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⡛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢛⣻⣿⣿⡿⠟⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠉⠉⠀⠀⠉⠭⠛⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⡤⠤⠖⠀⡇⠀⠀⠤⠄⣠⠂⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠁⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠄⠀⠈⡏⠉⠓⠲⣄⠀⣀⠔⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢀⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠤⠤⠤⢠⡆⠀⡄⠀⣄⣀⣠⠄⠰⠂⠀⣈⡀⣀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⣸⠁⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠐⠀⠀⠀⢀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠴⠒⠉⠀⢀⣀⣀⣀⣠⣬⣿⡶⢦⠤⡦⢴⣾⣥⣤⣤⣀⣀⠀⠀⠀⠀⢀⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠉⠉⠉⠋⠀⠈⠀⠀⠻⠿⠿⠟⠛⣿⠋⣿⠃⠀⠀⢠⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠇⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡼⠁⣸⠃⠀⠀⡔⣻⣾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠈⢦⡀⠀⠀⠀⠀⠀⡂⠂⠀⣼⢃⡼⠃⠀⢀⡞⣗⡾⠋⠂
⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠁⠀⠀⠈⠷⣄⠀⡀⠀⢀⣒⣲⠞⣡⠋⠀⠀⣀⣶⣿⠟⡠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣱⠙⣾⠿⣿⠁⣾⠁⠀⣠⠴⠟⣁⣀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠉⠀⡤⠞⠁⠈⢩⡏⠀⢻⠖⢻⡄⢷⠶⠯⠤⣤⣷⣼⣡⣿⣟⣹⠄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣀⣈⠷⠀⠀⢠⡿⠀⠀⣾⠃⠈⡇⠈⣇⠀⠀⢧⣷⣿⣧⣤⣿⣉⣇⡀
⠀⠀⠀⠀⠶⠶⠞⠒⠒⠚⠛⢛⣛⣓⣒⣿⣇⡀⢀⣏⢀⣀⣇⣀⣹⣦⣤⣤⣿⣧⡦⣯⣬⣭⣯⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠉⠉⠉⠩⠽⠿⢷⣶⢺⣶⣿⣭⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠷⠾⣿⣵


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⢻⡅⠈⢇⡴⠛⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠈⣛⣓⣚⣓⡰⢦⣴⣚⣵⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠻⢿⠿⠛⠿⠷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⣀⢀⣀⡄
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⣠⡝⠒⠶⠶⠶⠶⠶⠖⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣌⠙⠟⠁
⠀⠀⠀⠘⠿⢿⣷⣶⣦⣤⣤⣴⣶⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠙⠗⠲⠶⠶⠶⠶⠲⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠘⠀⠇⡇
⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓⠂⠀⢠⡿⡀
⠀⠀⠀⠰⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠠⢦⣉⣉⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠻⢦⣬⣭⣭⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⣤⠾⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠒⠀⡇⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢒⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠔⠁⠀⠀⠀⣹
⠀⠀⠀⡀⡄⠀⠔⠀⠀⠀⠀⠀⢀⠀⣒⣒⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠃⠀⠁⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠘⠄⡄⢠⡀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠠⡤⠔⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠤⠤⠼⠃
⠀⠀⡀⠀⠇⠰⡐⢄⠀⠈⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⣄⠀⠠⢄⡀
⠀⠈⠀⠀⠀⠀⠰⡀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠐⡀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠡⠀⢸⠀⠀⠀⠠⠃⠀⢣⡀⠀⠀⠀⠀⠠⠃⠀⠘⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢀⠂⠤⣀⣀⡪⠅⠀⠀⠠⣑⠠⠤⠤⠭⠕⣂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⠒⠊⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢘⠁⠀⠈⠩⡀⠀⠠⠂⣀
⠀⠀⠀⠀⠀⢸⠀⠂⢸⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠄⠀⠀⠀⠈⠀⢆⠢⡅⠨⠐⠤⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠂⢀⠄⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠐⠀⠀⠀⣠⣄⠀⠀⢠⡤⡀
⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄⠠⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⢄⠠⠀⢄⠤⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠟⠃⠈⠓⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡒⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠓⢤⠀⠀⠀⠀⡀⢠⠀⠀⠜⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠓⠚⠉⠀⠁⠁⠀⠁⠂⢀⡤⠀⠀⡔⠢⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⡀⠀⠠⠔⠚⠀⠀⠄⠫⠤⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠁⠀⠀⠀⠀⣰⠶⠡⠀⠀⠀⠀⠀⡀⠀⠶⠦⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠅⢐⡠⣌⠀⠤⡀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠠⢭⠐⠂⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⡌⠙⣆⣆⡤⠞⣓⠒⠢⢄⠀⠀⣀⠀⢠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⢉⢉⣩⠞⠁⢰⠒⣏⠑⡢⣠⣇⡎⣡⠤⠶⣐⠒⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⡴⣴⠋⡜⣡⢋⢧⠀⠸⡯⠤⠴⣂⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢠⡿⣌⣛⡻⢢⠤⠻⠂⠂⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠳⣄⢀⣠⣾⣷⣦⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠏⢰⣈⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣬⣁⣤⡈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢉⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠁⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠁⠀⠒⠀⠀⠀⠈⠂⢄⠈⠈⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠁⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⣤⠰⠂⠸⠆⠿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠿⠿⠚⠛⠛⠛⠋⢹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⢡⠊⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⠿⡿⠻⠿⠿⠿⡿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣷⣾⣷⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⢴⣦⣶⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣴⠀⢲⣿⣿⢿⡺⡿⠻⠟⠙⠛⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣰⣦⣻⡿⢿⠃⠹⠂⠀⠉⠉⠀⠀⠀⣠⠖⠋⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡱⠒⠤⢤⣀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⢤⣻⡇⠀⠀⢀⠀⠱⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣤⡐⢄⠉⡇⠀⠀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⢠⠇⡉⠉⠽⡟⢤⣀⢂⣴⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣞⡋⠀⠀⠀⠀⣷⢲⣻⢷⣩⠏⡩⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢈⡵⠦⢤⣠⠜⠉⣾⣱⡺⠃⣰⡴⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡏⠀⠀⠀⠀⠉⠐⠃⠛⠁⡰⢫⣽⠃⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢇⠀⢓⠦⠤⢄⣀⣀⣀⡼⠁⠈⠀⡆⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁⠀⠀⢀⡜⢸⠀⠀⠀⠀⢠⣿⣀⡠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠘⣆⢸⠀⠀⠀⠀⠸⣿⡠⠤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣷⠄⠀⠀⠀⠘⣼⡀⠀⠀⢀⡀⢻⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠛⠋⠀⠀⠀⠀⠀⢻⣦⡀⠀⢀⡴⣿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠑⠴⠋⡠⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣸⢔⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢉⣾⣷⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠛⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢿⣀⣮⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠛⢻⠋⣷⣄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢣⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣄⠙⣿⣦⡄
⠀⠶⣿⣼⣿⣿⣿⠟⢀⣠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⠁
⣶⣶⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠛⢯⣿⣿⣿⣿⣿⣿⣧⠀⣦⣾
⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠏
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣤⣀⣀⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣶⣶⢈⠉⠉⠙⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⣉⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣧⡸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠿⢿⣿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣷⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠈⣿⣿⣿⠈⢿⣿
⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿
⠀⣾⣿⣿⠟⣿⣿⡿⢿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡆⠀⠸
⠸⡿⠻⣿⢰⣿⢰⡇⠈⢿⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣸⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢰⣿⣿⣆⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⢻⡀
⠀⠀⠀⢀⣿⠘⣻⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⣾⣿⣿⣥⣄⢸⡇
⠀⠀⠀⢸⡏⢀⢼⣿⢿⣿⡿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣯⡥⢼⡇
⠀⠀⠀⣾⠇⠀⠈⠁⢸⡏⠀⠈⠃⢀⣤⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠰⡛⠛⢹⢿⡏⠙⠿⠀⣷
⠀⠀⢀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡾⢿⡿⠶⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⢴⢾⣿⣿⡅⠀⠁⠀⠚⠈⠁⠀⠀⠀⣿
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡂⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠰⣶
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠠
⠀⠀⠅⠀⠀⠀⠀⣶⡶⠸⡆⠀⣸⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⢦⡀⠀⢻⠟⠀⠙⢦⣽⣥⠾⠁⠀⣦⠀⢀⣇⠀⢰⠆⠀⣦⠀⢀⣆⠀⠀⠀⢠⡄⠀⣠⣇⠀⣾⠀⠀⢰⣀⠀⠀⣀⠀⡀⠀⠀⢀⣄⠀⢀⡆
⠀⠈⠀⣀⣻⣋⣠⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢦⣙⣟⣤⠟⠀⠀⠘⢦⣙⣟⣠⣄⡀⠀⠹⣦⣙⣯⡴⠃⠀⠀⣙⣿⣤⠴⠋⠀⠙⠶⣤⣘⣟⣠⠞⠁
⠀⠀⢸⡍⣿⣿⠀⠀⠀⠀⢀⣼⠋⠻⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠙⠂⠀⠀⢹⣿⡇⠀⠀⣞⠹⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡇⢀⡀
⠀⠀⠀⠙⣿⣿⠀⠀⠀⢠⡾⠁⠀⠀⠙⣧⠀⠀⠀⣿⢻⡅⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠙⠀⣿⠻⣆⠀⠀⠀⠀⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡿⡏
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⠀⠀⠐⠠⠌
⠀⠀⠀⠀⢠⡾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣵⣄⣀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⣿⢋⡿⠋⠙⠁⠀⠀⠀⠀⠀⢐⣯⣽⡛⣿⡏⠛⠁⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠳⠀⠘⠀⢰⡃
⠀⠀⠀⠀⠀⠀⣿⡌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⠛⢷⣄⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡄⠄⠀⣠⣤⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⡀⠀⢈⣻⡆⠘⢿⣿⣿⣿⣷⡿⠁⠀⠂⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠟⠁⠀⠀⠀⠀⣾⣿⡄⠀⠀⣾⣿⣇⠀⠀⡌⠻⣷⣶⣸⣿⣿⣏⣀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⣦⣤⡻⢿⣋⣠⡄⢟⢆⢸⡿⣿⣻⡟⣿⣿⣿⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠐⢿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣯⡿⣧⣬⣭⣥⣴⡶⠟⠋
⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣶⣿⣿⣟⣯⣴⣮⣥⣼⣿⢿⣿⠀⣿⣿⣿⣿⡀
⠀⠀⠀⢙⣛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢛⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢸⣿⣻⣷⣸⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⡷⠶⠶⠾⠀⠀⠸⠶⠶⢶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢸⣿⠙⣿⣹⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠿⣿⣿⣿⢸⣿⢸⣿⢹⣿⢻⣿⠟⠘⠿⠘⢛⣿⠟⣿⡏⠁
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⣿⠁⠉⢸⣿⢸⣿⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⢸⣿⠀⠀⠈⠉⠀⠉⢸⣿⢸⣿⠀⠀⠀⠀⠐⠿⠃⠿⠃
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⠿⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⢠⣤⣖⣿⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣦⣤⡄⡀
⠀⠀⠀⠀⢿⣿⣯⣿⣿⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠈⠙⠻⣯⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣭⣵⣶⣿⣿⣿⣿⣿⣶⣦⣤⡀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣮⣭⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠙⣻⣿⣿⣟⠋⠉⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣾⣟⡁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣶⣶⣶⣶⣾⣿⠿⠋⠙⠿⣿⣷⣶⣶⣶⣶⣶⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠶⣶⣶⣶⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠽⠛⠛⠛⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣽⣦⣤⣄⣀⡀⠀⠀⠀⠀⠿⠇⣠⣀⣀⣤⣦⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⠿⠿⠿⠿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⠀⠀⠀⠀⠘⢿⣿⣿⣭⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠄⠀⠘⠀⡇⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠐⢦⡀⠀⠀⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠶⠶⠦⠀⠠⠁⢸⡆⢀⠀⡿⠹⠹⡇
⠀⠀⠀⠀⣀⡀⠀⠀⠈⠛⠀⠈⢿⠀⢻⣆⠀⣤⠀⠀⠚⠻⠶⠄⢌⡀⣿⣷⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠉⠛⠛⠉⢀⣤⣤⣤⠀⠀⠀⢀⡙⠀⠹⣷⠀⠀⠚⠿⠿⠦⠑⣌⢰⡇⠀⠀⠀⠀⢀⣤⣤⢴⣂⡘⠿⠿⠂⣤⢆⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣴⣶⣶⠶⣠⣤⣍⡑⠲⢤⣀⡀⠐⠶⣶⣶⣬⣿⠀⠀⠀⢸⣶⣿⣿⣅⣺⣿⡿⢴⣶⣶⡝⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠛⠛⢛⣩⣶⣶⣶⠎⠙⠳⢦⣄⡀⠀⢸⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⣂⠈⢉⣋⢰⣿⣿⡆⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢉⣴⣾⣿⡿⠟⠉⠛⠦⣼⠀⠀⠀⢸⢸⡏⠙⢿⠸⢿⡿⠃⢿⣿⠇⣽⣿⣄⠈⣻⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣠⣤⣶⣶⣶⣿⣾⣶⣷⣾⣿⣷⣿⣮⣴⣾⣶⣠⣶⣶⡄⠛⢿⣟⡘⠿⡿⢇⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠙⠛⠁⠀⠻⣿⡿⠀⠀⠈⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢩⣭⣥⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢴⣾⣿⣿⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣍⡛⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣯⣽⣿⣿⣍⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⡻⠿⠿⠃⠀⠀⠀⠀⠀⢶⡄⠈⠛⠀⣠⡖⠀⠀⠀⠀⢿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣷⣮⡭⢥⣶⣷⡄⠀⠀⠀⠀⠀⢀⣬⢋⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣾⣷⡜⣿⣿⠀⠀⠀⠀⢠⣿⠁⠀⠙⣷⠀⠀⠀⠀⢀⣤⣄⣛⣛⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿⡄⠀⠀⠀⠀⠻⣦⣤⡾⠋⠀⠀⠀⠀⣿⣿⣿⠟⣛⡻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⠿⠇⣿⣿⠃⠀⠀⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⢐⣿⢟⣵⣿⣿⣿⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⣶⡆⣿⣿⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣄⠀⠈⢵⣿⣿⣿⡿⢋⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣦⡙⠿⢋⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢸⣿⣿⡇⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⣴⢿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣼⣿⣿⣧⢹⣿⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣶⣮⣭⣴⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⣛⣛⣡⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣩⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⢤⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠃⠠⢻⡧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠉⠙⠛⠈⠉⢧⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠼⠁⠀⠀⠀⠀⠀⠀⢻⣥⠤⠖⠶⠚⠋⠉⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠖⠒⠒⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠞⢱⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠠⣿⣦⣄⡀
⠀⣀⣀⠠⠤⠀⠂⠈⠁⠀⠀⠀⡰⠁⠘⣿⣿⣿⡆⠀⢀⠃⠀⠀⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣿⣿⣿⣿⡄⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠹⣿⣷⡂⠀⢸⡄⠀⠀⡀⣀⠀⠀⣿⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢸⣿⣿⣇⠀⣽⡿⠀⠀⠈⡇⠀⠀⢱⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣾⣿⠇⠀⠀⠀⡇⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢀⠟⣇
⠀⠀⠀⠀⠀⣿⢿⣆⠀⢀⡞⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢹⠈⣿⣧⡼⠰⠖⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⡉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⢸⡄⠘⡟⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⣿⣟⡛⢿⣦⣄⢳⡀
⠀⠀⠀⠀⠀⣸⠷⠶⠷⡶⠶⠶⢧⠀⠀⠀⠀⠀⠀⠀⢠⢾⡟⠿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⡁⢠⠌⠛⢿⣦⡘⣿⣆⢧
⠀⠀⠀⠀⠀⣿⠀⣀⡀⢇⠠⠤⠀⠳⡀⠀⠀⠀⠀⠀⣿⢸⠓⠲⠬⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠡⡤⠀⠀⢀⣿⣧⣸⣿⡿⠷
⠀⠀⠀⠀⠀⣿⡀⢇⡹⠾⠀⠀⠀⣰⠁⢀⠀⡀⠀⢀⣹⡌⡀⠠⠆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠁⠄⢠⠈⢉⣹⢿
⠀⠀⠀⠀⣠⡏⣧⠾⢧⣀⣀⠀⠈⡗⢰⣾⠿⢛⠿⣿⡀⣷⡷⠤⡄⠀⢈⡻⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⡿⠋⠛⣩⢾⡆
⠀⠀⠀⢸⡿⠧⠿⠟⣓⠲⢿⣿⣿⡛⣻⣿⢀⣿⡃⢘⣿⣿⡇⠀⠘⠒⠉⢱⣼⣿⠿⣣⠽⠿⢅⡀⠀⠀⠀⣼⡇⠀⢰⠇⠀⢳
⠀⠀⠀⢸⠇⢀⣴⡾⠿⣷⠀⢻⣿⣧⢼⣿⠈⣿⣡⡾⠃⣿⠇⠀⠀⠀⠀⢸⠁⠀⢾⡵⣣⢀⣀⣿⠓⢦⣀⣿⡇⢠⠏⠀⠀⢸⡆
⠀⠀⠀⢸⠀⣾⣿⠁⠀⠘⣇⠈⣿⠸⣿⣿⣄⠈⠁⠀⢀⣻⡀⠀⠀⠀⠀⡟⠀⠀⠈⣿⣵⠟⢿⣧⠀⠀⠈⠛⢷⣏⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡟⢹⣿⡆⠀⠀⠸⡄⢹⡇⢻⣿⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠻⡤⠤⢤⣿⠇⠀⢸⡏⠳⣄⠀⠀⠼⠘⠆⠀⠀⣼⠁
⠀⠀⠀⠈⠀⣿⣿⣧⠀⠀⠀⢻⡈⣧⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠇⢀⡼⢟⠢⡀⢸⡇⠀⠈⣷⣶⣤⣤⣤⡤⠾⠿⣧
⠀⠀⠀⠀⠀⣿⣿⢿⡀⢀⣴⡾⢳⡟⢙⣿⠭⣶⠄⠀⠀⠀⠀⣴⣙⣺⡝⣾⡟⠉⠀⠈⡆⠁⣾⡇⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣿⡟⠈⣧⣾⡟⠀⢸⡇⠈⠿⠾⣿⡆⠀⠀⠀⠀⢷⠶⢽⡃⣿⡇⠀⠀⠀⠉⢠⣿⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣿⠃⠀⠸⣿⠃⠀⢸⣧⣀⠙⣭⣽⠃⠀⠀⠀⠀⠈⠉⠉⢹⡏⢧⣀⠀⢀⣠⠟⠈⣿⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢙⣦⡀⢸⣿⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠉⠉⠉⠀⠀⠀⠸⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣿⠛⣾⡏⣿⡿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⣇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡏⠀⣷⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠷⣦⣠⡄⢀⡄⣠⣄⣴⠾⠋
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⣟⢉⣹⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣀⠀⣀⣀⣠⣤⠶⠞⠀⠀⠀⠸⡿⢿⠛⠛⣻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⠛⠛⠛⣿⡏⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⡏⢹⠉⠉⣗⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠛⠋⢹⣇⣾⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⢈⡇⠀⡇⠀⠀⠀⠀⠀⢀⡇⠈⡇⢰⡇
⠀⠀⠀⠀⠘⠛⠒⢶⣤⣤⣤⣤⡴⠖⢻⡟⠛⠋⠉⠀⠀⠀⠀⠀⣀⣤⣖⡛⢳⣾⡵⠛⢷⣤⡇⠀⠀⢤⣶⣾⣭⣤⡜⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⠿⠿⠟⠛⠛⠃⠀⠀⠀⠐⠛⠛⠛⠛⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣛⣋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⢿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣜⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⢀⡤⣀⣶⣞⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣿⣾⣿⣧⣤⣠⣤⣤⣄⣀⣴⣴⣯⣿⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⣷⡶⠶⢶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠘⠆⢠⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠠⣄⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣤⣤⡀⠀⠙⠀⡆⠀⠀⣿⠿⠿⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠙⠆⠀⠹⣆⠰⣦⡀⣀⡀⠀⠰⠶⣦⡄⢄⠃⢸⡇⡟⠀⠀⠇
⠀⠀⠀⠒⠛⠓⠂⢀⣀⣀⣀⠀⠈⠀⠙⠇⠘⢿⡀⠀⠰⢶⣶⣦⡑⣜⠁⡇⠀⠀⠐⠀⠀⣀⣀⡠⠂⠰⣯⣶⠀⣀⢄
⠀⠀⠀⠀⠀⠀⠐⠛⠛⢉⣡⣤⣤⡄⣀⣘⣒⠮⣥⣀⠀⠀⢤⣶⣶⣌⣳⡇⠀⠀⢰⣤⣾⣿⣏⢼⣿⣷⣠⣤⣤⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠰⠾⠿⢛⣭⣴⣶⣬⠙⠳⢦⣄⠉⠉⠁⡇⠀⠀⢨⣿⣿⣿⡿⠿⣿⡁⠘⠛⠛⢰⣶⣶⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢉⣴⣾⣿⡶⠎⠙⠦⣤⡇⠀⠀⢸⠸⣿⠙⣷⠸⣿⣿⠂⣿⣿⡎⣹⣯⡀⢙⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⣠⣤⣴⣶⣿⣣⣵⣶⣿⣶⣷⣶⣦⣤⣾⣦⣠⣶⣷⡌⠻⢿⣏⠸⢿⡿⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠛⠛⠁⠀⢿⣿⠗⠀⠀⠹⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⣀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣤⣶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠠⢉⠳⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣴⣿⣶⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡤⣿⣿⣟⢳⣦⠑⡈⢳⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣛⠀⠀⢡⡌⠹⣿⣷⡜⢿⣿⣿⣿
⠀⠀⠀⢀⣴⠞⡁⠔⠸⣿⣿⣾⣿⡧⠈⠄⢻⡀⠀⠀⢸⣿⣿⣿⡿⢋⣵⣾⣿⣇⠀⢀⠇⠀⢸⣿⣿⡌⣿⣿⣿
⠀⠀⢀⡞⠡⠊⠀⠀⠀⣹⣿⣿⡿⣃⠀⠀⢸⡇⠀⠀⢸⣿⣿⡿⢡⣿⣿⣿⣿⣿⠖⠁⠀⢀⢾⣿⣿⡇⢹⣿⣿
⠀⠀⠸⣷⣅⣠⣤⣶⣿⣿⡿⠛⢿⣿⣿⣾⠿⠁⠀⠀⢸⣿⣿⣇⠘⠉⠙⠋⠉⠁⠀⢀⣴⡃⠀⠉⢉⣡⣿⣿⣿
⠀⠀⠀⠙⠿⠿⠿⠿⠛⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡎⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⣴⠻⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣦⣄⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠉⢉⣿⣿⣿⣿⡿⠛⠛
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠛⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠔⣹⣿⠟⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿
⠀⠀⠀⠀⢿⣏⠀⠀⠀⠸⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠀⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠀⠀⠀⠀⠐⡍⠀⠀⠋⠃⠀⠀⠀⡤⠠⠀⠀⠀⠀⠨⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢤⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠦⢤⡄⠀⠀⠀⡀⠀⠀⠄⠉⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⣄⠀⠀⠀⣠⠐⡀⠀⠀⢀⠔⠁⠐⠀⠀⠀⠀⢀⠄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢄⠁⢂⡤⠐⠀⡆⠀⠘⠉⡐⠂⠤⡰⢶⣀⠈⠒⠃⠀⠂⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠚⡌⠀⠀⠀⠐⢡⠀⠀⠀⠁⠀⠐⠚⠀⠐⠐⠃⠀⠊⣀⠀⠀⠘⠁⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⣀⠠⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢁⣀⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⡀⠀⢠⠀⠀⠀⠑⣀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠉⠀⡀⠈⠈⠀⠀⠀⢀⡤⠤⠀⠉⠀⡀⠙⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠛⠛⣿⡟⠛⠃⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣄⠀⠀⣠⢄⠈⠠⠤⡀⠀⢀⡴⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣀⠱⡎⡀⠀⣀⡄⠀⠈⡞⢫⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢰⠃⢸⠀⠀⢸⠁⠹⠀⠀⢸⠀⢳⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡌⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠛⡍⠦⣬⠀⠀⢸⠀⠘⠀⠀⢠⠴⢃⠟⠉⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣘⣄⡞⠀⠀⠀⠀⡀⠀⠀⠘⣦⣎⣀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⡇⡇⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠠⣾⠁⢹⠆⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⣷⣶⣿⣿⣶⣶⣦⣞⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⠀⡾⠟⠛⠋⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⣴⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⢷⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠶⡞⠉⠁⢸⣯⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⡶⢈⣷⣼⣦⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⣤⡾⠖⠉⠀⠀⠉⠓⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡶⠉⠁⠀⠙⢢⣄⣴⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣶⡾⣶⣶⣿⠉⢿⠶⣤⣀⠀⠀⣀⣼⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠋⠀⣠⢻⣿⠀⢸⠀⠀⠉⠻⣿⣿⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠿⣯⣱⠀⠀⠈⡜⣿⡄⠘⡄⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⣿⣶⣾⢛⡿⠿⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣿⠉⠑⠚⠻⢷⣇⠀⢣⣼⡧⣄⣿⠀⠀⠀⠀⠻⢹⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⠀⠙⠛⠁⠀⠈⠛⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢸⡀⠀⠀⠀⠀⠉⢷⠘⣿⡃⠀⠙⣧⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡄
⠀⠀⠀⠀⠀⠀⢀⡎⢿⡀⠀⠀⢧⠀⠀⠀⠀⠀⢸⡄⣿⠃⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠁⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⢸⣧⣻⠄⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⡆⠀⠀⠹⣦⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⡀⠀⠀⢿⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣆⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠈⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢹⣧⠀⠀⢼⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢷⡀⠀⢸⡄⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡇⠸⠈⢧⡀⢾⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡏⠈
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⣇⠀⠀⢳⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⣾⠁⠀⡇⠀⠀⠀⢳⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⣦⢤⣾⡇⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⣾⣿⡇⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⣿⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣿⡟⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⣠⠞⠁
⠀⠀⠀⠀⣠⣿⣿⡖⣲⠦⠄⡀⠀⠀⠀⠀⠸⡆⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣤⣤⣤⣤⣤⡼⠟⠁
⠀⢀⣴⠞⢙⣿⣶⣾⡦⣍⠂⠀⠀⠀⠀⠀⠀⢻⡀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⢐⡽⠻⢤⣾⣏⢀⣿⠀⠈⠣⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⣀⣀⣀⡀⠀⣀⣄⣤⣤⡤⠴⠶⠞⠁
⣿⠁⠀⣼⠟⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠿⠉⠉⠉⠉⠿⠿⠿⠟⠛⠉⠉
⠉⠳⠾⣿⡀⣀⣾⠃⢀⣀⣤⣤⡴⠶⠚⠛⠉
⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁