While Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠟⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⣤⠁⠈⠀⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠀⠀⡀⠰⣙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⠀⠀⠀⠁⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⣴⠛⣄⠀⠈⠁⢀⣀⣠⣶⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠘⣅⠀⠈⢑⠚⠉⠁⢀⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢻⣆⠀⠀⠈⠙⠒⢺⠀⠀⢻⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣸⠹⣧⡀⠀⠀⠀⢸⠀⠀⠈⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⠀⠘⢷⡀⠀⠀⢸⠀⠀⠀⠸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⠢⠔⠩⢍⠲⡄⢸⠀⠀⠀⠀⡇
⠀⠀⠀⡠⠤⢤⣤⣤⠤⠤⠤⠤⢄⣘⣄⠀⠀⠀⢀⡆⢍⣒⡿⣧⠬⢦⣀⠀⣠⠃⠀⣀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⢥⣖⡉⠀⠀⣀⣀⣀⣀⣀⣈⣙⠪⡉⠉⠉⣉⣉⣉⣁⣈⣀⣀⣄⣉⣓⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⡎⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡠⣔⡲⠍⠛⠀⠀⠀⠀⠀⠑⠦⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠭⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⣞⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣀⣒⣀⣀⣀⣀⣀⣈⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢚⠞⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡭⠴⠛⠟⡤⢦⡶⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠒⠓⠾⢧⣀⡀⡁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠗⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢂⠁⠀⣠⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠉⠙⠲⢤⣀⠀⠀⣰⠏⠀⢀⠈⠒⠞⠁⠀⠈⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢁⣤⡴⢇⠀⠀⠀⠀⠀⠀⠈⠉⠻⢱⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⣀⠤⠒⣻⠎⢕⠀⡕⠲⠢⠤⠤⠀⠀⣠⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠱⡄
⠀⠀⢀⣠⠶⠛⠁⠀⠀⠐⠎⢦⠀⠀⠀⠀⠀⠉⠳⡤⢀⠀⠁⠀⢀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⣄⠀⠀⠀⠙
⣴⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠶⡑⢄⠀⠀⠀⠀⠀⢹⠆⠉⠓⠄⣀⣴⡏⠀⠀⠀⠁⠀⠀⠀⠀⡰⠁⠀⠀⠱⡀
⠀⣿⢶⣄⠀⠀⠀⠀⠀⠀⠱⡄⠈⢦⠑⣄⢀⠔⣂⠼⠀⢀⣤⠞⢋⡼⠁⠀⠀⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⢰⠁
⠀⢻⠀⠉⠳⣦⡀⠀⡆⡄⠀⠘⢆⠀⣉⠭⠓⠋⣀⡴⠞⠛⠒⠚⠉⠣⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠃⠀⠀⣢
⠀⢸⠀⠀⠀⠈⠙⢦⡀⠸⠀⠀⠈⠉⢀⣠⠶⠛⠉⠀⠀⠀⢀⡴⠊⠀⠈⠙⠲⠤⣀⡀⠀⢀⡄⢀⠀⠼⢥⠀⣀⠴⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣤⣾⠋⠁⠀⠀⢄⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠋⠙⣧⠔⠀⡐⠋⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⢻⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠋⠁⢀⣀⣤⠻⠤⠜⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⣾⠀⠀⠀⠠⣴⠖⠋⣿⠏⠉⠉⠉⢹⡏⠁⣿⡄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⣽⡀⠀⠀⠀⠈⢇⠀⣿⠀⠀⠀⠀⠘⡿⠀⢻⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢠⠃⢳⡀⠀⠀⠀⠘⡆⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⢱⣿⠀⠀⠀⠀⢠⡇⠀⠸⣧
⠀⠀⠀⠙⢦⡀⠀⠀⠀⢘⡇⢀⣀⡼⠷⠦⢶⡞⠀⠀⠀⢳⡀⠀⠀⠀⢿⠀⠀⠀⠀⠘⠃⠀⠀⣿
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡟⠉⠁⢀⣀⣀⣸⠇⠀⠀⠀⠀⢳⡀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⢠⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣶⣶⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡖⡀⠀⠀⠀⠀⠉⠳⣶⡤⢤⠄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡥⣤⠄⠀⠀⠀⠀⠈⠳⡄⠂⠂⠈⠀⡈⠉⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠖⠈⣷⠀⡰⣖⠤⡒⠚⠶⣆⡴⡭⠓⠁⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠐⠓⠒⠀⠲⣀⡘⢠⠋⣿⠋⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠉⠂⡤⢀⠀⠀⠈⠀⣾⠠⠃⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⠛⠿⣦⠘⣆⠸⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠠⠀⠁⠀⠹⠀⠀⠙⢤⠀⠀⠀⠇⠀⠀⠀⢀⣄⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⣦⣴⠀⠀⠀⠢⡀⠀⠠⠀⢀⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠠⠀⢥⠤⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠀⢠⠆⡈⢸⣩⣽⠀⠀⠀⠀⢀⡐⢣⠀⡔⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⣿⣿⠁⠀⢁⠇⠀⠀⠀⠀⢩⡷⠍⡜⢷⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⠸⠆⠀⢸⠀⠀⠀⠀⠀⡘⡄⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⢶⠆⠀⡀⠀⣀⡅⠈⠀⠇⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣈⡶⠀⠉⣛⡟⠇⠀⠀⠀⠀⡞⠊⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡎⠐⠒⠛⠖⠀⡀⠀⠀⠀⢂⣶⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠠⠄⠀⠇⢀⠆⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠳⠀⠀⢰⠞⢣⣰⡰⢍⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣾⠶⠿⠽⠟⠛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⢐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠛⣶⠿⢄⢀⡀⠀⢀⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⡉⠉⠑⡶⣷⠟⠓⠛⣶⣛⠃⠛⠿⡷⡢⠂⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠒⠙⠛⠀⠐⠾⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣶⣶⣶⣶⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⢀⠀⠀⡇⠉⠁⠐⢾⣿⣿⠿⠛⠛⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⡄⣧⣀⣀⡀⠀⠈⢻⠀⠀⠀⠀⢀⣀⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡘⡁⠀⠈⠓⠀⠀⠀⠀⢰⣬⡗⠐⡦⠴⣿⣿⣿⣷⣶⣴⣶⡾⠿⠿⠿⠶⠂
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⢿⠅⠈⠀⠀⢀⠀⠀⠀⠀⣻⢸⠀⢸⠁⠒⠆⠘⠛⠉⠉⢀⡀
⠀⢠⣶⣽⣿⣿⣿⣿⣿⢡⠀⡁⣂⠀⠀⢀⠈⠀⠀⣀⣄⣠⣄⡀⠸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣄⣀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⠸⠀⡏⠄⠀⠠⢄⡀⠀⢈⠀⠀⠢⡉⠙⠿⣶⣤⣄⠀⢀⣀⣠⣤⣴⡶⢶⣾⣟⣿⣿⣭⣿⡿⠶⢶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡛⠀⠀⠈⠀⢯⠚⠳⢂⡟⠉⠀⢸⠹⡹⠟⠋⠉⠘⠻⠿⡿⠟⠛⠋⠉⠉⠁⠀⠀⠄⣸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠵⡀⠀⠀⠀⠑⠀⠈⢠⣄⣠⣼⣷⣦⣄⡀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⢀⠀⠂⢰⠎⠉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠑⠠⠤⣈⠉⠛⠷⣦⣅⡀⠀⠀⠁⠀⠀⢀⢀⡠⠤⣶⠖⠉
⣼⣿⣿⣿⣿⣿⠿⠿⠟⠉⠙⠛⣻⣿⣿⣿⣿⣶⣦⣌⡂⠀⣀⠀⠀⠀⠉⠑⠳⢖⠻⠿⠷⢦⡠⠒⠋⠉⠀⠐⠆⠉
⢿⣿⣿⣿⣿⣿⠏⠀⠉⠈⢁⠠⡿⢫⣝⠿⠛⠛⠻⠿⣿⣾⣿⠛⠶⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠐⠀⣀⡼⠂⠀⠁
⠸⣿⣿⣿⣿⣿⣧⠀⠀⠂⠀⠍⢁⠜⣿⣿⣶⣶⣶⣶⠿⢿⣿⠀⠀⠀⠁⠙⠲⣤⣀⣀⣀⣠⣤⣴⠶⠖⠚⠛⠚⠓⡄
⠀⣿⣿⣿⣿⣿⡟⠀⠒⢠⣁⣈⣡⣤⣿⡟⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⣿⣿⣿⣿⠟⠃⠀⠀⠀⢀⣿⡿⠋⠉⠉⠐⠀⠀⠀⠤⢴⣶⣾⡟⠲⢤⣀⠀⠁⡤⠀⠄⠀⠀⠁⢀⠀⠁⠀⠁⠈⢁
⠀⢻⣿⣿⣿⡄⠀⠀⠀⠂⠈⠻⢀⠀⠀⠂⠀⠀⠀⠂⠀⠀⡉⠁⠆⠀⠀⠈⠙⠢⣄⡀⣃⣀⣤⡤⠴⠖⠂⠀⠐⡦⣄⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⢁⣀⠠⠆⠀⠀⠉⠋⠁⠀⠀⡀⠀⠀⠀⠀⢹⡀⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠍⠒⠤⡀⠀⠀⠀⠀⢈⣢⡄⠰⢖⣿⡿⠿⠟⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⢀⣤⣬⣑⣴⣶⣯⣭⡤⠖⠚⠉⣀⠀⢀⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠳⢍⠿⠿⢛⣛⡥⠤⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⢀⡆⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⢀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣜⣷⡠⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢻⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⣿⣽⣷⡐⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣟⣿⣷⡸⣿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⣧⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⢹⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣡⣹⣎⣿⣯⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⢿⡄⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⡟⢻⣿⣷⣿⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⣿⣯⠹⠻⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣭⣭⣿⣿⢿⣿⣼⣿⡷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣶⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠖⡠⠀⠀⠐⠂⠐⠒⠀⠀⠒⠀⠀⠀⠚⠙⣻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠢⠬⢭⣿⣿⡿⠟⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⡤⠒⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠢⢤⡲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣶⣿⣶⣤⣤⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠉⠀⠰⠄⢉⡓⠰⠙⠻⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠐⠀⠀⠀⠸⡋⠀⠱⠤⣄⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⡁⠀⠀⠀⡀⠀⠀⡀⠀⠀⢰⠋⠁⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠲⠶⠶⠮⠍⠁⠀⠀⣠⡄⠀⠀⠘⠀⢸⠀⠀⠀⠀⠐⠀⢈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠁⠀⠀⢀⣀⡐⠒⠒⠶⣶⣾⣿⣿⡇⠀⢠⠀⠸⣿⢠⠇⢘⣡⣴⣀⣤⣥⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠈⢀⣠⣤⣶⠶⠛⠋⠉⠁⠀⠀⣰⣿⡿⢩⣿⠃⡠⠁⠀⠀⣡⠏⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠅⠈⠙⣴⡾⠟⠋⠁⠀⠀⠀⠀⢀⠄⠀⢹⡿⠟⠀⡾⠟⠊⠀⠀⢠⠞⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⠄⠰⠊⠀⠀⠀⠀⠀⢀⣾⣿⣾⣷⣤⡤⠔⠂⠀⠁⠤⣄⠘⢂⠔⣡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡏⢀⠄⠀⠐⢆⠀⠀⣠⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣇⠀⠀⠀⠒⠀⠄⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⢻⣿⡁⡄⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡘⢶⡄⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢛⣿⣿⣿⣿⣷⡙⢦⡀⠁⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⢀⣠⡚⠖⠋⢹⣿⣿⣿⠟⠻⣆⠻⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠃⠌⢩⣥⣄⡀⠀⠀⠰⣶⡍⠉⠀⠠⢼⣿⠟⠁⠀⠀⠹⣧⡀⠀⠀⠉⠀⠘⡀
⠀⠀⠀⠀⠀⣴⣿⡿⠟⠛⠶⠦⢤⣬⣉⣛⠲⣄⡀⠀⠙⠦⠀⣠⠞⠁⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠐⠄⠴⠦⠀⠙⠛⠛⠛⠒⠛⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠻⣦⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣌⡁⠐⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠲⠘⢀⠓⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠲⠀⠀⠐⠰⠐⠐⠀⠀⠒⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⢄⡠⠔⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⡠⣀⡠⠜⠢⠤⠊⠑⠄⠄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠐⢄⡔⠰⠂⠒⠕⠀⣀⣢⠀⠀⠀⡠⠀⠈⠲⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣚⡀⠀⠢⣸⣿⠟⠀⢒⠈⡀⣴⣶⣷⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣦⠼⠢⠤⣰⣒⠪⢶⠈⢱⠈⠉⢉⠎⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠯⠬⢽⠦⢒⣊⠭⠻⠁⢀⡀⠔⢳⠒⠀⠉⣉⡝⠁⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠀⢠⠁⠀⠀⠙⠒⠋⠁⠉⠀⠀⢀⡆⠉⢀⣈⠙⢿⠍⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠃⠀⠀⠀⠀⠀⠀⠀⡧⠒⠈⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠐⡏⠉⠉⠈⠐⡄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠉⠁⠀⠀⠒⡖⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡰⠁⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡰⠁⠀⡇⠀⠀⡈
⠀⠀⠀⠀⠀⠀⣰⠒⠄⢰⡁⠀⠀⠠⠄⠀⣇
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠬⠆⠀⠷⠴⠦⠬⠼⠂

⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠠⠤⠖⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⣀⠀⠀⠀⠀⠤⠄⢒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢀⡸⣿⠛⠛⠳⠖⠚⠛⠛⣯⠀⠀⣀⣴⣶⣶⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣦⣤⠟⠀⠿⠀⠀⠿⠀⠀⣿⢀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣉⡀⠀⠰⣆⠀⠃⡰⠂⢠⡏⣾⣿⣿⠿⠟⠹⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⣄⣀⠉⢉⣀⣴⣋⠀⣿⡿⠀⣴⡄⠀⣠⡝⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠻⢭⣭⡭⠿⠋⠉⣳⣿⣧⠀⡈⠠⠄⠉⠁⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⢀⢠⡶⠒⠒⠲⠦⣤⠞⠉⠰⣿⣿⣦⡙⠢⠄⠚⢀⣤⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⣦⣹⣿⣷⠲⡄⠀⠀⢹⡆⢀⣠⡴⠶⡏⠙⢿⠶⠛⠋⣷⣭⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣸⠃⠈⠁⠀⣟⠀⠀⠈⣷⠛⠁⠀⠰⣧⡴⠾⣄⡀⣰⠇⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠶⠋⠀⢀⣀⣼⠻⠧⠀⣠⠷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠋⠁⢻⠀⠀⡴⠃⢀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣀⣠⡴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⠶⠿⠦⢤⣀⣨⣭⠿⣇⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⣯⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣼⣯⣄⡜⠷⡇⠀⣈⣙⡦⣤⡈⠉⠉⠳⡤⢤⡄⣼⡿⠶⢾⣷⠟⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣻⣾⡃⠘⠇⡸⠀⠀⣗⠚⣿⠍⣰⢫⡴⣤⡀⣿⢃⢾⡟⠛⠁⠀⠀⠀⣀⣤⡴⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠤⠭⠬⠥⠄⣴⣁⠤⠤⠽⠿⢭⡼⢡⠣⡜⡼⢩⢃⢎⠞⣿⠶⢤⣤⠴⣏⠁⠀⢀⣀⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠴⠶⠶⠶⢤⣼⣤⣤⠶⠶⠶⡾⠭⠿⠿⠿⠿⢧⣎⣿⣀⣥⠴⠶⠶⠶⠿⠾⠛⠛⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠚⠋⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠔⡁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⠋⡀⢀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠢⠉⠔⢊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⠀⢀⠄⠀⠀⠀⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡫⠄⠀⠀⠾⢀⠀⠀⠁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠘⠈⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣠⡤⠀⠀⠀⠀⠀⣤⠀⠀⢸⠄⠒⢄⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⡠⠀⠀⠀⡄
⠀⠀⠀⠀⠰⠂⠟⠃⠀⠀⠀⠀⠠⠂⠀⠑⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠢⡀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠴⠀⠇⠤⠠⠆⠀⠇⠠⠤⠤⠤⠤⠀⠀⠾⠀⠠⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠿⣱⣄⠀⣤⣤⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢾⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠅⠐⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⣄⠠⠀⢤⠄⠠⢠⡄⠠⣄⠀⠀⠀⢤⠄⢤⠀⠤⠄⠀⠀⠀⠀⢄⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠒⠂⠀⠂⠂⠒⠂⠀⠀⠒⠂⠀⠀⠂⠀⠂⠐⠀⠀⠀⠐⠒⠒⠒⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠈⠉⠁⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣬⣴⣾⣯⣵⣦⣴⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣖⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠿⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠠⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠄⠀⠀⣼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⠄⠀⠀⠀⠀⠀⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠉⠀⠀⠁⠈⠿⠒⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣀⠀⠄⡀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢸⣿⣷⣦⣈⠁⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠿⠛⠿⣿⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣽⠛⠶⢦⠀⠀⠀⡀⠀⢀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉⠀⠀⠀⠿⣤⠀⣀⠀⠀⠈⠉⠉⢙⡛⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠐⠒⠒⠒⢉⣉⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⣼⣃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠈⢻⠿⠿⠿⠿⣷⣦⣤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣶⡖⠉
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠉⢻⣿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢉⠃⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⣆
⠀⠀⠀⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⠖⠁⠈⠀⠀⠀⠀⢣⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠃⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠛⠙⠓⡖⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡤⢶⠁⠀⠀⠀⠢⡄⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⣀⣀⡸⣠⣤⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠛⢆⠀⢠⣌⣡⠟⠰⡌⠙⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⣀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡖⡒⠒⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣴⣦⣿⣤⣄⡁⠐⢢⠀⠀⠀⠀⣬⡎⠈⣁⠉⠀⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢠⠊⠙⠉⡉⠏⠳⠉⢛⣾⣀⣼⠁⠀⠀⠀⠈⠣⢄⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⣰⢫⣟⠛⠛⠒⠉⠓⣄⣸⠀⠀⠀⠄⡫⠀⠀⣸⠏⡽⠃⠀⠀⠀⠀⠀⡴⣋⠀⢉⡲⣄
⠀⠀⠀⠀⠀⠀⠸⣍⠀⠀⠐⠀⠄⢀⡾⠉⠢⠀⢀⡀⢀⠤⠚⠁⡰⠁⠀⠀⠀⠀⠀⡎⠊⠘⡀⢸⠀⡋⠳
⠀⠀⠀⠀⠀⠀⠀⣈⡵⢦⢤⡔⠂⢹⣁⡄⠀⠀⠴⣕⠝⠁⠀⢠⡃⠀⠀⠀⠀⠀⠀⠙⠢⠤⡇⣼⡮⠂
⠀⠀⠀⠀⢀⡴⠊⡀⠀⢉⣄⡼⠖⡉⠄⣇⠀⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⠀⠀⠀⣸⠦⣒⣦⡄⠀⠉⠒⠉⠀⠘⡟⠀⠰⡀⣿⡀⠀⠀⠀⡧⠤⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡏⠑⢲⢟⠁⠀⢠⡀⣀⢤⣐⠃⢀⡤⠟⠜⠷⠖⢂⣖⡇⠀⠀⠈⣱⠀⠀⠀⢀⡇⠀⠀⢻
⠀⠀⠀⠰⡇⠀⢸⠀⠀⠹⢍⣁⠀⢸⡀⠽⠁⠐⢀⢤⡁⠖⣁⠴⠁⠀⡠⠚⠁⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⢸⣁⡱⡌⠓⠒⠒⠚⢹⢸⢧⢸⠁⢀⠀⢰⠃⠀⠀⠀⠀⠀⠉⠀⢺⠏⡏
⠀⠀⢸⠻⢧⢸⠚⠀⠀⠀⣿⣣⠏⣏⢸⠹⠉⠁⢸⢸⢸⢸⠘⡿⣸⢸⡆⠀⠀⠀⠀⠀⠀⢀⣸⡀⡇
⠀⠀⠈⠂⠸⠖⢂⠄⠀⠸⠷⢧⡂⣇⣀⣃⣆⡃⢸⢸⠺⠬⢐⣣⠁⣉⣓⠀⠀⠀⣀⠤⠤⢉⣁⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠀⠀⢀⣔⡩⠋⠀⠀⠀⣤⠞⠚⡚⠃⠂⠉⣀⣔⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡜⢠⣄⡀⠀⡀⣀⣤⢀⠇⠊⢱⠴⣑⡇⠳⢆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠁⠈⠀⠑⠊⠀⠀⠈⠉⠀⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⣠⡔
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠦⣤⣤⣠⠞⠛⡅
⠀⠀⠀⠠⣄⣎⡾⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠒⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠠⠋⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠌⣤⡤⣄⡀⠀⠀⠀⠑⢄⢀
⠀⠀⠀⢤⡄⢶⢹⡙⣿⡷⠤⠔⠂⠒⠐⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠘⢙⢆⠀⠀⡀⠠⠁⠄
⠀⠀⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢧⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠉⠉⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⣀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢶⡆⠀⠀⢲⣶⣿⣶⣶⣶⣶⣶⣶⣶⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀⠀⠀⣲
⠈⠳⣦⣶⣾⣿⣿⣿⣿⡛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣿⡿⠟⠓⢲⡶⠋
⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⣀⣠⣤⣶⣾⣷⣦⣀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠾⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢬⣓⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣊⠠⠋⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠐⠀⠀⠀⠀⠻⠀⢹⠖⠑⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢼⡗⠀⠀⠀⠉⢹⠛⣀⡖⣠⠅⠀⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠀⠀⠀⣈⠀⠀⠀⣇⠀⠽⠖⣬⣓⡞⢉⠀⠰⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠀⠀⠀⠈⡀⢠⣌⣃⣄⣲⣿⣿⣷⣷⠉⠿⣷⢀⣠⣤⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠰⡏⠢⢭⣿⣿⣿⣻⣿⣿⣿⡶⣾⠊⢉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠐⠄⠀⠈⠙⣤⠚⣷⡋⠀⢿⣿⡖⢻⣿⣿⣿⡯⢏⡥⠄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠎⡜⠃⠘⣧⢷⣽⣥⡀⠐⠷⣿⡿⣿⣽⢿⣟⣿⡓⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠴⠃⡄⢀⠀⠀⢀⣽⣧⢆⣛⣰⣆⣿⣿⡿⠣⠀⠀⠹⣟⣿⣿⣿⣿⣷⣮⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠃⢇⠀⢀⠀⠸⠀⣄⣧⠌⣷⣸⣾⣿⣿⣿⡏⠀⠀⢀⠔⡀⠈⢿⣿⣿⣿⣿⣏⡆⣤⠁⠰⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠘⠀⡠⣘⠀⣠⣦⢷⡈⠀⠛⠁⠉⠛⠿⠟⠋⠁⠀⠀⢀⠊⢀⠀⡚⠈⣿⣿⣿⣿⡿⣿⡛⠡⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣤⠀⢻⣶⡓⢾⣾⠗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡀⠘⢠⣿⣿⣿⣿⣷⠯⡗⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⣿⣾⣥⡿⣷⣿⠃⠀⠀⢀⠀⢀⡀⠀⠀⠀⠠⡊⠆⠀⠀⢉⣧⠇⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣉⣟⢉⣶⣯⣯⣵⣻⣿⣿⠅⠀⡀⠀⠀⠒⠈⠨⠁⡀⠀⠀⡐⠂⠀⠀⠠⣿⣞⣾⣿⣿⣿⣿⡛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⢛⣿⢯⣿⠏⠀⠀⣤⠀⠆⠢⠀⠀⣂⠁⠠⠀⠀⠀⢑⡂⠐⣿⣥⣯⣭⣿⣅⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠇⠐⣿⣿⣿⣿⣿⣿⠛⠁⠀⣴⣾⣽⣻⠈⠱⢈⡐⠀⠂⠨⠅⠢⠈⠄⠧⣷⡜⢫⣿⣯⣅⣾⣿⢯⣻⢇
⠀⠀⠀⠀⠀⢀⣰⣿⡿⣿⣿⠻⡟⣉⣀⣀⣸⣿⣿⣿⣹⣧⣀⣀⠐⢢⠠⢀⠈⢀⠢⣵⣼⣿⣣⣽⣿⣿⣿⣿⣿⢿⡿⣦⢅
⠀⠀⠀⠀⠀⠂⠙⠀⠷⠞⠟⠻⣿⣿⣿⠻⣿⡟⣿⡿⣭⣿⣽⣿⣤⣄⣠⣤⣄⣭⣴⣿⣿⣷⣿⣿⣿⣿⣧⠙⣯⢼⠟⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠤⣽⠹⠏⢿⣶⣿⠟⣿⣿⣿⣿⠛⢿⡿⡿⠿⠋⠉⠃⠈⠓⢦⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠉⢹⡛⠢⢹⣾⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣧⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣄⡐⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠻⠻⠶⠷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣶⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⢸⡏⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⢹⣇
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠏⠁⢹⡆⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⣀⣹⣀⣠⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⠰⠿⠒⠓⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡶⠾⠿⠿⠿⠿⠷⠿⢶⣦⣿⡇⠀⢸⣿
⠀⠀⢸⣇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣶⣶⣶⣶⣶⣶⣶⣾⠟⠉⠁⠀⣸⡏
⠀⠀⠘⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠊⠉⠑⠄⠀⡀⠀⣠⣦⠞⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣾⠗⠀⠀⠀⢠⠈⠛⠿⢾⡿⠋⠀⠀⠘⣧⣼⠿⠒⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣶⣠⡔⢿⠗⠢⢀⠀⡠⠚⠓⡀⣠⣰⣧⣼⣄⣀⠔⠻⠢⣍⣍⠀⠀⢏⡻⣷⣠⣠⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⣀⣀⣀⣈⣀⣀⣀⣙⣿⣿⣿⣿⣿⣿⣁⣀⣀⣿⣿⣆⣀⣀⣹⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⠥⠬⠄⠍⣨⢀⠤⣄⣠⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣈⣈⣉⣉⣈⣀⠤⠤⠤⠭⣶
⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⣶⣶⣶⣶⡆⢠⣶⣯⠯⠭⠉⠉⠉⣷⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⡾⠉⠉⠉⠉⠁⣽⣶⣶⣶⣶⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⡇⢨⣿⣿⠃⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡌⢆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠬⠿⠿⠿⠿⠥⠬⢹⡙⠒⠒⠢⠤⠤⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡗⠤⠤⠤⠀⠐⢻⠟⠿⠿⠿⠛⠋⣀⠣
⠀⠀⠀⠀⠀⠀⠈⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠛⠂⠀⠀⠀⢸⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⡇⠀⠀⠀⠰⠓⢸⣶⣶⣶⣶⣶⣶⣦⣵⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠕⠒⠤⣄⠀⠁⠒⠠⢄⠛⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠋⢙⣠⠴⠃⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⢠⠐⠿⠿⣿⣿⣿⢻⠛⠉⠙⢻⢷⣤⣤⣤⣤⣤⣼⡖⠁⠀⠀⠀⠀⠀⠀⢠⣾⣧⣤⣤⣤⣤⡤⠞⠉⢻⢻⣿⠿⠿⠿⠿⢧
⠀⠀⠀⠀⠀⠈⠓⠒⠂⠉⠉⣿⡸⡀⠀⢀⡼⣸⠏⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⡴⡄⠀⠀⣸⢠⡟⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣵⣭⡾⠋⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣭⣭⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡈⠑⢄⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⢀⡀⠀⢀⡤⠀⠀⠑⠄⠂⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⡄⠀⢠⣠⠐⠒⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠃⢈⠉⠻⠿⠁⠀⢀⡀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠛⠀⠀⠀⠀⠦⡕⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠡⡀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⠀⠀⢾⠀⠅⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠀⠀⠄⢠⡗⠤⣠⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠢⠀⢆⠣⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢜⠸⣤⣀⠀⠀⠀⠘⠸⠇
⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⠀⠀⠀⠀⠁⢢⠀⡀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡀⠀⡇
⠀⢠⡜⠙⠘⢄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢇⣿⠤⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁
⠀⢸⡆⠀⠀⡇⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⢢⢸⠀⠀⠀⠀⠈⠷⠠⠤⣄⠈⠢⣄⠀⠀⢠
⠀⠈⡇⠂⠀⢹⡸⡆⠀⠀⡇⠰⡀⠀⠀⠀⠈⠆⠀⡀⠀⠀⠀⢸⠦⣄⠀⠀⠀⠈⠋⠑⠦⢷⡀
⠀⠀⣇⢸⠀⠘⡇⡇⠀⠀⢡⠐⢿⣄⠀⠠⠀⠈⢎⠁⡠⠂⢀⠄⢀⡠⠓⢦⣄⡂⠀⠀⠀⠀⢻⡍⠋⠐⠒⠒⡀⠀⢤⢀⠀⠀⢀⠴⠀⠀⠄
⠀⠀⢻⠘⠀⠀⣇⢸⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠈⢧⡀⣴⠡⠒⠁⠀⠀⠀⠀⠉⠉⢁⠖⠛⠚⠛⢍⠀⠐⠒⠂⠀⠔⠋⠓⠴⡒⠒⠦⠤⠦⢤⣤⣤⡀
⠀⠀⢸⡀⠄⠇⢹⠸⡄⠀⢠⡇⠀⠀⠈⢣⡀⠀⠀⣰⠯⠇⢤⣀⣀⣠⣤⠤⠀⡶⠖⠁⠀⠀⠀⠖⠌⣃⠀⠀⠀⠀⠀⢀⠤⠶⠍⡀⠀⣀⡠⠖⠀⠀⠞⠃
⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⣸⠉⡁⠒⠒⠒⠓⣄⠀⠷⠀⠀⠀⢀⠀⠀⢠⡄⠀⡴⠤⠤⠒⠃⡚⠁⠀⠀⠀⢀⡀⢄⡨⡀⠄⣂⡥⠖⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠸⡀⠀⠀⡇⣿⢠⠇⠀⠀⠈⠉⠀⠀⠈⠻⣦⠤⠤⠴⠷⠒⠛⢛⣗⡈⠀⠀⠂⠠⠄⢉⡑⠄⠤⣚⠵⠚⣉⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢧⠀⠀⠇⢸⡏⢀⠔⠁⠀⠀⠀⠀⠀⠀⡟⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⡁⠄⠒⡀⢈⣉⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⠄⠀⣀⠤⠔⠒⠒⠤⣀⠁⠀⠉⠉⠙⠓⠲⠶⠤⣤⣄⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠄⠁
⠀⠀⠀⠀⠸⡏⣷⠼⡌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⠑⠒⠲⠦⠄⣄⠠⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⡇⡇⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⡇⣷⠚⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⣅⠠⣿⢸⡀⠀⠤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢠⠃⠁⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⢠⡇⣿⠀⠿⢸⠿⠿⠷⢶⣴⣄⣤⣤⣤⣭⣭⢭⠼⠲⢤⣀⠀⠦⣤⠀⢀⡤⠚⠁⣰⠃⠀⡀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠐⠋⠹⠀⢨⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠨⣗⣀⡤⠞⠉⠀⣠⠞⠁⠀⠸⡇⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⡇⠰⠋⠉⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⣶⠒⠂⢈⠁⠀⠀⣀⠞⠁⠀⠀⠀⠀⡇⢸⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⡞⢠⡇⢸⢀⣀⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⣤⣀⣀⣀⣀⣀⣀⣰⠷⠘⡇⠟⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣛⡃⠀⡀⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠦⠀⠀⠀⠀⠀⣤⣀⡕⠀⡽⣧⠀⡿⢀⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣦⢀⡆⣺⡀⣖⡒⠒
⠀⠀⠀⠀⠀⠀⣶⠚⣋⠉⠁⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡿⠀⠉⠻⠁⣌⡽⠃
⠀⠀⠀⢰⠀⡆⢸⡆⡏⢴⣄⠛⢻⡋⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣀⠀⡠⠀⢀⠴⣄
⠀⠀⠀⠻⠚⣏⠀⣷⠛⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣀⣀⠀⠀⠀⠉⣿⠁⠠⣶⣯⢼⣗⣻⠎⣻⠇
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⣛⣉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⢷⣶⣄⠀⠀⢀⣀⡓⠆⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⠄⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠉⢉⡇⢰⡟⠿⠛⣦⠀⣷
⠀⠀⠀⠀⠀⢀⡤⠀⠀⣨⣽⠆⠀⠀⠀⠀⠀⣀⢰⠇⠀⠙⢦⣤⡴⠋⠠⠼⢧⡀⢀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡞⢀⣤⣄⠀⠀⠀⠀⠀⠀⢰⡏⠉⠟⠀⠤⣴⣄⣠⡀⣄⠀⠀⠀⢹⠟⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣙⢲⠀⠀⠀⠀⠀⠈⣉⡷⠂
⠀⠀⠀⠀⢸⡁⠻⢀⡟⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠹⣇⣠⡿⠘⠿⣶⡶⠟⠀⠀⠠⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⢮⡿⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⠀⠀⠀⠈⠻⢷⣄⣀⣀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢰⣟⣓⡾⠁⢠⣄⠀⠤⡄⠀⢸⠂
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⢶⣿⣷⣀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣹⠃⠐⠛⠛⣠⡴⠋⠀⣠⠴⠊
⠀⠀⠀⣶⠛⠉⠉⠉⠉⠀⠀⠀⠀⣀⣀⣤⠤⠤⠽⣿⣿⣿⣦⡀⠀⠀⠀⢀⣀⣤⡾⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠇⠀⠀⠲⣌⡉⠉⠁⠈⠁⣈⡽⠆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⢸⡿⠿⣯⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣰⠏⠀⠀⠀⠀⠀⠉⠓⠦⣄⡈⠛⠲
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠙⠶⠤⣤⡤⠞⠁⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⣿⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⣤⠶⠲⣾⢃⣾⡴⠒⣆
⠀⠀⠀⣿⠀⢀⣴⡿⠋⠤⠤⠤⠤⢤⣤⣤⢤⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣦⣀⣀⡇⠀⠀⣼⠟⠁⠀⣠⠏
⠀⠀⠀⣿⣴⣿⠟⠁⠀⠚⢛⣻⣿⠾⠛⣦⡀⢹⡧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣠⠶⠛⠉⣿⣄⣼⠃⢀⣠⠾⢯
⠀⠀⠀⣿⣿⡏⠀⠀⢀⡶⠛⠉⢀⣀⡀⠈⣷⣴⣷⣼⣇⠀⣠⠴⠖⠛⠓⠶⢤⣄⡀⠀⠘⠃⠀⠀⢰⣷⣯⠞⠛⠉⠁⣠⠏
⠀⠀⠀⢿⣿⣧⡀⠀⣿⠀⣤⣤⣤⣭⠟⠋⠉⠀⠀⠀⠉⠻⣇⠀⠀⠀⣠⠴⠖⠛⠛⠲⢤⣄⡀⠀⠀⣿⡿⣦⠶⠖⠛⣹⠷⠦⠤⣤⣤⣤⡤⠤⠤⠤⣶⡶
⣠⣤⣤⡼⣿⣿⣿⣿⠟⠛⠛⠋⠁⠀⠀⠀⢀⣠⡴⠖⠚⠓⠻⣷⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣾⡿⠻⢶⣶⣛⡿⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⠛⠒⠒⠛⢻
⢻⠀⠀⢠⣴⡞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⣈⣡⣤⣤⣴⣶⡾⠟⠛⠉⢉⣉⣭⣿⣿⡗⠀⢸
⢸⡇⢶⣾⣏⡉⠓⢶⣤⣤⡤⠖⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⣟⣳⣦⠤⠶⠶⠶⠶⠶⠶⠤⠤⠤⢤⣤⣤⣿⠿⣟⠛⠛⠛⠛⠾⠿⠟⠛⠛⠛⠛⠛⠛⠋
⢸⡇⠀⠙⠦⣝⠳⢦⣈⣽⣀⣀⣀⣤⡤⠶⠖⣛⣋⣉⣭⣥⣴⣟⣛⣛⣹⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⢸⡇⠀⠀⠀⠈⠙⠦⣬⣭⡭⠽⠶⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠘⠷⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⢤⡤⠤⠤⠤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠀⠈⠉⠉⠉⠉⢍⠀⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡤⣤⠄⢀⡀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠐⠂⣠⠤⢤⣤⠤⠀⢤⠀⠀⢶⣀⠈⢆⠀⢣⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠒⠂⠀⠀⠁⠀⡄⠀⠀⡀⡀⠁⠀⢀⣀⠀⠀⠀⠁⣂⠤⣵⠌⠀⠓⠮⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣵⠀⠀⠀⠀⠀⠀⠀⠰⠣⡄⡏⠃⠀⡔⠁⠀⠙⡄⠀⠈⠁⠀⠈⠂⠀⠀⠀⢀⣰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠱⢿⣆⠔⠀⠀⠘⠿⠆⠔⠁⠀⠀⠀⠘⢡⠀⡔⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢦⠐⠒⠀⠐⠂⠡⡈⠆⢇⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⢠⠀⠀⡂⢁⠜⠀⠀⠀⠀⠀⠀⠁⢐⣄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠤⠤⠦⡀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠉⠂⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢉⠀⡰⠉⠀⠀⡀⠂⠀⡠⠂⣈⠄⠀⠀⠈⠑⢄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠡⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠈⠀⠒⠒⠀⠉⠀⠀⠀⠁⠔⠃⠀⠀⠈⠉⠉⠉⠁⠈⠉⠈⠉⠉⠉⠉⠉⠂⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠒⠷⣵⣀⡀⠀⠀⠀⠀⡀⠄⠔⢊⠳⣄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠐⠂⠤⢀⣑⠈⡆⠈⠉⠈⠰⠆⠉⠀⠀⣤⡨⠶⠤⠤⠤⠒⠉⠉⠉⠁⠒⠒⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠁⠒⠒⠒⢄⠀⢀⠂⠀⠀⠠⠄⠀⠀⡠⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢀⣉⠉⠉⠉⠉⠉⣛⡒⠲⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡖⠁⠐⢼⠀⣀⣀⡀⠀⡤⠔⠐⠒⠒⠀⠉⠉⠀⠀⠁⠒⢤⣠⣤⣤⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠠⠆⠀⠀⠀⠀⠀⠈⢁⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡍⣒⠀⣈⣉⡀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠙⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣗⣪⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⣤⠀⠀⢀⣠⣥⣀⡝⣎⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⢣⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢀⣿⣷⠐⠚⢿⡗⢉⣉⣭⠵⠒⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠣⠅⠀⠀⠈⠀⠀⠀⢢⣀⡀⢀
⠀⠀⠀⢸⣿⣿⡇⢦⢞⡃⢐⣿⠶⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢙⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⢸⡿⠿⣷⢿⣿⣯⡜⠿⡿⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⡀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠐⣂
⠀⠀⠀⠈⢻⡀⠛⢛⣿⣈⠻⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠐⠄⠀⠀⠀⠰⠸⠦⡄⠀⠴⢀
⠀⠀⠀⢀⣼⡷⣦⣀⣙⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⢤⣀⠀⠀⢀⣴⣾⣠⣿⢀⡌⣸
⠀⠀⢠⣾⣿⡝⠳⢽⣋⣏⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠒⢿⣿⣿⣏⢁⠼⠋⠥⠂
⠀⠀⢸⣿⣿⣿⣦⠸⢿⣿⢸⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠶⠒⢀⠀⠉⢀⣀⣬⠔
⠀⠀⢸⣿⣿⣿⣿⣦⣸⣿⣄⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⣍⡙⠛⢿⡿⠢⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠦⠦⠠⠙⠻⠿⠿⠛⠒⠂⠀⠀⠀⠹
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣠⢶⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠓⠛⠆⠁⠈⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⢯⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⢧⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠵⠋⠉⠹⡄⠀⡠⠤⣄⡟
⠀⠀⠀⠀⠀⠀⢇⠀⢠⠀⠀⠀⠀⠀⢰⣿⠞⠉⠉⠓⢿⣾⠀⠶⣽⠄
⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⢸⠸⡄⢠⠶⣄⠼⠿⠄⠈⣯
⠀⠀⠀⠀⠀⠀⠀⡆⠸⡄⠀⡀⠀⠀⠸⡄⠹⡴⠋⠀⢀⣀⣶⠆⣏⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠥⠞⢹⣤⠛⢦⣀⣀⢱⣎⣁⡄⠀⠈⢢⠞⣰⠋⠉⠀⠉⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢴⠦⣌⡉⠉⠉⠀⣀⣀⣠⡵⠞⣹⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠉⠉⠉⠉⠉⠓⠤⠔⠊⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠔⠛⠉⢸⡇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣠⡀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⢸⡗⢆⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⢠⠋⣇⠀⠹⣄⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⡇⠈⢣⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠛⠒⠛⠲⠦⢽⣆⡀⠀⣎⢢⣠⠴⠖⠚⠋⠁⢀⢀⣿⡀⣀⠀⠀⢸⡄⠀⠀⢀⡤⠴⡄⠀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠠⠤⠤⠴⠶⠶⠤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⢎⢻⡛⠓⠒⠒⠛⠉⣩⠟⣽⠛⠁⠀⠀⣟⢉⣩⠿⠀⢀⡏⠉⠉⠉⠉⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠐⠺⢧⣳⡀⠀⠀⣠⠞⠁⠀⠋⠀⠀⠀⢀⡟⢸⡏⡆⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⣄⣀⣀⣀⣧⣀⠀⠀⠹⠻⣿⣷⠀⢸⣁⣀⡀⠀⠀⠀⠀⣰⠏⠀⢸⢻⣿⡄⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡼⠁⠈⠙⠓⠒⠈⠁⢸⣛⠉⠀⠀⠉⠙⠲⣤⠞⠁⠀⣤⠴⢦⣿⣧⠴⠿⡇
⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢤⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡄⠀⠀⠰⣶⡇⠀⠀⠀⠸⡖⠒⠒⠲⡆⠀⡇
⠀⢀⣠⣔⣋⠁⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠈⠓⠛⠒⠒⠒⠒⠛⠋⠉⠉⠉⠓⠓⠾⠦⠴⠋⠀⠀⠀⠀⠀⣧⢤⣤⣤⠇⠀⡇
⠀⠀⠀⠀⢩⡏⠉⠉⠈⠉⠁⠈⠉⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣤⠤⠴⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠒⠲⠾⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⡀⡤⠀⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⢣⡀⢀⡀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠄⡀
⠀⠀⠀⠀⡏⠔⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠢⣀⠑⢄⡀⠀⠀⡀⠄⠐⢈⡁⠤⠀⠐⠐⠉⢔⠂⠀⠀⢀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠂⠠⢄⡀⠀⠀⠀⠀⠁⠢⣀⣰⡆⠀⠈⠈⠁⠀⡒⠋⠀⠀⠀⣀⠠⠀⠐⠈⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⠀⠀⠀⢀⣀⡠⠤⠀⠀⠈
⠀⠀⠀⠀⡀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⡄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠒⠀⠉⠀⢸⠀⠘
⠀⠀⠀⠀⠂⠸⠈⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠠⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣯⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢄⡀⠀⠀⣀⣀⢰⠀⠔⠐⠺⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠿⠁⠀⠀⠀⠀⠀⢀⠈⠀⠘⠋⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⠃⠀⠀⠀⡀⠀⢠⣶⠙⠀⢰⠆⠀⠀⡟⠁
⠀⠀⠀⣨⡤⠶⢤⣤⡤⠤⠤⠶⠶⡶⠒⠒⠒⠒⠶⠶⠾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠾⠃⠀⠀⣧⡀⠀⠀⠀⡀
⠀⠀⣾⠁⣇⣠⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰⠁⢹⡄⠀⠀⡇
⠀⠀⠹⣄⠀⣀⡷⠋⠉⠉⠉⠙⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠦⠤⠞⠀⡴⠋⢹⠚⢳⡀
⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠀⣀⠀⠀⠀⠀⣀⣤⣞⣁⣠⣤⡤⠾⠃⢐⠊⠀⠀⡇
⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠄⠂⠐⡖⠉⠀⠈⢋⠉⢉⠍⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠐⠊⠀⠀⠀⠀⢠⠀⠀⠀⠘⡤⠶⠤⠤⠤⠴⠶⠖⠚⠋⠉
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⠑⠒⢲⠖⢣⡀⠀⠀⢀⠀⠀⠈⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⠙⢤⡴⣭⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠘⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣠⠞⠁⣸⡄⠀⠀⢿⠀⠁⢸⡇⡀⠠⠀⠀⠂⢉⡀⠄⠀⠐⠈⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠶⢾⣥⠴⠟⠁⣇⠀⠀⠿⠀⠓⠈⣁⣠⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⢸⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠃⠀⠀⠙⠅⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠀⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠠⠒⢲⠀⠒⠢⢄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣇⣀⢰⠇⠀⠀⣴⡁⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠⣿⡇⠀⠎⠀⢸⠁⠈⡦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠈⢁⠀⠦⡀⠾⠢⠔⠓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢘⠐⣿⠃⠀⡀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⠀⠀⠹⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢣⠀⠑⠢⠀⠀⡀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⢺⡇⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠒⠤⠤⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣸⣤⣠⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢉⠀⣀⠈⠋⠀⣤⠶⠞⡹⣫⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢉⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠾⢣⣾⠿⣷⠀⣼⠃⢸⠀⣽⣿⠷
⠀⠀⠀⠀⠀⠀⠀⡠⠖⡲⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡈⠋⠁⢀⢿⡄⢸⠀⣿⡟⣰
⠀⠀⢀⠀⠀⠀⠸⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠛⠲⠪⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠤⢄⠀⠀⠀⠀⠀⢰⡩⠉⠑⠂⠺⠀⠋⠙⡇⠙⠉⠁
⠀⡔⠁⣠⠁⠠⠀⠂⠀⢷⠤⢀⡀⠀⠀⠀⢀⠴⠺⡄⠀⠠⠷⠄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣀⠾⡀⠀⠀⠀⠀⠀⠈⠐⠢⣤⡤⠭⠔⠊
⠀⡱⠖⢀⠀⢀⡠⢴⠒⡞⠀⠀⠉⠉⠀⠀⠀⠀⡄⢻⡀⠀⠆⣠⠈⠒⠤⢄⣀⣀⣤⡞⠂⠑⢊⣀⡴⠃⠉⡱⠀⠘⡏⢻⠠⢯⠛⠳⡄
⠸⣤⢊⡠⠒⠁⠀⠈⠊⠒⠀⠀⠀⠂⠀⢲⠊⠀⠉⣆⢣⡀⡃⠱⠄⠱⠠⡀⠉⠀⠀⠱⢲⣄⡰⠉⠈⠉⠈⠀⠀⠀⠱⠚⢸⠿⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠁⢻⢇⠀⠀⠀⠱⡀⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠃⠸⠇⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠙⠨⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣝⠒⠒⠂⠤⠤⠤⠤⡔⠱⠒⠀⠀⠒⠒⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡟⠰⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡇⠀⠀⠀⠀⠀⠀⢀⣤⠥⢤⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢠⠛⠁⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢿⡶⠒⠒⠒⠃⠾⣶⠶⠶⣿⠟⢛⣛⡓⠈⠁⠐⠒⠂⣄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣷⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢱⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠇⡴⢀⡄⣠⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣾⣧⣾⣤⣇⣞⣰⠋⡼⢡⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⡋⠉⠉⠉⠙⠛⠻⢷⣿⣠⡏⢰⠇⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⣟⠦⠙⠦⣄⢀⡀⠀⠀⠀⠀⠉⠻⣧⣼⢋⠜⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠉⠓⠦⠀⠀⠀⠀⠈⠙⢯⣰⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣤⣬⣍⣻⣄⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢾⣿⣿⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣏⣿⣿⣟⣿⣿⣆⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣿⣿⣿⣿⣿⢿⣿⠉⢉⣼⣿⣿⠏⠀⠀⡼⣿⣿⣥⣌⡻⢦⡀⠀⢨⣼⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⣿⣿⡿⠃⠀⠈⠛⣩⣿⣿⠃⠀⠀⢴⠁⠸⡿⣿⣿⣿⣾⣿⠖⣤⡏
⠀⠀⠀⢀⣀⣠⣼⣿⣿⣿⣿⣿⣿⡶⠂⠀⠀⢠⣾⡿⠁⠀⢀⠀⡟⠀⠀⠻⣌⣴⡟⠉⠁⣠⣿⣠⣄
⢠⣴⠾⡿⠉⣿⠃⢠⡿⠋⣿⣿⣷⣿⣁⠀⢀⣾⠟⠀⠀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡏⣩⣿
⢸⡏⢠⠃⣸⠇⠀⡿⠁⣸⣿⣿⣿⣏⣰⣾⣿⠋⠀⠀⠀⣉⣙⣿⣂⢀⣤⣶⣦⣤⣤⣴⣿⣯⣾⡿⠃
⢸⣤⣾⣴⣿⢀⣼⣃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃⣰⠟⢁⡼⠉⢩⡿⠻⣶⣶⣶⣶⣤⣤⣤⣤⣤⡤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣝⢿⣷⣍⠻⣝⢿⡻⣿⣟⣿⡿⠁⣰⠃⢀⣾⠁⣠⡟⢀⣼⣿⣯⡻⣮⡻⣿⣿⣝⠻⣝
⢸⣿⡿⣿⡻⣿⣝⢯⡻⣟⢿⣝⢝⢮⡳⣍⠳⣕⣮⡳⣝⢮⡻⣝⢿⣾⣿⣴⣿⣧⣴⣿⣶⣾⣝⠻⣟⢿⣮⡻⣮⠻⣿⢷⣌
⣸⣿⣝⢮⡛⢮⡻⢷⣝⢮⡳⣝⠳⣝⡮⡪⡻⣮⡻⣯⡪⡳⣝⠢⣕⢝⢿⣿⡷⣍⠻⣿⢯⡻⣯⡳⣮⡳⣌⠻⣮⡳⣌⠳⡝
⣿⣗⢍⠳⣝⢶⣝⢦⡙⢷⣌⠪⡑⢌⠻⣮⡢⡈⠻⣮⡙⢮⡪⡓⢬⡳⣝⢮⡻⣮⡳⣮⡳⣝⢮⡻⣮⡻⣮⡳⣌⠻⣌⢳⣌
⣿⣝⢷⡕⢌⠳⣝⢷⣝⢦⡙⣷⣌⠢⣑⢮⠻⣮⡢⡨⡻⣦⡱⣌⢦⡙⣮⡢⡙⢮⡻⣮⡻⣮⡳⣝⢮⡻⣮⡻⣮⡳⣌⠳⣝
⢿⢿⠳⡝⠢⠑⠌⠳⡙⢷⡝⢮⣿⣷⣜⢦⡑⣌⠻⣾⣮⣯⣿⣾⣷⣽⣮⣿⣮⢪⡻⣮⠻⣮⠻⡎⢣⠙⣿⣿⣮⡻⣜⢷⣌

⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢰⡄⡀⠀⠉⢁
⠀⠀⠀⠀⠘⠀⠀⠐⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠀⠂⠈⠉⠑⠠
⠀⠀⠀⠀⠈⢦⡢⠤⠼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⢀⠇⠀⡄⠀⠐⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡧
⠀⠀⢀⣀⡀⣤⠤⣤⡇⠀⠀⢠⡔⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⡶⠛⠋⠙⠮⠳⢤⡄⠀⢸⡀
⠀⠀⠲⠬⠭⠥⡴⠁⠀⠀⠱⣞⠈⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠊⠔⡒⠨⡷⢎⠃⠢⡘⡄
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⡄⠀⢣⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⢏⠆⠚⡴⢡⢾⡄⠣⠑⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠠⠐⠒⣀⣱⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠁⠀⠀⠀⠃⡾⡎⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡆⠀⢲⠆⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⣷⠀⠀⠀⠀⠀⣇⡘⡄⠀⠀⠀⢸⡲⢄
⠀⠀⠀⠀⢰⠉⢑⣄⡠⠗⣄⠀⢀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⠀⠀⠀⠀⣸⠀⢀⣹⡀⠀⠀⠀⡇⠈⢣
⠀⠀⠀⠀⢸⠀⡈⠁⠀⠀⠈⠓⢉⡀⠀⠉⠒⠤⠀⠀⢀⠀⠀⠘⠁⠀⠀⡸⠀⠀⠀⠰⢁⡉⠁⣀⡽⢦⠀⠀⠙⠄⠈⣇
⠀⠀⠀⠀⡆⣰⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠲⢤⡇⠈⡇⠀⢀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠈⠉⠁⠀⠀⢳⡀⠀⠈⢦⠸⡆
⡤⠤⠤⠴⣁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⡟⠀⢸⠀⢀⠃⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⢣⣹⡄
⠙⠶⠄⣀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⢀⠆⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢦⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⢳⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣯⣿⣷⣈⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣽⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠟⣉⠉⣩⠍⣋⣷⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢋⡉⢉⡭⣹⡇
⠀⠀⠀⠀⢀⠤⢤⣤⡤⣶⡾⢛⡀⢅⠓⢘⢰⣶⣿⣿⣩⡾⣿⠀⠀⠀⠀⠀⡠⠤⢤⡤⢤⣶⡟⣃⠨⡘⠆⡉⣾⣿⣿⡇
⠀⠀⠀⠀⠈⢦⡀⠙⣿⣿⢉⣈⣓⠠⣄⠘⢿⣿⣟⣵⡿⢀⣿⠀⠀⠀⠀⠀⠹⣄⠈⢻⣿⡟⣁⣙⣃⢤⡀⠹⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠉⠣⣿⠇⣹⣞⣿⣿⣿⣦⠘⡽⠉⠉⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⠙⢼⣿⢁⣷⣿⣿⣿⣷⡤⢋⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣭⣭⣵⡿⠉⠉⡳⠔⠀⣠⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣩⣽⣭⣾⠏⠈⢉⠦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⣅⣀⠀⢈⣀⠀⢈⠿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣤⣬⣀⡀⠀⣁⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣶⣿⠋⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠷⣶⣶⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇
⠀⡎⢱⢰⠀⠀⣶⠀⡎⠱⢰⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⡏⢹⢸⠀⢰⠥⡆⡇⢀⢸⠛⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠐⠒⠶⠤⠀⠀⢠⠴⠶⠓⠂⢻⣿⣿⣷
⣤⡭⠨⣬⢥⢬⢤⢥⡭⡯⠬⠀⠬⠀⣤⣤⠀⠀⠀⢀⣿⣿⣿⣿⣿⠰⢟⠛⠒⠀⠀⠀⠘⡷⠆⠀⠈⣿⣿⣿⣧
⣿⡇⠡⢿⠈⡘⢸⢘⡁⡇⣿⠀⡇⢘⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠿⠇⠡⠸⠤⠇⠿⠸⠇⠇⠿⠰⠧⠈⠿⠿⠀⠀⢀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇
⠀⢰⠀⡄⠀⠐⠀⠀⡔⢠⠀⡄⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣀⣀⡀⠀⠀⠀⠰⡆⠹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⣠⣾⣷⣄⠀⠈⠻⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠖⠊⠈⣻⣿⣿⣿⣷⣄⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⡿⠟⠛⠛⢋⣤⠀⠀⠉⠻⣿⠁⠀⠀⠀⠀⠀⢠⡴⠯⠽⠋⠉⢈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣒⡿⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣄⠀⢀⣠⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠖⠒⠛⠋⠉⢙⣦⠤⠶⠶⠶⢤⣀⣤⠶⠛⣿⡤⠟
⠀⢠⡤⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⢀⢀⣠⠶⠛⣩⣴⡖⠋⠁
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠀⣀⣤⠶⠛⣉⣤⡶⠋⠁⠀⠹⣆
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣛⣉⣤⢶⢾⣿⡟⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣠⣯⣀⡿⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠸⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠉⠓⢦⣤⣀⣀⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣀⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⢶⡏⢿⣿⣿⢿⠛⠋⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣶⡦⢯⣿⣷⣄⡀⠚⠟⠀⣰⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡌⠹⣿⣿⣿⣶⣶⣾⣿⣿⣿⠿⠯⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣆⠈⠙⢿⣿⣿⣿⣿⣿⣿⡲⣄⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⠏⠙⢻⣿⣶⣀⣀⣉⠙⢛⣻⣷⣹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠟⠀⠠⠘⢋⣿⢿⣿⣿⣷⣿⣿⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⢻⣿⣿⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠸⠛⢛⣿⠟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣧⣀⣐⣾⣂⠉⠀⠀⠀⠀⠀⠶⣤⣤
⠀⠀⠀⠀⠀⠀⡰⠂⣀⠀⠀⠀⠀⠀⠀⠰⠁⣼⣿⣯⣙⣛⣿⣬⣵⣦⣤⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠁⠹⠡⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⣌⡽⠏⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⣿⠋
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡛⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣟⣀⣤⠄⣀⣀⠀⠀⠀⣀⣀⠽⣧
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣅⣀⠀⠀⢩⡏⣁⠀⢠⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⢀⡘⠛⠛⠛⠛⠛⠛⠒⠀⠀⠸⠷⠾⠿⠿⠿⠿⣷⡘⣆⠀⠀⠀⠀⣡⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠄⠀⠼⠿⠟⠋

⠀⠀⠀⠀⠀⢀⠖⠒⠒⡄⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⡞⠁⢀⣠⠜⠀⠀⠙⠆⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠘⡴⠊⠁⢀⣀⠤⠒⠒⠙⡄⢠⣴⡶⠶⠶⠶⢶⣶⣶⣶⣶⡶⠶⢶⣤⡀⢰⠉⠉⢱⡀
⠀⠀⠀⢸⡤⠒⠉⠉⠁⠀⠀⣀⣡⠃⣿⡇⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⠀⠀⡸⠛⠋⠙⠒⢁⡷⣄
⠀⠀⠀⠀⠳⢄⡠⠤⣔⠒⠩⡅⠀⣇⢀⣹⡆⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⠔⠓⠈⠉⠉⠉⠁⠀⠈⣇⠀⠻⠃
⠀⠀⠀⠀⠀⡔⢱⡄⠉⠀⣤⡶⢋⣉⣉⣁⠬⢍⠑⠲⢄⠁⠂⠐⠀⢀⠾⠒⠒⠒⠒⠒⠶⠶⠶⠖⠋⣎⡁⠒⠦⡀
⠀⠀⠀⠀⢸⢠⣿⡏⣦⡜⢁⡠⠞⠀⣄⠀⢠⡄⠉⠑⡀⠑⣄⠀⠀⢸⡤⠤⠒⠦⠔⠦⠄⠤⡤⢤⢤⠿⢿⣿⣦⠘⡆
⠀⠀⠀⠀⢸⠈⣿⣇⡟⢀⡇⠀⠂⠀⠠⣀⡠⠀⠀⠁⠉⡆⠸⡄⠀⢸⠀⠀⠀⢐⠄⠀⢀⡄⠀⢀⢼⠓⢦⢹⣿⡇⠨
⠀⠀⠀⠀⠀⠣⡈⠻⡇⢮⠀⠐⠀⠀⡴⠒⢄⠀⠁⠀⠀⢱⠀⡇⠀⢘⡀⠀⠀⠀⡀⠀⠀⠀⢸⠍⣰⠛⢩⡍⠙⢧⡃
⠀⠀⠀⠀⣠⣤⠈⠲⡇⢸⠀⢀⠀⠀⠳⢤⣜⠬⠤⠐⢒⣛⣒⣛⣛⣛⡓⠒⠠⠤⣉⣒⠉⠀⢈⢳⢯⠀⣄⠀⢠⡄⢳⣀⡀
⠀⠀⠀⠀⠈⠉⠀⣀⠱⡀⢱⡀⠈⠀⠡⣏⠠⣶⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆⢈⡶⣋⢭⣍⠺⡳⢌⡉⠀⢀⠎⠐⣇
⠀⠀⠀⠀⠀⡎⢙⢁⣻⠉⠙⢺⡦⠤⣀⡏⠉⠒⠊⠭⠭⠭⠿⠿⠿⠯⠭⠭⠍⠒⠊⠁⡏⣁⡼⠀⢓⢰⠀⣉⡉⠉⠓⠚⠁
⠀⠀⠀⠀⠸⡤⠤⠏⠀⠰⠄⠀⢳⠒⠨⢿⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢰⠋⠉⠀⡠⢃⠎⠀⠻⠋
⠀⠀⠀⠀⠘⡄⠀⠀⢤⠀⢀⡆⠀⡇⠀⠀⢣⡀⠀⠀⠈⢁⠀⠀⠀⢀⡈⠁⠀⠀⢀⣗⣒⣒⡩⠔⠋
⠀⠀⠀⠀⠀⠘⠢⣄⣀⣀⣉⣀⡠⠃⠀⠀⠀⠙⢤⣀⠀⠀⠙⠒⠒⠉⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣿⡟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡋⠙⢫⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢜⣿⡁⣀⡤⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣾⡟⠀⠉⡀⣿⣿⠻⢧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢉⠉⢋⣵⠀⣤⣴⣿⠻⠿⠏⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠛⠟⡅⠄⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡉⣻⣿⠀⠀⢠⠤⠤⠤⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⢘⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⠣⣬⡙⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢻⡗⠚⡇⠀⠁⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠛⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠽⠟⠉⠵⠘⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⠖⠁⠀⠸⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣹⡝⠷⣶⠶⠇⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠨⡟⠛⠁⠨⠟⠗⠶⠒⠗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣜⣠⣤⣤⡄⠀⠀⡀⢸⠒⣦⠧⠀⠤⠀⢀⣄⣀⡀⠀⢨⡟⢦⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢳⣤⣶⣾⡵⣿⣍⢠⠲⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠂
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⣶⡀
⠔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡈⠀⠀⠀⠀⠠⠀⡀
⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒
⠀⠀⠄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⠄⠀⠀⢀⡤⠴⠓⠹⠓⠗⠲⠦⣀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⡀⡀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠁⢀⡈⠀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠄⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⣀⣠⠄⠄⠀⠁⠀⠀⠀⠠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢀⡌⠀⠀⠀⠐⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤
⣿⣷⣾⣭⣭⣭⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠳⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣀⣀⡠⠄⠀⠈⠛⢶⡶⢄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡴⠊⢿⡏⠀⠀⠀⠀⢠⣴⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⡤⢾⠀⠌⠀⠀⠀⠀⠀⢤⡀⠐⡶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡞⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠛⢂⡔⠊
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠾⢯⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠐⠋⢀⡀⠀⠀⠸⡆⠀⠀⠹⡀⠀⢈⣦⡀⠀⠀⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣧⠀⠀⠀⢳⡔⢯⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀⠀⣼⠻⢳⣼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢧⡸⣇⣠⠞⠿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⢀⡔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡎⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠾⡇⠀⠐⠶⠀⢸⡇⠀⠀⠀⠀⢶⣄⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢘⣧⣀⣤⣿⣎⣀⣀⣀⣤⠶⠾⠷⣤⠤⠒⠒⠛⠽⣿⣦⡀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢹⠀⠀⠀⠀⠀⠀⠦⣝⠢⣄⠀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢀⣼⣥⣀⣀⣀⠀⢬⣽⠞⠉⠛⡦⠠⠀⣀
⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣤⡤⠤⠖⠚⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⡩⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠊⠉⢉⣢⣀⣀⢀⡔⠊⠁⠢⠤⢄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢀⡀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⡇⢀⡼⠒⠉⠁⠀⠀⠀⠈⠉⠃⡱⠄⠀⠀⠁
⠀⠀⠀⠀⢠⠊⡠⠤⠤⠞⠁⠘⣍⠥⠤⠬⠬⣵⣄⠀⠀⠀⠀⢠⠘⠁⠀⠀⠀⠀⠀⠀⡔⠉⠠⡀⠀⠘⠆⠀⠉
⠀⠀⠀⠀⡆⠀⢣⠀⢰⠊⠑⠒⠃⠀⠀⠀⠀⠈⠪⣆⡀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⣦⡌⠁⠐⠘⡀⢀
⠀⠀⠀⠀⠃⠀⠀⠙⠁⢘⠢⢄⣤⡀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠈⠑⡦⠤⠤⠤⠤⢐⡒⠒⠉⡄⠡⠀⠊⠀⡏⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠠⠄⠈⠁⠒⠒⠒⠆⠘⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠁
⠀⠀⠀⡸⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⢀⡉⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣠⠀⠀⠀⠀⠐⠒⡪⠒⠁
⠀⠀⠚⢥⠀⠀⠀⣠⣧⡀⠀⠀⠢⠀⠄⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⡁⢀⡀⠀⣤⡐⠉
⠀⠀⠀⠈⠓⢱⡼⠀⢁⠑⠐⢶⠦⣀⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠳⠔⠁
⠀⠀⠀⠀⠀⢸⢃⠀⠘⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡄⠈⠀⠀⣄⠀⢀⠀⠂⠄⢀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠁⡄⠑⢄⠀⠉⠚⠥⠶⡀⢀⠁⢲⠤⣀⠀⠐⢀⠀⠀⠀⠀⠀⠠⢃⠀⠀⠀⠁⠀⡠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠐⠠⢠⠾⠋⠑⠊⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⢀⢉⡀⠀⢒
⠀⠀⢰⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠒⠃⠀⠀⠀⠁⢀⠈⠀⠈⠁⣀
⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠬⣀⡠⠬⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⡀

⠀⠀⢤⡄⣤⣤⣤⢠⣆⡤⠶⣄⡀⢦⠀⠀⢀⡀⡤⠀⢤⡀⢠⣤⡄⠀⠀⠀⠀⠀⠀⢠⣴⢠⣄⢘⢻⣶⡤⠖⡴⠆⢢⣀⡀
⠺⠴⠊⢳⣿⠛⣱⠋⠀⠘⠛⠉⠉⠀⠀⠀⠉⢰⠗⢀⣨⠿⠈⠛⣁⠀⠀⠀⠀⠀⠀⠋⠃⢀⡴⢛⠅⠀⠉⠉⠀⠀⠀⠀⠿
⠀⠀⠀⠀⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠏⠀⠐⡂⠀⠈⢧⡀⠀⠀⢠⠂⠀⡶⠋⠐⠃
⠀⠀⠀⠀⠘⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⡆⡀⠘⠃⡃⡀⢀⡀⡤⠀⡼⠀⠀⣠⠆⠄⠰⠂⠀⠐⠳⠃⡀
⠀⠀⣠⠶⣦⠀⢱⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤⢹⢿⣿⣙⡟⠳⠵⠀⡈⢉⠁⠀⠃⠀⢰⠏⠴⠀⠀⠚⢻⣆⢓⠀⠅
⠀⠀⠀⢶⣿⢠⠘⡀⠒⠒⠀⠤⣴⣿⣧⣾⡤⠏⡊⠁⡟⢿⠀⠀⢐⠠⣀⠈⠑⠢⢄⡈⠀⠀⠰⠄⢠⢯⡍⢿⠁⠀⠘
⠀⠀⠀⠀⢻⡆⠀⡇⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⢸⣇⣿⣤⢶⡴⢤⠀⠑⢀⡀⠀⠙⠄⠂⣐⡦⠀⠀⣱⢁⣴⣮⡀
⠀⠀⠀⠀⢸⡧⢀⠣⠀⠀⢈⣯⣿⣿⠀⠀⠂⠀⢰⣶⣿⣿⠏⠀⠀⢀⡀⠀⠀⠀⠀⣀⠔⠀⠘⠛⠦⡄⣝⢿⣻⠟⠁⠢⣄
⠀⠀⠀⢀⠃⢏⢸⣄⠠⠀⠸⣾⣿⡇⠀⠁⢀⣾⣿⣿⣿⡧⠂⠀⢠⡴⠥⢄⣒⣴⣿⣧⠠⢀⠀⠀⢠⣞⠉⣤⣁⠀⠐⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⢿⢿⠁⠁⢠⡯⣿⠷⠾⠟⡡⣾⣟⡏⠡⣴⣾⣿⠛⢣⡏⠀⠀⠁⠀⠁⡛⠀⠉⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣷⣬⣿⠀⢠⣿⣿⠁⠀⣠⣾⣿⡿⡉⣰⠃⠁⢸⡇⠀⠸⡇⢀⠐⠀⡱⠔⠢⠴⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣿⣿⣶⣿⣿⣧⣤⣶⡿⠃⢉⡇⡇⠁⠀⠀⢸⢣⠆⠀⣷⠈⠁
⠀⠀⠀⠀⠀⠶⣿⣿⣯⣿⣿⣿⣿⢿⡿⢺⢷⣿⠿⣽⡄⣼⠈⠁⠀⠀⠀⠋⠌⠀⠀⢿
⠀⠀⠀⠀⠀⣤⣿⣧⣯⣿⣾⣷⣾⣿⣿⣻⣶⣿⣿⣿⣵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠞⠓⠛⢫⣿⡿⢿⣿⣾⡿⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢛⣾⣯⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠉⠑⠻⠉⠟
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢰⡉⠀⠀⠀⠀⠤⠤⣀⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠔⢉⠀⠂⠙⠛⠁⠀⠀⠈⣤⠦⣠⠤⣙⠲⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⡨⢄⣀⢤⠂⠀⣤⣀⠀⠀⠀⠀⠙⠲⢌⢳⡘⢆⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠚⠃⣰⢋⡖⠁⠈⢉⠔⣩⠞⠻⠂⠀⠀⠀⠀⠀⠀⡀⠹⠀⡇
⠀⠀⠀⠀⠀⣠⠎⠁⠇⠘⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡀⠒⠚⠦⡀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠅⠀⢠⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⢿⡀⢳⠀⡆⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠃⠀⠀⠀⣜⠀⢀⡌⢀⡞⠀⣀⡀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠁⢠⠛⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⡾⠎⢠⠄⢀⣾⠃⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡘⢆⠧⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⣵⡖⠈⡀⢰⠇⢠⠎⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡵⠁⠀⠚⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡸⠋⢀⢈⣧⠀⡄⠀⠀⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣫⠴⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢉⢀⠈⣴⠙⠄⠹⠀⡄⠇⠈⠃⠀⠀⠶⢶⠂⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⣿⠇⠀⠀⠤⠀⠀⠀⠀⢸⡴⠖⠛⠋⠛⠶⣤⣀⡀⠔⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⡐
⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠂⠀⡄⠁⠀⠀⠀⠒⠶⢄⠀⠀⠀⠀⡌⠀⠀⡤⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢡⡀⠀⠀⠀⠁⠀⢠⠶⡘⠙⣄⠀⠀⠀⠀⢀⡤⠓⠢⢀⡔⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠆⠈⠁⠀⣆⠀⠄⢠⣈⠊⠀⠀⣟⢰⣂⠀⢰⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢀⢠⡆⠀⡇⢀⠴⣿⠇⠀⠀⢰⡿⠀⠿⡀⣭⠒⠂⠤⠄⢀⣀⣀⠎⠀⢀⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠸⣯⣷⡘⠋⣿⠀⢻⡄⠀⠀⢰⡅⠀⠀⢁⠀⠀⢃⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⢧⠙⣇⠈⠳⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠠⡏⠀⢸⠀⠀⡀⠀⣆⠀⠀⠃⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⢼⠀⡖⠁⣸⠙⠀⣦⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⣿⠀⣌⢁⡦⠀⣉⡳⣾⡆⠠⠤⠤⣄⡀⠀⠀⠀⣀⣀⣤⢼⡇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢸⡇⢀⣿⢰⡽⠃⠀⠀⠀⠈⡍⠁⠀⠈⢹⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⠁⢿⡁⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡀⠀⢈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠈⠶⠒⠒⢺⡏⠉⠙⡃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠘⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⣸⠤⡆⠿⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠾⠋⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣾⣿⣿⣶⣶⣶⣶⣿⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣾⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣏⣛⣿⣿⡇⠙⠛⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⣍⠻⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡜⠀⠙⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠐⢿⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡇⠀⠀⢸⣄⠀⠀⠀⠀⠀⣀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠱⣄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠃⠀⠃⠀⠈⠁⣤⡀⠻⡿⠿⠲⠆
⠀⠀⠀⠀⠀⠀⠀⡶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣷⣤⠑⠒⡄
⠀⠀⠀⠀⠀⠀⣘⡦⠔⢀⠂⠆⡀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠙⢠
⠀⠀⠀⠀⣠⣿⣣⣴⣶⣎⠀⡲⢌⠀⠀⠀⠀⠀⠀⠈⠀⣿⣷⡈⢳⣄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡟⢿⠟⠋⠁⠀⠀⠙⢦⠀⠐⠀⠠⠀⠀⠀⠀⢻⣿⣷⠀⠹⣆⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡄⠀⠀⠀⠀⠘⣿⣿⣧⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡅⢀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢠⠀⠀⠀⠀⠀⠀⠛⠓⢷⡀⠀⠀⣠⠀⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣸⠀⠀⠀⠀⠀⢿⣿⣿⡆⠈⣆⡀⠀⠀⠀⠃⣈⡉⠒⠃⠀⠺⠅⢈⡗
⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠇⠀⠀⠀⠀⠀⠸⣿⣿⡿⠞⠁⠙⣦⠀⣠⠔⠉⠑⠂⠄⠀⢥⠀⣺⠇
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠸⠋⠀⠀⣰⡄⣃⠸⠤⢠⣴⠇
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢰⣏⣠⣽⣤⠶⠘⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠟⢉⡴⠋⠀⠀⠀⠀⠀⠘⣀⣴⠋⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢀⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⠀⠀⠀⢠⣼⡏⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣻⡀⠀⠀⣾⠸⠇⠀⠀⠀⠀⠀⢀⣴⠋⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⣿⣧⡀⠀⠛⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠻⢿⣿⣄⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠛⢚⠻⠿⠶⠦⠞⠁⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣨⣄⣀⣀⣀⡀⣤⣤⣶⣶⣾⣿⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⠿⠿⠿⠿⠛⠛⠇⠛⠛⠛⠉⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣈⣁⣤⣂⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠑⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣶⣦⣄⡀⠀⢀⡑⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠒⠀⠀⠀⠚⣿⣿⣿⣿⣿⡿⠙⠛⠛⠛⠉⠙⠙⠻⠟⣷⣤⡈⠢⡀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡷⠁⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⠃⠀⠀⠀⣴⣶⣦⡀⠀⠀⢀⢟⣻⣦⡈⠂⡁⢀⡄
⠀⠀⠀⠀⠀⠀⢀⣴⢟⠟⠀⠀⠀⠀⠀⠀⠀⠁⢼⣿⣿⡇⠀⠀⠀⠀⠘⠻⠿⠇⠀⠀⢀⣿⣿⣿⣿⣆⠸⡀⠻⡀
⠀⠀⠀⠀⠀⢠⡿⣭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢺⣿⣿⡆⠈⠀⠉
⠀⠀⠀⠀⢠⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣤⠽⠿⣦
⠀⠀⠀⡠⢁⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠝⠉⠉⠉⠙⠶⣄⣀⠀⠀⠠⠀⠺⠤⢀⠀⠀⠀⠀⠤⠏⠟⠻⠂⠤⣤⣀⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣓⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠋⠚⠛⠒⠒⠒⠒⠀⠘⠊⠐⠶⠀⠀⠀⠀⠰⠲⠶⠶⠦⠶⠶⠒⠀⠀⠀⠰⠒⠐⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣿⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠉⠉⠀⠀⠀⠀⠤⣀⠀⠀⠢⢄⠀⠀⠀⣀⢴⡄⠀⢀⠜⠓⢄⠀⢠⠴⡄⠀⠀⠀⠀⢽⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠑⠰
⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠈⠦⠴⠋⠁⢈⣀⢴⠁⠀⢰⠀⠆⠈⠉⠁⢀⡄⠀⠀⢸⠄⠐⡄⠐⠀⢄⠀⠀⣀⠸⠖⠛
⠀⠀⠀⠀⡠⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠩⠑⡀⠀⠀⠀⠀⢰⠉⢁⠀⠈⠑⠃⢠⠃⠀⠀⠀⢨⠀⠀⠀⠩⣧⠀⡼⠀⠀⡤⠓⣈⡀⡆
⠀⠀⠀⠀⠱⡗⢀⡁⠀⠠⢀⡀⠀⣀⡠⠔⠊⡁⣤⠤⣰⣮⠑⠂⠀⠀⠴⡿⢦⠠⠴⠤⠀⠤⠀⣱⠀⠀⠀⠈⠳⢀⢤⠀⠠⢋⠔⠁⠉⢰⡁⠔⣡⠀⢠
⠀⠀⠀⠀⠀⠏⡋⠙⣁⣤⡀⠀⠀⢀⣠⠀⢾⠜⢃⠴⣪⡝⠀⠀⠀⠀⠀⠐⠀⠉⠐⠀⠀⡀⠀⠀⠑⠀⠀⠀⠀⣤⠊⠉⢈⢄⠢⠀⠀⠀⠐⡋⢣⢣
⠀⠀⠀⠀⠀⠳⡸⠦⢄⠁⢌⢐⣲⠞⡃⢀⠡⢚⢃⠤⠊⠀⢀⠀⠀⣶⣀⡀⠀⠠⡀⠀⠀⠀⠐⡄⡰⠀⠀⠀⠀⠧⡀⠀⠈⠠⠴⠃⠀⠀⢀⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠤⣑⠒⠰⢀⠰⠂⡐⠄⠉⠁⠀⠀⠀⠀⠀⠀⠓⠀⠨⠄⠀⡓⠀⠀⠤⠄⣾⠀⠀⠀⠀⠠⡀⠈⢆⠀⠀⢀⣀⠀⡠⠋⠀⠀⢠⠃
⠀⠀⠀⠀⠠⡉⢉⠂⠀⠀⠉⠀⠀⢀⡉⠀⠀⡤⠶⠒⡚⠋⢹⡆⠀⠱⠤⣄⣀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⣀⡀⠈⠀⠈⠉⣨⢃⢉⠊⠀⢀⠔⢀⠆
⠀⠀⠀⠀⠀⠈⠀⡠⠔⠒⠁⠈⠐⢶⠬⠀⠀⢷⢨⢣⢍⡰⣙⡇⠀⢠⠀⠀⠈⠈⠈⠉⠙⢡⠋⠀⠀⣠⠖⠛⠁⢀⡀⠐⠈⣵⡿⠃⠀⠀⢁⡠⢃⠎
⠀⠀⠀⠀⠠⠢⡌⠀⠀⢀⠀⡄⠀⠀⠃⠀⠀⠘⠓⠋⠉⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠞⠁⠀⢀⣜⣁⠀⠀⡴⢁⠁⠀⢀⡴⠋⠀⡊⠀⡀
⠀⠀⠀⠀⠀⠉⢃⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⢀⠀⠠⡊⠢⠊⡤⠀⢂⠀⠀⠀⠀⠀⠀⡜⠀⠰⠋⠀⠀⠀⠘⠁⠀⠘⠉⠀⢡⠀⡰⠏⡀⢰⠀⡀⠀⡈⠳
⠀⠀⠀⠀⡠⢠⠈⠢⠀⠀⠀⠀⠀⠀⠂⠤⠐⢀⠇⠠⣎⡠⣈⡢⠀⠀⠙⠂⠠⠤⠀⠊⠀⠀⠁⠠⠀⠒⠒⢠⡀⠀⠀⠀⠀⢸⣴⠀⢠⡀⠀⠂⠉⠀⠀⡰
⠀⠀⠀⠀⡇⠈⠂⠒⠀⠀⠀⠀⠀⠀⠀⢤⡒⠋⠀⠀⠈⠀⠈⢀⡠⢄⡀⠀⠀⠀⡄⠴⠀⠀⠀⠇⠀⠐⠀⠀⣠⠀⠘⠠⠀⠀⢏⠀⠠⠉⠈⢢⠀⠤
⠀⠀⠀⠀⠈⠒⠀⢲⠀⠐⠋⠠⠀⠐⠢⠤⠤⠘⠀⢠⠤⡀⠀⠊⠀⠀⠰⠀⠀⠀⠃⡤⠋⠀⠐⠀⠀⠀⠒⠈⡀⠀⠀⠡⡑⢀⡂⠀⠀⠑⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⠇⢰⢀⠆⠀⢠⠤⡀⠀⠰⠀⢀⠀⠀⠀⠀⠀⡄⠀⡤⢄⠀⠀⠀⠓⠂⠀⠐⠒⠈⡁⠀⡀⠀⠰⠈⠃⠀⡐
⠀⠀⠀⠀⠀⠀⠃⠀⣀⠀⠀⠀⠁⣀⡀⠈⠛⠁⠀⠀⠀⠘⠄⠤⠀⠀⠀⡇⠀⠀⠘⠀⠀⠸⠀⠄⠀⠀⠒⢉⠁⠀⠁⠀⠴⠧⠒⠈⠀⠀⢀⣠⡆⠀⣀⠂
⠀⠀⠀⠀⠀⢠⣂⡡⠤⠐⠒⢆⡰⠀⠀⢠⢭⡁⡄⠠⣀⠀⣀⣀⡅⠀⠀⢁⠘⠀⠀⡀⠀⡧⠤⠄⠀⠤⠐⠇⠀⡔⠄⡦⠀⢰⠀⣠⢠⢤⣦⠁⠿⠹⠍⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡦⡠⠂⠉⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⣀⠔⠁⠀⠑⠰⠁⢠⡀⠝⠀⠀⠦⠢⠕⠀⠘⠋⠃⠈⠈
⠀⠀⠀⠀⢀⣸⣥⠀⠠⠤⠄⠠⢄⠀⠤⠀⠀⣠⣄⠀⠠⠄⠀⠈⡀⠀⠀⢸⠀⠀⢠⣒⡆⠀⠀⠀⠇⠀⠀⡠⡴⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡀⢲⣄
⠀⠀⠀⠀⣨⣀⣀⣀⣀⣠⢀⣀⣀⡄⠒⢠⣀⡤⣬⠀⠀⠀⠀⠀⡇⠀⠀⠠⠄⠀⣀⠉⠀⠀⠆⠘⠀⠀⣸⠎⠀⠸⣏⠤⠤⣆⠀⠀⠀⠵⠈⠀⡇⢩⡆
⠀⠀⠀⠀⠾⢀⣀⣀⣈⣈⠉⣀⣀⣁⣶⣁⣤⣉⡴⡄⠀⢸⠀⠀⠤⠤⠤⠐⠂⠂⢝⠀⠀⠀⠀⠀⢐⣐⢫⣤⢤⢼⣬⣀⣀⣈⣏⣉⣇⡄⠐⠒⢾⠀⠓
⠀⠀⠀⠀⡆⡠⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠈⠀⢀⡘⠀⠀⣀⣤⡀⢉⠩⠉⢀⣀⡀⠀⡀⢈⠍⡈⠉⠀⠀⠉⠀⠀⠀⠠⢲
⠀⠀⠀⠀⣄⢑⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢤⡇⠀⠀⠙⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠿⣀⣸⣶⠴⠤⠬⠤⠥⠴⢦⡵⡃⢈⠆⢠⠐⣠⠀⠀⠀⠆⠙
⠀⠀⠀⠀⡇⠉⠛⠖⠠⠄⣒⡒⠉⠉⠁⠀⠀⠎⠆⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣉⣀⣀⠀⠀⠀⠀⠀⠜⡈⠉⢀⠃⡠⢻⠀⣠⡖⠓⠐
⠀⠀⠀⠀⣇⠀⠀⠘⠀⠃⠑⠁⠀⢜⢀⡩⠀⣀⠠⠎⠀⠀⠀⠑⠀⠤⠀⠀⠀⠂⠀⠀⠀⠲⠀⢄⣀⠄⠨⠥⠤⠄⢈⠁⠉⢉⣉⡁⠄⠁⢸⠋⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠉⠢⢀⣀⡅⠤⠀⠐⠐⠀⠉⠀⠀⠀⠀⠐⠾⡑⠒⠶⠤⠤⠤⠆⠐⠀⠀⠀⠈⠤⠴⠄⠃⠒⠂⠀⠀⠈⠉⠉⠁⠀⢁⣀⣀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢶⡀⠀⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣀⣀⣸⣇⣀⣠⣾⣧⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠈⣻⡄⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠻⣦⡾⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠉⠉⣽⠉⠁⠈⢿⣀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠞⠋⣷⠀⠀⠀⠚⠛⣿⠉⠉⠀⠀⣠⠿⢧⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠾⠋⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠟⠙⠻⢦⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⠀⠀⠉⣷
⠀⠀⠀⢸⡇⠀⠀⠀⣠⠾⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠶⣤⣀⠀⠀⠀⢠⡟
⠀⠠⣤⣾⡦⠶⠶⢶⡿⠶⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠉⠛⠶⣦⡟
⠀⠀⠀⣿⠀⠀⠀⠈⢷⣤⣠⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⣀⣶⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢀⡀⣹⠟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢠⣞⣹⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⠃
⠀⠀⠀⣿⠀⠀⠀⢠⡾⠋⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⡴⢫⡾⠁
⠀⠘⠻⣿⠛⠛⠛⢿⡛⠛⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⣠⠄⣸⠏
⠀⠀⠀⣿⠀⠀⠀⠈⠳⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠾⣁⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢀⡴⢛⣶⣶⠏⣸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠶⣤⣄⡀⡴⢎⠙⣧⡞⠁⢘⣿
⠀⠀⠀⣿⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⠳⣦⡿⠋⠀⠘⢯⡿
⠀⣀⣀⣿⣀⣀⣀⣼⣋⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⡰⣾⡇
⠀⠉⠉⢻⡉⠉⠉⢿⡉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡶⠛⠁⠀⠀⠀⠀⠀⢀⣤⣾⠁
⠀⠀⠀⢸⡇⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⢴⣋⢼⡟
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠂⢀⡜⢡⡔⠁⢠⣴⠛⢤⡔⣩⡾⠁
⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣀⣴⢛⣴⣋⣳⣜⣣⣜⣃⣈⣶⣋⣨⣷⡼⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡠⣤⣴⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠛⠋⠉⠁⠐⠒⠀⠀⠈⠏⠀⠀⠀⠀⠀⠘⠟⠋⢉⣀⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠐⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣀⢀⣠⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⢷⣄⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠳⠄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠐⠀⢄⡀⠀⠀⠀⠀⢀⠛⡿⡿⠟⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠶⠋⠀⣠⡾⢳⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢋⣠⣤⣤⣶⣤⣽⣿⣶⣄⡀⠀⣿⣿⡏⢠⣶⢹⣿⣿⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⣠⠞⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⡀⠀⢘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣼⣿⣿⡿
⠀⠀⠀⢀⡞⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠢⠒⠒⠀⠀⠀⠀⠀⠀⣴⣄⣀⣼⣿⣿⡿⠁
⠀⠀⢠⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠋
⠀⠀⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⢠⣿⣿⡿⠿⢿⣯⣄⡀⣀⣀
⠀⣸⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⡿⠃⠀⠀⠘⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣿⠠⣎⣼⡿⠋⠀⠀⠀⢿⣿⣿⣿⣿⣷⣦
⢠⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣦⡘⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣱⠟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧
⡼⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡜⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠰⠗⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢠⣧⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⢸⠿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣇
⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⢰⣿⣿⣿⣿⣿⣿⣿⡿⠐⠏⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣦⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡱⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡿⣿⣥⣀⠀⡠⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡟⣿⠏⠀⠉⠘⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⣿⣿⣿⠟⠉⣿⠀⠀⠀⠀⠳⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⣛⣥⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿
⣿⠟⠁⠀⠀⢻⡄⠀⠀⠀⣴⡿⠡⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣄⢀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠻⠄⢀⠴⠀⠀⠀⠠⠴⠀⠀⠀⠀⠉⠈⠁⣿⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⡿⢿⣿⣿⡿⢿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢰⡇⠈⣿⣿⠁⢸⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣷⡘⡇⣰⣿⣿⣆⢸⢃⣾⡇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣿⣿⣼⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣧⣿⣿⣿⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⠉⣉⢙⡛⢩⣭⣭⣉⠉⢉⣭⣭⣍⡉⠉⢩⣭⣭⣿⠋⣉⠉⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⢸⡇⢸⠀⠀⠙⣧⢸⡇⠀⠈⢻⡄⢸⠀⠀⠀⠀⣿⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⣿⢸⡇⢸⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇⢸⠛⠛⠃⠀⣿⠈⠳⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠟⠸⠇⠸⠦⠴⠞⠃⠘⠷⠤⠶⠋⠀⠻⠦⠤⠤⠀⠟⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⠲⠆⠀⠀⣰⣆⠀⠒⠳⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢰⠏⠹⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⢀⡀⢠⡿⠶⠶⢿⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠠⠄⠀⠀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠈⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⢁⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣴⣿⣿⣿⡿⣿⣿⣿⢏⠩⠟⢿⣿⣿⣿⣆⠱⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⣸⣿⣿⣿⣿⣷⣼⠟⠻⣦⣿⡞⣾⣿⣿⣿⣿⡇⢿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⣿⣿⣿⣿⣿⣝⢿⡷⢀⣸⣿⡟⣿⣿⣿⣿⣿⡏⠁⣅⡀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⢀⢹⣿⣿⣿⣿⣿⣿⣥⣼⣯⣵⣾⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⢶⣦⣎⢿⣿⣿⣿⡿⠿⠿⠻⣻⠿⠿⣿⣿⣿⣿⡟⣰⢠⡍⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢆⡿⡛⠻⠮⠛⠩⣄⠒⠂⠀⣠⣌⡁⢀⣠⡬⠻⢋⡴⣫⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢿⣶⣦⣐⠀⢀⠀⡀⢀⡈⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⣀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣷⣀⠙⢿⣯⠙⠻⠦⣥⣄⣀⣀⠀⠀⢀⣀⣬⣤⠼⠶⣛⣿⡿⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡁⠀⠀⣼⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣏⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢽⣿⣿⣿⣿⣿⡁⣤⣙⠻⠟⣫⠀⢺⣿⣿⣿⣿⣿⡅⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠀⠀⣿⠏⠹⠙⠁⠀⠀⠀⠈⠀⠀⠀⠀⠟⢻⠟⠿⣧⠁⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣴⣀⣦⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠉⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⢤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⠄⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢒⡶⠀⢚⣁⡀⠖⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠐⠻⡍⠀⠀⠾⠱⠀⢿⡗⡈⠀⢬⣽⡟⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⢀⣤⠀⢡⡤⠐⠥⠔⠀⠀⠉⠄⠠⠀⠠⠀⠀⠉⣶⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠍⣀⠚⢈⡬⣭⠇⣸⠐⠋⡑⠃⠀⠐⢲⡄⠀⠈⠳⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⠟⠀⣠⠤⠔⢲⠁⠢⡂⡏⠒⠀⠄⣀⠐⠮⠟⢠⡄⠱⠂⠊⠀⠚⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠤⣠⠄⢀⡀⢿⣉⠆⠀⣮⡵⡼⣷⡀⣀⢰⠘⡣⠊⠀⢾⠷⠄⠀⠞⣷⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠼⠙⣦⠀⢠⡀⣄⡚⠁⢹⡄⠛⣠⠋⠀⠙⠀⠀⠀⠘⠀⣀⠿⣀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢼⣗⢀⠀⠀⠒⢰⣲⡄⢀⣌⡩⠉⢳⣰⠇⠀⠐⠂⠀⣶⠲⡴⠏⢀⡀⠓⠱⠋⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠇⠀⠐⡄⣾⣭⠇⠀⡽⠷⢀⡌⢿⡈⣀⠀⠘⣇⠛⠚⢀⠀⠃⠃⡀⠀⢰⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠈⠓⠒⠚⡻⠟⠛⡛⠷⢮⣷⡄⢰⠀⢢⠈⠉⢻⡆⢠⣾⡇⠀⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠤⠤⠀⢊⡿⢳⠄⠁⠀⠀⠙⣿⣎⠀⣀⣄⠀⢨⡇⠀⠘⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⣤⠤⠈⠓⠋⠐⠏⡷⢀⣤⠘⣿⡔⣷⢲⠄⣼⣤⡤⠞⢤⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣙⣛⡆⡤⠀⢠⣚⣓⣂⣀⠠⢹⣷⢈⣥⠎⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠉⠉⠙⠈⠉⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠈⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠼⠿⠿⠿⠤⠤⠤⠀⠜⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠳⡀⠀⠀⢀⡴⠾⠂⠀⠀⠀⠙⠳⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⠀⠀⡤⠂⠀⢀⡇⠀⠀⠀⠰⠛⠳⠀⠀⠀⠚⠙⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠼⡁⠘⠓⢸⣷⡀⠀⠀⠀⠀⠀⢀⣤⣘⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣆⢸⣿⣷⡄⠀⠀⣠⣶⣿⣿⣿⣿⣦⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿⣿⣿⣶⣾⡟⠛⢻⣋⣉⣉⣙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⠋⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠙⠃⣼⠀⣴⠀⠀⠀⠀⠀⠀⠈⠉⢉⣿⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣤⠀⢠⡏⠀⣿⣷⣦⣄⣀⣀⣀⣠⣴⣿⣿⠀⠻⡁⢀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣤⠀⠀⠀⢀⡞⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢣⠈⠀⠀⢸⡄⠀⠀⢀⡠⠤⢤⡂⣉⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⠀⢾⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⡆⠀⠀⢀⢷⠀⠀⡏⠠⢤⣬⣏⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠰⠀⠀⡀⠘⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠇⠀⠀⠉⠘⡆⠀⡇⠀⢲⣊⠡⠐⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠂⠀⠀⠀⠀⠁⢀⠸⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣺⠐⠂⠀⠀⠀⢻⠀⡟⠀⠈⠈⢀⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡀⠀⠀⠀⠀⠀⠉⣼⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣆⠀⢀⠀⠈⠈⣧⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠁⠀⢠⣤⠀⠶⠀⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠀⢸⠀⠶⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⣿⠀⠀⠀⢹⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡟⠄⢸⠀⢀⡎⠘⢦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⡟⠀⠀⠀⠈⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⠇⠀⢸⠀⡜⣀⠀⠀⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⣇⠀⠘⠂⠀⢹⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡀⠀⢸⠸⠁⠉⠀⠀⠄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠀⠀⠁⢠⡟⠁⠀⠀⠀⠘⢷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡏⠁⠀⣾⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⢸⡇⢀⠀⠀⢀⡀⠘⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⠀⠀⠀⢸⠀⠈⠁⠀⠐⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠴⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⡏⠘⠁⠀⣼⠷⢶⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡁⠀⠀⣀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠀⠀⠘⠃⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⠀⠶⢥⠤⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡀⡀⠀⠀⠀⢀⠀⠡⡘⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠄⠄⠀⠀⠀⡶⠀⠐⢂⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠐⠂⠀⠀⠀⠀⠆⠀⠀⠁⠈⢀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠭⠄⠀⠬⠁⠀⢀⣀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠚⠐⠀⠀⠈⠁⠀⠉⠈⠉⠈⠁⠀⠈⠀⣻⣿⡦⡉⠀⣄⠇⠀⠀⠀⠈⢉⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣩⡭⢷⣤⡤⠠⠸
⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠺⡁⠀⠉⠑⣦⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⠇⠀⠀⠀⠉⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡖⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣠⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⣴⡾⠤⣄⡀
⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡰⢏⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠙⠛


⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⠀⠤⣀⠀⡄⢁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠈⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠉⠁⠀⠀⠀⣀⠀⠙⢐
⠀⠀⠀⠀⠀⠀⣤⠂⠌⠀⠙⠀⠀⢀⠘⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠛⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡈⠓⠤⢀⣀⠤⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠀⠉⣀⠀⠠⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢿⠀⢀⢤⡏⣀⠼⡄
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠉⠓⠲⠋⠁⠰⣦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡃⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠈⡀⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣍⣉⣉⣉⣁⠀⠀⠀⠀⡤⠀⠐⠄⠀⡠⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡀⡀⠓⠠⠄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⠤⠤⠴⠤⠶⠀⠀⠀⡜⠀⠀⠀⠘⠀⠀⠤⠄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠉⠀⠐⣀⣴⠒⠒⣈⠙⠀⠀⠀⠘⠛⠁⠀⠀⠈⠀⠀⠈⠀⠀⠉⠛⠓⠂⠘⠃⠀⢀⡐⢶⠂⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣌⢀⣨⠥⠛⠛⠛⣧⡄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡴⠖⠻⠽⢷⣒⠐⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠆⡀⣀⣀⠐⡃⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠇⡄⡀⢀⣈⣉⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠈⠀⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⠀⣦⠀⣻⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡐⣂⡌⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠉⠉⠉⠙⣇⣻⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡇⠋⡋⠉⠉⣛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣙⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⡀⠈⠲⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠⢈⠄⡡⠎⡸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠀⠀⠌⡠⢀⡔⢡⠎
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠤⠤⠒⠒⠈⠉⠁⠉⠀⢀⠄⠀⠀⠀⡀⠀⠀⣠⢋⡰⠁
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠤⠄⠐⠂⠁⠉⠀⠀⠀⣀⠠⠠⠀⡠⣡⠎⣠⠎⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠄⠐⠀⠉⠉⠀⠀⠀⠀⢀⢮⡼⣇⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡤⠤⠤⠐⠒⠀⢀⠀⠀⢠⠏⣠⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠀⠀⠀⠣⣎⡴⣣⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠄⠂⠀⢂⡀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⠀⠀⠒⠐⠛⠁⠀⠀⠀⣀⣀⣄⣄⠄⠀⠠⢒⣋⣋⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⣀⢀⡀⠤⠄⢒⣊⣉⣥⠤⠤⠶⠒⠶⠾⠛⢛⣚⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⣮⡶⠶⢻⠩⢍⠍⣁⡬⡥⠴⢶⠒⠾⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣏⣤⡴⠖⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢀⠀⢉⣀⣀⣀⠀⠠⠀⠀⠄⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠰⠊⠉⠑⣄⣀⣀⢠⠒⠉⠐⠤⠤⡀
⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⢀⡀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⡁⢠⠜⠊⠁⠀⠀⠀⠀⠈⠉⢎⠢⠀⠀⠈⠄
⠀⠀⠀⠀⢠⠋⡠⠤⠤⠎⠁⠸⡭⠥⠤⠬⠭⡵⡄⠀⠀⠀⠀⡄⠛⠁⠀⠀⠀⠀⠀⠀⠊⠉⢀⠀⠀⠳⡀⠀⠑
⠀⠀⠀⠀⡆⠀⢣⠀⡰⠊⠓⠒⠀⠀⠀⠀⠀⠀⠨⣄⡀⠀⠐⢅⠀⠀⠀⠀⠀⠀⠀⠘⠤⢐⡤⠉⠁⡀⢡⠀⡠
⠀⠀⠀⠀⠃⠀⠀⠉⠁⠈⠢⢄⣄⡀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠈⠒⠦⠔⠤⠤⠐⢒⠐⠊⢁⠂⠤⠐⠁⢈⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠇⠀⠠⠀⠈⠁⠒⠃⢒⠒⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⢸⠀⠀⠀⠀⡌
⠀⠀⠀⡈⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀⠠⠀⠀⠀⠀⠑⠂⠝⠊
⠀⠀⠘⢤⠀⠀⠀⠐⢥⡀⠀⠀⠂⠤⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠍⠀⣀⠀⢤⡄⠈
⠀⠀⠀⠀⠁⢱⡜⠀⢀⠑⠚⢶⠤⠀⠝⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠖⠈⠒⠊
⠀⠀⠀⠀⠀⠸⠁⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡆⠘⡀⠀⢄⠀⢸⠀⠑⠠⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡄⠀⠀⠀⠆⠐⠄⠀⠈⠘⠀⠲⡀⢀⠈⢲⠠⣀⠀⠐⢀⠀⠀⠀⠁⠀⠘⡠⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠠⢤⠞⠁⡉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⣀⠀⠀⡂
⠀⠀⠸⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠂⠁⢀⡄
⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣦⠉⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡙⠄⠀⠀⠀⠀⠀⢀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣗⢦⡀⠀⠀⠙⣷⣤⣀⣀⡈
⠀⠀⠀⠀⠀⠀⡴⠋⠹⣆⠈⠣⡈⠀⠀⢀⡏⠀⠈⠁
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣆⠀⠈⠢⡀⠸⣿⠂
⠀⠀⠀⢠⣾⡥⠂⠀⠀⠀⠀⡀⠠⠶⠈⠲⢻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠻⣀⢀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠟⠁
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠢⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠃
⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢹⣯⠓
⠀⠀⠸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠍
⠀⠀⠀⢇⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠃
⠀⠀⠀⢸⡆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢀⣀⡤⠂⠀⠀⠀⠐⡄⢀⠐⣩⠉
⠀⠀⠀⠀⢿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣠⡼⣿⠉⠉⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠘⢿⡄⠀⠀⠀⠀⠚⢩⡇⣻⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⡄⠀⠀⠀⢀⡾⠀⢻⡇⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠳⣝⠀⠀⠀⡸⠁⠀⢸⢃⣠⡶⢿⠉
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⡧⣤⣀⡁⠀⣀⣼⠟⠋⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠛⠾⠟⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠛⠛⠷⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠁⠀⠛⠷⣄⢠⣀⠀⠀⠀⣠⣤⣤⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⠻⠾⠛⢋⣀⠀⢹⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠛⠉⠀⢘⣹⣃⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⣥⠄⠀⠀⠀⠀⡀⣠⣾⣻⣿⣿⣛⣉⣵⢿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠞⡻⢶⠟⠀⠀⠠⣤⣄⠀⢸⣤⡃⠿⢹⠋⠙⠛⣿⢠⡟⣿⢩⣿⡏⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠙⣦⣙⡁⠀⠀⠀⠀⠉⠁⠀⠀⠹⠁⠀⣿⠀⠀⠀⣿⠸⣇⣿⢸⣿⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠉⠛⣄⠀⠀⠀⠀⠀⠘⠶⠆⠀⠀⣯⡴⣷⣤⣿⠀⣙⣿⠟⢻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⡀⠀⠀⠀⠀⠀⠀⢠⠟⠁⢻⣿⣿⣴⡟⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣷⡶⠶⠶⢿⡿⢷⣦⣴⡟⠉⣿⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⢷⣄⣰⠟⠁⠀⢈⣿⠃⠀⠉⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⢻⡟⠀⠀⠀⠀⠀⠀⠘⡇⠹⣧⡀⠀⠀⠀⠙⠋⠀⠀⣠⡾⢹⠷⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠘⣿⣶⣤⣀⡀⠀⢀⣴⠟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣿⠀⠉⠛⠻⠿⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣴⠏⠀⣠⡶⠛⠳⢦⣤⣤⣤⡤⢶⡟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢀⡾⣿⠀⣴⠏⠀⠀⠀⡞⠀⠀⣾⠀⠀⠇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠃⢻⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⢻⡟⠛⠻⣟⠛⠛⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢈⡿⠀⠀⢻⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⠀⢀⣾⠃⠀⠀⣼⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⢛⣿⠃⠀⠀⠀⣿⢶⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠀⠈⢷⣄⠀⠀⠀⢸⡇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡜⠳⣦⡾⠿⣇⠀⠀⣸⣦⣄⣀⠙⣷⡶⣶⣄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣴⡾⠃⠀⠀⢿⣦⣭⣽⣿⣯⣤⣤⡿⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣿⡀⠀⡴⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⣴⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡖⠃⡮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣸⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢒⣧⠀⠀⠠⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⢿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠧⠦⠔⠂⠉⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢸⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⢀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⣀⣠⣤⣴⣶⣿⣿⣿⡿⢃⣷⢠⡠⠀⣿⠀⠃⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣰⠀⠀⡀⢰⡇⠀⠘⠂⢠⡀⠀⣐⣒⣔⠒⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⡛⠀⣸⣿⡐⠠⢰⣿⡄⣈⡩⠉⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⣿⡤⠎⠉⢻⠥⡤⢀⠀⢞⠑⠚⢉⢡⠘⠛⣓⠿⠛⠫⠁⠡⠈⡀⠪⠩⣖⢄⢀⠈⠉⡃⠶⠸⠿⡧⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠰⠋⢉⠀⣠⢶⡁⢀⠈⠁⠀⠈⠐⠀⠾⠖⠒⠉⠀⢀⡠⠄⠳⠀⢰⣇⠀⢄⠀⠉⠑⠙⢸⡧⣄⠀⠐⠈⡶⠚⠁
⠀⠀⠀⠀⠀⢀⠀⠘⠫⣀⠔⠂⠀⠀⠠⠀⡀⣿⢠⡆⠈⠒⠂⠀⠀⠀⢠⠀⠂⠁⢀⠀⢠⡇⣸⣿⠀⠀⠁⠀⠀⠒⠠⠒⠉⡆⠀⠀⡇
⠀⠀⢄⠀⡠⠀⠀⢀⠀⠀⠀⠀⠁⠈⣀⣀⣀⡻⠊⠁⢀⣄⠄⠀⠀⠀⣼⡆⠤⣎⣀⡠⠼⠷⠿⠛⠓⠉⢋⠄⠂⠁⠀⠀⠁⡇⠀⠀⠃
⠀⠀⠋⠠⠦⠁⡂⠠⠀⠈⠁⠂⠀⠻⠿⠿⠽⠿⠂⠀⠀⠀⠀⠈⠙⠀⠿⠷⠈⣀⡀⠪⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠶⠄⠘⠀⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠃⠀⠠⠦⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠭⢜⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠖⢻⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠀⡀⠀⠀⠀⢘⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡶⠶⣺⣿⡿⡿⠻⣻⡿⢛⣻⣿⠿⢟⡿⣿⠿⢟⣿⢿⣿⢿⣿⡿⣿⠿⣻⢿⡿⣿⡿⡿⠟⡿⢛⡿⣿⣿⠁⠀⠠⠀⠀⠀⠀⢠⣿⢸
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⡽⠋⠨⠚⠁⠈⣠⢞⢕⡴⡫⠟⢁⠔⡩⠚⣡⠖⠋⠁⣠⠞⠉⠩⠚⠁⢀⠕⠋⠀⠁⠈⠀⣠⢞⠉⠜⣻⡇⡀⠀⠀⠀⠀⠀⠀⢾⣿⢸
⠀⠀⠀⠀⠀⢀⡾⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠊⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣨⠖⠁⢠⡾⣻⠁⠀⠀⠀⠀⠀⠀⠐⣸⡿⢸
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣼
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢡⣿⡀⠀⠀⠀⢀⣠⣾⠟⣹⡇⣿
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⣻⡷⢀⡴⠾⠿⢋⣩⣴⣿⡇⣿
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⡿⠻⠟⠟⠟⠉⠉⢿⡇⣿
⣿⣷⢄⣤⣀⣤⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡿⣻⡟⣹⣿⣿⣿⣿⣵⣿⣫⢞⣿⣿⣿⣿⣿⣿⠞⢡⣾⢏⢞⡾⣿⢯⠟⡠⡃⢁⡼⠃⠀⠂⠀⡰⣠⢦⠆⡴⢀⠄⡠⠂⡀⡠⠂⡴⣰⡶⢣⣾⡿⠁⣰⢿⡇⣿
⣿⣿⢋⣞⡽⢃⣿⣿⡿⡟⡵⣫⣻⡿⡿⣿⡿⠋⠁⡴⡻⡣⣣⢟⡼⡳⠃⡴⠁⣠⠞⠀⠀⠄⢀⣜⡵⠃⢀⠞⢡⠊⠐⢀⠎⠄⢁⡞⢝⠊⠀⣱⠋⠀⣼⣯⣿⡇⣿
⣿⣣⢯⡞⣠⢏⣼⣟⢌⣞⣝⡽⣓⢟⣾⠝⠁⢠⣞⣟⣕⡽⣣⢟⠜⢡⠎⠀⡴⠋⠠⠀⡀⣠⢮⠞⠀⣠⠋⡰⠁⠀⡴⢣⠂⣠⠋⡠⢁⠌⡼⢱⠀⣼⣶⡿⠍⡇⣿
⣿⣧⠋⢜⣵⡿⣿⣿⣫⣪⠎⣠⠎⡪⠋⠀⣰⡿⣫⠞⢪⡞⡱⡃⡴⠃⢀⠞⢁⠔⢁⢜⡴⣳⠋⢀⣔⠁⠈⠀⢀⠞⡱⢁⡜⢁⡼⣡⢣⣞⠾⢁⢮⡿⢃⢎⠜⡇⣿
⠿⠷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣜⣙⣃⣚⣛⣑⣃⣐⣋⣚⣚⣚⣀⣀⣁⣀⣋⣀⣉⣚⣙⣁⣀⣈⣡⣤⣾⣥⣭⣈⣰⣋⣀⣋⣀⣁⣁⣀⣁⣃⣘⣡⣡⣬⣬⡇⣿
⠀⠀⣿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⣶⣻⠀⢴⣿⣿⣿⣿⡿⡿⠿⢟⡿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⡿⠿⢿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠿⡿⣿⣿⡿⠿⠿⠿⣧⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠎⠐⣤⢤⠄⡐⠀⠻⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣇⠀⡔⠀⠀⠀⠈⣏⠁⠂⢌
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠠⠀⡴⣤⠀⠀⢸⡄⢸⣆⠀⠀⠀⣿⠀⠐
⠀⠀⠀⠀⠠⠆⠑⢤⡀⠀⠹⠙⢟⠚⣟⣶⣾⣠⠏⠘⠖⠂⠀⣿⣤⠷⠦⢄⣤⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⣶⡟⠾⠿⣿⡯⣼⣠⠞⠉⣴⣿⣧⣄⣀⡬⠄⠀⡠⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠴⢿⣿⣦⣤⣿⣷⠟⣷⡄⣾⣿⡟⠀⠁⢀⠠⢀⣤⣧⡇⣴⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠡⠂⠀⠉⠁⠙⣿⡿⣷⣾⣿⣿⣿⣿⣶⣶⣾⢾⡿⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⢬⣄⣀⣿⣿⣽⣿⣿⡿⣿⣿⠋⠉⢺⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠞⠙⣿⣿⣿⣿⣿⣀⣿⣿⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢹⣟⡏⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠏⠜⠒⠧⠸⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⡇⣿⢸⡟⢻⣆⢾⠚⠛⢠⡾⠙⣦⢰⡇⠀⣿⠘⢻⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿⡇⣿⢸⠇⢸⣟⣾⠶⠆⢸⡇⠀⣿⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⠇⣿⠸⠦⠼⠏⢿⢤⣤⠘⠿⢴⠟⠸⢧⢤⠍⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⣶⢶⠶⣦⣤⣤⣀⣀
⠀⠀⠐⠺⠭⠉⠉⠛⠒⠛⠋⠉⠉⠛⢿⣷⣿⣾⣟⢻⣿⡿⢷⣧⡿⣯⢾⠞⣫⢖⡶⠚⠛⠂⠈⣄⣀⣈⠛⠺⢿⣾⣛⣿⣿⣿⣿⡿⣦⡠⣴⣶⡦⠤⠤
⠀⣀⣀⣠⣆⡉⣉⣠⠀⢤⣤⣀⣀⣀⡈⢿⠬⢶⣺⢷⠛⡀⢜⣛⠁⡁⣙⢄⣚⣋⡸⠥⣶⣶⡭⠒⡦⣄⠭⣤⡙⢟⣿⢛⣛⣛⣷⡿⣿⡟⢄⠈⠉⠉⠫
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣤⣤⡥⠁⠨⠐⡢⠋⠡⠖⡋⠙⠊⠀⠀⠄⢛⣁⠀⠀⡫⠃⠀⠴⣣⡈⠀⠈⠃⠈⠈⠋⢉⡉⠉⠘⣾⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠉⠔⠂⠻⣁⠠⢒⠅⠠⠄⠀⠀⢀⢔⠋⠀⠀⠠⢂⠀⠀⠸⣣⠄⠀⠀⠀⠀⣀⠰⠏⠑⠔⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣀⠵⢂⠞⡡⠴⠴⠆⠠⠋⠀⢀⠴⠋⠰⠀⠀⠀⠁⡠⠤⢴⠇⠀⡠⢶⠀⠘⠁⠀⠀⠈⣠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⡠⠋⢀⠁⠀⠀⢀⡔⠀⢀⣖⠁⠀⠀⢀⣤⠤⡪⡕⢠⠎⠀⢸⡕⢁⡶⠀⠀⢠⢮⠎⣂⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣵⣯⣽⣶⣿⢽⣯⡴⠑⣡⠜⠄⡤⠀⣩⠀⢀⢕⢵⠆⠀⠀⠈⠀⠀⡴⠃⠈⣠⠀⠀⢠⠎⠀⠀⠰⠋⠉⠀⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠻⡍⣹⣏⣿⣽⡂⢨⠋⠂⡠⠚⠀⠔⡁⠐⡕⡡⠂⠀⠀⡴⠒⠟⣡⠄⠘⠁⠀⢀⡴⠁⡴⠀⡀⠀⣠⢞⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢲⣤⢲⠟⡺⠿⣽⡅⠛⠀⠀⢒⠥⠀⠞⠁⠀⠸⠁⡜⠠⡪⢖⡀⠀⢀⡤⠀⠀⠘⠛⠀⠘⢡⠞⠀⠀⡡⠋⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣲⣦⠿⣟⠍⡈⠐⢼⡄⠀⣣⠎⠀⣂⠖⠀⠀⠊⠀⠀⠁⠘⠀⠀⠵⠊⠀⠀⠀⠀⠀⢰⡫⠂⠀⠀⠞⡡⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁⢀⣧⠀⠀⣇⠀⠁⠀⠈⠁⢀⣀⠀⠀⡔⠀⢀⡶⡄⠀⣪⠄⠀⠀⡠⠐⠁⠈⠀⠀⠀⠀⠈⣀⠄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣿⣶⠁⠀⠀⠀⠀⡠⠂⡀⢀⠀⢰⠠⠋⠀⢠⠪⡀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣚⢅⡴⠁⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣿⣟⣬⣿⠀⠀⠀⠀⠀⠀⠀⠐⠡⠄⠁⠠⢞⠁⠀⣪⠅⠀⠉⠀⠀⠀⠀⠀⢀⡠⠞⢀⠕⢫⠂⢬⠟⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠐⠃⣀⣀⠀⠀⠀⢀⠔⠂⠉⠀⠾⠃⢀⡜⣴⣿⣾⡵⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⠀⠀⣀⡤⠤⠤⠤⢀⣀⠀⠀⠀⠁⠀⠠⠚⠁⡼⠋⡰⠂⠁⠀⠀⠀⠀⠀⠀⡞⢜⣵⣿⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣋⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⢀⠀⢤⠀⡢⠄⠐⠈⠉⠉⠉⠉⠙⠚⠤⣌⣋⢈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣹⣠⣏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿

⠐⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠠⠄⠀⠀⠇⠀⠉⠸⠀⣿⣿⠂⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠢⠉⠀⡆⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢿⣻⢛⡿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢤⣄⠀⡏
⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⠀⠀⠀⢻⢿⠀⠀⠀⠀⠀⠀⠀⢴⢰⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⢹⣿⣆⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣟⣋⡿⠥⠼⡄
⢸⡀⠀⠀⠀⠀⠠⠄⠀⣀⡀⠀⢀⣀⣀⣄⣀⠀⣠⣤⡤⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠻⢿⣿⣴⠆⠠⢺⠔
⠈⠀⠀⠀⠉⠉⠐⠈⠉⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡀⢡⣦⠀⠀⠘⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠸⡆⠀⠀⠀⠀⠀⠐⡃⢸⡅⠀⠀⠀⠀⠀⠐⠔⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⠘⠀⠀⠸⠧⠾⢤⠤⠘⠏⣸⣅⡈⡌⢀⡀⠀⠀⢀⣆⡴⣈⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠶⠀⠲⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⢶⣒⣶⣮⣖⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣽⣷⡿⠿⣷⣶⣶⣦⠄⠀⢰⠀⠶⠢⠈⠉⠛⠻⠿⠯⠥
⠘⠀⠀⠀⠀⠤⠀⠀⢐⠠⢤⠀⠀⠀⠀⠀⠀⢉⣤⣦⣶⠴⠚⢚⣯⣥⣶⣶⡾⠛⠍
⠸⢀⣀⣀⣀⣱⣾⣿⣿⣿⣟⠀⣠⢀⣠⣤⢠⣘⣁⣉⣛⡛⠛⠿⠉⠉⠉
⢸⣿⣿⣿⠟⡻⢿⣿⣿⣿⡛⣷⠺⠶⢶⣶⣾⠿⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⣀⢠⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠉⠉⠉⠀⠈⠀⠀⠪⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢠⠦⣀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢀⣀⠠⡤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠐⡡⠤⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢠⢊⡙⡆⠰⡉⠀⡌⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡀⠀⠀⢡⠁⠀⠉⠉⠁⠀⠈⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠝⠒⠂⠴⣁⠈⠙⡲⠞⠀⠀⠀⠀⢀⣀⡤⠚⠊⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⠀⠀⠀⠀⠀⠑⢼⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⠾⡞⠠⠄
⠀⢀⣀⠄⠀⢰⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠠⢄⣀⠀⠀⠀⢠⠄⠎⣄⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠠⠄⡀
⠀⡍⠁⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠁⠀⠁⠙⡎⠉⡠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣒⣤
⠀⠡⠀⠀⢸⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣐⡂⠀⠀⢀⡀⠀⠀⠀⠀⣧⢀⠲⠁⠄⠊⠁⠀⠀⣀⡠⠤⠔⠒⠈⠉⠁⠀⠀⡔
⠀⠀⠀⠀⢾⠀⠈⠙⠓⠶⣤⣀⡀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠙⠋⠩⠤⢄⣈⠤⠐⠂⠀⠈⠁⠀⠀⠀⠀⢀⣀⣀⣤⠤⠞
⠀⠀⠀⡖⣘⣆⣀⣀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠁⢢⡄⣀⣀⣀⣀⠀⠀⢀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠂⠉⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠉⠊⢍⠑⠦⣄⣀⣀⡀⠀⢀⣀⣤⡄⢠⠀⠀⠀⠀⠀⠉⡟⢺⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡁⠒⢠⠄⣀⠈⠃⠈⠉⠉⠉⠉⠉⠀⡏⠁⡌⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡇⠀⢸⠀⠀⢹⠉⢒⠐⡖⠂⣪⠈⣿⠁⢀⣋⣁⡒⠒⠲⠤⢴⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡇⠀⢸⠀⠀⣾⠁⠀⠀⡇⠀⢻⠀⡷⠒⠋⠁⠀⢱⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⡇⠀⢸⠀⠀⣟⠀⠀⠀⣇⡀⢸⠀⣧⣤⣤⠤⠶⠛⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠧⠤⠞⠀⢠⡿⠀⠀⠀⠈⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠘⢯⣠⠇⠀⠀⠀⠙⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢀⡠⠤⠐⠊⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣢⣔⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣍⢻⣯⡛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⠻⣿⣮⡂⠹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠛⠉⠙⠓⠆⠀⠀⠙⢿⣷⠀⠘
⠀⠀⢀⣀⡀⢀⠀⠀⢀⡀⠀⣀⢠⠀⡀⣀⠀⠀⠀⢸⣿⡇⢀⣠⣀⡀⠀⠈⠻⠋⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠸⣿⣇⠀⢰
⠀⠀⢸⠴⣇⢸⠀⠀⡎⣷⢰⡁⠀⠀⡷⡅⠀⠀⠀⢸⣿⣿⡟⠛⠛⠛⠀⠀⠀⠀⠀⣎⣹⣯⠴⠋⠀⠀⠀⠀⠀⠻⢿⣠⠈
⣀⣀⣘⣒⣁⣘⣒⣘⣀⣈⣃⣙⣲⣀⣡⣜⣀⣀⣀⣼⣿⣿⣧⣤⠶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣿⣿⣑⡽⡟⣯⡹⣹⢙⣟⣿⣿⡆⣿⢸⣿⣿⣿⣿⣿⣿⣷⢻⡟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡇
⠿⠷⠽⠾⠿⠾⠧⠿⠾⠿⠿⠿⠷⠿⠬⠽⠿⠿⠿⣿⣿⣿⠘⣷⠀⠀⠰⣤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠆⢂⠀⠠⠀⠆⡄⢠⢰⢠⠠⢀⡄⠄⠀⠀⣿⣿⣿⡇⢻⣆⠀⠀⢉⣤⣴⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠉⣀⣁⣁⣀⠉⠈⠈⠈⠀⠀⠀⠀⠀⣿⣿⣿⣧⢸⣿⣆⠀⢼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⢻⣿⡄⠠
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠘⣿⣿⣆⠀⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⢀⠎⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣦⣤⣤⣴⣶⣿⣟⣁⣀⣀⣀⣀⣀⣀⣈⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡞⢛⠏⠉⠉⠉⣈⠉⠉⠋⠭⡷⡲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡟⢀⣀⡤⠄⠀⠈⠳⠤⠄⠀⠀⠈⢌⣒⣛⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣹⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣾⣿⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⢼⣾⣧⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣰⣷⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠁⢀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⢸⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⡇⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣳⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣫⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡃⣾⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡴⢚⣿⡿⠿⠛⢉⡉⣰⣟⣶⣼⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⠀⠀⣠⡼⠁⣹⠏⡝⠀⠀⣾⢁⣿⣿⠛⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠀⣽⣿⣿⡿⠠⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣼⡿⠃⠀⠀⠀⠀⠀⢿⣿⣿⣟⣿⣿⡿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡟⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡧⣯⣿⣿⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆⡻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣅⣀⡀⠀⠀⠀⠀⢠⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣼⢿⣿⣿⣷⣤⣼⣿⣷⣭⣿⠛⠗⠁⠀⠀⠀⢀⣨⣉⣺⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠝⠛⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣷⢆⣤⡄⠁⠻⠋⡶⢂⣤⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⣌⠀⣨⣉⠀⠸⠤⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣤⣤⣶⣄⢈⠛⠁⡈⢉⢻⣿⣿⣿⣿⡿⣿⣿⣾⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⠿⠿⢿⠆⢤⠠⠚⣺⣿⣿⣿⣯⢽⣿⣿⢣⣽⠿⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠕⠈⠉⠛⠛⠛⠛⠛⠋⠀⢀⣤⣀⣒⡶⣦⣶⣿⣿⣿⡟⠻⠞⠿⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣿⣿⠿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⢀⣤⣶⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⢿⣿⣿⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣦⣼⣿⣿⠟⠋⠁⠀⠀⠘⢿⣿⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠛⠻⣿⣿⣿⣿⣶⣄⡀⣠⣴⣶⣶⣶⣦⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⣿⡀⣿⡟⠀⠈⠙⠿⣿⡿⢋⣥⣤⡈⠹⣿⡆⠀⠘⣿⣿⡿⠛⠉⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠠⣾⡿⠇⠈⠁⠀⠀⠀⢸⣿⡇⣾⣿⣿⡇⢀⣿⣿⣶⣶⣬⣭⣤⣄⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⣤⣶⣶⠀⠀⠀⠀⠈⢿⣧⣙⠛⣉⣠⣾⣿⣥⣴⣿⠛⢉⡉⠙⢿⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠿⢿⣿⠃⢰⣿⣿⣆⠈⣿⣇⡀⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⣴⣶⡀⠘⢿⣿⣧⡀⢸⣿⡀⠹⢿⠿⠃⣰⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣿⣿⡀⠀⠀⠀⠹⣿⣿⣄⠀⠛⠋⠁⠀⠻⣿⣶⣤⣴⣾⡿⠋⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠿⢿⣷⡄⠹⣿⣷⡀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣤⣶⣶⣦⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠛⢉⣠⣤⣤⡈⢻⣿⡄⠙⢿⣿⣄⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣰⣶⣶⠆⣻⣿⣿
⠀⣠⣴⣾⡿⠿⠛⢉⣠⣴⣾⣿⣿⣿⣿⡇⠀⢿⣿⡄⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⣿⣿⠀⣠⣿⣿⠏
⣾⣿⡟⠁⣠⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉⢀⣤⡀⢿⣿⡀⠀⢀⣹⣿⣿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠟⢻⣿⣷⣬⣥⣶⣿⡿⠃
⢻⣿⣧⠘⣿⣿⣿⡿⠿⠛⠉⣀⡀⣴⣷⡜⠷⠟⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⢿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣥⣀⠀⠀⠙⠻⠿⠿⢛⣭⣶⣤⣄
⠈⢿⣿⣧⠈⠉⠁⠀⢰⠶⡜⢯⡿⠈⢉⣥⡀⣾⣿⠌⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⣿⣀⠀⠀⢈⣿⣿⣿⣿⣽⣷⣤⡀⠀⠀⣰⣿⣿⣧⣛⣻
⠀⠈⢿⣿⣷⣶⣿⣷⠈⠛⣃⡀⣴⢳⡜⠿⠟⠈⢉⣤⡌⢿⣿⣿⡿⠿⢿⣿⣧⣴⣿⣿⣿⣿⣧⣤⣽⣿⣭⣿⣽⣽⣽⣯⣯⣴⣾⡿⣵⢿⡿⣿⣿
⠀⢀⣿⢋⡾⣹⣿⣥⢦⡔⢯⡿⠈⢛⣡⡀⣾⣷⡌⠻⠋⠈⢿⣇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⠞⣱⢋⣾⣿⠃
⠀⠸⣿⠈⠰⢫⣾⡟⠛⣃⡀⣴⢶⡜⢧⠿⠈⢉⣤⡄⠀⠀⠈⣿⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⢿⣿⣿⢟⡽⢻⡞⣵⣿⡛⠻⣷
⠀⠀⠹⣷⣄⣘⣿⣷⡐⣯⣿⠈⠛⣁⡀⣾⢳⡜⠿⠟⢀⣠⣶⠿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢰⣿⣧⡞⡵⢫⣾⣭⣭⠁⢠⣿
⠀⠀⠀⠈⠛⠛⠻⣿⣷⡀⠀⠀⠘⢯⡿⠈⢉⣠⣴⡾⠟⠋⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣇⠀⣼⡏⠙⢿⣷⣿⡷⠶⠒⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⠀⣿⣿⠾⠛⠉⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⠛⢻⡿⠃⠀⠀⠀⠸⣿⡿⠛⢿⣟⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣷⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣤⣠⣄⣤⣤⣀⣄⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣴⠀⢀⣠⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡄⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⢱⣶⠿⠿⣿⣿⣷
⠀⠀⠀⢀⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡶⡏⢀⣲⣾⣿⣿⡆
⠀⠀⠀⣼⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⡿⣿⣷⣄⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⠶⠂
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣟⡻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠇
⠀⢀⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣀⣀⣀⣀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⠇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠈⣿⣿⣿⠫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣿⣿⡏⠀⣹⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠙⢹⣿⣷⣿⣿⣿⣿⣿⣿⣿⢓⣋⠍⢿⣿⣿⣿⣿⠏⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⣺⡦⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣭⡴⣺⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣭⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣯⣫⣽⡿⣉⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢀⣡⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣾⣋⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠆
⠀⠀⠀⠿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣻⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠙⢻⣿⣿⠶⠋⠀⠙⢀⠀⠙⣿⡇⠹⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡀⠀⠟⠉⠀⠈⠀⠛⠛⠿⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⣄⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡿⠿⡟⠛⠃⠉⠛⠂⠁⠀⠈⢉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠉⠀⠀⠀⠀⢠⣤⣤⣤⣶⣿⣿⣿⠿⠟⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠯⡄⢤⡀⢀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⣀⡤⠤⠒⡚⡉⠉⠀⠀⠀⠘⠒⠃⠈⠓⠸⠰⠮⠄⠀⠀⠀⠀⠀⠀⠀⠩⣷⠸⢼⢊⣳⡅
⠀⠀⢠⢠⠸⣤⡧⣄⠳⡟⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠩⣁⢀⠄⢠⢭⡀
⠀⠀⠙⠹⠄⠯⠙⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠹⠀⠲⠟⠺⠿⠊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣐⣶⡀⠀⠀⠨⠭⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⢀⡏⢸⠁⠉⢹⠂⡞⠟⢳⡀⡏⠁
⠀⠀⠀⢀⡔⠀⠐⠚⠁⠀⠀⢠⠤⣼⠀⣈⠓⠚⠁⠘⠓⢤⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠠⠤⣀
⠀⠀⠀⣼⠰⢋⡆⠀⠀⠀⠀⢻⠀⠀⠈⠻⡋⣱⠹⣦⣤⠞⠀⢀⣅⡀⠀⠀⠀⠀⠀⠀⡾⢷⡾⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠙⠻⢤⣄⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⣸⢯⡾⠁⣤⠀⢤⡀⠸⠀⡀
⠀⠀⣠⠤⠤⠶⠖⠒⠒⠛⠉⢉⣙⣿⣷⣤⡀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⢀⡏⣰⠃⠨⡁⠶⠯⠀⠖⠭⡄
⠀⠀⢻⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠉⠉⣽⠇⠀⠀⣴⠾⣏⢸⡇⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⠀⠈⠑⠢⣄⡐⠻
⠀⠀⣿⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⡄⠀⠈⠓⠲⠖⠋⠀⠀⠹⣇⠀⠀⠀⠀⣀⣰⢃⡏⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⣿⠀⢀⣼⢟⣁⣀⣀⣀⣀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⢰⡏⠉⣻⠞⠋⢹
⠀⠀⣿⣴⡿⠁⠐⢒⣶⡶⠻⣅⠹⡧⣄⠀⠀⠀⠀⠀⠀⠀⡴⢋⡩⠟⢋⣧⣴⣃⣠⠴⡇
⠀⠀⣿⣿⠀⢀⡴⠋⠀⠠⣤⣼⠗⠓⠿⣤⠔⠋⠉⠙⣓⣦⣄⡈⠀⠀⢸⣿⡏⣉⣠⢴⣁⠀⠀⠀⠀⠀⠀⣀⡀
⣀⣀⣸⣿⣷⣼⠷⠿⠟⠛⠁⠀⢀⣠⠤⠬⢷⣄⠔⠋⠁⠀⠀⠉⠓⠦⣤⣿⢿⣭⣤⣾⠉⠉⠉⠉⢉⣩⣭⠿⠥⠤⢶
⣿⠀⢠⣴⣛⠁⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣭⣤⣴⣶⣶⣛⣋⣉⣭⣿⣶⠂⢸
⣿⠘⢿⡧⣌⡛⢶⡞⠉⠀⢀⣀⣠⠤⠤⠶⣻⣿⡿⣿⣯⣭⣭⣭⣿⣿⣶⣶⣶⠿⠽⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸
⢹⠀⠀⠈⠳⢭⣛⣳⠿⠿⠿⠒⠒⠛⠋⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡰⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠡⡐⢌⠢⡀⣷⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢷⣿⡵⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠂⢄⣿⢠⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠴⠖⠋⠑⢦⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠘⠛⢻⢧⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣸⣷⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢣⣀⣇⡟⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⡌⢇⣯⠾⠒⠁⠀⡠⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣲⠆⠘⠁⠈⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⢄⡰⢧⠐⢛⣦⠀⠀⠀⡇⢰⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⢹⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣻⠀⠀⠀⣷⢸⡀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠚⠘⠂⠀⠀⠀⠀⠀⠀⠀⡟⠐⡏⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣷⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⢠⠀⡀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⢰⠇⢸⠇⠀⠀⠀⣠⢞⡵⠒⠕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣴⡇⠀⠀⠈⡇⢷⠀⠀⠀⡀⠀⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⡴⢛⡵⠛⠀⠀⠀⠠⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣧⢸⡀⠀⠀⠀⠀⠒⠀⠀⠚⠀⠰⠀⠀⠀⠀⠀⠀⢸⡇⢨⡇⠀⠈⠓⣯⠰⡋⢶⠟⣱⠀⣸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠀⠀⠀⠀⠀⢸⠈⣇⢀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⣼⠀⠀⠀⠀⠹⡀⠈⠠⠞⠁⢀⡏
⠀⠀⠀⠀⢀⡤⡶⠋⠀⠀⠀⢀⣤⣓⠂⠤⡀⠘⠆⢀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡤⠶⠃⠀⠀⠀⠀⠀⠙⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠠⡏⠈⠃⠤⢤⣴⠷⢍⠫⠤⠴⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⢻⡴⡿⠁⣲⡄
⠀⠀⠀⠀⠑⠶⣤⠤⠤⠴⠶⠾⣰⣶⡾⠟⠁⢀⣀⣀⣀⣀⣀⣀⣈⣧⣶⣖⣒⣒⣒⣿⣁⣀⡀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢠⠗⣆⠓⣦⠒⢠⢄⡴⡄⣠⠀⠀⠀⠀⠀⠀⠀⠊⠇⠋⠊⢆⠀⢀⡄⢐⡛⠐⠀⠉⠐⠠⢠⢄⡀
⠀⠀⠀⠀⠀⢠⠀⠠⠃⠀⠀⠀⠀⠀⠀⢠⡏⡼⠒⢋⢓⢖⢼⢸⣷⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡀⠯⠖⠀⣠⠴⣃⡤⠤⢤⡞⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⣃⣀⣃⣰⣋⠁⠀⠀⢀⡞⠰⡳⠠⣋⠇⠈⢦⠀⠀⠘⢶⣖⣂⣀⡀⠀⠀⠩⠆⠠⠃⠋⠃⡇⠀⠈⠉⠉⠉⠁⠀⠀⠘⢧⡴⠒
⠀⠀⠀⠀⢺⡭⣤⣤⣤⡤⠤⣿⠀⠀⠈⠒⠶⠒⠒⠶⠒⠖⢺⡀⡀⣴⡟⠴⠶⠖⠒⠒⠒⠒⠒⠟⠛⠉⠙⠃⠀⣀⣤⠤⠖⠊⠀⠀⠀⠀⠈⠳⣀
⠀⠀⠀⠀⠀⣣⡀⠀⠀⠐⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢯⠓⠉⡤⠀⣀⡜⣀⠀⠀⠀⠀⠀⠙⠫⡉
⠀⠀⠀⢰⠏⠁⠭⠖⣋⠻⠭⠆⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⣚⣨⢴⣺⠿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢿⠏⠩⠁⠀⠀⠀⠀⠤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢰⢠⠄⡄⠀⠂⠈⣄⡉⠉⠫⠿⠫⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡠⠄⠰⢏⣟⣯⣒⣂⣀⡀⠘⠃⠤⢠⣤⡤⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠒⠊⠉⢠⠤⣞⣠⢟⣎⣞⣿⣶⡶⢄⠂⣠⠀⢀⠀⠀⠌⡀⠹⢎⣳⡽⢂⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⢀⣀⠐⠄⠿⠟⣻⡿⡿⡿⢿⣞⢴⣏⠀⢣⢀⡒⣆⢠⢦⠁⢀⠀⢁⡵⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢠⣾⣽⣿⣷⡁⢪⢼⡟⣙⣗⣯⣿⣄⠛⢀⡀⢾⠘⠢⠌⠻⣃⠂⢉⠟⠒⣚⢁⢀⣠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣭⣾⣼⣿⢾⡿⠿⣷⡽⢃⢋⡟⣉⡤⢻⠧⠈⢳⣄⣠⡀⣥⡄⠁⠀⠀⣁⣠⢿⠁⢐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣮⡹⡾⣿⣾⣏⣯⠸⣿⣣⢁⣸⡟⠡⠆⠀⠲⡀⢸⠛⠾⣛⠚⣷⢀⡀⢦⠘⣧⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⡿⣷⣿⣷⡿⣳⣾⢋⡔⢁⣀⠤⠒⠒⠒⠶⣭⣿⠀⣃⣼⣙⣿⠟⢦⣈⡿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣭⢷⣺⢹⠟⣷⣿⡷⣯⣴⣿⠁⠀⠀⠀⠀⠀⠘⣷⣦⣮⡘⠺⣏⡔⡐⣾⣛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡭⢽⣋⠿⣬⣻⠿⣿⣽⡿⠀⠒⠀⠀⠀⠀⠀⢻⢋⣢⡑⢾⡋⠞⢻⠯⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣵⣿⢿⣿⣿⠾⣾⢽⡼⣿⡏⠁⠀⠁⠀⠀⠀⠀⢀⣿⡘⠷⣻⠮⢯⡟⢛⠒⣅⣐⣋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣟⣿⣟⣛⡾⢺⣿⠇⠀⠀⠀⠀⠀⠀⣀⣼⡿⠉⣻⠆⡜⣦⣌⡿⢉⠀⢿⣧⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣤⠀⠀⠟⡷⣸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠋⢀⣼⡅⢾⣷⢄⣿⡷⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⣾⡇⢆⣱⢸⣿⠀⠀⠀⠀⠀⡠⠂⠀⠐⠿⢿⣿⣾⣻⡫⠫⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢹⢟⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠘⣷⡿⣿⣧⠧⢘⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⠸⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣏⣔⣁⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣉⣻⡟⠦⣄⣀⠀⠀⠀⢀⣷⡀⣀⠔⢹⢹⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢄⡿⣉⣙⣿⣙⡿⢋⣻⣿⠤⠧⢜⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⡟⠁⣠⠸⢿⡮⣌⠈⠑⣦⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡼⠁⠄⣮⡀⠁⠳⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠁⠀⠉⠒⠒⠚⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡟⣸⣿⣿⣿⢿⣿⠃⠙⣧⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣏⣠⣾⢿⣾⣿⣿⣿⣷⣟⡀⠻⣿⣇⣈⣤⣴⣞⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿
⡿⠉⢢⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠘⢷⠀⠙⠛⠟⠛⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠀⠀⠀⠀⠀⠈⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣶⣌⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⠋⠉⠒⠀⠀⠀⠀⠀⢀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠎⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣇⡼⢻⣿⣿⣧⡀⢰⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠁⣤⡤⠄⠀⠠⠄⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠤⣦⣄⠀⠀⠈⠙⠛⠻⣿⡎⠿⣿⣿⣿⣿⣾⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠘⣷⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⢤⣤⣙⣿⢿⠷⠀⠀⠀⠀⠀⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠋⠀⢠⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣼⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡴⢦⣤⣀⠀⠀⠀⠀⠀⠀⣩⡿⠀⠀⠀⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢣⣤⣬⣿⠟⠋⠘⠺⢿⣿⣿⠃⠀⠀⠀⠁⠀⠚⠙⠻⠟⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⢾⣧⣀⣴⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠠⣄⡂⠈⠀⠀⠀⠀⣀⣠⢴⣶⣶⣷⣶⣿⣿⣿⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⠚⠓⠿⠿⠛⡿⠟⠛⠛⠋⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠙⠛⠿⡟⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⠋⠀⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠞⠋⠛⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠀⠀⠀⠈⠏⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣿⣿⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠢⠀⠘⣿⡄⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠚⠃⠀⠀⢀⣈⣽⠇⠀⠀⣠⠀⠀⢠⡟⠀⠀⠀⡇⢹⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠦⣴⣿⡇⠀⠀⠀⢁⣳⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣶⣾⣶⣿⣿⣿⡇⠀⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠟⠉⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡃⠂⠀⣀⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⡄⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢹⣗⡀⢰⠆⠀⠀⠀⠀⢰⣿⣿⣿⣿⠂⠀⠀⠀⢀⢾⣠⣿⣿⣿⣿⣿⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣷⣬⣶⣴⣾⣿⣾⣷⠞⠀⠀⠀⢠⢀⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣯⣝⣻⡿⣧⠀⠀⢀⣀⣀⣾⣿⣿⣿⣿⠀⠀⣠⠾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁