Cup Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣕⣂⠲⠶⠶⠿⠶⠷⠶⠶⠶⣶⣖⣒⣲⣶⣿⣷⣾⣿⣿⣮⣤⣬⣭⣀⣀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⣿⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⡿⠀⣸⣷⠖⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠃⢠⣿⡛⠈⠁⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⣾⠏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣿⣿⣧⣲⣏⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣽⣿⣿⣿⣶⡿⠋⠁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣄⣀⡂⢤⣤⣤⣄⣾⣿⣴⣀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣾⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠰⡓⠚⠦⠘⠙⡟⠛⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣦⣴⣦⣬⣤⣤⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣳⣤⣃⠉⠀⠀⠉⠻⠿⠿⢿⣿⣯⣭⣽⣿⡿⠽⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣤⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣘⣛⣿⠯⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⢒⣒⣒⣶⣶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣎⠡⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡢⣀⣀⣉⠉⠀⣚⣹⣧
⠀⠀⠀⠀⠀⢸⠇⡟⠛⠛⠿⠯⢽⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣭⡭⠿⠖⠛⠋⠉⠀⣿⡆⢀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢸⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣀⣀⠀⠠⠄⠐⠂⣿⡷⠋⣩⠴⠖⠲⢦⣍⠻⣶⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⡺⠭⠭⠆⠀⣿⠱⣿⠃⠀⠀⠀⠀⠙⡆⠘⣿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡫⠭⠄⢀⣿⣤⠟⠀⠀⠀⠀⠀⠀⡿⠀⡿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣒⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠃⣼⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣀⠒⢀⣿⠃⠀⠀⠀⣀⣠⠴⠚⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠥⠴⠀⣼⣏⣤⠶⣚⣋⣩⣴⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡍⠁⣰⣿⡯⣲⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⠀⣰⣿⣨⣼⣋
⠀⠀⢀⣀⡤⠶⠒⠛⠛⠹⣇⠀⠀⠀⠀⢲⣶⠀⠀⠀⠄⠀⠀⠠⠤⢔⡒⢢⣿⠉⠉⢉⣽⣿⣿⣷⣶⣤⣄⡀
⡰⠚⠉⠀⠀⠀⣀⣤⡄⠀⢿⡆⠀⠀⠀⠈⠭⠭⠭⠍⠀⠨⣭⣥⣴⡷⠶⣺⡿⣶⣶⠾⠿⢯⣍⠀⠈⠁⠈⢻⡆⠀⠐⠀⠀⡀
⠡⣄⠀⠀⠀⠀⠁⠁⠁⠀⠈⠳⢦⣤⣌⣉⣉⣉⣉⣉⣉⣉⣭⣥⣴⣶⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠈⠙⠳⢦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣠⣤⣤⣤⣠⣤⣤⣴⣶⠿⠛⠁⠀⣀⣈⣉⣉⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠾⠭⣭⣍⣛⣛⣛⡛⠛⠛⠒⠒⠒⠒⠒⠛⠛⣛⣛⣉⣩⡭⠶⠟⠋⢁⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠁⢀⣀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠤⠄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂⠀⣁⡀⠀⠤⠀⠤⠤⠤⠤⠤⠄⠠⠤⠀⣀⡉⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠤⠤⠤⠤⠄⠀⠀⠀⠀⠠⠤⠤⠤⠄⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⡆⠀⢁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⣥⠀⢸⡩⠹⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠘⡕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡠⢊⠤⠊⠄⠘⢆⠈⠢⣰⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀⠤⠄⠀⢹⡲⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠘⠠⠂⠀⠀⠀⠀⠀⠑⣤⣀⠁⠚⠒⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠖⠉⠀⠀⠀⠀⡘⠋⠢⠣⠈⡄
⠀⠀⠀⠀⠀⠀⢀⠹⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠀⠀⠲⣄⡸⠄⣀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⣉⢀⣁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠰⢁⠃
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⡀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⣈⣉⣉⢉⣁⣀⠤⠔⣂⡩⠕⠊⠉⠉⠛⠋⠉⠀⠀⠀⠠⠔⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠉⢁⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⡠⠐⢈⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠐⠀⠈⠁⡐⠂⠠⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠶⠾⠛⠛⠉⠉⠀⠐⡀⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠀⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣟⣛⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀
⠀⠀⠀⠀⠀⢠⡔⠂⠀⢀⣀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢀⢺⣶⣶⣾⣿⣿⠿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣷⣶⣶⣿⣥⣴⣾⣯⣭⣯⡙⠿⠿⣿⣷⣦⣄⠀⡸⢀⡆⠀⢀⡠⠤⠄⠠⣄
⠀⠀⠀⠀⢸⣦⠙⠿⣟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣿⣿⠟⣡⣾⣧⣶⢛⡀⠀⠀⠠⢤⣷
⠀⠀⠀⠀⠘⠿⠀⠂⠚⠙⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⣴⢞⣿⣿⣿⣷⡏⠁⠀⠀⠀⠀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢛⣛⣛⡛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠍⠡⠖⠀⡰⠊⣿⣿⣿⠍⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣿⣿⢁⣿⠛⠋⠀⠀⠀⠀⠀⡄⢀⠆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢿⣿⢻⡏⠀⠀⠀⠀⢀⡠⠊⡠
⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡟⠀⠀⣀⠠⠖⢋⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠋⣁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣾⣼⠿⠟⡫⡶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⣤⣠⣤⣤⣄⣤⣤⣤⣤⣴⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠁⠈⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡷⢄⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⣈⣙⣆⠀⠀⠸⣶⡊⣢⠀⠀⠀⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠃⠀⠀⠙⢢⣿⡇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠸⠏⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣤⣤⣤⠀⠀⠀⣀⣀⡀⢤⠤⠠⠄⠀⠀⣀⡀⣰⣿⣶⣦⠙
⠀⠀⠀⠀⠀⠀⠀⢰⣤⢷⡀⠀⠀⠀⠀⠀⡔⣿⣿⣿⣿⣿⡃⠀⠀⠀⠠⣤⣠⣀⣴⣶⣶⣶⣶⣤⡀⢛⣉⣽⣿⠿⢡⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⢷⡄⠻⣄⠀⠀⠀⠀⠉⠠⣉⠙⠛⠋⠾⠿⠿⠤⣦⣤⡄⢉⡿⢶⣦⡌⠻⠿⠟⠚⠛⢋⣡⣴⣿⣷⡿⠵⠾⠍⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠲⢶⣶⣶⣿⣿⣿⣿⣿⡿⠛⠓⢤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠳⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢠⣼⣿⣿⣟⣿⣡⡾⠀⠀⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢣⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢫⠡⢿⠃⠀⠀⣠⠆⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠱⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⣤⣴⠏⣾⣿⣶⠟⠁⠀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣛⠋⢸⣏⠋⢀⣒⣨⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠑⣀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣟⡛⠛⠿⠛⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠜⢻⣿⣌⠳⣜⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠄⠐⣰⣿⣿⣿⣿⣷⠶⠦⢄⡀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡴⠋⠍⠀⠀⠀⢀⠉⠓⢼⣧⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠌⣤⣾⣿⣿⣿⣿⣿⣿⢺⡤⠂⠙⢢⠀⢽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡄⡐⠀⢺⠃⠂⠀⠀⠀⣠⠎⣴⠄⠀⠙⢿⣿⢦⠉⠐⢶⣶⡶⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠛⠈⠀⠀⠀⡠⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⢀⡀⠀⠘⠓⢾⠗⣒⣱⠀⠀⠙⣇⠉⠈⠀⠂⠢⣌⠉⠉⠀⠉⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⣀⡞⢻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⠀⠉⡉⠁⠀⠀⠀⠙⠐⣀⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠁⠀⢀⠯⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣤⣤⣄⣀⣉⣀⣀⣐⣺⣿⣷⣶⣤⣤⢘⣋⣉⣉⠉⠀⣀⣀⠀⠀⠀⠠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠊⡟⠿⢛⠷⠛⠛⠋⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣠⡾⠃⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠘⣿⣄⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠘⢿⣦⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠈⢿⣇⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠠⠟⠁⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠔⠒⠒⠒⠒⠒⠀⠉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠀⠐⠒⠒⠒⠒⠢⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢉⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⡟⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢦⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠎⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣀⠴⠋⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠓⠋⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠵⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡫⠤⠴⠤⣖⣚⣉
⠀⠀⠀⠀⢀⡤⠒⠒⠉⠉⣀⣤⣤⣶⣶⣶⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣶⣶⣶⣶⣤⣤⣀⠉⠉⠒⠲⢤⡀
⠀⠀⠀⠀⠈⠢⢄⣀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃⠀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠦⠤⠤⣍⢉⣉⣙⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣋⣉⡉⢭⡤⠤⠴⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⣈⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣁⠤⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⣴⣾⠿⣛⣛⠭⠭⠭⠍⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⠙⣛⡛⢿⣷⡄
⠀⣿⡟⠻⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠾⠛⣿⡇
⠀⣿⡇⠀⠀⠈⠀⠀⠐⠓⠛⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠟⠛⠒⠊⠁⠀⠀⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣠⣤⣴⣶⣶⣶⣦⣤⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠉⣁⣤⣤⣤⣉⠙⢿⣷⡀
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠛⠉⠉⠙⠻⣿⣆⠹⣿⡄
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣾⠏⠀⠀⠀⠀⠀⠀⠘⣿⡆⢿⣧
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸⡿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢠⣿⠇
⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢠⣾⠏
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣴⡿⠋⣰⣿⠋
⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢀⣠⣶⡿⠋⣡⣾⠟⠁
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⣾⠿⠛⢉⣤⣾⠟⠁
⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣍⠀⢀⣠⣶⡿⠛⠁
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣶⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣾⠿⠿⠛⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣯⣧⣦⣼⣧⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡸⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠐⠂⠐⠚⠑⠁⡀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⡇⢀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠋⠩⠶⠂⡀⠀⠀⠀⠄⠠⠭⠥⠤⠄⢀⡁⠂⠈⠀⠉⢠⣖⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⣿⣿⡶⢶⢶⣦⣥⣤⣭⣭⣭⣭⣤⣤⣭⣤⣾⣶⣶⣾⣦⣽⠖⠉⣀⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠍⠉⠿⣿⣿⣿⡿⠅⢺⠅⢸⠅⢀⡤⠖⠒⠶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢁⣭⡭⢭⣿⣖⣾⠅⢐⡡⢸⡴⠋⠀⠀⠀⠀⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⣈⣭⡭⠵⣖⡋⠀⠊⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠥⠀⠐⣒⣛⣩⡭⠄⢈⠁⣾⠀⠀⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣈⢋⣩⡭⠥⠶⣒⠀⠄⠀⣟⠀⠀⣀⠤⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣒⣚⣛⡭⠅⠒⡁⠀⡯⠌⠀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠭⠭⠷⢖⣚⣋⠀⠂⣸⣡⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢘⣫⣭⣭⡷⠠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣌⠠⠄⠀⠀⠀⠀⡀⢀⣀⠒⣀⣐⣲⣿⣿⠯⠀⣩⣾⣁⣀⣀⣀⣀
⠀⠀⠀⣠⡤⠶⠒⠚⠉⠿⠉⠀⠀⠤⠤⠤⠿⣗⣦⢄⠀⠀⠀⠐⠀⢛⣀⣿⣿⣿⠿⠭⣴⣾⡿⠿⠿⠿⣯⣭⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⣠⡀⠀⠰⠅⠀⠀⠀⠀⠐⠄⠀⠈⠓⠤⢤⣄⣀⣀⣀⣉⣍⣱⣶⣶⣦⣴⠶⠋⣠⡤⠂⠀⠠⠀⠀⣈⡭⠆⢀⡄⣈⣷⣽
⠀⠀⠀⠉⠳⢯⣝⣓⠒⠶⠦⠤⠤⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣭⣤⣄⣀⣀⣤⣶⣶⣶⣯⡤⠤⠶⠶⠒⢚⣉⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠞⠛⠛⠛⠋⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠙⠛⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣤⠴⠶⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⠦⣤⣀⢉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠲⠦⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤⠶⠞⠛⢹⡇
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⣧⠀⠸⣇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠹⣆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⢦⣀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠈⠠⠄⠄⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡴⠚⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡀⠀⠈⠉⠙⠒⣤⠂⠀⠀⠀⠀⠀⣤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⣶⡇⡻⢿⣿⣿⣷⢆⡀⠈⠁⠈⣥⠀⡘⡄⣴⠄⠙⣿⡆⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡙⢾⣿⣿⣷⣦⠉⣿⢿⣿⠀⠐⣬⠠⢄⣢⣴⣷⣿⣿⣦⣷⡿⠃⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣬⣙⡛⠿⠿⢶⣶⣾⣷⣶⣶⣾⣿⣿⣿⠿⠿⠛⠋⠁⠀⠐⠒⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣮⣇⠘⠍⠛⠛⠛⠓⠶⠶⣤⠬⡭⠥⠤⠔⠒⠀⠀⠈⠀⠀⠀⣠⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠄⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⠢
⠀⠀⢀⠀⠈⣻⠟⢽⣷⡿⣿⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠛⣎⠂
⠀⠀⡄⠸⡀⠈⣷⣾⢿⣾⣏⣿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠈⠹⣿⡿⣿⣿⠀⠈⣿⡿⡷⣶⣤⣤⣀⣀⣀⠠⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣤⣶⣤⣿⣿⣷⣶⠟⢂⡄⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣤⣤⡤⠀⠀⠀⢠⠖⡐⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠈⠻⠿⣿⠶⠀⠙⠛⠃⠉⠀⠀⠀⠀⠀⠀⠛⠿⠃⠀⠀⠹⠛⠿⠓⠀⣠⠔⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⣖⠯⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⡄⠹
⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠐⠀⠄
⠀⠀⠀⠀⠀⡇⡀⠐⣆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠐⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣧⢡⠀⠹⣆⠀⠐⠁⠀⠀⢒⡀⠐⣶⠀⠀⡔⠀⠀⠢⠑⠀⠀⠀⠀⠀⠀⢐⡊⠦⠂
⠀⠀⠀⠀⠀⢻⣎⢆⠀⠘⣦⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠹⣎⣦⠀⠈⣧⠀⠀⢀⠔⠒⠚⠛⢻⠛⠛⠛⠚⠉⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠘⣯⢣⠀⠸⣇⠀⢿⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠆⠀⠘⣇⢇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠃⠀⢽⣿⠉⣇⠀⡷⡏⢭⠿⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡜⡀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠘⠧⠐⠓⠚⠛⠃⠈⠓⠃⠙⠋⠧⠜
⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⡇⡇⣂⡬⠤⡤⠤⣄⠀⠀⢀⣀⠀⢀⢀⡀⠐⠄⠀⠀⠒⠀⠄
⠀⠀⣀⠀⠀⠀⠺⠿⠦⡄⠃⠑⢉⠀⡤⠙⠁⢸⠄⠀⡀⢸⠇⢀⡸⠃⠄⠁⠈⢿⣆
⠀⠀⡏⠉⠁⠐⠒⠒⠀⠉⠉⠉⠀⠀⠀⠀⠀⣾⠀⢀⠃⡜⠀⠀⠉⠈⠀⣀⣠⣤⣀⣀⣀⠀⣀⣠⠖
⠀⠀⢹⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢀⠠⠊⠀⠀⠀⠀⡰⠋⣡⠞⠉⣹⣋⣩⡿⡽⠁
⠀⠀⠀⢳⡍⠌⠀⠀⠀⠀⠀⠒⠶⠤⠀⢀⠞⠉⠁⠀⠠⠀⠀⠀⢰⣡⠷⠛⢛⣉⣉⢉⡩⠋
⠀⠀⠀⠀⠙⠷⣤⣀⣀⠀⣀⢒⣠⡤⠄⠀⠀⡀⠀⠀⠘⠀⢀⡴⠛⠧⣤⣠⣤⡬⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠃⠁⠀⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠤⠀⠅⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠐⠀⠀⡒⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⣿⣿⡿⠛⡛⢿⣿⣿⠀⠀⠀⣥⣤⣤⣤⣤⣤⣤⡄⠀⠀⠃⠀⠀⠀⠀⠀⠘⠀⠀⠍⠉⠉⠉⠉⠩
⠀⠀⠀⠀⠀⢻⣿⠀⣾⡿⠀⣿⡟⠀⠀⠀⢹⣿⢁⣴⡦⢹⣿⠃⠀⠀⢸⣿⠟⣩⡻⣿⡇⠀⠀⢠⣤⡤⠤⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣾⣿⡇⠀⠀⠀⢸⣿⣦⣉⣡⣾⣿⠀⠀⠀⢸⣿⣜⠟⣡⣿⡇⠀⠀⢸⣯⠰⠗⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⡶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⠛⠻⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⠾⠟⠋⠀⠙⠟⠛⠛⠻⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠻⣷⡀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣸⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣰⡟⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣾⠏⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣴⠟⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠟⠋⠁⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⢨⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢨⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣼⡿⠁⠀⠀⠀⣼⡿⠁⠀⠀⠀⣼⡿⠁⠀⠀⠀⣾⡿⠁⠀⠀⠀⣾⡿⠁
⠀⠀⢸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃
⠀⠀⢿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠘⢿⣧⡀⠀⠀⠘⢿⣧⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⡿⢋⣭⣭⣙⣿⣿⠟⠛⢿⣿⡟⣫⣍⣻⠟⣩⣙⣿⠟⠻⢿⣿⠿⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⠁⣿⣿⣿⣿⣿⠃⣿⣿⡆⠻⠃⠿⢿⡿⠸⠿⣿⠁⠚⣛⣸⠇⠚⢛⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡄⢿⣿⣿⡿⢻⣧⣙⣛⣡⣿⠇⣿⣾⣿⢸⣷⣿⣌⡛⣋⣿⣆⡛⢋⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣀⣤⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣀⣀⣀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠠⠀⢀⡀
⣿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⣿⣻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡔⢵⠄⠀⠒⠒⠒⠒⠒⠒⠒⠂⣤⢱⠄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣼⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠁⠀⠁⠀⠐⠙⠒⠒⠒⠒⠂⠈⠀⢸⠑⠈⡆
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠌⡸⠁
⣿⣿⣿⣿⣿⣯⣝⢿⣿⣿⣿⣿⣿⣿⣿⡟⣭⣏⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡠⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠶⡈
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢄⠘⢄⡈⠑⠒⠠⠤⠤⠒⠁⡠⠃⢀⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠀⠒⠁⠄⠈⠠⠶⠊⠑⠂⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡞⠀⢰⠇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣼⠁⢠⡏⠀⢠⠏⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⠇⢰⡇⠀⡾⠀⢠⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣛⡋⠉⣼⠀⣸⠀⢸⡇⠀⡾⠀⢠⡏⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⣾⣫⡥⠄⢀⣿⢀⣹⢀⣸⡄⣀⡇⠀⣸⠀⠀⢬⣙⠲⣬⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣬⠙⠃⣰⠯⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡙⠢⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⣉⣁⣤⠶⣛⣵⠰⠿⢿⣉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⣛⣉⣭⣥⡶⠿⣛⡥⠘⠛⠛⣾⡇⢸⡇
⠀⠀⠀⢀⣀⣤⠶⠚⠋⠸⣿⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣉⣩⣭⣶⠾⢟⣫⠄⠉⣒⡴⣿⡷⣸⠇
⠀⣠⡶⣛⡵⢒⡭⠀⠀⠀⢻⣝⡓⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⣩⣭⣥⡴⠶⢟⣫⠅⣾⣿⣵⡾⣫⡼⠋⠶⣄
⣼⠋⣾⢋⡾⢁⡶⢋⡤⠀⠀⢻⣭⣁⠀⠀⠀⠀⠀⠀⠠⠤⢤⡤⠤⠴⠶⠾⣛⣻⢏⣾⣛⣯⣷⣞⠉⡀⢌⢳⡘⣷
⣿⠘⣿⡘⢧⡘⢧⣘⠦⣄⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠐⠒⠒⢒⣚⣛⣿⣽⣧⣾⡿⣿⡿⣿⡽⣧⢹⡌⡇⠇⣼
⠈⠳⣌⡻⣦⣝⠳⠯⣟⣒⠀⠀⠀⠀⠙⠷⣶⣤⣤⣤⣤⣭⣭⣭⣭⣿⣿⣿⡿⣻⣿⡇⣿⡇⣿⢧⠏⡼⠀⣡⡾⠃
⠀⠐⠚⢛⣶⣭⣟⡳⠦⢬⣁⠀⠀⠀⠀⠀⠈⠉⠛⣛⣒⣒⣚⣛⣿⣭⣭⠶⢾⣻⣵⠾⠋⠜⢁⣀⣤⣶⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠐⠚⠛⠛⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣒⣒⣲⣶⣶⣲⣟⣿⣯⣭⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⡭⠭⠭⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⠶⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣭⣭⣭⣭⣽⣯⣭⣭⣍⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠤⠴⣒⣚⡭⠭⠭⠭⢭⣉⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⣠⠴⢚⡩⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢆⢸⡆⢒⠢⢤⣄
⠀⠀⠀⠀⠀⢀⣤⡤⠀⢀⣤⣶⣶⣶⣿⣷⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠉⠛⢮⡹⣄
⠀⠀⠀⠀⠀⠈⠙⠃⠀⠻⣿⠛⣹⠟⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠳⣄⠀⢻⢸
⠀⠀⠀⠀⢀⣤⣶⣃⠀⢰⠃⣼⠁⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⢹⠀⠀⡸
⠀⠀⠀⢀⣿⡿⠛⠉⠀⠸⣄⠙⢿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣴⠚⠉⠉⣉⡿⠊⠀⠀⠀⠀⢸⠀⢀⡟⢀⡴⠁
⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠘⣷⣄⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣧⠴⠞⠉⠀⠀⠀⠀⠀⠀⢸⡷⢋⣴⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢦⠉⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠶⠚⠁⠀⠀⠀⠀⠀⠀⢸⡇⢈⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢉⣉⡉⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣨⠇⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠓⠦⣤⣄⣀⣀⣀⣠⣤⠴⠞⠋⠉⠀⣠⡴⠛⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⠔⠀⣴⠋⠀⠀⠀⠀⠁⠀⠀⢀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠃⢄⠀⠙⠳⠶⠶⠶⠶⠒⠛⠛⠉⠉⠀⣀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠭⠁⠒⠒⠒⠒⠀⠀⠀⠀⠂⢊⣉⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠲⠶⠖⠒⠒⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣋⣩⣥⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠤⢤⣍⣙⡓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠿⠖⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠓⠶⣌⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⡿⠋⠁⠀⠀⣀⡤⠴⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠶⢤⣀⠀⠀⠙⣦⠹⡆
⠀⠀⠀⠀⠀⠀⣾⢸⡇⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⠳⣄⠀⠀⠀⠀⠀⠈⠙⠦⣄⡿⢀⡇
⠀⠀⠀⠀⠀⠀⡿⣎⠳⣏⡁⠀⠀⠀⠀⠀⠀⢠⠶⠚⠋⠉⠉⠉⠀⢠⠖⠲⣄⠸⡆⠀⠀⠀⠀⠀⢀⣠⢟⣡⠟⣇⣠⠤⠶⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⢿⠈⠳⢬⡙⠶⣤⣀⡀⠀⠀⢸⡄⠀⠀⠠⡞⠛⠛⠋⠀⠀⣹⢠⡇⠀⣀⣠⡤⠖⣋⡴⠋⣡⠞⠉⠀⠀⣀⣀⣀⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠓⠶⢬⣉⡛⠒⠶⠿⢦⣤⣤⣽⣦⣤⣤⣤⡤⠿⠾⠛⠋⣉⣠⠴⠚⠁⠀⠀⣯⣀⣠⠴⠚⠉⠀⠀⠈⠙⢦⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠙⠓⠶⠶⠶⠶⠒⠋⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⣄⣀⣀⣀⣠⡤⠴⠚⠉
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⣄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣟⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⠁⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⠙⠲⠦⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠚⢉⣡⠼⠃⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠳⣝⡲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⢭⣛⡒⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⢖⣚⡭⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠾⠭⠭⠭⠭⠭⠭⠵⠶⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠒⠀⠚⠛⠘⠋⠉⠉⢉⣽⣿⡷⣶⣴⣤⣤⣞⠙⠗⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠆⠀⠀⠀⡀⠀⢶⣁⣀⣤⣴⠀⠀⡀⠂⠐⠛⢉⠛⢿⣿⣶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢰⡃⠀⠀⠀⠀⠈⣄⠀⣉⡂⠘⠻⠿⢦⣄⠈⠣⣼⣿⠟⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠦⢭⣀⠲⠤⠄⠀⠁⠀⠐⠀⣤⠀⢀⣝⣉⣽⠿⠛⢡⣾⢙⣿⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠄⠀⠉⠉⠉⣠⡭⠀⠚⠾⠀⠼⢻⣾⣿⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢛⣸⣏⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⣉⣉
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠁⠀⢀⣽⡧⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣿⠯⣍⠛⠟⠷⢦⡀
⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⢠⠏⢺⡛⡻⣦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⢖⣽⣿⣗⣫⣏⡀⠀⠠⠈⢿⡄
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠉⣺⣹⣟⠩⣻⠟⣶⣾⣤⣀⡀⠀⡀⢀⢀⡘⣀⣴⣾⣯⢒⣿⣨⣿⣿⡴⠀⠀⠀⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⡥⢀⡌⠉⡙⠛⠟⠿⠿⠿⠿⠿⠟⢻⣿⣏⣯⣵⡙⠉⠀⠀⢀⣀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠀⠺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣁⣄⠀⣴⡚⠇⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠒⠓⠒⠒⠲⣬⠷⠶⠶⠎⠛⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢉⡛⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡗⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢁⣠⠔⠒⠉⠉⠉⢉⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⢁⡴⠋⠀⠀⠀⠀⠀⡾⠋⠉⢳⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣰⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠲⠾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⣿⠁⢀⣴⠟⠛⠳⣄⠀⠙⢦⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⡞⠉⠇⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢿⣇⠘⣇⠳⠄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢿⡆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢿⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢸⠃⡼⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⣤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠺⣿⡁⢀⣤⠤⠶⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢀⣽⠇⢀⣿⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠤⠤⠤⠤⠶⠒⠒⠒⠛⢉⣡⡴⠟⠉⣿⣯⣥⣌⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠉⠉⠀⠀⠀⢸⠟⠁⠀⠘⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⣴⡃⠀⠈⠓⠠⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⣲⠄⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠈⠛⠦⠤⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⢤⣤⣀⣀⣀⣀⣀⣀⡀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢠⡄⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠠⣿⠁⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠹⣧⡀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣼⠇⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣁⣀⣀⣠⣤⣀⣀⣀⣈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⣠⣴⠶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀⠀⠈⠙⣷⣀⣤⣤⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣯⣥⢶⣤⠙⣷⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⢻⡇⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠂⣾⡏⠁⠀⢀⣿⠃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⣨⡿⢀⣠⣴⠟⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡿⠈⣿⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠄⠀⠀⠀⠹⣷⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⣿⠷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣿⣦⡀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣿⠀⠀⠀⣠⣼⣿⣥⣀⡀⠀⠀⠀⠀⠀⠐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠁⠀⠈⠛⠷⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟⠋⠀⠀⠉⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡴⠶⣄⠀⠀⠀⠀⣠⣦⣄⠀⢀⡟⠁⠀⠀⠀⢸⡇⠀⣴⢶⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠸⣇⠀⢀⡿⠀⠀⠀⠀⢻⡀⣽⠀⢼⣷⠄⠀⠀⠀⢸⡇⠀⣷⡞⢀⡸⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣤⡶⠏⠀⢷⡾⠛⠀⢿⡶⠟⠁⠀⠀⠀⠀⠈⠛⠃⠀⠀⣧⠀⠀⠀⠀⠈⠇⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⣠⠞⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⢁⣤⠶⠛⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣴⠟⠁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠸⣯⡀⠐⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠈⢿⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⡿⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣾⠋⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⡭⠁⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣉⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢼⣃⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣭⣿⣳⣶⣤⣙⣦⢸⡷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢸⣧⣄⡀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⣀⣴⣾⠁⠈⢻⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⠇⠀⢀⡼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢩⡟⠀⣠⠞⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⠛⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⢷⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⡞⠉⣼⢋⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⡿⠀⣼⣧⢞⣋⣉⣭⡭⢿⣷⠀⠀⣀⣀⣤⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⣧⣼⢃⣿⡯⠭⠤⠶⠖⣻⠏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣝⣛⡋⠻⠦⠤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠚⠉⠉⠉⢉⣉⣉⣀⡀⠀⠀⠀⢀⣀⣀⣉⣉⣉⣉⡉⠙⠓⠒⣤⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣞⠁⠈⠉⠓⠶⠮⢭⣉⣉⣠⣤⣤⠉⠉⠉⠉⠈⠁⠁⠐⣂⣦⣤⠤⠼⠒⠋⢁⡽⠋⠉⠉⠉⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠤⢤⣀⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣠⠤⠖⠂⣿⡠⠚⠉⠉⢳⠘⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠉⠉⠛⠛⠒⠂⠀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠁⠀⠀⢰⠀⡿⠁⠀⠀⠀⡼⠈⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠓⢄⠀⢰⡖⢦⡀⠀⠀⠀⠀⠀⠀⡎⢰⠃⣀⣠⠴⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠠⣟⠍⠓⢦⣌⠳⣮⣧⠀⠳⣤⣇⢀⣀⠀⠀⠀⠀⢠⠟⠋⣁⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠙⠛⠛⠉⣩⠿⣿⠛⣻⠿⡟⠉⠀⠀⠀⠀⣠⠏⠠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡾⠁⡴⠃⢰⢃⡸⠁⠀⠀⠀⣠⠞⠉⠓⠦⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠙⠋⠀⠀⠈⠉⠁⣀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠛⠛⠒⠒⠛⠿⠩⠭⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⠀⣀⢀⠀⣀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣴⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠰⠤⠤⣤⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣣⠖⠋⠁⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⠴⢀⡀⠀⣤⠀⠀⢀⠀⠀⠈⢹⣶⣀⠀⠀⠀⢀⠀⠀⠸⠆⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣯⠀⢀⣴⣾⡇⢸⠟⣻⣿⣿⣿⡿⣀⠀⢈⠁⠂⢴⡦⠀⡉⠤⢸⡆⠀⠈⣿⣷⢸⠖⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠙⢧⣿⣿⣿⣷⣆⠈⢛⣿⡿⣿⡇⠀⠂⣈⢢⡄⠒⣠⢶⣶⣟⣿⣧⣨⣧⣿⠏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣦⣍⡛⠻⠿⣿⣷⣮⣥⣌⣥⣷⣬⣤⣭⣴⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⢠⢠⠔⠋⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠙⡟⠿⣷⣶⣤⣤⣌⣉⣙⡛⠛⠛⠛⠛⠛⢛⣛⡋⠉⠉⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡄⠑⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠙⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠄⠚⠁⢀
⠀⠀⠀⠀⠠⢂⣼⣿⣿⡿⠿⣿⣿⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠶⣄⠢⡀
⠀⠀⠀⠀⠀⠁⢻⡟⠫⣾⣿⡯⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢷⡌⢂
⠀⠀⠀⠀⢸⠀⠀⣶⣶⣾⣷⣾⣟⣻⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡄
⠀⠀⠀⠀⠈⠀⠂⠈⠛⢧⣼⣿⣿⣿⡇⢈⠉⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠿⠟⠣⠀⠀⣿⣿⣽⡝⣻⣟⣛⡛⢛⠻⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡹⢁⠇
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⣴⣾⣿⣿⠿⠿⢿⣿⣿⡁⣀⠔⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢠⣶⣶⣶⣗⠂⠀⠀⡀⢠⠟⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠈⠻⠿⡿⠟⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠉⠙⠛⠛⠛⠁⠀⡠⢖⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠽⣒⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⣚⡋⠉⠉⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠶⠶⠦⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣀⡠⠤⠒⠒⠉⠉⠁⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⠉⢸⣋⣁⡈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠲⠶⠶⠶⠶⠖⠒⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⡿⠀⠀⡽⠀⣼
⠀⣠⣄⠤⣄⣠⠤⢤⣀⡀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠞⠀⡴⠃
⡼⠁⠊⠀⣏⠀⠈⠀⢀⡼⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⢁⡴⠊
⠳⣄⣀⡼⢭⣉⠉⢉⡡⠴⠒⠢⢄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠖⠁
⠀⡞⡡⠂⠀⡸⢲⣋⡔⠂⠀⢀⡼⠀⠀⠀⠀⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠓⠒⠋⠁⠀⠈⠳⠶⠒⣋⡤⠖⠒⠛⠉⠉⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠤⠤⠀⠀⠀⠄⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠐⠀⠀⠀⠀⠈⠓⠶⠶⠤⠤⠴⠶⠶⠖⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⣊⣩⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠯⢭⣽⣓⣒⣒⣒⣒⠒⠒⠒⠒⣒⣒⣛⣉⣉⣉⣩⡭⠤⠴⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢹⡆⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣁⣀⣈⣁⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣠⠴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣠⣿⢶⣚⡛⢶⡀⠐⠄
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⣿⠉⠛⠛⠿⠾⠿⠿⢿⣿⣿⠿⠿⠿⠿⠟⠛⠛⢹⡟⠋⠉⣹⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣠⡴⣿⢤⡀⠀⠀⠀⠀⠀⣼⣁⡤⠞⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⠀⠋⠀⡇⠀⠀⠀⢀⣼⣯⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣹⣦⡀⠀⠀⠀⢿⣤⣤⣤⡇⠀⠀⣠⣾⣉⡁⠀⠀⠀⠀⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⠉⠀⠀⠙⠳⠶⠶⠶⠶⠦⠤⠶⠶⠛⠁⠀⠈⢉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠶⠶⠶⠤⠤⠤⠴⠶⠶⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣯⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢠⡄⢰⢰⡟⠋⣳⠀⠀⠀⣾⢳⡄⣸⣃⠀⠀⠀⣿⢰⣿⠇⢰⣻⡇
⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠴⠀⣧⠾⠀⣧⡴⠋⠀⠀⠀⠙⠞⠁⢸⡇⠀⠀⠀⠿⠈⠛⠚⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡤⣤⡀⠲⠟⠛⠛⠒⠓⠦⠤⣤⢶⣄⣄⣀⡀⠀⡠⣂⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣭⠥⠌⠉⠉⢀⣀⣤⡰⡸⢿⢡⠤⠄⠈⠉⠉⠉⡉⢻⣶⣽⡅⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢺⣯⠹⢻⡷⠄⠀⠰⣾⡿⡿⠻⠆⣮⠚⠅⠀⠀⠀⠠⣾⠜⢿⣟⣄⣗⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣚⣓⠀⠀⠉⠀⠐⠊⠣⠀⠁⠀⣀⣤⣀⣤⡤⠞⠩⡝⠏⠬⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⢘⠛⠁⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠐⠂⣠⠄⠈⢄⡀
⠀⠀⢠⡿⠀⠐⠸⠞⠷⡀⠀⠀⣱⡂⢘⣰⢂⢠⢴⡦⣄⠀⠀⠾⡟⠀⠀⠀⠀⠈⣖⣲⠒⠀⠘⠛⣦
⠀⠀⢿⣤⣄⠀⢾⡿⠻⠄⡀⠀⠉⠛⠠⠴⢺⡔⠿⠳⠃⣯⡞⠋⠉⠀⠀⢰⡯⠀⠀⠀⠀⠖⣾⣾⠟
⠀⠀⠈⠻⣥⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠁⠀⢀⠀⠀⡈⡉⠀⠀⣀⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠉⠙⠲⠲⠤⠤⢄⣀⣘⣅⣀⡐⠆⢲⣲⣶⣶⣒⣊⠯⠤⠴⠖⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣄⣉⣀⣤⣤⠤⣤⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⣠⡶⠃⠀⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢻⣧⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠈⢿⣷⡄⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠹⣿⡄⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⠿⠁⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠖⠒⠒⠚⠛⢛⣛⣋⣉⣉⣉⣉⣛⡛⠛⠛⠒⠒⠒⠶⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣞⢉⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠷⢶⣶⣤⣄⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⠶⠾⠿⠛⠛⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⠖⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠉⢧⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠊⠁⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠚⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅⣀⣀⣀⡀
⣠⠴⠒⠚⠉⢉⣩⣤⣤⣶⣶⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣶⣦⣤⣤⣍⡉⠉⠓⠒⠦
⠳⢤⣀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⣀⡤
⠀⠀⠈⠉⠛⠒⠶⠤⠤⣭⣉⣉⣉⣛⡛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⢛⣛⣉⣉⣩⣭⠤⠤⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⣍⣉⣉⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢉⣉⣉⣩⡭⠴⠚⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡸⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠐⠂⠐⠚⠑⠁⡀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⡇⢀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠋⠩⠶⠂⡀⠀⠀⠀⠄⠠⠭⠥⠤⠄⢀⡁⠂⠈⠀⠉⢠⣖⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⣿⣿⡶⢶⢶⣦⣥⣤⣭⣭⣭⣭⣤⣤⣭⣤⣾⣶⣶⣾⣦⣽⠖⠉⣀⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠍⠉⠿⣿⣿⣿⡿⠅⢺⠅⢸⠅⢀⡤⠖⠒⠶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢁⣭⡭⢭⣿⣖⣾⠅⢐⡡⢸⡴⠋⠀⠀⠀⠀⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⣈⣭⡭⠵⣖⡋⠀⠊⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠥⠀⠐⣒⣛⣩⡭⠄⢈⠁⣾⠀⠀⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣈⢋⣩⡭⠥⠶⣒⠀⠄⠀⣟⠀⠀⣀⠤⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣒⣚⣛⡭⠅⠒⡁⠀⡯⠌⠀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠭⠭⠷⢖⣚⣋⠀⠂⣸⣡⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢘⣫⣭⣭⡷⠠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣌⠠⠄⠀⠀⠀⠀⡀⢀⣀⠒⣀⣐⣲⣿⣿⠯⠀⣩⣾⣁⣀⣀⣀⣀
⠀⠀⠀⣠⡤⠶⠒⠚⠉⠿⠉⠀⠀⠤⠤⠤⠿⣗⣦⢄⠀⠀⠀⠐⠀⢛⣀⣿⣿⣿⠿⠭⣴⣾⡿⠿⠿⠿⣯⣭⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⣠⡀⠀⠰⠅⠀⠀⠀⠀⠐⠄⠀⠈⠓⠤⢤⣄⣀⣀⣀⣉⣍⣱⣶⣶⣦⣴⠶⠋⣠⡤⠂⠀⠠⠀⠀⣈⡭⠆⢀⡄⣈⣷⣽
⠀⠀⠀⠉⠳⢯⣝⣓⠒⠶⠦⠤⠤⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣭⣤⣄⣀⣀⣤⣶⣶⣶⣯⡤⠤⠶⠶⠒⢚⣉⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⣠⡴⠶⠚⢋⣉⣭⡤⠴⠶⠒⠒⠚⠉⠉⠙⠋⠉⠉⠉⠉⠉⠐⠐⠐⠂⠀⠀⠀⠀⠀⠀⠉⠁
⣾⠁⠀⣴⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⣿⠀⠈⢧⣄⡀⠀⠀⢤⣤⡰⣖⣶⣖⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⢤⣄⣀⣠⠖⠀⠀⠀⢸⡏⠉⠉⠉⠙⠛⠦
⣿⠀⠀⠀⠈⠙⠒⠿⠿⣿⣯⣭⣭⣍⣀⣈⣉⣙⣛⣛⣛⣛⣉⣻⣛⣙⣽⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⣸⣧⣤⣤⣄⡀
⢿⠀⠙⠛⠓⠶⢤⣤⣄⣀⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣤⡤⠶⠒⠀⠀⠀⡉⠀⠀⠀⠈⠻⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠃⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠂
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⠎
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣐⣀⣴⣞⣉⣀⡀⡠⠋
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⠿⠿⠛⠻⠟⠿⠛⠛⠫⠭⠍⠉⠙⠛⠓⠒
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣱⠚⠉⠁⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠶⢦⣶⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⢿⣿⣿⣿⣿⣷⠶⠞⠁⠀⣤⡶⠖⠛⠛⠒
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠒⠶⢦⣬⣍⣉⣛⣛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣥⣤⠴⠖⠀⠀⣿⠶⠒⠲⣆
⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣸⠂
⠀⠀⢀⡾⠋⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀
⠀⠀⠸⣧⣄⣀⠀⠈⠙⠛⠻⠶⠶⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡞⠁⡠⠊
⠀⠀⠀⠀⠉⠛⠿⣶⣶⣦⣤⣀⣀⣀⣀⣀⠉⠉⠉⠉⠉⠀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⣁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠻⠿⠿⠿⠿⠟⠓⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠉⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣤⣤⣄⣠⣠⣄⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠼⠟⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣓⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢉⡛⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡗⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢁⣠⠔⠒⠋⠉⠉⢉⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⢁⡴⠋⠀⠀⠀⠀⠀⡾⠋⠉⢳⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣰⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠶⠾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⣿⠁⢀⣴⠟⠛⠳⣄⠀⠙⢦⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⡞⠉⠇⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢿⣇⠘⣇⠳⠄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢿⡆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢸⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⣤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠺⣿⡁⢀⣠⠤⠶⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢀⣽⠇⢀⣿⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠤⠤⠤⠤⠶⠒⠒⠒⠛⢉⣡⡴⠟⠉⣿⣯⣥⣌⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠉⠉⠀⠀⠀⢸⠟⠁⠀⠘⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⣴⡃⠀⠈⠓⠠⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣲⠄⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠈⠛⠦⠤⠀⡀⠀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⢤⣤⣀⣀⣀⣀⣀⣀⡀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⡎⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⣤⣶⣶⣺⣶⣒⣺⣖⣺⣒⣒⡲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠑⠒⠈⠉⠀⠀⠀⠀⠈⡇⢠⠃⠀⡇⠀⠉⠉⠙⠒⠮⣭⣒⢤⡀
⠀⠀⠀⠀⠀⡾⢏⠀⠀⠀⠀⣀⣀⣀⣀⣀⢠⡼⠥⣯⠤⣼⣀⣀⣀⣀⣀⣀⠀⠀⠈⠛⢟⡄
⠀⠀⠀⠀⠀⡿⠦⣑⡒⠤⠭⣭⣀⣀⣀⣀⠈⠂⠀⠁⠀⠃⠀⠀⣀⣀⣀⣉⣉⣤⣤⠶⢾⠇
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠉⠉⠓⠒⠒⠒⠛⠉⠭⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⣸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡟⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣿⣤⠤⢤⣉⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠙⠛⠒⠒⠒⠶⠶⠦⠤⠴⠶⠶⠶⠒⠒⠒⠛⠉⠁⠀⢀⡏⠀⠀⠀⠘⡆⠈⡆
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣠⠞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣠⡤⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡦⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠶⠶⠖⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠐⠒⠈⢉⣉⣉⣁⣀⣀⣀⣀⣉⣉⣉⡉⠀⠒⠂⠤⢤⣀⡀
⠀⠀⠀⠀⢠⠊⠉⣠⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⡀⠈⠑⡄
⠀⠀⠀⠀⢸⠀⠀⢥⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⡶⣿⣶⣶⣶⢶⣶⡦⡤⣤⣀⣀⠜⠀⠀⡏⠉⠉⠉⠒⢄
⠀⠀⠀⠀⢸⠀⢄⡀⠈⠉⠙⠛⠀⠦⠤⠤⠤⠤⠬⠭⠭⠭⠽⠛⠟⠛⠛⠋⠉⠀⢀⠀⠀⡧⠤⠤⢄⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⠒⠠⠤⠤⠄⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⠔⠒⠒⠉⠁⠀⠀⡇⠀⠀⠀⢣⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡌⠀⢀⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⠀⢀⠎
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⢄⡀⠒⠒⠒⣋⡠⠤⠴⣁⣀
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡩⠐⠂⠉⠉⠈⠁⣀⣀⣀⠀⠉⠈⠉⠐⠒⢬⠑⡄
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠲⠶⢿⣛⡛⠛⡛⢿⠿⠿⣿⣿⣿⠶⠖⠚⠀⠤⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠒⠒⠤⠤⠤⠀⠈⠉⠉⠀⠀⠀⠤⠐⠒⠉⠀⡷⠊⠙⣆⠈⡇
⠀⠀⠀⠀⠀⠀⢀⠜⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠈⠑⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡤⠋⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢒⣠⠤⠤⠤⠀⢀⣀⣀⠠⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⣉⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠩⠜⢂⣒⣒⡒⡒⠒⠂⠐⠒⠒⡒⢒⣒⡐⠣⠭⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⣤⡄⠀⠒⡆⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⣠⠆⠀⠀⠘⠧⠴⠞⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡞⠁⠀⠀⠀⠀⠠⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠹⣷⣤⡀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣀⠀⠀⠑⢄⠀⠀⠄⠐⢀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⠓⠀⠁⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢀⠠⠀⠤⠔⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠄⠛⠀⠀⠈⠀⠈⢀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠒⠀⠀⠀⢊⣁⣌⠀⣼⡏⠻⢿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣬⣴⠄⢸⡁⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣝⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠚⠀⢹⣇⡤⢙⢶⣉⣉⣉⠙⠛⠛⠛⠓⠲⠚⠛⠛⠛⠛⠉⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠀⠈⠉⠑⠢⢸⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠄⢀⠀⠀⠈⡛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠡⠐⢮⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣴⣶⣶⣦⣾⣷⣄⡙⠻⠷⣤⣤⣀⣀⠀⣀⣴⠞⠋⠀⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⣿⣿⡟⠉⢻⣛⠶⢶⣾⠿⠟⠋⠉⢱⠄⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢠⣠⣾⣤⣄⢹⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣀⣠⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡰⣾⣿⣿⣿⣿⣿⣷⣿⣽⣛⣻⣿⣿⠿⠟⠉⠛⠁⠀⠀⠀⠀⠀⠀⠤⣤⣔⠒⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⡶⠾⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣙⡛⠻⠿⣿⣿⡿⠿⠿⢿⡿⠿⠿⠋⠭⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣿⣿⣿⠿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣽⡃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⣤⣴⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⠀⠀⠀⡼⠉⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⢀⠞⠀⡄⠀⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢸⣠⣋⣀⡀⢿⠞⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⢸⡿⠿⢋⡴⠁⠀⣀⡤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⣄⢸⢀⡴⡫⢄⢀⣎⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⡗
⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠛⡟⠁⢸⡄⠀⢹⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⡠⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠾⠿⠟⣇⠀⢸⠃⠀⠸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠒⠲⠺⡦⠶⠶⠷⠴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢀⠴⠊⠱⢄⡈⢇⢄⠀⠀⠀⠀⠀⠠⡖⢉⣉⣉⠉⢳⡄⡇⢄⠀⠀⠀⡼⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢡⡇⡆⠀⠀⠀⠉⠉⠻⢷⣀⣀⣀⣀⣀⡏⠉⠉⠉⠉⢹⠁⠘⢢⣴⣤⠞⠁⠀⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⣌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠹⡰⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠄⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠛⠳⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠘⠛⠛⠃⣀⣀⣀⣠⣤⣬⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠭⠭⠿⣛⣛⣛⣛⣛⣛⣛⣛⡻⠯⠭⠽⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⢟⠿⠋⠁⠤⠀⠠⡈⠉⠁⠒⠦⢍⡚⢕⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⣿⡕⠁⡢⠔⠈⠀⠀⡨⣃⠀⠀⠀⠀⠐⢌⠓⣝⢷⣬⠷⠐⠠⣄
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⣾⠏⡄⡜⠁⠠⠚⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠑⠈⠆⢿⣦⠴⠒⢘⠂
⠀⠀⠀⠀⠀⢀⡞⡔⠁⠀⢸⡸⠘⠈⠀⠀⠈⠉⠀⠀⠀⢤⠀⡈⢁⢠⠈⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⢷⢠⢀⠀⡀⡀⠀⠈⠢⠠⢀⡌⠀⢀⠀⡀⠸⠀⠀⠀⠀⣇⣿⣥⣀⡀⠀⠀⠉⠐⠆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢻⡸⡌⠆⢆⢰⣔⢄⡀⠀⠀⢀⠁⡴⣡⢘⠔⠁⠀⠀⡜⡸⣹⣿⡋⠙⠍⠟⡖⠦⠤⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣧⡱⡄⠀⠀⠙⠻⠏⣛⠲⣦⡁⣴⠿⠃⠀⠀⢀⠜⡰⢡⡿⠉⠁⠀⠀⠐⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⣜⢦⡈⠓⠤⣀⠀⠉⠂⢑⠃⠀⠀⠀⢐⡡⢞⣴⡿⠁⠀⠀⡀⠀⠀⣸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢮⣓⠦⢤⣉⣉⣀⣀⣀⠤⠴⣒⣩⠖⣫⠟⠁⠀⠀⠀⠁⠀⠀⡿
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠁⠹⡷⣬⣍⡓⠒⠒⠲⠒⠒⠚⢋⣡⠴⠛⠁⠐⠐⠀⢀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠡⡡⡈⠢⣙⠻⠛⠓⠚⠚⠛⠛⠉⠁⠀⠀⠀⠊⡠⢠⢊⠄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠊⡂⠀⠑⠀⠄⡀⠀⠀⠀⠀⢀⠠⢐⠠⢊⠔⠱⠃⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠊⠐⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢽⣶⣦⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⢴⣒⡭⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠯⠽⠿⠿⠿⠖⠒⠋⠉⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠺⣦⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⣧⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣠⡾⠃⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣴⠟⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠘⣷⡀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠘⠷⠀⠀⠀⠹⠆
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⢾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡷
⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠈⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠙⠛⠿⢿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠶⠦⢤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠒⠶⠦⠤⣤⣤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⠤⠤⠬⢥⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣭⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠚⠭⠿⢿⣛⣛⣛⣛⣛⣿⣿⡿⠛⠿⠍⠑⠁⣠⠤⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⢠⠥⠒⢲⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠃⠀⠤⠶⢚⣡⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⢒⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⣼⠶⠞⠫⠥⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⢈⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠀⢀⡀⠀⠀⠀⠀⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠈⠀⠰⢔⣤⣌⣓⣦⣤⣤⣤⣤⡤⠤⠴⠒⠁⣠⠴⠓⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⠤⠤⠴⠶⠶⠒⠛⠋⠉




⣿⣟⣟⣿⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣛⣿⣟⣛⣻⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣦⠀⠀⠀⢀⠀⢀⡀⢤⣃⢐⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠞⠋⣹⠀⠀⠈⠀⠸⣆⣀⠍⢀⠀⠀⠀⠀⠀⣞⠙⠀⠁⠐⡄⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠂⠉⠁⠀⠀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢐⣷⠂⠑⠂⢲⡄
⠀⠀⠀⠠⠖⡊⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠇⠤⠋⢿⡢⠀⠀⠈⡇
⠀⠀⠀⣴⣏⠔⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠒⠂⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠠⠜⠁
⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢠⢾⠕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⢄⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣄⠀⣠⠀⠀⠀⠀⠀⠀⡏⢱⠍⠛⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢫⢁⠀⢀⢀⡀⠀⠀⠀⠀⠳⠚
⠀⠀⠐⠢⣎⣍⢐⡒⡁⠀⠀⠀⠘⢸⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣮⣯⠔⠁⡕⠩⠀⠀⠀⢤⠀⢠
⠀⠀⠀⠀⠀⡨⠃⠈⠈⠀⠀⠀⠀⠡⢇⠀⠀⠉⠁⠉⠀⠀⠀⠀⠉⠁⠀⠀⣰⡏⠀⠀⠀⠀⠁⠀⠀⠘⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⡤⠤⠤⠤⠀⠀⡠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢷⣦⣤⣤⣴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⢀⠔⠊⠁⢀⣈⣑⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡠⠊⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⢼⢸
⠀⠀⠀⠸⠀⠒⡲⠋⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢲⣤⣾⣀⣠⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⠛⠢⠀⣄⡘⡘
⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠈⠛⢿⡯⢚⣿⣥⡄⠀⠀⠀⢴⠒⠫⣰⡀⣬⡀⢠⠿⢈⡴⠁
⠀⠀⠀⠀⠀⠸⠂⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣱⠿⣷⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠤⠭⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠃⠀⢀⢀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠦⠀⠀⠀⣀⠀⠄⠀⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠂⠀⠀⠉⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠁⡀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⡔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⣁⡈⢁⡄⠣⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠔⠋⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⢄⠐⢢⠀⠊⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⡦⠋⠙⠓⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⠑⠤⠄⣀⣀⠀⠀⠀⡑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠒⠀⠉⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣠⠂⢀⡠⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡁⠂⡀⠐⠉⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⢶⣾⣛⡛⢣⡍⢉⠁⡨⢋⣥⣒⣛⣛⣓⡶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⢀⣀⣠⣤⣧⣼⣦⣸⣤⣡⣵⣾⣿⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢂⣤⣤⣟⠳⣄⣀⡹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠻⣛⣛⣭⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⣿⠋⠈⢿⠀⣿⣿⣟⡛⢻⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⢿⡀⠀⠸⣇⠘⣿⡉⠈⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣹⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡑⢄⡀⠻⣄⠘⣧⡀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣍⡀⠻⢦⡈⠃⣄⢦⡙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⡿⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⣷⣦⣀⡉⢦⣼⣿⣿⣧⣀⠀⠀⠀⢀⣴⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡴⠶⠺⢭⡼⣿⣯⣭⣙⣛⣛⡛⠻⠒⢾⣛⣛⢱⣟⣒⣂⠂⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣿⣿⡟⠁⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣙⣿⣿⡽⠿⢿⣿⣿⣶⣶⠌⠳⡀
⠀⠀⢠⣀⣲⣶⣤⣤⣀⡙⢿⣿⣿⣿⣿⣷⣄⣈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣟⣉⣉⡉⠉⠉⠀⠀⠲⠿⠿⠟⠛⢁⣠⡾⠁
⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣏⠙⠛⠭⣛⠛⠻⠟⠿⠶⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠤⣌⣁⠂⢀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡷⣶⣤⣴⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣉⣉⣉⣹⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠠⣬⣼⣿⣿⡟⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠆⠙⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⠿⠿⢛⣛⡛⠉⣉⢿⣉⣽⣿⡿⢷⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⠟⠋⠁⠠⠶⠋⠀⣿⡶⠋⠉⠛⡟⠉⠀⠹⣮⣿⣿⣿⡿⠏⣙⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⣰⠃⠀⠀⠀⢠⡿⢧⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠋⠀⠀⢸⣷⡻⠿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠰⢇⠀⠀⠀⠀⠀⠛⠿⢯⣢⠀⢀⣴⣾⠟⠋⠁⠀⠀⠀⢀⣼⡿⠁⠀⠀⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆⠈⠀⠘⢄⠀⠀⠀⣀⣀⣀⠀⠈⠀⢺⠋⠀⠀⢀⣀⣠⣴⠾⠛⠋⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⠑
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣿⣛⠀⠀⠀⠀⠀⠁⠀⠉⠀⠐⠲⠬⣷⡄⠀⣀⣴⡿⠟⠋⠉⠀⠀⢀⣀⣀⣤⣴⣿⣿⣿⣿⣿⡿⢁⣼⣤⡀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣿⣿⣶⣌⠉⣤⣬⣷⣾⠷⠶⣾⣷⣦⡣⣾⠟⢁⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣜⣿⣾⣭⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠾⣷⣶⣾⣤⣤⣀⡐⢉⣻⢿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠟⠋⠁⠠⠀⠀⢈⢩⡿⠇⠀⠐⠚⠛⠛⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠄⠐⠆⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⡿⠁⠀⠀⠀⠀⠀⠀⣠⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣿⠁⠀⠀⠀⠀⢀⣠⠞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣀⣀⡤⠴⠚⠉⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠴⠃⠀⠀⠀⢀⣀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⢠⣤⠦⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠉⠙⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣄⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢲⣷⠶⢶⢶⣦⣤⣶⣶⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉

⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠂⢀⠀⡠⠂⡠⣩⠃⠀⠀⠀⠀⢣⠡⡀⠀⡀⠀⠠⠚⠉⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠭⢙
⣤⣴⢆⠂⠀⠀⠀⣠⠄⠐⢐⣤⣤⡀⠀⠀⠑⠈⢣⣔⣪⠖⠁⠀⠀⠀⠀⠀⠀⠑⢔⡂⣤⠋⡐⠁⠀⠀⣀⣠⣄⠤⠀⠀⠀⠀⡀⠠⡤⣆⣀
⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠄⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢙⠀⠀⠀⠀⠀⠉⠉⠨⠀⠀⠀⢠⠈⠀⠀⠈⠉⠙⠃
⠀⢀⣴⠁⠀⠈⣶⠁⠀⠀⠤⡀⠀⠀⠀⠀⠀⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡈⠂⠀⠀⠀⠀⣠⠄⠀⠀⣧⡓⠀⠄⣬⠠⠀⢀⣀
⠉⠀⡅⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡎⠈⠃⠀⠐⠀⠀⡁⠀⠀⠀⠻⠀⠀⠀⠀⠃⠐⠒⠒
⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⡘
⠀⠀⠀⠳⣀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠀⠄⠀⠰⡁
⠀⢀⠠⣂⠜⠉⠉⡉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡚⢻⠳⠌⠀⠀⠀⠈
⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠠⠄⠉⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣴⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣶⣷⣶⣶⣾⣷⠀⠀⠐⡇⠀⠈⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⡇⠀⠀⢰⠃⢂
⠀⠀⠀⠀⠀⡂⠀⢒⡂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⢴⣶⣶⣥⣶⠄
⣤⣴⣗⡄⠀⠀⢤⠠⠅⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣿⣿⡏⢴⡎⣿⣿⣧⠀⠀⢸⢸⣿⣿⠟⠁⢠⣾⣶⣶
⣿⣿⡿⣿⠦⡀⠀⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣶⣿⣿⣿⣧⡀⣼⣸⠟⠁⣠⣶⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⡇⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣵⣷⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⠯⠗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣀⣤⣀⣼⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣿⣷⠈⠀⠀⠀⢘⣻⣋⡿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣦⣄⡤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠖⠢⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢆⠀⠘⠄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠂⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢻⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⢧⡀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⢀⣀⠽⢦⡀⠳⣖⠂⠈⠀⠀⠉⠉⠉⠉⣉⣉⡉⠁⠀⠀⠉⠉⠉⠁⠉⠐⣒⣤⠴⣿⠴⠚⠋⠉⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠛⠲⠶⠦⠿⣤⣬⣦⣤⣤⣤⣤⣤⣤⣾⡥⠤⠭⠷⠾⠿⢷⡖⠒⠒⠛⠋⠉⠀⠀⣯⠴⠖⠛⠳⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⠇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⠔⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡏⢠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠶⠶⠶⠶⠶⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠉⠉⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⠶⠤⠤⠤⠤⢤⣀⡀⠀⠉⠛⢦⡀
⠀⠀⠀⢀⡴⠟⠉⠀⣀⡠⠤⠔⠒⠚⠋⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⠇
⠀⠀⢰⡟⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠦⣤⣄⣀⣀⣀⣀⣀⣸⣀⣀⣀⣠⣤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣾⠏
⠀⠀⠈⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠴⠶⠖⠛⠋⢉⣠⠾⠃
⠀⠀⠀⠈⠳⣌⠉⠙⠛⠒⠒⠲⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠖⠲⠦⢶⣶⠖⠚⠛⠋⠉⠉⠁⢀⣀⣀⣠⡤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠤⢤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠳⣤⠶⠖⠒⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⢦⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⢀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣦⣦⣤⣼⣘⣶⣾⣿⣆⣿⣿⣷⣰⣶⣶⣶⣶⣦⣤⣨⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉






⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣠⣀⣀⡀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠖⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣼⣤⣴⣤⣠⣀⠀⠀⠀⠀⠀⠀⢀⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⣀⡄⣀⠀⠰⡿⢈⣉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠋⠉⠀⠴⠛⣈⣿⠆
⠀⠀⠀⠀⠀⢰⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⡈⠃⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⠒⠒⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣿⣿⠁⣀⡸⣤⡀⠀⠀⢰⣧⡄⢠⡀
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣆⣀⡀⠀⠿⠟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣿⣿⠟⣿⣝⢳⣴⣦⣴⣦⡈⠙⠁⣰⣧
⠀⠀⠀⠀⠀⠈⠈⠛⢿⣿⣿⡿⠀⠀⠀⠂⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⣦⣌⣲⣆⣥⣾⣿⣿⣾⣿⣿⣿⣿⣿⣦⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⢀⡀⣻⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢦⣿⣇⠛⣿⣿⣿⣿⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⢿⣧⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡡⠤⠤⠀⠀⠀⠀⠈⡻⠀⠻⣧⡈⠛⣿⣿⣯⡘⣿⣿⡟⠿⣽⡻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠹⢿⣷⡴⢿⡇⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⣄⣤⡀⢹⡇⠀⠀⠀⠉⠁⠀⠀⢀⡠⣦⣦⡀⠐⠂
⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⠿⣿⣿⣿⡿⠟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣿⠟⠻⣧⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠉⠓⠰⠤⡄⠴⢿⡿⠿⢿⣿⣆⠀⠈⠿⣦⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀










⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣟⣉⣽⣶⣶⣶⣛⣋⣀⠀⣀⠀⠀⢠⣼⣿⣭⣤⣀⣀⠀⠉⠉⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⠇⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣣⣴⣿⣷⣦⣀⡀⠐⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠁⠒⠲⠬⣭⣍⣛⣛⣛⣻⣿⢻⣟⣛⣛⣛⣛⣛⣿⣯⣭⣿⣿⠟⠛⠁⣼⣿⡟⠛⠛⢢⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⠛⢇⠀⢠⡿⠋⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⡳⠤⣀⡾⠁⠀⠀⢀⡴⠏⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⡍⠰⢂⡼⠁⠀⠠⢜⣋⣵⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⠛⠿⣯⠉⠠⢀⡾⠁⣀⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠙⢁⣼⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣋⡠⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⡟⢀⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣿⣅⣀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢰⣿⣿⣿⠟⠋⠀⠀⠈⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⣀⣀⣀⣀⣈⣉⣭⣵⣿⣿⢷⣿⠖⢻⡶⠂⣀⡤⠀⡃
⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⠿⠿⢻⣿⣿⣿⣾⣿⣷⠶⠿⠟⠛⢉⣀⣼⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣀⣀⣤⣤⣴⣾⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠙⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣤⣤⣴⣶⠶⠶⣶⣿⣿⣿⠿⠶⠶⠶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⢿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠶⣦⣤⣀
⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠈⠛⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉⠀⣀⣤⣄
⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠾⠛⠋⠉⠀⠀⣀⣤⠞⠋⢸⣿
⠀⠀⠀⠀⠀⠈⠙⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠉⠀⠀⠀⢸⣿
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣼⡏
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⢤⣤⣀⣴⣿⠋
⠀⠀⠀⠀⣀⣠⣄⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⠛⠷⢶⣤⣀
⠀⢀⣴⡾⠟⠉⠁⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠈⠙⢿⣦
⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠚⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠙⠻⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠷⠶⢶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡶⠶⠶⠾⠟⠛⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⢆⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢠
⠀⠀⣠⣄
⠀⢠⡇⠀⠑⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀
⠀⢸⡧⠀⡀⡅⢳⡀⠀⢸⢣⠀⠀⠀⠠⣴⣮⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣹⣿⡿⠲⠄
⠀⠘⣇⠀⠙⣃⡸⡇⠀⡇⡀⢧⠀⠀⣤⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⣡⣶⡏⣄⣀⠠⠤⢤⡀
⠀⠀⠈⢯⡀⠘⣇⠁⢸⡇⣿⠃⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⡷⠞⠛⢻⠎⡇
⠐⡶⢶⡀⠈⠒⠛⣄⢵⠘⣿⣀⡿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢸⠀⠀⣠⢞⣼⠃
⠀⠘⢆⠛⠶⣾⠿⣿⣿⣷⡿⠛⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⡾⠒⢉⣴⠞⠁
⠀⠀⠀⠑⠮⣀⣀⠠⠔⠋⠑⢦⣀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠯⠀⣼⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⠻⠶⣶⣄⡀⡛⢿⣦⣄⣀⡀⠀⣀⣠⣴⣿⣿⣷⣿⣢⠸⣶⡖⠮⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⠀⠀⠈⠀⠓⠉⠒⠮⠭⠭⠿⠿⠿⠭⠭⠽⠟⠋⠁⠉⠑⠋⠁⢀⡫⠌⠀⠀⠀⠐⠂⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠤⣄⢀⣀⣀⣀⣤⣀⣀⣄⡀⣤⣄⣀⣠⣠⣤⠤⠤⣴⠶⠚⠛⠦⣄⡇⠀⢀⣤⡄⠐⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠾⠿⠿⣯⣿⣶⣿⡿⠾⠿⠿⠛⠻⢥⣀⠀⠀⠀⠀⠀⣈⣠⠀⣿⡇⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠙⣿⡶⠚⠉⠉⠀⢸⡟⣿⡃⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠦⣿⣧⠀⠀⢀⣠⡼⠟⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⠚⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠤⠤⠀⣒⢒⣒⣒⣀⣀⡀⠀⠀⠀⠀⢄⢐⣒⠒⡒⡒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣬⣤⣶⠒⠀⠅⠁⠩⠚⠉⠈⠁⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠁⠀⠈⠘⢶⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠿⢿⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠠⠄⣀⣪⡵⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡙⠟⠿⠷⠶⣿⣥⣤⣤⣠⣄⣀⣠⣤⣤⣤⣭⣥⣬⣶⢶⡶⠶⠛⠛⠋⠉⡴⢒⡸⠁⣀⣤⠄⢀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠋⣿⠀⠠⣄⣀⡠⢈⣡⡭⣫⣖⣡⠴⠒⠒⠒⢦⡌⢻⣅⢲⣆⡒⠋⡁⠘⠼⠂⠇⢹⢫⠀⠀⠐⢕⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠉⠀⠀⠀⠈⠽⠿⢯⠘⠉⠑⠃⠀⠀⠶⡄⠀⠈⠀⠻⣶⠳⠍⢢⡝⡄⠐⢤⡆⠈⠆⠀⠀⠀⠀⢩⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠄⢦⠄⣠⡌⣄⢆⣠⡄⠋⠋⡠⠛⢲⡗⠻⠗⠀⠓⢈⣆⡗⢷⡄⢴⠇⠓⡈⠈⠁⠀⠀⠀⠀⠀⢽⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠁⠖⡨⣀⠹⢿⡉⠀⡠⢶⡄⠷⢴⡀⢛⣂⠀⠀⠈⡏⠁⠀⡇⣽⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡄⠀⣷⠀⢸⡇⠰⡦⢼⣿⡿⣸⡛⢤⣿⣷⠄⠀⠇⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⣧⠀⢸⡇⠀⠱⡏⡳⢖⡉⣡⠾⠀⠰⣶⣦⣴⣦⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠄⡸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⣭⣄⠸⣧⣄⣀⠀⠙⠉⠈⢀⣹⣿⡶⠟⣍⠉⣯⢄⣷⡾⠀⠀⡥⢄⠀⢀⠠⠊⠈⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⢀⣃⣀⣻⠜⠛⢿⣻⢹⢶⡐⣤⣤⣻⡿⣺⣷⣇⣏⣼⡿⣯⣸⣤⠀⠀⣦⠀⢀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⡀⠤⣀⣀⣐⡒⢨⣶⡀⠀⠀⢨⡻⣟⢡⣶⣄⠉⢻⣼⢢⠇⠘⠻⣿⠳⢾⡻⢶⡧⠤⢀⣴⣎⠁⠀⠀⣿⡩⣤⡴⠶⢯⣑⠢⣄⣀⡀
⠀⢄⣬⣵⡾⡛⠯⣍⡭⣺⢿⣕⡶⢁⠀⠠⡬⢟⣿⠚⢯⡝⣠⣞⢻⣻⣄⠀⣻⣿⣶⣾⡇⣈⣙⣢⣴⡯⠙⠀⠀⡠⠱⣚⣱⠾⠶⠶⡉⣏⡶⠾⠭⣝⢢
⠄⣼⢟⣇⡹⠀⠀⠀⡯⠉⠠⠤⠀⠈⢣⣀⠠⢧⣾⣿⣦⣧⡈⢁⡼⣻⣯⡀⠉⠉⢈⣧⡻⣿⡿⣿⠉⠁⠀⢀⠞⠁⠈⠁⠀⠀⠀⠈⠵⣃⠚⠻⡗⢾⡇
⠀⡻⠜⣏⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠉⠳⣦⡀⢹⣯⣽⣿⣿⣿⡏⠛⠓⠀⠲⢿⣸⣍⣾⣷⠋⢀⣠⠞⢁⢄⢄⡄⠀⠀⠀⠙⠒⠠⠛⢈⣀⠴⢋⡴
⠀⠈⠒⠮⢭⣥⣀⣢⠤⡀⢀⣀⡂⠤⠨⠄⠀⠀⠈⢢⣍⢻⡿⠋⠛⠧⠀⠀⠹⣿⣻⣽⣟⣿⡿⠾⠧⠃⣀⣐⠌⣐⡤⠈⠲⢖⣀⣢⡠⠌⣃⠴⠃⠁
⠀⠀⠀⠀⠀⠢⢉⠛⠳⠾⠭⠭⠶⢶⠶⢶⠴⠭⣭⣬⣭⣭⣥⣄⣒⠰⠄⣂⣀⣙⣐⣤⢠⣤⣬⣥⣶⣶⣶⣶⣶⣶⣶⠾⠟⠋⣉⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠟⠛⢉⣉⣠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠬⣄⣁⣐⠒⠀⠀⠠⠠⠤⠤⠤⠀⠀⠠⠤⠤⠄⠐⠒⢒⣂⣀⣩⡭⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠄⡰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠢⠀⠀⠚⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⡔⢊⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⣶⡾⠀⣼⠄⢨⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢺⣋⢅⣨⠟⠂⣸⡿⠿⣄⡯⢀⣾⣯⡙⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣧⣙⢁⣀⣴⣯⡾⣿⣿⣿⣿⣿⣿⣿⣷⣬⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠒⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣿⣿⡿⡇⣀⠤⢄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⠙⢿⣿⣿⣿⣿⣿⣶⣯⣭⣭⣭⣵⣾⣿⣿⣿⣿⠿⠋⢸⣏⡀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠸⡄⠈⢚⠙⠻⠿⠯⠭⠽⠿⠿⠯⠭⠽⠟⠛⠉⠁⠀⠀⡾⢉⣹⠶⠋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢠⢐⠂⠳⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⢀⣠⢾⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢢⠑⠄⠈⢳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠡⠐⠂⠉⠁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⢁⠀⢸⠶⠚⠒⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢔⢣⡷⠓⡟⠭⠗⡲⠤⣄⠘⣦⣥⣤⣤⣤⣶⡾⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⡕⠋⠀⢠⡿⠦⣄⡀⢈⠈⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠿⣽⣲⡦⣤⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠺⠿⠿⠭⠭⠭⠭⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⢠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠚⠛⠃⣐⠀⠀⠀⠀⠀⠀⠀⠀⠵⠤⠤⢬⡭⠤⠤⢼⣥⠃
⠀⠀⠀⠀⠀⢀⣀⠘⢠⣀⡀⠀⠀⠀⠀⠀⠀⠘⢷⠆⠀⠀⠀⠀⠘⢣⣶⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠀⡐⠉
⠀⠀⠀⠀⢠⣤⣀⣴⣠⣍⣀⠆⠀⠀⠀⠀⢀⣶⣦⠀⣤⣴⣦⣤⣤⣌⣥⣄⡀⠀⠀⠀⠀⠐⠓⠛⠚⠏⠰⠟⠖
⠀⠀⠀⠀⠉⠉⠉⣩⣈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠠⢂⡢⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠮⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠥⠵⠦⢠⠆⡀
⠀⠀⠀⠀⠐⠌⠀⠀⠁⢘⡱⠀⠀⠀⠀⠀⠀⠀⠎⠉⠃⣀⢀⣀⢠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠭⣍⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⠴⠶⠶⢲⡬⠀⠂
⠀⠀⠀⠀⠀⠶⠓⠃⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠘⠉⠋⡤⠆⠀⠀⠀⠀⠀⠀⠀⢰⠿⠓⠲⠖⠀⠟⠖
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠃⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠉⠐⠞⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡰⠈⠓
⠀⠀⠀⠀⠀⠈⠂⢄⠄⠠⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣍⡄⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢤⠷
⠀⠀⠀⠀⠼⠷⠂⠿⠶⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠾⠿⠿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣃⣚⡛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣄⣠⣀⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⠙⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠛⠟⠻⠿⠿⣿⣟⢛⢿⣿⣿⣛⣛⣭⣥⣤⣤⣤⣤⡴⠶⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⡿⠇⢀⡀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣨⣿⣿⣿⣿⠦⠠⠑⣿⣷⠶⠦⠤⠤⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠉⢽⣿⣿⣿⣯⠅⠐⠊⣿⡟⠶⠶⠶⢶⣤⣈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⡦⢨⣿⣿⣿⣷⣧⣀⣀⠀⣿⡇⠀⠀⠀⠀⠹⡏⠱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⣴⣽⣿⣿⣿⡿⠁⠀⠠⣿⡃⠀⠀⠀⠀⢸⡷⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣽⣿⣿⣿⣿⡂⠀⠀⣿⠀⠀⠀⣀⣴⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠀⢤⣿⣀⡴⢦⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⣾⣿⣿⣿⣿⣅⠀⠀⢸⣿⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢻⣿⣿⣿⡏⠀⠀⣻⣿⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⠃⠀⢰⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢠⣀⠀⡀⠠⢙⣿⣿⣿⣿⠿⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠽⢿⣿⣋⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠚⠛⠛⠛⠾⠟⠟⠛⠛⠋⠉




⣿⣟⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⢳⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣤⡀⠙⣦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⠘⣿⣦⡀⠙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⠘⣿⣿⡄⠸⣿⣿⣄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢃⣿⣿⣿⠀⣿⣿⣿⣧⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢏⣼⣿⣿⠏⢀⣿⣿⣿⣿⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢛⣵⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⣭⣵⣾⣿⠟⠋⣁⣤⣾⣿⣿⣿⣿⡿⠃⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠿⢛⣉⣤⣶⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⢛⣉⣥⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢁⣠⣴⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⡿⠟⠋⣁⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢠⣿⣿⣿⣿⣿⣿⠟⢉⣤⣶⣿⣿⡿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣾⣿⡟⢸⣿⡿⢁⣶⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢿⣿⡇⢸⣿⢁⣿⡿⣋⡴⠖⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⢿⣷⡈⠿⠸⡿⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠛⢷⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠒⠠⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠀⠀⠀⠀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠻⣶⣶⣶⣦⣤⣤⣀⠀⠀⠀⢀⣠⠾⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠐⠒⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠒⠛⠛⠛⠛⠉⠉⠉⠀⠒⠊⠉⠀⢀⣰⣧⣠⡤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣶⡶⠾⠛⢹⠏⠀⠀⣀⣀⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⠀⣴⠋⠉⠉⠙⢷⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠁⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡥⠴⠂⠀⠀⠀⠀⣼⠃⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⠀⠀⠀⣀⣤⠞⢁⣴⠋
⠀⠀⠀⠀⣀⣠⠤⠤⠒⠛⠂⠙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠛⠋⠉⣀⣴⠟⠁
⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⣴⠾⠋⠉⠻⢦⡀
⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⢻
⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣷⣶⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣾⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠈⠙⠛⠶⠤⢄⣀⣀⡈⠉⠉⠉⠈⠛⠛⠛⠁⠀⠉⠁⠘⠛⠓⠒⠲⠶⠶⠖⠒⠚⠛⠛⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣀⣀⣠⠴⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣈⣋⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠞⠟⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣹⣰⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣀⣀⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠶⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣯⣉⣀⣀⠀⠀⢀⣀⣉⣽⣇⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠰⠾⠿⣿⣿⣿⣿⡿⠷⠆⠸⠿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢉⡷⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠛⠿⠶⠤