Organs Emojis & Symbols

⠀⠀⠀⢀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡐⣻⣾⡴⠀⠀⠀⠀⠀⣴⢿⣵⣦⡿⣦⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⣿⣿⣿⣷⠂⠀⠀⠀⢰⣿⣿⡇⢸⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠇⠸⣿⣿⡇⠀⠀⠈⠻⠿⢻⡏⠀⠀⠀⠀⠀⠈⠻⢷⣤⣴⠏⠁
⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢠⣴⣶⣶⣤⠴⠆⠀⠀⠀⣴⣿⣷⣄⣠⣾⣿⣦⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⣵⢈⣇
⠀⢹⠿⡯⠁⠀⠀⠀⠀⠀⠀⠘⢿⡏⢿⡿⢹⡿⠃⠀⠀⠀⠀⣿⣿⡳⡟⠀⠀⠀⠀⠀⠀⢻⠋⢚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠟⠻⡽⠁⠀⠀⠀⠀⠀⠻⢿⠟⢣⠀⠀⠀⠀⠀⠀⠈⠻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣷⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠸⣴⣿⣿⣿⣿⣦⠇
⠀⠀⠀⢏⠸⠇⡸⠀⠀⠀⠀⠀⢀⡀⠀⣸⣿⡿⠀⠀⠀⠀⠀⣬⡼⢧⣵⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡿⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⢶⣝⣫⡶⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣄⡀⢀⣠⡀
⠀⠀⠀⢹⡅⢈⡏⠀⠀⠀⠀⠀⠠⣀⣹⣏⣀⠄⠀⠀⠀⠸⣯⠛⠛⠛⠋⢹⠃⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠄⣒⢸⡇⣐⠠⠀⠀⠀⠈⣿⡄⠀⣀⣠⡿⠀⠀⠀⠀⠀⣿⠋⠙⣿
⠀⠀⠀⣸⣇⣸⣇⠀⠀⠀⠀⠀⠇⢒⠝⠛⡒⠸⠀⠀⠀⠀⠉⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡠⠚⠉⠉⠀⠁⠉⠑⢄⠀⠀⠀⠀⡠⠂⢂⣸⠂
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⢡⠀⠀⠌⠀⠀⠸⡯⢶⠀⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⡄⠀⠀⠀⠀⠁⢀⣀⠌⠀⠘⠀⠀⠀⠐⠁⠘⠀⠀⠀⠠
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠳⣈⠱⢃⠶⠂⠐⠋⠀⠀⠀⠃⠀⠀⠀⠐⠀⠰⣀⠀⠀⡰
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠀⠀⠀⠁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠢⡀⠀⠀⠀⠀⠀⣀⠴⠔⠒⠀⠀⠀⢤⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠤⣄⣀⠇⠀⠀⢸⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⡔⠁⢀⠔⠉⡆⠀⠀⠀⠀⠀⠀⠌⠒⣀⠀⢢⠀⠆⠀⠄⣀⠀⠀⢀⠌⠀⠀⠀⠇⠀⠀⠂⢂⣰⡂
⠀⠸⢄⠀⠈⠢⡀⢁⠀⠀⠀⠀⠀⠀⡀⡴⠁⢀⠴⠀⠈⠢⠀⠠⠭⡉⠘⠀⠀⠀⠀⢀⠀⢀⢼⣄⣾⠁
⠀⠀⠀⠉⠒⣠⡈⢾⠀⠀⠀⠀⠀⠀⠋⠀⣎⡁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂⠁⠀⠙⠛
⠀⠀⠀⠀⢸⡻⢀⡆⠀⠀⠀⠀⠀⠀⠰⣄⣘⡻⠀⠀⢀⠴⠲⡀⡄⠒⡆
⠀⠀⠀⠀⠀⠉⢰⠁⠀⠀⡔⢢⠀⠀⠀⡇⠀⠀⠀⠀⠀⢡⠤⢹⠟⠙⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣰⠀⠀⡰⠁⠈⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⢃⠀⠀⠀⠀⢠⠊⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠇⠀⢠⠁⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠤⠰⠂⠁⠙⢇⠀⠀⠀⠀⡇⠀⠀⢀⣋⢠⠂
⠀⠀⠀⠀⠀⢰⠀⠠⠃⠀⠀⠀⠀⢂⠀⡘⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⡀⠀⠀⣸
⠀⠀⠀⠀⣀⠜⠀⠇⠀⠀⠀⠀⠀⡘⠀⠱⣀⡀⠀⠀⠀⠈⣆⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠈⠁⠈⠁⠔
⠀⠀⠚⣉⢁⡤⠖⠁⠀⠀⠀⠀⠀⠘⠠⠀⠤⠜⠀⠀⠀⠀⠀⠙⠒⠠⠤⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠄⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⢀⢀⠐⡼⠓⠟⠀⠆⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠉⢠⠓⠿⠄⢀
⠀⠀⠀⠀⠀⠇⠉⢀⠀⢀⣄⡤⢠⠵⠰⣤⠜⠣⠀⠀⠀⠀⠠⠂⢀⣴⡿⢿⣿⠹⢻⠷⢦⣴⣦⣦⣴⣶⡆
⠀⠀⠀⠀⠸⠂⢴⣀⣀⠘⠡⠒⠀⠆⡀⢀⠐⠈⠆⠀⠀⠀⣴⣶⠟⠡⠶⣶⣦⡶⣤⡤⠄⠐⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡈⣙⣶⠶⡶⣦⡤⠄⠀⠀⠀⢠⡞⠙⠶⠖⠃⠈⠁⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠒⠒⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣸⠀⢤⠀⠀⠀⠀⠀⠀⢀⣔⡀⡴⣜⡆
⠀⠀⠀⠠⠀⠠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠋⠀⠀⠁⠀⠀⠀⠀⠀⢸⠋⠷⠦⠤⣷
⠀⠀⢀⠀⠀⣸⡧⠞⠁⣠⡦⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢐⡄⡂⠀⠀⠀⠈⠃⠀⢸⡇⢽⡁
⠀⠀⢸⠆⠘⠋⠀⠀⢀⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣚⢤⠀⢢⠼⠁⠀⠀⠀⢺⡅⣀⠠⡌⢡⡼⠃
⠀⠀⠀⢶⣷⣁⡠⠀⠈⢻⣿⠀⠀⠀⣴⣞⠙⡿⢿⠿⡿⢿⣷⣶⠃⠃⠀⠀⠀⠀⣾⢠⡉⠂⢤⣴⠆
⠀⠀⠀⠀⠹⣿⡿⢀⠀⣼⠏⠀⠀⠀⠙⠛⠋⠛⠺⠦⠴⠿⠏⠂⠀⠀⠀⠀⠀⠀⠘⠯⡙⠋⠀⠁
⠀⠀⠀⠀⠀⠈⠛⠊⠚⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢠⠶⠁⢤⢀⣴⢆⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⣰⠇⠀⠀⠘⠟⣡⡿⡁⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠃⠀⠤⡶⠊⠁⠀⠀⠀⣀⠀⠰⠆
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⡾⢿⠸⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡟⠀⠀⠀⠀⣻⣤⠘⠀⠁⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⢿⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⣤⠂
⠀⠀⠀⢠⠀⢀⣤⡾⠋⠀⠀⠀⠀⠐⢦⣄⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡏
⠀⡀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃

⠀⠀⠀⠀⣀⡀⠀⠀⣠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠠⠤⣤
⠀⠀⠀⠀⢠⡅⣠⡾⠋⢉⡭⠃⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠘⣇⣍⠀⢘⢯⠈⢁⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⡁⡼⢃⠀⠀⢈⠹⣄⡁⡀⠀⠉⠀⡀
⠀⠀⠀⣠⡜⠿⠃⠀⠀⠍⠓⠮⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠟⢈⠀⠀⢉⠚⢿⡄
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢠⠞⡄⠒⠀⠀⠀⠀⠀⢄⠉⠀⠀⡀⠾⣇⠀⠀⠀⠀⢀⡿⠆⠀⠀⠀⢠⡄
⠀⠀⠀⠈⡛⠁⠄⠀⠐⢈⢀⡇⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠏⠀⠀⠀⠀⢐⢇⠀⠁⢀⡀
⠀⠀⠀⠀⠈⠠⡲⣄⢀⣈⠁⠉⠀⣼⠀⠀⠀⠀⠀⠀⠛⠜⠖⠁⠰⠀⠀⠀⠀⠈⠀⠐⠺⠘⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣌⢥⡆⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢼⠃⠀⠀⠀⠏⢰⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⡤⠀⠘⡅⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢠⢰⡄⠀⠀⠀⠈⠁⠀⠛⢦
⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣸⠀⠳⣄⠀⢀⣁⠀⠀⢠
⠀⠀⠀⠀⠈⠙⢶⣆⣀⣀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠉⠳⠀⠒⠂⠁⠁⠀⠀⠘⢦⡴⢒⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠛⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠁⡂⡇⡐⢈⣤
⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠒⣄⡀⠢⠀⠀⠀⠀⠒⠒⠀⠀⠀⣠⡡⠀⠁⣗⣇⡇⠈⠠⣨⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠣⠈⠲⠄⢠⣇⠀⠰⠂⠀⠺
⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⣤⠙⡃⠀⠀⠔⡢⠀⠀⠀⠀⠈⠓⢌⣐⠂⢄⣧⠄⢐⣀⠱⠛
⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⢠⡾⢄⣧⡶⠶⠊⠀⠀⠀⠀⠀⠀⣢⠙⣇⠁⠄⡇⠄⢃⡕⠁⣀
⠀⠀⠀⠀⢈⠃⡀⣴⡯⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⢣⠈⡹⠷⢉⠡⠃⣠⠃
⠀⠀⠀⠀⠀⠙⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠘⠙⠉⠹⠒⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣈⣁⡀⠈⡉⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⡤⢀⡂⡖⠩⠞⠉⠉⠁⠘⡅⠙⠂⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣄⡀⠆⠉⠛⠋⡡⠄
⠀⠀⠀⠀⠀⠀⠃⠈⠁⡄⠀⢠⣆⡶⠀⠴⠇⠺⣤⠒⠉⠂⠀⠀⠀⠀⠀⡰⠁⢀⣴⣿⠿⢿⣿⠹⠋⡟⠻⣶⣿⣶⣷⣶⣿⣷⠆
⠀⠀⠀⠀⠀⢼⠆⢠⡆⣀⣀⠘⣁⠔⠂⠀⠆⠀⠀⠈⠂⠰⠀⠀⠀⠀⢠⣿⣵⠿⠏⣴⣶⣶⣥⠶⣤⣤⠤⠀⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠰⠁⠂⢧⣦⣤⣤⣬⣁⣀⡆⠀⠀⠀⠀⣀⡿⢫⣤⣤⠶⠂⠛⠉⠹⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣷⣈⢭⣝⠿⠀⠀⠀⠀⠀⠀⠿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠄⢠⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣨⣧⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⠀⣴⣞⡷
⠀⠀⠀⠀⢀⠄⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠁⠀⢋⠀⠀⠀⠀⠀⠀⢠⡿⢹⡿⣦⣈⣯⡀
⠀⠀⠀⠀⠀⠀⢠⣧⣠⡾⠁⢀⣀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠠⡄⢠⠀⠀⠀⠀⠀⠷⠄⠁⢨⣆⠘⡷
⠀⠀⠀⢸⠀⢠⣾⠟⠉⠀⠀⣾⣿⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠀⠄⣀⠀⡇⠘⠇⠀⠀⣤⡴⠀⠀⠀⠰⢮⠉⣷
⠀⠀⠀⠘⢃⡈⣀⠈⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⢀⣀⣀⣄⣀⣠⣴⣮⣅⠰⢀⣽⡚⠁⠀⠀⠀⠀⢩⡟⢡⣤⠛⠗⠀⠿⠋
⠀⠀⠀⠘⠻⣿⣄⣴⠂⠀⠠⣼⣿⡇⠀⠀⢠⣾⣏⡉⢛⡛⣿⠻⣿⣿⣿⣿⡿⡥⠁⠐⠀⠀⠀⠀⣾⣧⣬⣉⣑⢶⣶⠿
⠀⠀⠀⠀⠀⠹⣿⣯⠄⣤⠀⣿⠟⠀⠀⠀⠀⠙⠿⠛⠩⠓⠶⠦⡴⠿⠿⠅⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠦⠣⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠑⢆⠀⣠⡷⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠀⢸⡿⢉⣴⣏⠇⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠛⠂⠤⢤⣴⠖⠙⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣴⣿⡏⣾⠀⠀⠀⠘⣇⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣾⢋⣹⣇⠲⡄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣡⣾⣿⣇⣼⠁⠄⠀⠀⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣿⠁⠙⠛⠿⠁⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠐⡞⠅
⠀⠀⠀⠀⢀⠂⢠⣤⣾⠟⠁⠀⠀⠀⠀⠂⠙⢷⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⣿
⠀⠀⠀⠀⠺⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣁⣌⣹⣁⣬⣉⣩⣍⣡⣿⣫⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⡀⠀⠸⠀⢀⡀⠀⠀⠀⠈⠑⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢈⠴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠸⠰⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⠀⠀⢠⠄⠂⠉⠀⠀⠀⠁⠀⠀⠁⠀⠙⢄⠀⠁⠉⢀⠐⠈⢰⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠐⡇⠈⡀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⢀⣀⠀⠐⢀⡩⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⡇⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡆⠀⡆⠀⠀⡀⢀⡱⠂⢁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢁⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⠀⠈⠀⠔⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⢀⣀⣠⣀⠀⠔⠤⢀⣀⣀⣀⣀⣀⡀⠄⠚⣄⣀⣀⠀⢸⠀⠈⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⢸⡀⠊⠁⠀⠀⠀⠀⠒⠂⠀⠙⠉⠉⠉⠈⠉⠃⠀⠀⠀⠀⠀⠘⡄⠀⢣⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣬⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠘⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠆⠀⠀⠇⢠⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠀⠀⢰⠀⠀⢱⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠀⢸⠀⢂⠀⠀⠀⠀⠸⠉⣉⣁⣈⣉⣉⣉⣉⣉⢁⣀⠇⠀⠀⠀⠀⠂⠈⡄⠀⠀⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡸⠀⠀⠀⡌⠀⠈⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠄⢈⠃⠀⠀⠀⢀⠃⠀⡇⠀⠀⠘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠃⠀⠀⢠⠁⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⢰⠀⠐⠂⠤⠤⠤⠀⠠⠔⠁⠀⠀⠀⠀⠈⠀⠀⢁⠀⠀⠀⢙⠀⠀⠀⠸⡀
⠀⠀⠀⣰⠃⠀⠀⠀⠈⢆⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠋⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠈⠀⠑⠈⠀⠀⠀⠆⠀⠀⠀⠀⣀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠁⠀⠀⠀⠀⠀⢣
⠀⠀⢸⠀⠀⠀⠀⢠⢶⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡶⡀⠀⠀⠀⢸
⠀⠀⢀⡏⠠⢀⡀⡎⠘⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡘⠀⢃⢠⢰⠈⢸
⠀⠀⠈⡇⠸⢰⢰⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠘⢸⠀⣆⠟
⠀⠀⠀⠘⣇⠺⠞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠿⡇⡟
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢰⡄⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⢻⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⠀⠀⠸⡆⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⢀⠀⠀⠠⠜⠀⠀⡆⠀⣠⠂⠇
⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠀⢆⠀⠐⢛⠀⠀⠀⠀⠘⠃⠀⠰⠁⠀⠀⠀⠀⢀⠀⠀⠀⣈⡼⠌⠀⠁⠀⠀⠀⠀⢁⠀⢰⣄
⠀⠀⠀⠀⠀⠊⡥⠀⠀⠀⠀⠀⠁⠒⠋⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠈⢳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠄
⠀⠀⠀⠀⠀⣉⠖⠂⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠖⡉⠄⠀⠀⢰⣎⠨⣿⣿⡏⢱⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⢐⣆⠁⠐⠃⠘⠀⢀
⠀⠀⠀⠀⠀⠐⠊⣡⠆⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢓⡆⠀⠀⠀⠀⠀⠀⠀⠄⢨⠀⠁⠜⠈⣠⠤⠶⠐⢂⡀⠉⢀
⠀⠀⠀⠀⠀⠈⠀⢨⢤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⢂⠀⠿⡇⢀⠆⠈⠂⠀⠀⠀⠈⣩⢥⡇⠀⠋⠀⠐⣊⣥⡤⠀⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡦⠆⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢸⠀⡾⡇⢸⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⠻⠳⣶⠺⣿⡥⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣟⣧⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠤⠴⠖⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣸⠋⠀⠈⢻⡀⠀⠀⠀⠀⠀⠃⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠐⠶⠒⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⢸⡀⠀⢹⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠐⡚⡠⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢘⡇⠀⠸⠀⢉⠀⠀⠀⠉⠐⢦
⠀⠀⠀⠀⠠⢈⣩⣥⠁⠀⠀⠀⠀⠙⠧⠤⠀⠠⠴⠒⠁⠀⠀⠀⠀⠙⠒⠤⠀⠠⠤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠛⢦⠈⠇⠀⠆
⠀⠀⠀⠀⠐⠀⠼⡤⠁⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⡀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢻⠀⠂⢠
⠀⠀⠀⡈⠑⠶⠓⠀⠀⠀⠀⠷⣤⡄⣀⠀⠂⠀⠈⡂⢁⢤⣤⢻⠀⠀⠀⠀⠀⡲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠈⢙⣃⠄⠀⠀⠀⠀⣀⠀⡍⠀⠉⠉⠉⠈⠀⠀⡤⠌⡁⠀⠀⠀⠔⠘⠃⣶⡦⠀⡀
⠀⠀⠀⢀⣩⣍⠀⠀⠀⠀⠀⠤⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠒⡇⠀⠀⠘⠀⢰⠂⠀⡴⡠⠞⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁
⠀⠀⢀⠀⠰⢾⡁⠀⠀⠀⠀⠴⠤⠅⠀⠀⢀⡀⠀⠀⢀⡃⢀⡇⠀⠀⢆⣀⣬⠀⢰⡶⠤⠄
⠀⠀⠀⢢⢴⠉⠀⠀⠀⠀⠈⢢⠃⢘⠀⠀⠰⢦⣁⡢⠈⠀⡨⠃⠀⠀⠘⠟⠁⠀⠀⠸⠢⠀⠀⢀⠄⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠂⠏⠂⠀⠀⡘⡇⠈⠉⠉⠀⠀⠀⠀⠀⠈⠣⠙⣤⣈⠄⠀⠀⠑⠀⠁⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣆⣢⣀⣀⡄⠀⡀
⠀⠀⠀⠀⡜⢀⠔⡈⢀⠚⡄⣀⢠⠰⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠛⡟⣻⣿⣿⠙⠛⢾⣤
⠀⠀⠀⠀⢇⠙⢉⣥⣨⠞⠁⢀⡴⢠⢄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢆⣯⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⠤⠀⠈⠉⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠉⠉⣴⠱⣈⣹⣧⣀⢵⡖⠃⠀⠀⠀⠀⠀⠀⠀⢀⡇⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠃⠀⠀⠀⠀⠜⡆⠀⠀⢀⡀⠀⡆⠀⠀⠀⠀⠀⣀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣄⠀⠀⠐⡇⠀⠀⢸⠁⠀⢑⠀⠀⠀⠀⠀⠩⣄⢸
⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁⠀⠀⠀⠈⠙⠢⠼⠃⠀⠀⡟⠀⠀⢈⠁⠀⠀⠀⠸⠂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡄⡤⠆
⠀⠀⠀⠀⠀⠀⢀⡏⠀⡔⠁⣤⣁⣀
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠐⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡃⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⠀⢠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⣦⡿⠣⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂⢧⠶⠲⠏⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⣥⣀⣠⡷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣹⠀⠀⡆⠀⠘⣄⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠶⣄⣀⠀⠀⠀⠀⡄⠀⣈⣤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠨⠋⠉⠉⠀⠁⠀⠉⠀⠈⠍⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠉⠢⠴⠂⢤⢤⠄⠰⠀⠀⠉⡀⠂
⠀⠀⠀⠀⠄⠄⠇⠠⢀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠏⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⡉⣤⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠒⠒⠒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄
⠀⠀⠀⠀⠀⢀⠾⠞⣡⣤⡤⣀⠈⢳⡀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠔⠠⠋⣠⠞⠁⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠎⡜⠉⠉⢢⣧
⠀⠀⠀⠀⠀⡌⠀⢸⣿⣿⣿⣿⡆⠀⡇⠀⠀⠀⠀⠀⠘⣤⠐⠁⡄⣀⠀⢁⡈⠳⣄⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢈⠟
⠀⠀⠀⠀⠀⠸⡀⠈⠿⣿⣿⠟⠁⢠⠃⠀⠀⠀⠀⠀⠀⠈⠙⢏⣜⣁⣀⣋⡀⠀⠬⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡘⠦⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢲⡶⢦⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⠈⠀⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢦⠀⣿⡀⡴⢲⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⡫⠋⣀⠕⣊⡴
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡾⠙⠀⠸⡾⠛⢷⡗⠀⠙⠳⡄⠀⠀⠀⠀⠀⠀⡞⠀⠁⠀⠀⠞⢿⠁
⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢰⠃⠀⠀⢇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢣⣠⠀⠀⠀⠤⢺
⠀⠀⠀⠀⠀⠀⠈⢆⣠⠞⠳⣀⡴⠁⠀⠀⠀⠀⠀⠀⣧⡄⠀⠐⢻⠀⠀⠀⣞⠃⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⢀⢔⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠗⠊⠁⠀⠀⠀⠈⠉⠲⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠋


⠀⠀⠀⠀⢀⡤⠖⠒⠲⠦⣠⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⢠⢪⠏⠉⠲⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠖⢛⡲⡄
⠀⠀⠀⠀⡌⠎⠀⠀⠀⠀⣻⠄⠀⠀⣠⠔⠃⠀⠀⠀⢰⠳⠋⡀⠈⡆⠀⠀⠘⣬⠔⠀⠀⢡⠀⠀⠀⠀⠀⠀⠑⢬⠁⠀⠀⠘⢼
⠀⠀⠀⠀⢡⠀⠀⠀⡠⠄⣹⢇⡲⠞⠁⠀⠀⠀⠀⠀⡇⠀⠀⢑⣾⠤⡀⡎⢠⠟⢦⠀⠀⡸⠀⠀⠀⠀⠀⣀⣀⢠⠋⠁⠀⠀⢸
⠀⠀⠀⠀⡸⢀⠴⠚⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠊⠙⡆⢰⡃⠸⣄⠀⢀⡴⠁⠀⠀⠀⠀⡯⡶⣄⠉⠁⠀⠀⡠⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠔⠁⢸⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠹⠓⠀⠉⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⣄⣀⣀⣤⡄
⠀⠀⠀⠠⠚⠉⢉⣍⠉⠳⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡿⢃⠋⡧⢸⡇
⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣤⢭⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⣀⢰⠀⣀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣾⣿⣼⣴⣿⣧⡀⠀⠀⠀⢸⠛⢿⠛⢿⠀⠀⠀⠀⠀⢾⠀⠸⠄⠀⠀⠀⢀⣶⣶⣶⣶⡄
⠀⠀⠀⣾⣿⣿⡇⢸⣿⣿⣷⠀⠀⠀⠸⢶⣏⣷⠞⠀⠀⠀⠠⠄⠘⠀⠐⠀⠄⠂⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠋⠁⠈⠛⠛⠛⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛

⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⣄⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⢸⠿⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢀⠏⠀⠀⠀⢰⣟⡆⢰⣻⡇
⠀⠀⠀⠀⠀⠘⠓⢺⠀⠀⠀⠀⠀⠀⠀⣤⣅⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⡆⠀⠀⠀⡆⣰⠟⠛⢇⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠉⠁⠀⠀⠈⠉

⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠘⢿⡿⢷⡾⢿⡿⠂⠀⠀⠁⠀⠀⣤⡄⠀⠀⠁⠀⠀⢀⣼⣿⣿⡀⠀⠀⠀⢸⡎⢰⡄⢹⡷
⠀⠀⠀⠀⠀⠙⠮⠔⠋⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠚⠻⣿⠟⠛⠀⠀⠀⠸⣷⣹⣇⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢡⠀⠀⠀⠂⡌⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢰⠀⠂⠀⠈⡆⠀⠰⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⣇⠀⠠⠀⠉⠀⠀⣠⣴⡞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠛⠀⢀⣠⣤⡶⠻⡏⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣾⡿⠁⠀⠀⠀⠉⠈⠉⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣿⠉⠀⠀⠀⠀⣠⣤⣤⣶⣾⣻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠏⠀⠀⠀⠀⣰⡿⠛⠻⢿⣿⣯⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⠀⠀⠀⠀⢀⣏⣠⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢿⠀⠀⠀⠀⢸⣷⠒⠒⠒⠘⠛⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠘⡆⠀⠀⠀⢨⣯⣽⣗⡒⢀⠀⢈⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠚⠀⠀⠀⠀⢠⣿⣦⣤⣶⣿⠋⠀⠈⠙⢈⢷⡀⠛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢸⡎⣟⠓⠛⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣤⣴⡶⣻⢿⠅⠸⣿⠂⠭⡄⠀⠐⠦⢠⣾⡷⠈⢶⠛⢿⣆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠙⣿⣿⣾⡿⠀⠀⠀⠘⢃⡀⠀⠀⠀⢵⠏⣸⣷⣾⢈⢳⣤⡻⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣿⣿⣿⣇⠀⣠⢆⡀⠘⡅⠀⢩⠝⢻⣀⠻⢘⣛⠎⢈⡳⡙⣦⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢿⡟⠓⢦⣄⠀⡀⠀⠀⠀⠈⠊⠃⠈⢡⣵⠏⢙⣴⡼⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⣿⡄⠈⠢⣄⣉⣉⠢⠀⠀⠀⠀⠀⠀⢹⣤⠋⡰⠦⠑⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠙⣿⣿⣶⣤⣄⣈⠡⡁⠀⠀⠀⠀⠀⠀⠘⣣⠎⣧⣗⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⡾⠀⡄⢸⣞⡟⠻⢿⣧⣄⣹⡛⠲⠄⠀⠀⠀⠀⠁⠈⡝⢻⡝⢂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⠇⠸⣧⣾⡏⠁⠀⠀⠈⠉⢹⣷⣤⣤⣀⠀⠀⠀⠀⠘⠁⣿⣅⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢽⠃⠀⠀⠀⠀⠀⢸⠏⠉⢻⣿⠟⢶⣶⣤⣤⣤⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣶⣾⡇⠀⠀⠀⠀⠀⣸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⡆⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢸⣿⠀⠸⣿⣇⠀⢹⣿⣄⣀⣹⣿⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠘⣿⣧⡀⢹⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣶⡄⠘⢿⣷⣄⣈⣿⣿⡿⠛⠉⣽⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡄⠉⠛⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣽⣿⡿⠟⠁⠀⠀⢰⣿⣿⣿⡟⢁⣶⣶⣦⠘⣿⣿⣿⣿⠀⠀⠀⠀⠉⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠘⢿⣿⠟⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣀⠘⢿⣷⣤⣀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⣠⣴⣿⣿⣧⣄⡀⠈⠉⢻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣉⠛⠿⣿⣶⣤⣀⣀⠈⠙⠛⠛⠋⠉⠀⢀⣀⣤⣶⣿⣿⣿⣤⡈⠙⠻⢿⣷⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣍⣛⠻⠿⢿⣿⣷⣶⣶⣿⣿⠿⣿⣿⣯⣄⠀⠉⠻⢿⣷⡀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣶⣶⣤⣤⣤⣤⣤⡄⠀⠈⠙⠿⣿⣆⠀⠈⣿⠇
⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠙⠉⠁⠀⠀⠀⠀⠻⠿
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡀⢰⣶⣶⣶⣦⢰⣶⣶⣦⠀⣶⣶⣶⣶⡄⣶⣶⣶⡄
⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⢸⣿⠁⢸⣿⡇⠀⣿⣿⠀⣿⡏⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⣿⡇⠘⣿⣷⣾⣿⠇⠀⣿⡿⠀⢻⣿⣶⣿⡿⠀⢸⣿⠇

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡆⢸⣿⣿⣿⣷⡰⣿⣿⣷⠀⢿⣿⣿⡀⠸⣿⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡀⢸⣿⡇⠀⣿⣿⠀⠀⢸⣿⡀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠘⠿⣿⣿⣿⠃⠀⢿⡟⠀⠀⠸⡿⠀⠀⠀⢿⠇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⣀⣀⣀⣀⠀⣀⣀⣀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡆⢸⣿⠟⢻⣿⡜⠛⣿⣿⠀⣿⡿⠛⣿⣧⠛⢻⣿⡇
⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣸⣿⡇⢸⣿⣄⣸⣿⡇⠀⣿⣿⠀⣿⣯⣀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠛⠛⠛⠛⠁⠀⠛⠋⠀⠘⠛⠛⠛⠋⠀⠘⠛⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⢀⠀⠀⢀
⠀⠀⠅⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠀⢈⡀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠹⣿⡯⠄⠈⡧⣼⠇⠀⠀⠀⠤⡀
⠀⠀⠀⠐⠄⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⡆⠀⠀⢯⠀⢦⠀⠀⠀⣴⡟⣷⡄⠀⠀⠀⢠⣿⣻⣷⠀⠀⠀⠁⠃⠀⢀⠇⠅⠀⢀⡀⠈⠉⠙⠐⢄
⠀⠀⠀⠀⠀⠀⡇⢰⠁⠀⢰⠂⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⡶⡵⠃⠀⠀⠀⠙⢿⠛⠀⠀⢀⢈⠀⠛⠛⠃⠀⠐⠀⠀⠀⡀⠀⡀⠀⠈⠀⠁⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠈⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⢱⠃⠀⠀⡁⠐
⠀⠘⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠘⠐⠀⠀⠀⢨
⠀⠀⠀⠀⠐⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠇⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⡬⣬⢀⡾⣄⠀⠀⠀⠀⠁
⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠀⠀⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠙⠋⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⡀⠠⠄⠀⠀⠂⠁⣄⠀⠀⠀⠀⠀⠀⣰⢧⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡰⢧⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄⡀
⠀⠀⠀⠄⠈⠀⢀⠀⢤⠀⠠⠀⠀⠀⠀⠀⠀⠀⡻⠞⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠷⢾⠐⠂⠒⡀⠚⠛⠛⠓⠺⠛⠟⠆
⠀⠀⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⢠⡀⢠⠀⠄⠀⠱⠢⠀⠨⠅⠀⠀⠀⠈⠃⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠐⠄⠀⠠⠀⢸⡀⠀⠀⠀⠀⠀⠐⠄⢰⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄
⠀⠀⠀⠄⠈⡀⠈⠠⠀⢀⠈⠂⠈⠰⠀⡀⢀⠀⢸⠃⢸⠀⠁⠁⠀⠀⠀⢸⠀⠀⠀⠀⠂⠠⠀⢧⠀⡀
⠀⠀⡀⠀⠀⢈⠀⠀⠂⡀⠐⢇⠈⠀⢀⠁⠀⠀⢸⡀⠈⠀⠀⠁⠰⠀⠀⢸⡀⠀⠀⠀⠀⢀⠀⣸⠀⠀⡀⠡
⠀⠃⡰⠀⠀⠂⠀⠂⠀⠀⠀⠰⠀⢲⠀⡃⢠⠀⣼⣇⢘⠀⠀⡄⢠⠀⠀⣼⣇⠀⠀⢠⠀⠀⢠⣿⡀⡇⡀⠀⢀⠀⠁
⠀⢠⢀⡡⣀⡈⡀⢀⢆⠀⢡⡀⠡⣸⣠⡇⣼⢸⡇⣸⣾⢸⡆⣇⣸⢸⡖⣇⣿⢸⡆⣇⡇⣶⢸⣁⣷⡇⣷⣴⣸⣰⡇⠀⠀⠐⠂
⠀⠀⠀⠀⣇⠀⣆⠀⣰⡀⢠⡀⢄⢸⠈⡇⢹⠀⣿⡇⢻⢸⡇⡇⢸⢸⠇⣿⣿⢸⡇⡿⡏⣿⢸⣿⡇⡇⡟⢿⢹⠹⡇
⠀⠀⠀⠀⣼⠀⢸⡇⢸⡇⢸⡇⢸⢸⠀⡇⢸⠀⣿⡇⢸⢸⠃⡇⢸⢸⠀⣿⡿⢸⡅⡇⡇⣿⢸⣿⡇⡇⡗⢸⢸⠀⡇
⠀⠀⠀⠀⠀⡇⡆⠁⡘⢰⢸⣧⠸⢸⠀⡇⢸⠀⢻⠇⢸⢸⠀⡇⢸⢸⠀⢸⠁⢸⡇⡇⡇⡘⠀⣿⠀⡇⡇⢸⢸⠀⡇
⠀⠀⠀⢇⠀⢹⡇⢀⠇⢸⡏⢸⡇⢸⠀⡇⢸⠀⢸⠀⢸⢸⠀⡇⢸⢸⠀⢸⠀⢸⡇⡇⡇⣿⠀⣿⠀⡇⠃⢸⢸⠀⡇
⠀⠀⠀⢸⠀⡘⡇⢸⠁⠈⣿⢸⡃⢸⠀⡇⢸⠀⠘⠂⢸⢸⠀⡇⢸⢸⠀⠘⠀⢸⡇⡇⡇⣿⠀⠛⠀⡇⣧⢸⢸⠀⡇
⠀⠀⠀⠌⠂⠏⠰⠸⠸⠰⠃⠀⠆⠸⠀⡇⢸⠀⠀⠀⢸⢸⠀⡇⠘⢸⠀⠀⠀⢸⠇⡇⠁⡿⠀⠀⠀⡇⠘⢸⢸⠀⡇
⠀⠀⠺⡶⠾⢿⣾⣷⡶⠶⣶⣶⠶⠶⣶⣶⠶⠶⣶⣶⡶⠶⢶⣶⣶⠶⢶⣶⣶⣶⡶⢶⣶⣶⣶⣶⠶⣶⣶⣶⡶⢶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⠀⠀⠂⣤
⠀⠈⠹⠷⠶⠾⠿⠿⠶⠶⠿⠿⠶⠶⠿⠿⠶⢶⣿⣿⣷⣶⣾⣿⣿⣶⣶⣿⣿⣿⣶⣶⣿⣿⣿⣧⣤⣾⣿⣿⣶⣶⣏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⠀⢸⠀⠀⡇⢀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⡇⢸⠀⠀⡇⢸⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⠐⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠄⠀⠀⠰⣄⠔⠁⠀⡇⢸⠀⠀⡇⠘⡇⠀⠀⠀⠀⠀⠀⢀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠄⠀⠠⠈⠀⡏⢀⠀⢰⡇⡾⠀⠀⡇⢀⡇⠄⠐⠀⠀⠑⠈⢁⠀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠁⡀⠀⠀⠀⠀⡇⠀⠀⠀⠟⢁⣀⣀⠙⠺⠇⠀⠀⠀⠀⠀⠠⠈⠀⠀⡀⠀⢧
⠀⠀⠀⠀⠀⢠⠎⠄⠀⠀⡀⢀⠉⠒⢄⡀⢘⣤⡤⡶⡖⠉⠀⠈⠓⡄⠀⢌⡀⢁⢀⣐⣃⢀⠠⠐⠀⠠⠀⢣
⠀⠀⠀⠀⢀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡩⠁⢰⠃⣇⠀⠀⠀⠀⡇⢳⠀⡀⠉⡁⠀⢀⠈⠉⠐⠄⠑⠄⠀⢣
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⣿⠀⠀⠀⠀⢷⠘⡆⢀⠁⠂⠀⠂⠀⢀⠀⡀⠀⠀⠀⠈⢇
⠀⠀⠀⡼⠀⠀⠀⠀⡀⠀⠌⠀⠀⢀⠀⠀⠂⣺⠁⢀⡏⠀⠀⠀⠀⢸⡄⢧⠀⠀⠂⠀⠍⠀⠀⠀⢀⠀⠀⠀⢀⠘⣆
⠀⠀⢰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⡰⠃⠀⣸⠁⠀⠀⠀⠀⠈⡇⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠸⡄
⠀⢀⠏⠀⠀⠂⠀⠀⠀⣀⠀⢐⣀⡠⣤⠞⠁⠀⢀⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢧⣄⡀⠐⠀⠀⢀⡈⠀⠀⢀⠀⠀⢳
⠀⡼⠀⠀⠀⡠⠖⠋⠉⠁⠀⠀⠀⠠⠃⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⡈⢇⠀⠈⠉⠉⠀⠁⠆⠀⠀⠀⠀⠈⡇
⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠂⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠈⠀⠀⠀⠃⠂⠀⠀⠢⡀⠈⠀⠀⢸
⣼⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⣈⡴⠃⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠂⠀⢂⣀⣠⡥⠤⠔⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠤⠤⣤⣈⣀⠀⠀⠀⠀⠂⠀⠠
⠙⣤⣤⣤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣤⣤⡴
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠤⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠂⡀
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⣠⠠⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⣷⠀⠀⠀⠀⠀⠀⠘⠀⡐⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠂⠀⢀⠀⠀⠀⢀⠀⠃⠀⠀⢰⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠈⠀⠐⣈⠈⠙⠁⠀⠀⠘⠄⠀⠠⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⠀⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠊⠀⠐⠆⠤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠑

⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢀⢈⠀⡀⠉⠠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⡨⠁⣙⠁⠇⠀⠁⠀⠀⠀⠀⠀⣸⠝⢂⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠐⠀⠀⠀⠀⠒⠃⢀⣄⠐⠊⠀⠀⠀⠀⠀⠀⠇⡠⢀⠂⠈⢀⠉⢀⠀⠀⠀⠀⠀⠴⠇⠪⠀⠀⠀⠘
⠀⠀⠀⡐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠊⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⠄
⠀⠀⠀⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⠠⠄⡀⢰⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⡴⠠⠐⠀⠀⠐⠀⠰⡀⠀⠀⠀⠀⠀⢀⠐⢀⡀
⠀⠀⠠⠁⠀⢀⡃⠈⢤⠀⠀⠀⠀⠀⠀⠊⠀⡤⠤⠄⠀⠡⠁⠀⠀⠀⠀⢀⠴⠫⠤⠐⠀⠀⠀⠀⠀⠀⠠⠤⡄⠑⡐⠁⡤⠄⠂⠀⠀⠀⠀⠂⠀⠀⢅
⠀⠀⢀⠀⠀⡉⠐⠈⠈⠀⠀⠄⠀⠀⠀⠀⠀⠡⡀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢦⠈⠗⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡈⠢⡀
⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠅⠀⠀⠀⠀⠀⠀⠀⢄⠀⠜⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢠⠉⠀⠈⡀⠀⠀⠀⠀⠀⠀⠨⣇⠀⣹⠅⠀⠀⠀⠀⠀⠀⠀⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡠
⠀⠀⠀⠀⠀⠸⢀⡀⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠊⠀⠀⠀⠀⠀⠀⠀⠄⡥⠤⣰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠈⠁⠄
⠀⠀⠀⠀⠀⠰⠂⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡷⠒⣶⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⢰⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠙⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⠀⠀⠀⠀⠀⢀⠔⠉⠈⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠆⠀⠴⡀⠀⠀⠀⠀⠀⠀⠨⠢⣐⠅
⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡻⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠒⠀⠖⠀⠀⠀⠖⠀⠀⡆⠀⠀⠀⠈⠆⠀⠀⠀⠀⡀⠌⠁⠉⢡⠂
⠀⠀⠀⠀⠀⢠⠀⠘⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⡰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡐⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂⠀⠂⠆⠀⠀⠀⠀⠐⠔⠂⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠿⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⢿⣛⠿⠭⡛⠳⠭⣩⠙⠢⢌⡅⡂⠖⠈⢒⠤⣠⠐⡢⠎⡭⢛⠱⠟⡟⠛⢿⢿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣴⣾⣿⣿⢿⣿⠟⠟⢉⢐⠤⢈⠐⠠⡃⠂⡤⣛⠖⡤⣈⠐⠢⠌⣁⠚⠴⣩⢐⡠⢈⡕⡃⠶⣬⢘⡱⠎⡙⠂⠤⣉⠿⣿⣿⣿⣷⣆
⠀⢀⣴⣿⡿⠿⢿⡿⡊⠂⠕⡛⠆⡬⣘⠰⠈⠂⠤⣉⠖⠡⢋⠶⢠⡜⣣⢖⣴⣫⣞⣰⣮⣕⣳⢶⣥⣛⠶⣨⡑⠃⢌⡁⠂⡅⠣⡬⡻⢾⣿⣛⣳⣄
⠴⠟⠛⠓⠁⠀⠀⠀⠀⠁⠀⠀⡁⠂⠁⡀⠖⣩⢘⡴⣯⠖⣫⣾⣭⣺⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣯⣵⣮⣑⠢⢅⡃⠤⡁⠆⡁⢢⣝⠿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣫⢖⣯⣾⣥⣾⣿⡿⠿⠛⠚⠋⠉⠉⠉⠉⠁⠁⠀⠉⠉⠙⠛⠿⢿⣿⣷⣮⣝⢢⡅⠂⡄⠣⠌⠱⣙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣢⣶⣿⠿⢛⢫⣉⣢⠀⣄⣠⣴⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣲⣦⣤⣀⠨⢙⢿⣿⣷⣎⡱⢌⡣⠌⡑⠆⡺⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⣟⠫⣒⠯⣂⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠪⡻⣿⣿⣤⡕⢊⡁⠂⡌⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⣪⠕⣪⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡊⠻⡿⣯⡵⢀⠠⢌⡑⠽⣿
⡀⢀⣠⣤⣤⣴⣶⣿⣿⠿⠟⡫⢕⣊⣴⣿⣿⣿⣿⣿⣿⣿⢯⣿⠋⣹⡟⠛⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⣟⣄⢹⣿⣾⡱⢌⡁
⠻⣿⣿⣿⣿⡿⠏⠋⠒⣫⣴⣶⣾⣿⣿⣿⣿⣿⢻⢙⢸⢘⢸⣿⡀⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇⣿⣿⣿⣿⣷⡟⠻⣿⣗⠬⣀
⠀⠀⠙⠲⡶⠶⢶⣷⣿⣿⣿⣿⣿⣿⣯⡟⣻⢹⢸⢠⢸⢸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⠁⣀⢻⢻⢱⣿⣿⣿⣦⣬⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠈⢑⡪⢵⡻⢝⣿⣞⢿⣿⣍⡿⡞⡞⡘⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣯⣿⣷⠏⠀⠈⡌⡜⡎⡏⡟⣿⣿⢿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢚⣭⠺⡻⣷⣅⠙⢷⣽⡄⠁⠙⠀⡀⠻⣿⣿⡯⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⡐⡜⠘⣜⣙⣼⣿⣿⡾⢾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⡳⢕⢪⡽⣷⣤⡈⠙⢦⣄⡐⠔⢀⠉⣽⢿⣾⢭⣛⣛⣛⣻⣿⣿⣯⣿⠟⠁⠀⠀⠈⣈⡼⣾⣯⢽⣿⡿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠒⠍⠊⢙⠟⣶⣄⡉⡙⠳⠦⣤⣀⡉⠛⠿⠿⢿⣿⠿⠿⠟⢋⣀⣠⡤⠔⠒⠋⢁⣢⠽⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⠀⠐⠠⣉⠂⠌⡻⢻⣶⣤⣄⣉⡙⠛⠓⠒⠒⠒⠒⠚⠋⣉⣁⣠⣤⠤⢖⠛⠥⢒⢣⠑⡊⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠠⢌⡰⢉⠒⠤⢁⠲⠭⢹⢻⠿⠿⣿⣶⣶⣾⣿⣿⡿⠩⠹⢒⠈⠨⠐⠐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⠎⡄⡐⠠⢉⠃⠂⠌⠌⠘⠐⡐⢐⣐⠀⠨⠨⠨⠐⡀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣡⢃⡂⠀⠨⠠⠃⢀⡂⡀⡀⡀⣄⠸⠈⠇⠂⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣘⣁⣃⣖⣰⣀⣂⣇⣂⣇⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⡟⠀⣿⠈⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠛⠛⠁⠀⠀⣹⠀⢹⡉⢻⢻⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠹⣆⠀⠳⣬⣄⣡⣈⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⣤⣴⣦⣤⡊⠀⢸⡃⠀⠀⠈⠉⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠰⢿⣿⡟⠁⠈⢂⣸⠓⢦⣤⡄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣤⣠⣀⣀⣤⣿⣿⡇⠈⠏⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠛⠉⠁⠀⠈⠉⠛⢷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣴⣤⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⡿⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⡇⠀⠀⣠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⣿⣿⣿⣿⣦⣀⠀⢐⣇⠀⠀⠀⠉⠉⢙⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⣸⣿⣿⣿⣿⣿⡿⢻⣿⡿⢸⢻⠀⢄⠀⠀⡈⢊⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠉⣿⣿⣿⣿⣿⡿⡿⠈⣧⣤⣼⡿⠿⠿⠇⢀⠀⠠⢤⠞⠈⠻⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠈⣿⣯⣿⣿⠏⠀⠿⠀⣼⣿⠊⠀⣦⣀⣠⡴⠋⡄⠂⠀⠀⠠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⡟⠀⠸⠀⢀⣽⠃⠀⠀⣿⣿⣏⣤⣚⠁⠀⠉⠉⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⢐⠀⣿⣿⣿⡇⠀⠳⠀⢸⣿⡀⠀⠀⣿⣿⣿⣿⠈⣁⠰⠠⠇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠤⡀⢸⠀⣿⣿⣿⠇⠀⠀⠀⢠⡝⠃⠀⠀⠀⠳⠹⣝⠺⡏⠂⠉⠀⢤⡗⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠉⠨⡮⢈⠭⣿⣿⣿⠀⢀⣠⣴⠏⠁⠀⠀⠀⠀⠠⠀⠂⠀⠘⡄⡌⠁⠰⠠⢉⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡠⡈⢀⡁⠂⢒⣿⣿⣿⣷⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠁⠃⠀⢳⠀⠦⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢰⠐⠡⢱⠀⣈⠂⣒⣮⣿⣿⣿⣿⣿⣞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⡎⢁⠕⠊⠓⠒⠊⡉⠀⠸⣿⣿⣿⣿⣿⣿⡷⡂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠀⠀⢰⠆⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠈⡟⠃⠐⡃⠢⠐⠏⠆⢨⡆⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠘⢄⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢶⣠⣁⢈⠑⠘⣴⣥⣴⣵⣿⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠂⠀⣀⣤⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠛⠛⠒⠛⠻⠟⠛⠛⠻⠿⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠿⠛⠛⠛⠛⠿⢿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⢋⣽⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⡙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠁⠀⢠⣿⠏⠀⠀⠀⢀⣤⣴⣶⡶⠂⠀⠀⠀⠀⠀⠹⣿⡄⠀⠈⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⢀⣿⡏⠀⠀⠀⣴⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠈⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⡀⠀⢀⣠⡆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⡿⠋⠛⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠻⣧⣀⣀⣾⣿⣿⣿⣿⠟⠀⠀⠀⣸⣿⠁⠀⠀⠀⢀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀⠀⠘⣿⣆⠀⠀⠀⠈⠛⠻⠿⠿⠟⠛⠁⠀⠀⠀⣰⣿⠃⠀⢀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣬⣻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣟⣥⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⡿⠗⠐⠂⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⡿⠛⠀⢀⠴⠉⠀⡄⠀⠀⠀⠀⠘⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠙⣿⣿⠀⢀⣰⠉⠀⠀⢀⣿⣦⡄⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐⠿⣿⣿⣿⢧⣤⣟⠀⠀⠀⢀⣾⣯⣿⠗⠀⣼⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣆⠀⠑⠂⣠⣾⣿⠋⠀⢸⣿⣿⣀⡴⠋⠁⠉⠉⠐⠿⣿⣧⣶⣖⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⢉⣿⣆⣀⡤⣿⣿⣿⣦⣴⣾⣿⣿⣿⠶⠖⠖⠒⠀⠲⢀⣽⣿⠿⢿⣿⣿⡏⢀⠀⠘⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣤⣶⡦⣤⣶⣿⣿⣿⣶⣾⣿⣿⣷⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠶⠺⣿⣿⡏⠉⢹⣿⡟⣉⣩⣽⣿⣮⠙⠉⢠⠞⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠩⣿⢷⣴⣾⣿⣇⢻⣿⣿⡟⠁⢠⣴⣿⡄⢀⣿⣤⣠⣿⣏⣿⣿⡏⠉⣻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠻⣿⣿⡏⠙⢧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⢷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣾⡇⣰⠟⣽⣧⠘⣴⣿⣿⡏⠙⣿⣿⣿⣶⡾⣿⠻⣲⡤⡀⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣿⣇⣾⡟⠉⣠⡾⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣧⣀⡀⠉⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣄⣀⡀⠀⠀⠁⢠⣾⣿⣋⣁⣸⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣅⣈⠛⢿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠟⣿⣿⡿⠀⠀⠀⠉⣙⣿⣿⣽⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣆⣀⣀⣥⢾⣿⠟
⠀⠀⠀⠀⠀⠀⠂⠁⣀⠸⣿⣿⣿⣿⡿⠃⠀⢀⡤⡶⠋⢉⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⠟⡙⠤⡀⠁⠄
⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⣈⣿⣿⡟⢁⠄⠈⠀⠀⣠⣴⣿⢿⣟⣿⠟⠻⣿⣿⣿⣿⣿⣿⣤⣾⣿⠄⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠻⢿⣧⣤⣤⣶⣶⣶⣿⠋⠀⣸⣿⣿⣦⣴⣿⢿⣿⣿⣿⣿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⠯⣿⣿⣿⣿⠟⡛⠻⣷⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣶⣿⣿⣿⣿⡏⠉⣿⣿⣿⣉⣾⢟⠁⠀⠀⢠⠍⠙⠀⠐⠀⠸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡛⢻⣿⣿⣿⣿⣷⣶⣿⣿⡗⠁⠁⠈⠁⣄⠀⡀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠉⡐⠋⠛⢻⠻⢿⠇⢀⣰⠀⠀⠂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡁⢸⡟⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⠀⠀⠀⠀⠀⠛⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⠀⠤⠘⣿⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢦⡀⠀⠀⠀⠰⡏⠃⠀⠀⠀⣿⠀⢀⡶⡫⠭⠉⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡆⠀⠀⠀⡷⠀⠀⠠⠀⡇⠀⣿⠈⠀⠀⠀⠈⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠀⠰⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⢨⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢀⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⠀⡇⢸⠃⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢰⡈⡏⠀⠀⠀⠘⣧⠀⠀⣼⡃⠀⠀⠀⣷⡍⠀⠀⠀⠀⠀⠘⠇⢀⠶⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⣻⠶⠿⡀⠐⠠⠖⠸⡇⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠀⠀⠀⢿⣆⠀⠀⢀⡼⠋⠀⠐⠆⣀⠟⠀⠀⠘⢦⡀⠀⢀⡴⣱⠇⠀⣠⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⡆⠀⠀⠀⠘⣏⠓⢾⠏⠀⠀⠀⣠⠞⠁⠱⣤⡀⠀⠈⠙⠲⠞⢡⡿⠂⠁⠀⢀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠐⣲⢶⣷⣶⠶⠒⠲⠓⠚⠁⠑⢀⣴⣿⠁⠀⠀⠀⢈⡗⢦⣀⠚⠀⠀⠸⣇⠀⠀⠀⣏⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣀⣆⠂⣼⠇⢸⡆⠀⠀⠀⢸⠃⠀⠻⠀⠀⢠⣤⣬⣙⡛⢾⡋⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⠟⢻⡆⣿⠀⠀⣷⠀⠀⠀⣾⠀⠀⠀⣆⢠⢿⢦⡀⠈⠉⠉⠻⡤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠉⣱⣾⣡⠴⣻⢅⡏⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⡷⡜⠀⠙⠻⣶⣀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠂⠈⠁⡹⠟⠁⠀⢠⡟⣸⠇⠀⠀⠈⡇⠀⠀⡇⠀⠀⢀⣦⠥⠀⠀⠀⠠⠈⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣾⠶⠋⠀⠀⠀⠀⡏⠀⠀⣇⠀⠀⠁⡉⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣤⠤⠖⠂⣨⠟⠁⠀⠀⠀⠀⠂⣼⠃⠀⠀⠹⣆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠈⠳⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣄⣀⡀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣄⠈⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡟⠀⣴⣿⣿⣿⣿⣿⠟⠋⢁⣀⣤⣤⣄⣀⠈⠛⢿⣿⣿⣿⣿⣦⠀⢻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠀⣸⣿⣿⣿⣿⡿⠁⢠⣶⣿⣿⣿⣿⣿⣿⡿⠆⠀⠙⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣶⣴⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣽⣿⣿⣿⡟⠀⣼⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣷⣄⡀⠙⠛⠿⠿⠿⠛⠋⢀⣤⣾⣿⣿⣿⡿⠁⣰⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⠟⠀⣰⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⢛⣿⣿⣿⣿⣿⠀⡛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⣰⣿⣿⡿⠛⠻⣿⠀⣷⡀⠈⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣏⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢈⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠹⣿⣿⣷⣤⣴⣿⠀⡿⠁⠀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣬⣿⣿⣿⣿⣿⠀⣴⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠞⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⣡⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣌⠲⣤⣀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢃⣾⣿⣿⠟⡛⠿⢰⣶⡆⠻⢛⡻⣿⣿⣷⡜⢿⣷⣦⣀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⢃⣿⣿⣿⣇⠺⣿⣾⡿⠿⠿⣷⣿⡿⣨⣿⣿⣿⡈⣿⣿⣿⣷⣄⡀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢸⣿⣿⡏⣭⣼⣿⢃⣶⣿⣷⡌⢿⣷⣬⡙⣿⣿⡇⢹⣿⣿⣿⣿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣇⡛⢻⣿⡘⢿⣿⡿⢃⣾⡿⢛⣁⣿⣿⡇⣸⣿⣿⣿⣿⣿⡇⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⡏⣴⣿⣿⣶⣦⣶⣿⣿⣧⢙⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⠡⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⡘⢿⣿⣿⣮⣡⣦⠸⡿⠏⣴⣌⣡⣾⣿⣿⢋⣼⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣌⢻⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⡟⣡⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢻⣿⣷⣝⢿⣿⡿⢋⣴⣿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠂⣿⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⣿⠄⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠿⠃⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⣻⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣉⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠀⡀⠀⢘⠇⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠑⠢⣇⠇⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠱⢟⣼⣶⠒⢐⠂⠀⣾⣇⠀⠀⠀⠀⢠⢰⠃⣼⠂⢀⡀
⠀⠀⠀⠀⠐⠐⢑⣤⣶⡀⠀⢀⣀⠀⢈⣽⣁⡴⠏⣠⣄⡈⠹⣦⣄⠀⠀⠸⣾⠆⣟⣠⡟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠙⠛⠴⠋⠉⣟⠋⢠⣾⠿⠛⠳⢦⣀⡈⠝⠳⣦⣄⣈⣿⣿⠒⠊⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠐⠊⢿⣁⣄⣀⠠⠔⢫⣼⡟⠁⠀⠀⠀⠀⠈⠉⠻⢤⣜⠿⣿⢿⠧⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢩⠴⠖⣻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣎⡀⠈⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠉⣤⠖⠀⠰⢏⡰⢫⡝⣿⣦⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣷⣎⢿⣷⢄⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡟⡜⣹⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠉⣸⣶⣶⣙⣿⣇⣀
⠀⠀⠀⠀⠀⠀⢃⠔⠀⢁⢰⣿⠿⣽⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢧⣸⣿⡇⣼⡇⢫⠉⢦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠼⡸⡧⢷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠸⠇⢷⠸⢳⡘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠘⠃⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⠁⠀⠠⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠼⣆⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀
⠀⠀⠀⠅⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠙⠀⠀⠈⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⢀⠊
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢳⡰⢃⡠⠄
⠀⠀⠀⠀⡀⢭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠓⢄⠀⡄⠀⠀⠀⠀⠀⠀⣸⢗⠁
⠀⠀⠀⠀⡑⢄⠃⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠖⠚⠿⣧⡀⠀⠀⣀⠤⣺⠃⠈⠑⠂⠀⠒
⠀⠀⠀⠀⠙⢀⣲⡀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠀⠙⢦⡠⣿
⠀⠀⠀⠀⢤⣄⡘⠧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⡇⠀⠀⠀⠀⠘⣷⠀⠖⠊⠁
⠀⠀⠀⡸⠨⠴⠀⠀⠀⣶⠤⠄⠀⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢡⡸⡟⠓⣶⢀⡼⠸⠀⠀⠀⠀⠀⣰⠲⠂
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠟⠠⡀⠀⠀⠀⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⢉⣻⡊⢱⠉⠀⠀⠦⠀⠀⠠⠒⣧⠆
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠒⠙⠉⠀⠉⢦⡀⠀⠀⠀⠂⠤⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⡶⠶⢄⢀⢄⠰⣤⣀⣠⣂⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⠀⠃⡸⠉⠈⡀
⠀⠀⠀⠑⠂⣄⠀⠀⠀⠀⠀⡏⠀⠀⠀⠠⡀⠆⠀⠀⠀⠀⠀⠒⠈⠁⠸⣇⠤⠒⠣
⠀⠀⠀⠀⠀⠉⠲⢤⡄⠀⡰⠇⠀⠀⠀⠀⠑⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⡱⠤⣄⣤⣾⠁⠀⠀⠀⠀⠐⢤⡄⠀⠼⠀⢀⠀⡤⠘⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠔⠉⠀⠬⢇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠘⠀⠀⠀⠀⢈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠜⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢠⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠛⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣴⣖⣊⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡵⠶⠶⠞⠛⠛⠳⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣷⣦⣄⣀⡀
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠻⠶⢦⣄⣀⣴⠞⠋⠉⠉⠀⠉⠉⠙⠳⣦⡀
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⢠⡿⠁⠀⠀⠀⠀⣠⡞⠁⢠⡴⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠈⢷⡀
⠀⣾⠁⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⣀⠀⣼⠃⠀⠀⠀⠀⣴⡿⠋⠀⠀⠙⢿⣦⠀⠀⠀⠈⣷
⢸⡇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⣀⣀⣨⡿⠟⠛⠛⠉⢻⡏⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠋⠉⠉⠉⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀
⢹⡇⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠛⠷⠟⠻⢶⣄⣀⠀⠀⠘⣧⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣸
⠈⢿⡀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⠛⠛⢻⣆⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡿⠃⠀⠀⠀⢠⡟
⠀⠘⣷⡀⠀⠀⠀⠀⠹⣦⡀⠀⢶⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣰⠟
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣤⣤⣤⣤⣤⣤⡶⠾⠛⠉⠈⠛⠶⣤⣤⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡶⠤⣤⣤⣤⣤⡤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⢀⣿⣇⠀⠀⣀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⡀⠀⢸⣦⣾⡿⣿⣦⣾⠀⠀⡀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢤⡆⣉⣻⣶⠟⡟⠁⠀⠀⠙⡟⢷⣾⣋⣁⣦⠄⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠐⣤⠽⣺⠟⢡⡾⣻⠀⡇⠀⠀⠀⠀⡇⢸⡿⣦⠙⢷⡚⢥⡔⠈⢧⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⡀⠐⠙⢠⠿⡐⢻⠂⡇⠀⠀⠀⠀⡇⢺⠓⣸⢧⠸⠑⠀⡈⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢰⠃⠐⣧⢤⡿⠒⢚⠟⢸⢷⢬⠀⡇⠀⠀⠀⠀⡇⢨⢴⢿⠈⢟⠒⠺⣧⢤⡗⠀⢳
⠀⠀⠀⠀⢀⠏⠀⠚⠃⠘⠁⢠⡞⡆⠉⠈⠧⠀⡇⠀⠀⠀⠀⡇⠠⠏⠈⠁⡞⣦⠀⠙⠀⠛⠂⠈⢧
⠀⠀⠀⠀⡼⠀⠀⠤⣼⠤⣴⢋⡇⠹⡖⠄⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠔⡾⠁⣏⠳⡤⢼⡤⠄⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠐⠚⢁⣼⠁⡼⣧⠀⠟⠦⠀⡸⠁⠀⠀⠀⠀⠹⡀⠠⠞⠇⢠⡿⡄⢹⣄⠙⠒⠀⠀⡇
⠀⠀⠀⠀⣇⠀⠀⢉⠟⠈⠃⠰⣃⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⡱⠀⠋⠘⢏⠁⠀⢀⡇
⠀⠀⠀⠀⠈⠢⠤⠤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⠤⠊


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⠤⠤⣤⡴⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⡰⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠳⣼⡃⠀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠀⣀⡀⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣇⡠⣤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠐⠀⠀⠀⠀⠏⠀⠀⠀⢠⠀⠀⠀⠁⠁⠀⠤⠃⠾⢀⠀⠀⠂⠠⡤
⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⢀⠁⠀⠀⡴⢼⣀⡴⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠈⠀⠙⣄⣀⣠⣀⣁⢦⡀⢀⡓⠂⡍⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⢰⠐⠀⠀⠐⠒⠉⠡⣀
⠀⠂⠀⠀⠁⠀⠀⠈⠉⠀⠀⠉⠄⠁⠀⡀⠀⠐⠀⠀⠀⠒⠀⠢⠄⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠀⠁⠀⢀⡄⠀⠶⠖⠈⠉⠂⠤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢂⠀⠀⠀⡅⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢒⠥⢤⠀⣀⡠⠋⠀⠀⠀⠀⣠⠚⠃⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠉⠀⠐⠉⠝⠠⠀⠀⠀⠀⠰⠂⠂⢦⠀⠀⠀⠈⢹⡃⠀⡀⠀⠀⠀⠀⣀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠶⠤⠄⠀⠈⢂⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠛⠁⠀⠈⠀⠀⠀⠀⢣⠀⢀⡤⠞⠉⠁⡀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠋⠈⠠⣇⠂
⠀⠀⠀⠀⠈⢙⡏⠀⠉⠓⠀⠀⠀⠰⠂⠀⠀⠀⠀⢠⣀⠤⠔⠁⢀⠀⢤⣀⡤⡄⠀⠀⠘⠀⠀⠀⣀⠙⠀⠀⠻⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠨⣜⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠄⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠉⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣀⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⣿⣿⢻⡶⣶⡾⠟⣿
⠀⡴⠋⠁⠀⠀⣤⡀⠈⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢏⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⠱⡁⣰⢶⢸⣿
⠐⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⣆⡙⠃⡌⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠸⣿⠛⠻⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻


⠀⠀⠀⣀⡀⠀⡇⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⡠⠐⠢⡄⠀⠀⢠⠔⠒⢄
⠀⢀⣾⣿⣯⠀⡇⣼⣿⣿⣀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡜⠀⠀⢀⡷⠄⠠⢾⡄⠀⠀⢡⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄
⢀⣿⣿⣿⣿⡟⠳⣿⣿⣯⣿⡆⠀⠀⠀⠀⠀⣹⠀⢀⣿⡄⠀⣿⠀⠀⠀⠀⡆⠀⠀⠃⢹⠀⠀⡏⢘⠀⠀⢠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣼⣿⢿⣿⣿⡇⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⠿⣿⣤⣿⡿⠟⠀⠀⠀⠀⠈⠂⠐⠁⡏⠀⠀⢸⠈⠂⠐⠃⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠿⢿⡾⠿⠟⠁⠀⠛⠿⢿⣞⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⣠⣴⣶⣶⣶⣶⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠄⢀⠈⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠈⢻⠄⠀⠀⠀⠀⠀⢴⣷⣲⠀⣘⣾⡦
⠀⡿⠟⠛⠛⠋⠉⢉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢰⣟⣛⣲⠀⣒⣚⡫⠆
⠀⠀⠀⠀⠺⠶⠬⡍⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⢰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠻⡄⠀⠀⠀⠀⠀⠸⠀⢂⡟⠿⢿⡐⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⡿⠀⠀⠀⠀⠀⢫⡰⠋⠨⠤⠀⠹⣦⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣠⣤⣀⠩⠭⢀⣠⣤⡀⠀⠀⠀⢠⣄⠈⠂⠀⠀⠀⠀⠀⠐⠊⢀⣤⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣤⣤
⠀⢾⣿⣿⣿⣮⣥⣿⣿⣿⣿⠀⠀⠀⠈⠉⠒⠈⠀⢀⣀⣠⠀⠈⠒⠊⠉⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢰⣷⢻⢠⣀⡄⡍⣶⡆
⠀⠀⠙⢿⣄⠻⡟⢀⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠘⣿⡌⣼⣿⡇⢠⣿⠇
⠀⠀⠀⠀⠙⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⠿⠃⠈⠁⠀⠀⠀⠀⠀⣿⣷⣽⣿⣧⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⠀⠀⢸⠀⢸⠀⠀⢀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠑⡄⢸⠀⢸⠀⡔⠁⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠀⠸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢇⡀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠉⠑⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠉⠁


⣏⣈⣯⣯⣟⣯⣿⣽⣽⣽⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣢⣠⣄⣤⣀⣀⣀⠀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⣤⣠⣤⣷⣾⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣷⣧⣇⡏⡬⡬⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡉⠄⢀⠀⠙⠉⢳⢽⡽⣝⡟⡿⡿⣽⣱⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢬⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣯⣕⣥⣮⣢⣰⣝⢝⢜⢞⣞⡽⡽⣳⢯⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⠯⣳⣿⣽⣿⣿⡿⠛⢍⣩⣻⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣺⢾⣕⡳⣏⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠔⢪⠒⣽⢾⣻⣿⡿⠱⣪⣠⠾⠟⠉⠙⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣗⡽⣻⢟⡳⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠊⠐⠭⣚⢕⣿⠟⣡⣜⣿⠇⠀⠀⠀⢳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣗⣯⣾⡱⢎⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠊⠔⡰⡩⣿⡚⠁⠀⠀⠀⠐⠤⣤⡀⣉⢛⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡺⡻⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠝⠑⡃⠀⠀⠀⠀⠀⠈⠈⠑⠋⠻⠿⠿⣻⠿⣻⢿⡟⠛⠍⠁⠁⠑⠐⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠋⣀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊
⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠂⠒⠀⠒⠠⠤⣀⡘⢧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡶⠉⠉⠁⠀⢀⡀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡤⠞⠂⠹⣦⢄⣀⠀⠀⠀⠀⢀⣿⠿⠟⠒⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠈⢉⡶⠒⠒⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣟⢲⡇⠀⠀⠀⠀⠀⢹⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠙⡅⢸⢾⡁⠄⣀⠀⢀⣄⣸⠇⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⢾⠀⠀⠀⠀⠀⠘⢸⠀⠳⣄⢘⣾⠉⢙⢟⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣆⢀⠀⠀⠀⢀⣸⠀⠀⠈⠑⡎⠉⢻⠋⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠃⠀⣀⠌⠀⠀⢀⠀⣀⠗⠈⠙⠻⠗⠶⢤⣀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠐⠋⠁⠀⠀⠀⠀⣳⠇⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠒⣀⣀⣠⣴⠆⡰⣄⢠⡜⠛⠳⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⡇⠈⠙⡆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣀⡀⠘⡖⠀⠊⠐⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠚⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀⢰⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡞⠤⣤⠠⣔⡁⠘⠃⠀⠠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠋⠉

⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣀⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⠀⡼⠿⡿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣭⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣖⣶⣆⣿⣿⡿
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣏⣭⣽⣯⣁⣯⢩⡏⣹⣻⠻⢷⣿⣦⣴⣄⢠⣀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⡁
⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠼⠏⠹⠿⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣴⠿⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠻⠛⠛⣿⣿⣿⣿⣿⣿⡇⠉⠙⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠾⣛⣿⣿⣿⠉⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⣿⣿⣿⠏⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠁⢀⣠⣾⣿⡿⠛⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣂⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣠⣴⣖⣿⣿⣿⡿⣿⡛⠛⠈⠙⠛⠛⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣀⣀⢀⣀⡈⠉⠉⠁⠉⠀⠀⠈⠀⣀⠐⠾⠍⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠃⠊⠁⠀⠀⠈⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣯⣿⣿⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠑⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠸⣯⣼⣿⠿⠿⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣠⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢡⠀⠀⠀⠀⠒⠀⠀⢘⠀⠀⡿⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣦⠀⢣⠀⠀⢰⠀⢆⠀⢂⠐⣄⡇⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡀⠀⢣⠀⢸⠀⠀⡂⠀⢣⡞⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠰⣧⠂⢸⠀⠈⢢⠀⠅⡠⣞⠄⠐⠀⠀⠒⠠⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢂⡏⢀⠏⣀⣀⣨⣷⣎⣉⣡⣠⠤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠰⣄⠀⢳⠀⠀⠀⣠⠔⠊⠉⠉⠺⣷⡴⢓⡿⠿⠷⠿⠥⣭⣷⣌⣂⡀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⢀⠆⠀⢰⠃⢀⠄⣄⣴⣾⣿⡿⠉⠏⢲⡀⠀⠀⡇⠉⠙⠲⣌⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⢀⡼⠖⠒⠛⢶⣞⡛⠋⠉⠀⠀⠉⡦⠚⠉⠉⠉⠑⢼⣄⠀⠐⠈⠳⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡿⠁⠀⠀⢀⣀⡀⠘⣇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡆⠑⠤⢄⣀⣀⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣇⠀⠀⠀⡏⢀⡗⠀⣸⢠⣤⡄⠀⢸⡀⠀⠀⢠⠄⡄⣠⠇⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⠁⠘⢦⡀⠀⠈⠉⢀⣴⠋⠓⠫⠴⠖⠋⠙⠦⣦⣭⡼⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⡤⠴⠞⢭⣙⠓⢒⡚⠉⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡕⠁⠀⠀⠀⠀⠈⠙⠶⢬⣟⣲⣶⣶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢰⣼
⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠋⠉⠀⢀⣀⣀⣀⣀⡀⠀⠉⠙⠛⠿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⢀⣠⣶⣾⣿⠿⠿⠿⠿⠿⠿⣿⣷⣶⣤⣀⠀⠀⠀⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢠⣴⣿⠟⠋⠁⠀⠀⠀⢀⣀⣀⣀⠀⠈⠉⠛⢿⣿⣦⡀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣷⣦⣄⡀⠈⠻⣿⣦⡀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀⠈⢿⣿⡄⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠈⢻⣿⡄⠀⢿⣿⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠁⠻⠋⣠⣾⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠈⣿⣷⠀⠘⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢠⣿⡿⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢸⣿⡆⠀⢿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠘⣿⣷⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢸⣿⣧⠀⠀⠀⠙⢿⠗
⠀⠀⠀⠀⠀⠈⢿⣿⡀⣤⡄⠙⢿⣷⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⢀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠉⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⣼⣿⠃⠙⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⢀⣾⣿⠃⠀⠀⠀⠈⠻⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⡿⠟⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠘⠿⣿⣦⣄⡀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣀⣤⣾⣿⠟⠁⠀⠀⠀⠀⠈⠙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⠀⠈⠙⠿⢿⣿⣶⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣄⣀⠀⠈⠉⠉⠉⠉⠁⠀⢀⣀⣠⣤⣶⣿⡿⠟⠛⠛⠛⠛⠛⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢁⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⢀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠁⠀⠀⠉⠉⠛⢶⠔⠃⠀⢸⢸⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡰⠁⣠⠞⠛⠻⠾⠿⠷⡿⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡄⣄⠀⣰⠀⠀⢀⣀⠼⠊⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠉⠻⠶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡾⢁⣀⡠⠤⠤⣿⣿⣀⠿⠒⢉⣁⢤⠖⡃⠰⠓⠬⠙⢀⣀⠀⠘⠛⢋⣒⡒⠠⢄⡀⠀⠀⠚⠋⠉⠉⠉⠉⠙⠛⠳⢦⣄
⠀⠀⠀⠀⠀⣾⠉⢀⠽⠓⠒⡉⡸⠇⡠⠖⠛⠫⣦⠎⠰⠇⡏⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠈⠑⢆⣬⠷⠤⢀⣀⠀⣀⠀⢀⠤⠶⠶⠶⠾⠷⣤⣀
⠀⠀⠀⠀⢸⠇⠀⠀⠰⣄⢀⠟⣠⢊⢶⠇⠀⠀⠈⣆⡃⠃⡋⠉⠩⠈⠁⠐⠀⠀⢰⡄⠀⠰⠀⠀⠙⠓⠀⣠⣂⢀⣰⠦⠀⠂⠀⠐⠀⠀⠁⠀⠉⠛⢦
⠀⠀⠀⠀⣿⠀⡀⠠⡢⣙⢏⠜⡆⢳⠃⠀⠀⠀⠀⠟⢇⡄⡆⠀⠀⠀⠀⠀⡔⡇⢈⡧⡰⢡⠀⢪⠍⡪⠖⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠛
⠀⠀⠀⢀⡏⠀⢙⣒⣼⡏⡎⢀⡐⡜⠦⣀⣀⢀⣀⣘⡈⢄⠀⠀⠀⠀⠀⡐⠀⠇⣊⣘⠁⠫⠤⡂⣉⡹⠕⠂⠀⣀⣀⣤⣤⠶⠟⠋⠉
⠀⠀⠀⢸⠇⠀⠀⠀⠰⡘⡰⠋⠈⠹⣙⢶⣠⠴⠒⠢⢥⡀⠑⢄⠀⣠⢎⡴⠒⠪⠓⠀⠄⠄⣀⠈⠉⣁⣠⡶⠟⠋⠉
⠀⠀⢀⡟⠀⠀⠀⠀⣧⡇⡇⠀⠀⠀⢷⠋⠀⠀⠀⠀⠀⠈⢦⠈⡶⣡⡎⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠋⠁
⠀⠀⣼⠁⠀⠀⠀⣠⡏⡁⡹⣄⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠘⡰⢱⡇⡇⠉⣙⣻⣿⠟⠋⠉
⠀⣸⠃⠀⠀⣠⠞⠁⢿⡻⢠⠊⠑⠒⠦⠤⢄⡣⠤⢀⣀⠀⢀⢇⠇⢿⠻⣟⣟⣿⢇
⢰⡏⠀⢠⠞⢁⣑⣚⡿⠂⠁⠀⠀⠀⠀⠀⠀⠉⠛⢒⣛⢗⣾⣼⠀⠈⠧⣀⠀⠙⠊⡄
⣼⠀⢠⠀⠨⠀⠟⠝⠁⠀⠀⢀⣀⣀⣤⣤⠤⠶⠞⠋⠁⠈⡇⣿⣆⠀⢤⣄⡉⠒⠤⡘⡀
⣿⠀⡃⠀⢀⣈⣤⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⢿⡿⣄⠀⢣⠹⡒⢄⠀⠐⡀
⠹⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠸⠀⢸⣦⡀⠱⡳⡀⠁⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠃⡆⠀⠈⠛⡟⢄⠀⡑⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠇⢠⠁⠀⠀⠀⠇⠀⠢⡐⡈⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣼⣤⣤⣤⣵⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⢀⠀⠀⠈⡃⠐⡇⠀⠀⢰⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠈⢇⠀⠀⠇⠚⡇⠀⢐⣿⡿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡙⠀⠀⠀⠀⠀⢘⠆⠀⡇⢸⡇⢰⣷⣿⣏⢸⣝⢛⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⡔⣇⡀⠉⠈⠡⢾⣿⣷⣦⣬⠛⠸⢯⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⡆⠀⠀⠀⠀⠀⠘⠀⣿⣠⠞⠙⠂⣤⣀⡈⠉⢉⣤⣻⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⢯⠀⠀⠀⢀⣿⡿⣿⡦⠈⠛⢛⢉⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣄⠀⡀⠀⠀⠀⠃⢸⡀⠀⠀⢸⣿⢷⡂⣴⡗⠐⣭⡛⢿⣻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠙⢷⣀⡀⠀⠀⡀⢸⡇⠀⠀⠸⣿⣿⣷⣧⣼⡇⣇⠸⣯⠿⡿⣧⠆
⠀⠀⠀⠀⠀⠀⠰⠠⢪⠓⠀⠀⠀⠈⢙⠳⣼⣿⡿⠁⠀⠀⠀⢿⣿⣷⣿⠟⣁⣿⢀⢜⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠂⣬⣷⡀⠀⣀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢈⣿⣿⣷⢿⣿⢣⣿⠘⣏⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣼⠈⢳⣄⠈⠻⡓⠰⢴⣼⠀⠀⠀⠀⠀⠈⢾⡤⣿⣿⡏⣾⣿⣰⣜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠂⡀⠹⢶⠘⠪⣭⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣽⣽⣧⣿⡿⢻⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⣆⠙⠠⠈⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⢁⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⢀⡼⠒⠀⠀⠀⠢⠄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⡞⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⢀⠜⢀⡴⠚⠋⠉⠙⠓⢄⣠⣦⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⣇⡿⠶⠴⢶⠚⠁⠀⠀⠀⠀⠀⢀⡤⠂⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⠨⡝⣶
⠀⠑⢄⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠑⢮⣧⣾⠋⠀⠀⠀⠸⠀⠀⣀⣠⠴⠶⡿⡁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣩⣹
⠀⠀⠀⢉⡿⡿⡏⠀⠀⠀⠉⠙⢶⡖⠒⠛⢿⡐⢦⣤⣀⣀⠔⠊⠁⢀⣀⣀⡱⣳⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⢰⡛⠲⠇⠙⣦⡀⠀⠀⠀⠀⣷⠀⠀⣮⢧⡀⠈⠙⢽⡴⠖⠛⠋⠉⠙⠛⢷⣭⣟⢹⠿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣞⠉⠀⠀⠀⣼⠟⠀⠀⠀⠀⣿⢀⠠⡿⢧⣑⢦⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢩⣣⣈⠳⣤⣀⣀⣀⣀⣴⠟⠁
⠀⠀⢹⣧⠖⢶⡒⠋⠀⠀⠀⠀⣼⠿⠗⠊⠀⠈⢹⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠛⠉⠉⠙⠻⣯⡉⠁
⠀⠀⠈⢻⣄⡀⠀⠀⠀⠀⣠⡾⣃⣤⠶⠖⠒⠺⠿⣄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠈⢙⣒⠶⠾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣟⢶⣤⣀⣀⣀⣀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⡠⠞⠉⠀⡠⢾⡇⢸⠃⠀⣠⣤⣤⣄⠀⠀⠀⠀⣿⡄⠀⠉⠉⠉⠉⠉⠀⢿⡄⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢠⠎⠀⠀⢀⠞⠀⣸⠀⠈⣇⠀⡟⠀⠃⣈⣷⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⣒⠀⣤⣰⣿⡇
⠀⡎⠀⠀⢀⠎⠀⢠⡏⠀⠀⠈⠻⣷⣤⣤⠬⠟⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋
⢸⠀⠀⠀⠌⠀⢀⡾⠁⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠸⡆⠀⠀⠀⢠⡾⠁
⠀⢳⡀⢀⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢤⣀⠀⠀⠲⣤⣀⣉⣻⣶⣶⣿⣿⡿⠿⡿⠿⠿⠿⠿⠿⣷⣦⣄⡀
⠀⠀⠠⣄⡀⠙⠶⣤⣀⡙⠷⣦⣭⣛⣷⣶⣴⣿⣿⠿⠛⠋⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢝⠓⠦⣀
⠀⠀⠀⠈⠛⠷⣦⣬⣙⡻⠷⣶⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢣⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⣭⣭⣿⣶⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠴⠖⠚⠛⣛⣻⣷⣦⡀⠀⠀⠘⡄⠀⠀⠀⠈⠙⠻⠿⠟⠋⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⣉⣉⣻⣿⣷⣤⣀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⣀⡤⠾⠿⠿⣷⣶⣬⣔⣤⣀⣀⣀⣀⣀⠤⠒⢁⣀⣀⣀⣀⣤⣤⣤⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠛⠋⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉







⡶⣶⢶⠶⠶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠂⠀⠀⠀⠀⠀⠀⠈⣩⢝⡩⠿⠛⠛⠛⠟⠛⠛⡛⠛⠲⡒⠒⠲⠦⠤⢤⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢂⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⠏⠁
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡨⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡴⠊⠁
⠀⠀⢸⠇⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠴⠋
⠀⠀⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣢⠴⠚⠁
⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣧⠀⠀⠀⠀⠀⣢⠴⠒⠉⠉
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣿⣔⠠⢂⡤⠚⠁
⠀⠀⠸⡅⠀⠀⠀⣈⠤⣶⠤⢤⡤⣤⡴⡟⢰⠇⠈⠉⠉
⠀⠀⠀⢷⣄⠧⣠⠋⠀⢣⡤⠴⠖⠉⠀⠙⠋
⠀⠀⠀⠘⣿⢰⠃⢄⡴⠋
⠀⠀⠀⠀⠘⠾⠤⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣦⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠛⣻⣿⠿⠟⠛⠋⠉⠉⠀⠀⠻⠿⠟⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿
⣟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠙⠛⠻
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⣏⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢁⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣷⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣉⣉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣖⣯⣿⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠠⠂⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣧⠀⠀⠀⠰⢿⠠⢀⡞
⠀⠀⠐⠉⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢨⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀⠀⢺⡄⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠒⢂⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣀⢻⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠛⠥⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠤⢤⠤⠶⠎⠊⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣠⣶⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠢⣄⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⣤⠚⠛⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠑⢦⡀⠀⠀⠀⣠⣿⣿⣿⣒⣤⣀⣀⡀⠁⠀⠀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢤⣀⡀⢀⡞⡟⢧⣤⠖⠋⠉⠈⠉⠛⢿⣄⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⡆⠀⠀⠀⠀⠀⠙⡏⠀⡇⡾⡏⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⣼⡀⠀⠀⠀⠀⠀⠀⠃⢠⠈⡃⠀⢄⠀⠀⠀⠀⠀⠀⡿⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢹⣆⡁⠀⠀⠀⠀⠀⠂⡜⠀⢻⡀⠀⣽⣥⢦⣶⣇⡜⠅⣰⠖⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠠⠙⠂⠀⠀⠀⡴⣴⠁⠀⠀⠻⢆⢳⣯⠀⢠⠻⢁⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠆⣀⣤⡠⠄⠊⠀⠈⢠⠀⠀⠀⣼⠁⠁⢹⡿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢀⡀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⡴⠛⠋⠙⠻⣦⣧⣀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⢂⣀⣠⠀⣦⣸⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠏⠀⠀⠀⠀⠀⠈⠂⠙⢳⡀⠀⠀⣀⠔⠈⠁⠀⠀⠀⢲⣤⠓⠈⠉⣦⠀⠈⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡀⢦⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠵⠀⠀⠊⠀⠀⠀⢀⡦⠞⠁⢰⠤⠀⠉⠀⠀⣀⣴⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⢾⣶⣦⣤⣴⡾⠄⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢠⡶⠋⠉⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⠊⠳⣼⠶⠛⠛⢷⡞⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢸⡆⠠⠏⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣸⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠞⠢⣄⠀⠰⣇⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠋⠛⠛⢤⣠⣯⡑⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠇⣴⠏⠀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⢟⠁⣹⡗⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⡀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠏⠀⢸⠆⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢣⣄⠀⠀⠀⡤⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣽⣧⣹⡏⠀⠘⠻⠁⠀⠀⢀⣀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠲⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣿⠤⠤⣿⣿⠀⠀⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠁⠀⣏⠀⠀⠀⠀⠀⣿⠒⠲⣿⣿⠀⠀⠀⠀⠀⠀⠷⠀⠀⠴⠇⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡰⡞⠂⠀⢈⡏⢠⠀⠀⠀⣿⡉⢙⣿⣿⠀⠀⠀⠀⡄⣫⡅⠀⠙⢣⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⣹⠀⠚⣇⣈⡟⢁⡀⠀⠀⢿⠀⣹⣿⣿⠀⠀⢀⣠⡈⢯⢀⡞⠃⢸⡇⠀⣐⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⡙⢦⢿⠀⠀⠈⣾⡠⣽⠋⢀⢀⣿⠀⣼⣿⣿⣆⠀⡌⠙⣧⢬⣇⠈⠀⢸⡇⡞⢁⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣀⣄⣸⣧⣜⠘⣇⢀⢀⣭⡹⢅⠑⣫⠞⠃⣀⣼⣿⡿⢛⠷⣄⣉⠔⢢⢥⣀⠀⡾⠁⣤⡿⣠⢤⣀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡁⠸⠟⠛⠻⣄⠛⣴⡏⢀⣷⠴⠛⢁⣦⣷⣿⣿⣿⣷⣧⣆⠘⠙⠶⣯⡀⢹⡖⠃⣸⠟⠛⠻⣢⣄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠠⡄⣀⣼⣄⠛⡟⢋⠁⣄⣼⣿⡿⠟⠉⠀⠀⠙⠻⢿⣿⣦⣄⠈⡙⠿⡃⣤⣧⣄⠠⠀⠹⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠞⠉⠉⢻⡏⣽⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣾⣿⣆⣹⡏⠀⠈⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠀⢦⡀⠐⠿⢆⣀⠤⣊⣫⣍⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣥⣆⣑⠤⣀⡼⠛⠀⢠⠆⠀⢀⡎
⠀⠀⠀⠀⠀⠤⠀⢢⣼⣗⣠⣤⣻⣤⣾⠛⣤⠞⠛⣉⣾⡏⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢻⣷⡙⠛⠳⣄⢻⣦⣼⣍⣤⣀⣾⣧⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡽⠋⠁⣠⣾⡇⢀⣆⣽⡟⠀⢀⡖⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢳⠀⠈⣿⣏⣵⡀⢹⣶⣤⡈⠙⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠞⣭⠟⣡⣶⠟⣹⠋⢡⣮⣼⡿⠟⠶⢎⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⡽⠖⠻⣿⣦⣷⠈⠹⡏⠻⡶⡌⢻⣬⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠞⠀⢀⡀⠀⣀⢐⣵⠏⣼⠁⠀⣴⠋⡍⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠛⡦⠀⠘⣇⠛⣦⡀⡀⠀⢀⠠⠀⢃⠐⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⣀⣬⣵⡗⢦⡙⠁⢷⣿⠀⠁⠛⠄⠙⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠃⠘⠣⠈⠀⣿⢰⠈⢯⡽⣶⣯⣤⣄⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠁⠘⠛⡞⠉⢀⣿⣹⢆⠀⣼⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣧⣀⠸⣝⣗⠀⠉⠻⠙⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣴⠋⠛⠀⠙⢋⢝⣿⠙⢄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⢀⡼⢹⣿⣩⠙⠃⠀⠉⠹⣦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠏⢹⠀⡀⠀⡠⢪⡿⢹⡆⠨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣸⠟⢻⡡⠀⠀⣀⠐⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣈⣻⡿⣻⡏⠀⣱⣎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆⠀⣿⡏⣷⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠲⡦⠔⠺⠋⠠⣱⠾⢷⡀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣠⢏⢿⣔⠈⠙⠃⠲⢄⠐
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡠⣠⡾⣿⠀⠈⣟⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡴⢋⡟⠀⠀⣿⢷⣠⡄⣀⣀⣉
⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⢰⡏⠀⠃⠀⠀⢮⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠿⠀⠀⠉⠀⢻⠄⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠾⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠠⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠤⠴⠶⠶⠶⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⣀⡤⠖⠒⠛⠛⠋⠉⠙⠛⠛⠛⠓⠶⠶⣶⠖⠋⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲
⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢀⡴⠋
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣇⣀⣀⡤⠞⠉
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠙⠦⣄
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⡆⠀⣀⡴⠃
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⠉⠀⠀⠀⠙⠋⠁
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⣇⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁
⠈⠓⠒⠒⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣠⢻⣥⣀⡤⠅⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡖⡆⠩⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠐⢯⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⣶⣦⣄⡆⠀⣟⣴⡿⠒⢀⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠟⠋⠉⠚⠥⣐⡒⠛⠛⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢉⠠⠂⠀⠀⠀⠀⠀⠀⡄⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⣶⠻⣄⡀⠀⠀⠀⠀⠀⡠⡞⣰⣟⢦⡀⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⡰⣻⣦⠀⠀⣀⡤⠂⠀⠀⠀⢰⡿⢿⣗⡙⠀⠄
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢘⣼⣿⣿⣇⠀⠩⡤⠀⠀⠀⠀⣿⠃⣾⡇⠰⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⢻⣿⣿⠀⣀⠆⠀⠀⢠⡄⠋⢸⣿⡇⠀⠁⠀⠀⢰⣦⡄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣸⢻⣿⠀⠘⠇⠀⠀⠘⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣼⠇⠁⣾⡿⠀⣿⡄⠀⠀⢀⠀⠀⣼⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⢀⡿⠀⢰⣿⡇⠀⢸⣿⣤⣤⣿⠀⠀⢸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠛⠀⠈⠁⣰⢟⡟⠃⠀⠈⢻⣿⢹⡟⠀⠀⠈⡃
⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠁⢀⠀⠀⠀⣿⢘⡇⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠈⡄⢹⣦⣿⠟⢻⣾⠇⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠋⠀⡠⠀⠀⢐⠈⢻⡟⠀⢈⣿⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡴⠋⣤⣼⣧⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠈⠻⣿⡆⠸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⣿⡷⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⢸⠁⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⠀⠔⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠒⠀⠀⠀⠀⠀⠀⢀⡠⢄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⢀⡀⠀⠀⠰⠀⠠⠂⠀⠀⠀⠀⡐⡡⠊⠁⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠀⠉⠉⠀⠀⠈⠒⠤⠄⣀⣀⡀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⠟⠙⢧⣀
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⢂⠂⠀⠀⠆⠀⠀⠀⠀⣽⣄⠱⣄⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢰⠀⠀⠀⠛⢦⡜⠀⠀⠀⢠⠞⡡⠊⠳⣤⡑⠢⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠓⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠞⠁⠀⡠⢊⡽⠓⠚⠁⠀⢀⠄⠠⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⢀⢞⡴⠋⠀⠀⠀⠀⠀⠸⡀⢀⠆
⠀⠀⠀⢰⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⠀⠀⣀⡴⢋⠔⠁⠀⠀⡐⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠓⠐⠁⠀⠀⠀⠐⡇⠀⠀⠒⠒⠒⠂⠀⡦⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⢁⡰⠁⠀⠀⢠⢊⡞⠁⠀⠀⠀⢀⠤⠀⠐⠂⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢄⡀⠀⠀⢀⡠⡺⠁⠀⠀⢀⡼⠃⠀⢀⠔⠒⠒⠒⠒⠒⠁⠀⠀⠀⡔⢡⠏⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠑⠈⠀⠀⠁⠊⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡠⠒⠁
⠀⠀⠀⢀⠔⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢷⡀
⠀⠀⠰⡁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠎⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳
⠀⠀⠀⠈⠢⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠉⠢⡀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⡠⠀⠀⣸⠁⠀⠀⢀⢐⠌⠁⠀⠡⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠃⠀⠀⠀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⠑⣤⠚⠀⢀⡴⠋⠀⠀⠀⡑⠁⠀⠀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⢀⡾⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡨⠋⠀⢀⡴⠋⠀⠀⠀⠀⢰⡇⢀⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠸⡀⠀⡐⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠉⠈⠀⣠⠞⠁⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠟⠢⡀⠀⠀⠅⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⠀⢀⠃⠀⠀⠀⠈⠢⡀⠀⠀⢰⠀⠑⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣠⡴⠋⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢠⠁⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⡄⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠀⠉⠁⠀⠀⢀⠜⡄⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⢄⠀⠀⠈⠢⠀⠀⠀⢀⠔⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⠄⡀⠀⠁⣀⠔⠁⢀⡴⠞⠁⠀⠀⠀⠀⠀⢰⠒⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⠁⠈⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠀⠉⠐⣠⠄⠀⠀⠀⠀⠀⠣⠊⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠉⠀⠀⠀⠀⠔⠈⠐⠀⠀⠀⠈⠢⡑⠀⠀⠠⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠈⠲⣄⠈⠢⣄⠙⣤⡈⢦⡈⣢⡘⣦⡰⣄⢠⡀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⡈⠳⣦⣈⠻⣦⣝⠿⣬⠻⢦⠻⢦⠻⣎⢷⡽⣦⣷⡸⡄⣆⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣤⣈⠙⠶⣄⡙⠳⣤⡙⠲⣌⠳⣄⠳⣄⠳⡌⢢⡹⡌⢏⢿⣻⡼⣼⣜⡆⣄⡄⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡉⠓⠦⣄⡙⠲⢄⡙⠲⣌⠓⢤⡑⢌⠳⣌⢢⡑⡜⢌⢇⢣⢡⢱⢹⣯⣿⣯⣻⣷⣦⣰⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⣈⡙⠳⢦⣌⡙⠲⢤⣉⠳⢤⡙⠦⣙⠢⡙⢮⣳⣝⣬⢮⢦⣧⣣⣇⣂⡄⢨⡟⢿⢿⡿⣿⣯⣷⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⢤⣄⣉⡙⠳⠶⣤⣉⡓⠦⣬⣙⠲⣬⣓⣮⣷⣿⣿⣽⣾⣿⣷⣷⢳⡇⠟⢙⠽⠷⢦⣄⠠⠡⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢤⣄⣉⣙⠛⠶⢦⣬⣙⡓⠶⣭⣛⣶⠿⣳⣿⣿⢿⣿⣟⣿⣿⡻⢟⣮⢶⣍⡰⣠⠀⠉⠳⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣈⣉⡛⠓⠶⠦⣬⣍⣛⡲⣾⣭⣿⣿⢿⣷⣿⣿⢸⣿⣿⡟⠀⠙⠂⠈⠻⠷⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⣉⣉⣉⡛⠒⠒⠶⠿⣭⣿⣿⣿⣭⣿⣿⣿⣧⡋⣝⡛⢳⣦⡴⠀⠀⠀⠀⠈⢯⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣛⣓⣒⣒⠻⠿⢿⣿⣶⣬⣛⠻⠾⠻⠟⠋⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠤⠤⠤⠀⠠⠤⠭⠭⠭⠭⠍⠍⠙⠛⢛⣗⡶⠋⠏⢒⠻⠙⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠬⠭⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠿⢿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣬⣿⣵⡶⢾⣷⣶⡯⣿⣿⣶⢛⣿⣷⣶⣼⣷⣾⣿⣷⣶⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⢛⣫⡍⢉⡙⢻⣿⣽⣾⣽⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢿⡿⢯⣝⠛⠛⢋⢹⡃⣨⠑⣷⢩⡿⢉⢻⡟⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠙⠀⢸⢟⠿⢫⢹⢸⡇⢸⠀⠻⠾⣷⣴⢸⡁⢏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢉⣌⡇⢰⣦⡆⢰⣝⣼⣽⠴⠛⠁⣠⡄⠻⠿⡂⢚⣗⢛⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⣛⣀⣤⡭⠁⣈⣁⣀⣠⣴⡾⣶⣿⣿⣿⣿⡷⡞⣿⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⠃⠊⠻⣿⣿⣯⣵⣦⣙⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣏⣿⣿⣿⣿⣿⠿⣿⣿⣿⢠⠀⣸⣿⣿⡿⠿⠿⣟⣳⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⣿⣿⣿⣿⡏⠙⠺⣿⣿⡼⢀⡙⢿⣿⣷⣾⣧⠊⠑⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⡇⠀⡦⣩⣙⣷⣦⣄⠀⠈⣿⣿⣿⠊⢦⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣧⣄⣗⣿⣿⣿⣿⣿⣿⣶⣦⡍⣿⠀⣉⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣟⡯⣿⣿⣽⡟⡛⠿⣿⣿⣿⣷⣌⣄⢶⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣿⠉⢩⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣧⣿⣿⣿⣿⠿⣮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢦⣽⣋⣾⣸⡟⢸⣷⢿⣿⣿⢿⣿⣇⣿⣿⣿⢿⡿⣿⣳⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡉⠉⣹⡰⢤⢊⣫⡜⢸⣽⢸⣿⣿⣷⣾⣾⣧⣴⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠈⠀⠁⡼⢴⣿⣶⡎⠀⠀⠀⠈⠉⠉⠉⠉⡙⢻⣯⠟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⠄⣀⠀⣀⢙⠸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠠⠈⠀⠊⠁⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣿⡿⣿⡿⢿⡿⡟⠚⢻⣇⣴⣶⠴⡶⢒⠿⢴⡁⢀⡠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠠⢉⡇⣰⣏⠛⣠⣧⣾⣏⣿⢾⣤⣿⣽⣿⣷⣷⣯⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠟⠻⠛⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⡲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠀⠀⠀⠀⠀⠀⢸⠀⠠⢿⠀⠀⠀⠀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠰⠀⡀⠀⢸⡀⠀⠀⢸⡀⡴⣿⠀⠀⢀⣾⢿⣿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡅⠆⠀⠀⠀⠀⠈⢳⡀⠀⢨⡄⢅⡇⠀⢀⣴⣾⢿⠙⣿⣥⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠰⠀⠀⡀⠀⢈⠄⠀⠠⡁⠀⠸⡆⢨⡇⠀⣼⣿⡟⠿⡆⢻⣦⢉⣥⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠂⡀⠀⠀⠀⠈⠀⢀⡖⢻⣠⠀⣀⡀⠐⠶⠿⣿⣿⣟⣤⣌⠉⠸⠟⣼⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⡆⢀⢁⠀⠈⡀⠈⢀⡝⠀⣹⣄⣨⠿⠙⠳⢤⣀⣀⡀⠈⠉⢡⣤⠸⣿⣮⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢸⣇⠈⠀⠂⠁⠀⡀⠀⢀⠀⣻⠋⠁⠀⠀⠀⢸⣿⣿⣿⣦⡀⠸⡿⠿⠛⠛⢻⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⣿⠄⡈⠄⠀⠀⠀⠄⢀⡞⠹⡆⠀⠀⠀⠀⣿⣿⣇⡙⢉⣥⡄⢁⣐⢶⣿⣿⣷⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠈⢿⣤⡀⠆⠀⠀⠀⡘⠀⠈⣇⠀⠀⠀⠀⣿⣏⣽⡇⣾⠟⣡⡄⡿⠳⣌⢿⣾⣿⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢦⠀⠀⠠⠈⠷⣦⣌⡀⠐⢈⣄⣘⣿⠀⠀⠀⠀⢿⣿⣿⡿⢿⣷⣿⠇⣿⡀⡻⣷⣙⣝⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⣇⠑⡀⠂⠢⠀⠀⢩⠛⢶⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠋⣀⣿⠇⡐⢞⢛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⣘⣧⠀⠀⡀⠀⡃⠁⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡾⣹⣿⡋⣼⡇⠸⣎⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣯⠙⣷⡀⠌⠻⣶⠶⠆⣄⣰⣺⡇⠀⠀⠀⠀⠀⠀⢸⡽⣋⢻⣷⣿⡟⣸⣿⡇⢦⡹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣂⠈⢻⣦⣠⢤⡃⢞⢣⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⡿⣿⡇⣿⣿⣿⣯⣷⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢐⢀⣶⠄⠙⢷⣌⣤⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⣾⣫⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢧⡈⡑⠂⠐⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⡍⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⣤⣭⡄⠀⡉⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⣤⣆
⠀⠀⠀⠀⠀⢠⠎⣴⠹⠂⡜⠁⠈⠀⠈⠅⠈⠁⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣶⣴⣶⣦⣄⡃⠀⣉⠈⡐⠀⣀
⠀⠀⠀⠀⡄⣈⠂⠀⠐⠄⠠⣾⠼⠒⠒⠋⡘⠶⠋⠀⡂⠀⠀⠀⠀⢀⠋⢀⣴⠿⠝⠻⠿⠀⠋⠟⠙⠻⠿⣷⣿⡾⠿⠿
⠀⠀⠀⠀⠀⠺⠁⠾⣤⣤⠀⡡⠊⡉⠀⠋⠄⠂⡀⠁⠘⠀⠀⠀⠀⢸⣷⡿⠃⠰⠿⢿⣶⣟⣷⡶⠒⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⠍⣱⣞⠶⡶⢶⣦⠦⠀⠀⠀⠀⢀⡼⠋⠲⠾⠟⠁⠈⠁⠘⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠲⠶⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠖⢂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢇⠘⢣⠄⠀⠀⠀⠀⠀⠀⣾⠤⣦⡛⠓⠇
⠀⠀⠀⠀⠌⠀⢸⡀⢀⡤⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣯⠐⠛⢛⡚⢻⡆
⠀⠀⠀⡄⠀⢀⣾⠷⠛⠁⢠⣾⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢐⡈⠀⠀⢸⡕⠺⣇
⠀⠀⠀⡷⠀⠚⠁⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣚⠠⡄⠐⡦⠞⠀⠀⠀⠀⠛⣯⠀⣄⢤⡄⠁⣼⠇
⠀⠀⠀⠱⣾⣗⠁⣠⠀⠈⡹⣿⣿⠀⠀⠀⣠⡞⠛⢿⠿⡿⣿⣿⢿⣷⣤⡞⠑⠀⠄⠀⠀⠀⢸⡇⣠⠋⠆⣀⣤⡦
⠀⠀⠀⠀⠘⢿⣿⣿⠀⡀⢐⣿⠃⠀⠀⠈⠛⠿⠗⠺⠦⣴⣄⣴⣾⣟⠋⠊⠀⠀⠀⠀⠀⠀⠈⢯⡛⠷⠇⠈⠉
⠀⠀⠀⠀⠀⠈⠙⠷⡘⢣⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⢲⡀⣠⣾⢤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠿⢋⣠⣿⡁⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠁⠀⠤⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⡟⢹⠁⠀⠀⠈⣆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⢰⣟⡀⢷⠘⠂⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠼⠀⠀⠀⠀⢠⣾⣿⣧⣤⡀⠀⠀⠀⠀⢼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠
⠀⠀⠀⠀⠂⠀⠀⠀⢀⣼⠇⠈⠉⠛⠇⢀⡀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣛⠄
⠀⠀⠀⠀⣦⣀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠈⠛⢶⣤⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⣀⠀⠀⠀⡄⠀⠀⠀⢀⡴⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⡿⠦⠉⠀⠀⠀⠀⠀⠀⠀⣼⣣⠞⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣌⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢤⠀⠀⠁⠀⠉⠻⠶⠲⣤⡀⠀⠸⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢤⡀⠀⣄⠉⢻⡂⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠲⠉⠲⣤⡀⠎⠙⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⠦⣀⣷⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠘⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠖⢂⠀⠀⠀⡀⢘⣻⡷⣄⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣦⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⣉⠁⠾⣿⠿⣶⠀⠐⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠈⠇⠀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠈⠃⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⡀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠑⡄⠀⠀⠀⠀⠀⢤⢀⣀⡀⠠⡄⠀⢀⠀⠀⠑⠈⠠⠀⠀⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣄⠉⠀⠈⠓⢦⡀⠀⠀⠠⡱⣮⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢿⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠳⣆⠀⠄⠙⢢⡀⠀⢹⡿⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠈⠉⠛⢷⡀⠀⢷⠀⢼⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢱⣆⠌⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢻⡀⡙⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⢹⡿⠻⢏⣿⣿⠟⠃⢄⠀⠀⠀⠀⢸⣧⣶⡶⢶⣶⢶⣦⣀⣀⡤⠀⢀⣀⣀⣈⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣴⡏⠁⣴⣶⣍⢰⣶⣦⡬⣁⠀⠀⠀⣤⠀⠀⠀⠢⡽⡗⠀⠙⠿⣿⢿⡿⠛⠉⠛⠛⠛⢦
⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠉⠐⡤⢰⡄⠀⠀⠀⣿⠚⠀⠑⠘⢯⡟⠛⠀⠉⠉⠛⢼⡇⢀⡾⠁⠀⠀⠀⠀⠐⢶⡤⠀⠀⠀⢣⠈⠻⡶⠶⠤⠤⠄⣅
⠀⠀⠀⡪⠂⠀⠀⠀⠀⠀⠀⠀⠐⠤⢾⣷⢦⡀⠀⢯⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⡬⠟⠋⠀⠴⠶⣤⣀⠀⠀⠀⠛⠄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣯⣖⡀⠀⠘⣧⠈⠀⠀⠀⠀⠀⠀⢠⠖⠡⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠠⡀
⠀⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢅⠾⠛⠀⠀⠘⢧⡐⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⢻⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠉⠓⠒⠚⠛⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⣨⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢼
⠀⠀⠉⠁⠙⠶⣤⡀⠀⢀⣤⣴⠦⣢⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣀⠀⠂⣶⣷⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⣾⡤⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣈⣉⣉⣉⡑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⠲⡙⡄⠀⡄⠇⠀⠉⢢⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡞⠀⠀⠱⡜⢦⡁⠆⡠⠔⡡⠊⢣⠘
⠀⠀⣀⣀⣀⠠⠤⠤⠤⠄⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠰⡇⢘⠀⠀⠲⠤⠸⠎⠁⠉⣤⣈⡀⣀⠀⡆
⡴⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠈⠉⠉⢙⢖⣠⢦⡀⡶⠦⠤⠨⠀⡇
⡇⠀⠉⠛⠒⠶⠤⠤⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⣧⠼⡤⢴⠞⡛⠛⠁⢸⡗⡌⢦⠀⠀⣰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠠⡀⠀⠀⢢⢸⡁⠈⠢⣳⠔⢁⠜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢈⠑⠂⠘⢼⡇⠒⢊⣡⠔⠉
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢺⠃⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⡆⡞⠙⠲⠶⠶⠖⠚⠋⠉⣉⢳⡦⣄⡀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠋⠉⠛⠲⣄⠀⡇
⠀⡇⣇⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠘⡄⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⢇⢸⠀⠀⠀⠀⡎⠀⣠⠔⠚⡄⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁
⠀⢸⠸⡀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠆⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾
⠀⢸⡀⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢈⢶⠋⠉⠉⠉⠀⠈⠱⣄⠀⠀⠀⠀⢸⡇⡏
⠀⠀⡇⢇⠀⠀⠀⠀⢦⠢⡀⢀⡠⠔⠊⢸⠀⠀⢰⠒⠒⠒⠀⠈⢣⠀⠀⠀⢸⢸⡇
⠀⠀⣇⢸⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⣸
⠀⠀⢸⠘⡀⠀⠀⠀⠀⠀⠀⡴⠋⠒⠤⣈⢆⡄⠀⠀⠀⠀⠀⢰⢸⠀⠀⢠⡇⡿
⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⡼⠀⠀⣦⡀⠀⠉⢻⠦⠄⠀⠀⠀⠨⠞⠀⠀⢸⢡⡇
⠀⠀⠀⡇⢳⠀⠀⠀⠀⠞⠀⠀⡜⠀⠈⠓⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⢱⠸⠀⠀⠀⠀⠼⠀⠈⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡇⡿
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠈⠧⡀⠀⠀⠀⢀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡇
⠀⠀⠀⠀⡇⢷⠀⠀⠀⠀⠀⠈⠑⠢⠄⠡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⢷⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⢸⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠃
⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼
⠀⠀⠀⠀⠀⢹⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢠⡇
⠀⠀⠀⠀⠀⠈⢧⣌⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⢋⣠⡞
⠀⠀⠀⠀⠀⠀⠸⡏⠓⢦⣄⣉⡉⠙⠛⠛⠛⠋⣉⣉⣠⠴⠖⠋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣲⣂⣤⣤⣤⣤⣤⣤⣲⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠉⠉⠉⠒⠒⠦⢄⣀⣀⣀⣀⡀
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⡷⠶⠛⠛⠉⠀⠉⠉⠉⠉⠉⠁⠒⠒⠒⠤⣄⡀
⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣼⣿
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡥⠋⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠂⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣏⣿⡀⠀⠀⠀⠀⠀⠀⠀⣢⠖⠋⠁
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡯⠿⣿⡄⠄⠀⠀⢀⡤⠂⠁
⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠏⠓⠦⢤⠴⠋
⠀⠸⣄⡀⠀⠀⠀⢀⡀⢈⠴⠶⠖⠚⠛⠁
⠀⠀⠘⣇⠀⠀⠀⠎⣶⣇⣄⡈⠔
⠀⠀⠀⠈⢦⡀⠀⡼⣩⠏
⠀⠀⠀⠀⠀⠙⠾⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢠
⠀⠀⠀⡇⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣧⠀⠀⡠⠃⠀⠀⠀⡰
⡀⠀⠀⠈⠢⣀⡀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠁⠀⣿⠤⠊⠀⠀⠀⡠⠊
⢣⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⡇⠀⠀⠠⠖⠋
⠀⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠠⠶⠒⠒⠒⠈⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡴⠬⠿⠆⠀⠖⠦⢤⠀⠀⠀⠄⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢡⠀⠀⠀⡈⠀⠀⠀⠀⠐⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⣡⠀⢠⡳⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠚⠛⣮⢑⡀⠀⡐⠒⠀⢲⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠂⢀⡜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠲⠚⠦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⡞⠉⠙⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡤⢺⢁⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠰⡖⡏⢇⠀⠈⡥⡔
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡠⠔⡋⠀⠀⠀⠘⠘⣄⣧⡀⠗⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠘⢄⡀⠤⠚⠤⠐⠁⠸⠀⠀⠀⡖⠡⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⣣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⡄⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠉⠉⠁⠀⠀⠄⠀⠰⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⠅⠵⠶⡁⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠋⢰⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⡅⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡟⠉⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⡑⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠄⠈⠻⢾⡇⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣏⡃⠀⠀⠀⠀⠀⣀⣤⠚⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣰⢟⡉⠉⠈⢷⠤⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠁⠛⡀
⠀⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⠀⠀⡏⠀⠀⣼⣦⠀⠉⠙⢂⠀⠀⢀⠀⠀⢀⠆⠀⠀⠀⠈⣒⠀⠀⠀⠀⠀⠀⣩⠉⠁⠸⢴⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠇⠀⠀⠙⠃⡄⠀⠀⠈⢃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠦⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠂⠁⠀⠀⠀⠉⢢⡄⠃⠀⠀⠀⠀⢳⡀⠀⠀⠐⠄⡀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠦⢤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙
⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⣡⠀⠒⠶⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠈⠙⠂⠀⠐⢻⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⡄⠀⡠⠂⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠘⢤⡇⠀⠀⠀⠀⢀⣻⡇⣢
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡠⠒⠀⠉⠚⢧⡀⠀⠀⠀⢠⣵⣼⠃⠀⠀⠀⠀⣂
⠀⠀⠀⠀⣋⡄⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠀⢹⠞⠓⠠⠀⠀⠉⠓⣄⠀⠀⠀⢿⠁
⠀⠐⢠⡾⠉⠀⠘⠂⠄⡀⡀⢀⠀⠀⠒⡀⣏⣀⢀⠤⡄⢰⡄⢀⡄⡠⠀⠀⠀⠀⠀⠀⠐⠀⡀⢷⡀⠀⠀⠀⠀⡎⠀⠀⠀⢠⢖⠐⠀⢸⣧⠄
⠀⣇⠀⠀⠄⠀⠐⠀⠋⠉⠁⠉⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠈⠛⠓⠒⢀⣰⣿⠀⠀⠀⢆⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠚⠉⠁⠘⢦⣀⠀⠀⠀⠠⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠑⠘⠃⠀⠀⡀⠀⠱⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠙⡫⢛⣿⠁⠜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠉⠉⠛⠶⣄⡀⠀⠀⣿⣀⣀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠋⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡤⠞⠋⠉⠉⠛⠲⣄⣠⠞⠋⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢠⠟⠀⢀⡤⠶⠦⣄⡴⠚⢳⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⠀⠀⡾⠀⣤⡖⠋⣧⠀⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⡀⠀⠻⣄⣈⣀⡠⠃⠀⢠⠇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠳⣄⠀⠈⠉⠉⠀⢀⣴⠋⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠙⠒⠶⠖⠚⠋⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣶⣶⣶⣶⡖⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣉⣉⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡅⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣔⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣎⠳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⢥⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢤⡀⡽⠆⠀⠀⣿⣿⢥⣿⠀⠀⠀⣸⡇⠀⣦⣦⠂⢀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢑⠚⠓⠾⣿⣤⣀⠻⣇⡸⠃⠀⣿⣿⣾⣷⠀⣸⡇⠘⣧⣀⡯⣎⠀⣪⡛⢁⡠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡌⢸⡏⢀⠀⣀⣀⡚⣯⠠⠛⣡⡴⢀⣿⡿⢒⣷⠀⠨⠵⣝⡄⢯⠁⣣⢢⠅⠰⢊⡠⢸⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⣷⣤⣷⣼⣄⡉⠋⢩⣭⠻⠀⢯⣴⡾⠋⣉⣄⣹⠷⢄⡀⠸⠷⣼⡄⠇⣨⣶⣿⢁⡀⠚⡧⠾⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡩⠟⠛⡹⢿⣾⣿⣬⢶⣷⠛⢻⡧⠆⣠⡾⠋⠙⠻⣶⣤⣹⠻⠦⠾⠃⠘⠧⠤⣀⣨⠑⠋⢀⣤⡢⠂
⠀⠀⠀⠀⠀⠀⢀⡈⡀⠠⣀⣠⣀⡀⢉⢀⣴⣿⣋⣥⡶⠛⣰⠞⠉⠀⡄⠀⢀⣠⢍⠛⢿⡷⢸⡟⠛⣷⣤⣓⡠⣤⣼⡽⠷⠀⣶⣀⡀
⠀⠀⠀⠀⠀⣀⣉⣱⣳⣤⣐⣮⣽⣶⡌⣹⣭⠕⠛⢦⣶⠦⣧⣤⣶⡧⡄⠀⠈⣿⣮⣄⣨⣷⠘⠷⣄⠘⠻⣏⠙⢷⡴⢶⡒⠻⣏⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠉⣡⣠⠟⠡⠛⠩⢟⣵⣏⣠⣾⣿⣿⠈⠻⡏⠓⠀⠁⠀⠀⠈⣨⣩⡿⢻⣿⣦⣤⣝⠒⠈⣁⡄⠑⠀⠀⠀⢀⠘⠋⠂
⠀⠀⠀⠀⠀⠰⠶⡄⠈⢡⠀⠠⣀⡐⣉⣽⣼⣹⣿⣿⠿⣀⢀⣤⡔⠀⠀⠀⠀⠀⠁⠟⠃⣸⣾⢻⠸⣿⣯⠭⡻⢴⣾⣇⣀⣶⡯⠷⢾⠃
⠀⠀⠀⠀⠑⢤⣈⣙⢀⣻⣿⣶⣿⣿⠟⢯⣿⢋⣿⣷⢀⣇⣟⠛⠣⠀⠀⠀⠀⠀⠀⠀⣴⡃⣿⣸⣇⣿⠘⡆⠱⠠⠌⠉⠂⠀⢸⢷⡄
⠀⠀⠀⠀⠀⣤⡿⠛⠛⠉⡉⢽⣣⡏⢠⣾⠟⣻⡟⣿⢺⡾⣏⡀⠀⠀⠀⠀⠀⠀⠠⡞⢻⠃⣿⠻⡅⠹⢳⣧⡶⠀⣌⠙⠛⠃⠀⢀⡄
⠀⠀⠀⠀⢀⣠⣤⡴⢦⡷⢿⣿⣿⠈⠹⢃⣼⣿⡇⣿⡘⢰⣯⠀⠀⠀⠀⠀⠀⠀⠘⢃⣿⢸⡿⣾⠈⢀⡟⠃⡗⠐⠻⣷⣧⠴⣾⡻⣅
⠀⠀⠀⠀⠀⡤⣼⡖⠡⠶⣾⠧⠙⢂⣸⣿⡉⠛⡇⠿⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢸⡷⠈⣧⡈⠋⠀⡵⡯⠁⠘⢿⠓⡄⡉⠁⠠
⠀⠀⠀⠀⠀⢈⣤⣴⢶⣾⣟⠂⢠⠞⣸⡇⡅⠘⢣⣰⠏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣴⡏⠍⢱⠀⢷⠀⣝⡓⠈⣧⠈⠩⠆
⠀⠀⠀⠀⠀⠀⣠⠴⣮⣇⣬⠀⠋⠀⠉⠀⠉⡒⠀⠋⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢠⠣⠈⠀⡊⠑⠘⠘⠂⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠼⢇⡙⠌⣡⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⢾⡅⡄⠐⣒⠲⣄
⠀⠀⠀⠀⠐⠀⠀⠞⠳⠟⠆⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠊⠈⠟⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⣀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠦⠄⠀⠀⠀⠀⠁⠀⡟⠀⠀⠸⠀⠀⠀⠀⠒⠒⠒⠒⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⠁⠀⢀⣠⠴⠒⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣤⣄⡀⠀⢀⣀⡀⠀⠉⠛⠻⣦⡀
⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠠⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠞⠋⠀⠀⠀⢀⣀⣤⣶⠿⢃
⠀⢀⠙⠳⣶⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠾⣿⠟⠛⠉⠀⠀⠈
⠀⠈⠉⢉⠱⠿⢤⣭⣉⣉⣙⠛⢛⣛⢛⠓⠛⠛⠋⠀⠀⣓⠒⣛⠛⠛⠛⠛⠛⠋⣉⠉⠉⠉⠁⢀⡄⠀⠀⠀⠈⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⣾⠀⣀⠀⠂⠈⠉⠉⠘⠛⠛⠛⠛⠋⠉⠁⢀⠛⠉⠉⠉⠀⠀⠀⠀⠔⠊⠉⠑⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠚⠛⠋⢹
⠀⠀⠀⠸⠀⠈⠙⠛⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠾⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⢸⡇
⠀⠀⠀⠀⠀⣿⣎⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠀⠀⠀⠀⠀⣠⡴⠶⠾⠶⢤⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢻⡏⠓⠤⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠈⠀⠀⢐⡒⣤⠶⢷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠖⠒⠒⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠈⠃⠌⠉⣉⠝⠋⠉⠉⠉⣉⡉⠉⠙⠛⠒⢤⡀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡼⠃⠀⠀⠤⠀⠀⠈⠉⢄⡀⠀⠀⠀⠉⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠸⠀⠀⠀⣧⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣤⡤⠐⠂⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⡠⠤⠐⠛⠀⣀⣉⠤⠤⠤⠤⡴⠒⠓⠀⠀⠀⠠⠠⠀⠈⢁⡈⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⢾⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⣶⣄⣀⠀⠈⠛⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠘⠷⣤⣤⣀⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠦⠤⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣰⡾⣿⣿⣿⣿⣿⣿⣿⡿⢽⣶⡶⢭⣑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣯⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠊⠝⡲⣝⡲⣍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣱⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢝⢦⡑⢌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⡀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠙⣄⠳⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠿⠝⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢦⡀⠀⠑⡄⠈⢦⠙⡌⢇
⠀⠀⠀⠀⠀⠀⠀⡰⢁⠌⠀⠀⢠⠴⠚⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡡⠀⠈⢧⠰⡈⡆
⠀⠀⠀⠀⠀⠀⢰⠁⡞⠀⠀⢀⡏⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢿⡆⡇⢱
⠀⠀⠀⠀⠀⠀⡏⢠⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⢡⠀⢸⠀⠀⠀⠇⢡⠈
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠬⣇⢸⡆⠀⠰⡃⠈
⠀⠀⠀⠀⠀⠀⣇⠘⡆⠀⠀⢹⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⡇⠀⠀⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⠀⢸⡄⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠹⢿⣷⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠳⡘⢆⠀⠀⡷⣄⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡠⠴⢚⡭⠿⣾⣬⡲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⣰⣨⢭⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠴⠞⠁⢀⡾⠁⠈⠙⠷⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠛⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡴⠛⠉⠙⠲⢄⡀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⠔⠁⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠳⣝⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⢋⠕⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢝⡺⢿⣻⣿⣿⣶⣶⣶⣶⠾⣿⣛⠯⢕⣊⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠻⠭⠭⠭⠭⠭⠽⠗⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠱⡄⠀⠀⠀⠀⠀⠡⡀⠳⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢄⠀⠈⡀⢄⠀⢃⠈⣧⠀⠀⢡⢀⠤⠒⠒⢦⣀⠸⠖⠊⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠙⢆⡀⠑⠀⢀⠀⠃⠈⠢⢈⠓⣄⢸⣃⣀⣰⣴⢲⠇⢀⣀⡀⠀⢸⠉⠙⢓⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡉⠳⠆⣀⣀⣸⣄⣴⡶⠟⡶⠙⠿⠋⠀⠀⠘⣇⠘⠢⠃⢀⣼⢻⠉⠉⠀⠀⠉⠳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠊⣯⠎⠀⢀⠗⠓⠓⠠⡀⠀⣴⢼⠛⡲⠒⢉⣣⠏⠀⠀⠀⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠁⢀⢪⡏⠀⢥⠃⠀⠀⠀⠀⢱⠤⠒⠁⠀⢁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⠀⠀⠀⠠⣀⢐⣒⣤⠏⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⠊⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⣀⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠈⠑⠢⠀⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠐⠂⡀⠐⠂⢄⣀⣀⡀⠀⢀⠤⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠶⡄⠀⢠⡆⠀⠀⡔⠂⡄⠐⢀
⠀⠀⠀⠀⠀⠈⢦⠊⠁⠀⠀⠈⠲⠣⣂⠽⠀⠀⠙⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠘⠦⣀⣀⣠⠞⠀⠀⢸⠈⠂⠁⠀⠀⡃
⠀⠀⠀⠀⠀⠀⡎⠀⢴⠃⢂⣠⠋⠉⠉⠙⠢⡄⢠⣏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠘⣄⣈⣉⣡⠁⠀⣤⡀⠀⠀⢸⠟⠁⢉⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠈⠓⡀⢀⠔⠁⠀⠁⢢⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⢆⠐⠄⠉⠀⢠⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⢈⠀⢔⢐⡀⠀⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⡤⠟⠁⠈⠦⣤⣥⣤⠾⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⠃⠀⠀⠀⢀⣠⠶⠛⠉⠉⠀⠀⠉⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⡄⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠉⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠓⢤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⡰⣄⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠄⢀⡀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠈⠙⠲⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⢳⠀⢠⡇⠀⠀⠀⠀⠀⢀⠄⠈⠉⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⠇⢀⣀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣃⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠉⠉⠉⠉⣻⡇⠀⠀⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠄⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⡀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠊⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠘⢦⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣠⠄⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⢀⣎⣠⡤⠤⠤⠴⣤⣀⣀⣠⠃⠀⢀⣀⡤⠴⠮⠤⠄⠐⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠙⠛⠛⠛⠉⠉⠙⢄⠀⠀⠀⠀⠁⠈⠉⠉⠉⠛⢄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣈⣁⣉⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⠿⠥⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⣭⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠉⠀⠀⢠⠚⠛⠒⣲⠦⠷⠖⠲⠤⠤⣳⡖⠒⠽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢀⠎⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣠⠎⠀⢠⠏⠀⠀⠀⠠⠀⠀⣀⡤⠤⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠒⢦⣾⡽⠃⠀⣠⠧⠖⠒⠒⠒⠲⡴⠊⠁⠀⠀⣸⠃⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢷⡀⠉⠘⠂⠉⠀⠀⣾⠳⠒⠮⠭⠍⢹⣗⡟⠀⠀⡾⡰⠋⠀⠘⡀⣣⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣤⠑⢢⡽⠓⠂⠀⠀⠀⠀⠀⠉⠓⠒⢒⣲⠞⢿⠢⠤⠖⠋⠀⠀⠀⠀⡇⡵⣉⣀⣘⣦⣤⣀⡀⠀⠀⣲
⠀⠀⠀⠀⠀⠀⠈⡄⠘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⢦⠀⠳⡄⠀⠀⠀⠀⠀⢠⠁⣎⡡⠊⣻⠒⠚⡍⢒⣢⢖⡢⣄
⠀⠀⠀⠀⠀⠀⡠⠻⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⠘⢦⡀⠀⢀⡴⢃⠔⠪⠤⠴⠻⣧⣜⠇⣼⡇⢀⣿⠋⠁⠂⢤⣠
⠀⠀⠀⠀⢀⠞⠀⠀⢣⠀⠘⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠑⣖⣉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⠊
⠀⠀⠀⣠⠋⠀⠀⠀⠀⢇⠀⠈⢾⠛⠋⠉⠉⠉⠙⠒⠂⠀⠀⢀⡠⢧⡀⠀⠀⠀⠈⢣⡀
⠀⢀⠔⠃⢠⠃⠀⢰⠂⠘⡄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⣠⠔⢯⠤⠚⣿⢮⡀⠀⠀⢀⡇
⢀⠏⠀⢠⡃⠀⠀⢸⠀⠀⢱⣀⣀⠤⠜⢆⠀⢀⡠⣴⠋⠀⠀⢸⠚⠉⠀⠀⠉⠓⠶⠚
⢼⡀⠀⠀⢹⢷⠒⠚⠀⠀⠀⠁⠀⢄⣀⢸⠊⡹⢆⡧⠤⠤⠤⠼⠁
⠀⢟⠓⠤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⡇⠀⠁
⡠⠯⠍⡒⠇⢸⠙⠒⠦⠤⠤⠤⣀⣀⣠⡤⠤⡞
⠉⠳⢀⡀⠤⢏⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠑⡦⠀⠀⠀⠀⠀⠛⠋⣏⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⠀⠀⢀⣾⣤⣖⣟⣲⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⢀⣄⣐⣒⣒⣒⣀⣰⣂⣴⢦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣦⢽⣖⡛⠉⠉⠄⠢⠔⠈⠒⠂⢖⠠⠉⠉⠙⠒⢾⣭⣖⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣞⡯⣽⠖⠋⠁⠅⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⡄⠀⠀⠠⠉⠓⠂⢉⠓⠭⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢞⣯⣾⡏⠀⠀⠀⠀⠈⠂⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⣇⠀⠘⢦⠐⢈⠓⢍⡢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⣿⣿⣮⣽⡆⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠘⡄⠀⠀⡗⠀⠀⢀⠙⢮⣢⡀
⠀⠀⠀⠀⠀⠠⣠⣿⣽⣿⣿⣿⣿⣷⢯⣨⠞⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⡈⣧⢣⡹⣟⢆
⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⣧⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⢿⣾⣯⣡
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡟⠁⠈⠢⡀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⢀⠃⠀⢸⡏⢿⢿⣧
⠀⠀⣄⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣹⣿⠶⠤⠬⣦⣤⣀⣸⡀⣀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⠀⡠⠚⠀⠛⣏⢷⣣
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣖⡔⠀⠀⠈⠉⢉⡷⠚⠛⠻⠶⣦⣥⣀⠀⠀⠀⠀⠀⠰⠀⠀⢄⢀⡟⠀⠀⠀⠀⠀⣻⣞⣯⣆
⠀⢰⣿⣿⣿⣿⣿⢟⢿⢻⢻⣿⣿⣿⣿⡿⡧⣺⢉⣀⡀⠀⠐⠐⠁⠀⠀⠀⠀⠀⠆⠀⠉⠹⠥⣢⡀⠀⠀⠀⠀⠸⡽⠁⠀⠀⠀⠀⠀⢀⣧⢻⣿
⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⣕⡟⠈⠉⢉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⠹⡀⠄⠀⢠⢹⣀⠀⠀⠀⠀⠀⠀⠨⠿⢿⣟
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⡅⢿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠖⠋⠁⠈⠙⠒⠂⠀⠀⠘⠆⢠⡋⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣿⣷⣿⣦⣿⠂⡄⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⡇⣿
⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⢿⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠞⠀⡇⢓⠂⢄⠀⠀⠀⠀⠀⠀⠀⢠⢦⣡⣿
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡟⠲⠄⠄⡤⠤⠔⠢⠤⠀⠂⠀⠁⢀⣰⠃⠀⠀⡀⠘⡀⠀⠁⠠⠄⠀⠀⠀⠀⢐⣾⣿⣯
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⢴⡄⡀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠪⠃⠀⠀⡼⣇⠀⢿⡄⠀⠀⠀⠀⠁⠐⠒⠹⣾⣿⣻
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣽⣦⣹⣂⠀⠀⣠⡤⢶⣞⡫⠁⠀⠀⠀⠀⡇⠈⢾⠀⡇⠀⠀⠀⠀⠂⠂⢀⣟⣻⣟⡏
⠀⠀⠁⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣳⣍⣂⠤⠤⠷⠻⠍⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠇⠀⠋⠀⠱⢄⡠⠒⠦⠶⡤⢼⣿⣿⡝
⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣳⣿⢧⣝⡂⠝⠂⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⣰⠀⢰⠃⠃⠀⠀⢓⠀⠀⣠⣴⣿⡿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡭⠲⠐⣠⠎⠀⡀⠀⣠⠞⠁⠀⡏⠀⡆⠀⠀⠀⢻⣾⣿⣻⣟⡝⠁
⠀⠀⠀⠀⠀⠀⠙⢯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠁⢤⡤⠤⣸⠏⠂⠘⢰⡇⣼⠁⠐⢪⣾⣽⣿⣿⣻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡛⢦⣏⣄⡒⢀⣾⡾⢻⣾⣾⣻⡿⣻⢟⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣽⣾⣿⣿⣵⣿⣿⣾⣿⣿⣿⡿⣟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⢿⣿⡽⢟⠫⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣚⠫⠗⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣛⣛⣛⣛⣋⣉⣉⣓⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣶⣶⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣀⣠⣤⣤⣤⣤⣴⣹⣿⣿⣿⣿⣿⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡏⠉⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣧⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣛⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⠛⠒⠒⠶⠶⠶⠒⠚⣛⣿⣿⣾⣿⣿⣿⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠐⣿⡏⠉⠀⠀⠀⠀⠀⠉⠉⠙⠲⢽⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠼⡇⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠢⡰⢶⣦⣀⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡶⠹⠿⢁⣤⢷⡀⢀⣀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠷⡄⠛⠋⢁⢀⡝⣮⣽⠟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⡀⠀⠹⣿⣃⠁⣭⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢿⡇⣚⣷⣎⣍⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢶⢸⡧⠙⢃⣰⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣚⣓⣇⣈⣉⣁⣈⣉⣑⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⢠⢶⡀⠀⢠⡆⢀⡄⠀⢠⡄⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢀⣦⠀⠀⣴⡆⡰⢿⣠⠊⢧⡴⠃⠈⢷⡴⠃⠸⠏⢻⡴⠛⡇⠀⠀⠘⣧⠴⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠏⠘⣤⠞⠀⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⡤⡤⠀⠀⠀⣧⠀⢀⡞⢁⡞⠓⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠰⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠿⠁⣾⠖⢾⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠸⠏⠀⠀⠁⠀⢿⡀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⢀⣀⡀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠠⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣤⢤⣄⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢁⡀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣞⠀⡀⢀⣤⣿⠀⠀⠀⠀⠀⠀⢀⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣾⠋⢉⡟⠓⠿⠁⠸⠆⠀⠀⠀⠀⣠⠟⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣀⠀⢀⠜⠁⠀⠞⠀⠰⠃⠀⠀⠀⣀⡤⠶⠋⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⡻⠛⠶⠶⠾⠤⢤⡶⠶⠶⠞⣯⠝⠛⠳⣄⠀⢠⠃⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠊⠀⠀⢀⠊⠀⠀⠸⡇⡰⠚⢾⡇⠀⠀⠀⠈⠷⠃⠀⠀⠀⠀⠀⠀⢿⣀⡤⠶⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢲⡤⢀⣠⣁⡀⡇⠀⠆⠿⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⣁⠌⠀⠀⠀⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⢸⠷⣤⡀⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠙⠳⢤⣀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢦⣀⡎⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣀⣀⡤⠶⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣖⢐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠟⠁⠘⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢂⣴⣯⣵⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡟⠀⣶⣿⣿⣽⣦⣀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⢃⣄⣯⣾⣿⡿⣿⣿⣟⣿⣿⡇⠀⠀⠀⠀⠀⢸⢁⣾⡟⣿⣿⣽⡿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⡿⣢⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣾⣿⢷⡽⣱⣿⣾⠟⣿⣷⣽⣿⣿⣆
⠀⠀⠀⠀⠀⣼⡿⠁⢺⣈⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠘⠻⠷⠋⢀⣯⣤⣶⣿⣿⠿⠛⡙⢻⣿⣆
⠀⠀⠀⠀⣾⡿⠀⣠⣾⣿⣆⣸⣿⣿⡽⠩⢁⣤⠟⠋⠀⣀⡄⢐⡢⠄⣄⡀⠀⠺⣿⣿⣿⣿⠟⣁⣠⣾⣿⣆⢹⣿⣧⡀
⠀⠀⠀⣼⣿⠀⣼⣿⡿⢿⠛⠿⢿⣟⣀⡼⠃⠀⣀⣴⣾⠟⠉⠁⠀⠀⣀⣈⣳⣄⠈⠻⣿⣿⣶⢿⣟⣿⣿⣿⡜⢿⣿⣷⡀
⠀⠀⣸⣿⠇⢰⣿⣿⣿⣿⣿⣿⡷⠌⢁⠀⣠⣾⡏⠀⠈⠁⠀⠀⠀⠉⠉⠉⠁⠌⣷⣄⡼⣯⣽⣿⣟⣟⡺⢿⣷⣼⣿⣿⣧
⠀⢠⣿⠇⠀⠀⣿⣿⣿⣿⠟⢃⣤⣶⣟⣶⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣩⣴⡶⢶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⣸⣷⠂⠀⠀⣿⣟⣯⣴⢣⣾⣿⡟⢣⣾⠟⣩⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣶⣬⣻⢷⡄⠙⠻⢷⣿⣿⣯⡍⢸⣿⣿⣧
⠀⣿⡯⠀⠀⠀⣿⣿⣿⣧⣿⢟⣵⡇⢸⡟⣴⡿⠷⢤⣀⡀⠀⠀⠀⠀⠀⢠⢃⣹⣿⣿⡸⣷⣽⡆⣦⡙⣷⣦⣾⣿⣸⣿⣿⣿⡄
⢰⣿⡷⠀⠀⠀⠞⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⠁⠀⠐⠂⠀⠀⠀⠐⠐⡷⢆⣸⣿⣿⣿⣯⣿⡿⡇⣽⣷⡈⣝⠿⣄⢙⣿⣿⣿⣇
⢸⣿⣿⠆⠀⠀⢸⢻⣿⡿⢻⣷⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠙⠛⣿⡏⣿⣿⣿⢣⣿⣿⣧⠹⣿⡗⢹⣿⣿⣿⣿
⢸⣿⣿⡅⠀⠀⠈⣼⣿⣿⡿⣾⣿⣿⣿⣿⡿⠀⠀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣼⣿⣇⠈⢿⣏⡦⣿⣿⣿⢹⡯⣻⣾⣿⣿⣿⣿
⠘⣿⣿⣷⡄⠀⠀⠈⢪⣿⣽⡯⣿⠟⠉⣸⣧⠀⠀⠀⠀⠀⡀⢀⠎⠀⢀⣠⣾⣿⣿⡈⠀⠀⠈⠛⠻⠛⠿⣈⣱⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⣳⣽⣿⣏⣤⣾⣿⣿⣷⣶⣴⣶⡦⠊⠥⠀⠐⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⠄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣦⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠛⢿⣯⣿⡟⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⢀⡭⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⢰⣶⣴⣦⣀⣾⣶⢴⣶⣶⣆⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣰⣶⣿⣿⣿⣿⣻⣾⣿⣻⣶⣾⣿⣿⣿⠿⠟⠛⠟⠻⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣺⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠛⠋⠀⠛⠚⢛⠛⢉⣉⣡⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⣻⣯⣿⡿⢿⡿⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣾⣿⣿⡿⠿⣿⣿⡿⠷⣿⣦⣌⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⣽⣿⢣⣾⣿⣿⣿⡿⣷⣌⠙⢿⣷⣶⣿⣦⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣴⣿⣿⡇⣿⢴⣿⣿⣿⡇⠈⣿⡄⠀⠈⣿⣿⡽⢿⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢻⣿⣟⣿⣶⣿⣿⣿⣿⣿⣿⡴⠶⠿⠿⠟⠻⠞⠭⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠟⠺⠿⠷⠃⠔⠚⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢋⣉⣉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⣀⣀⣤⣤⣤⣀⣀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢟⣩⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠔⠉⣠⣴⡾⢟⣿⣿⣿⣿⣿⣿⣿⡻⢷⣦⣄⠈⠣⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣴⡿⠛⢉⣠⠀⠀⠀⠀⠀⠀⠀⢠⣈⠙⠻⣷⣮⣿⣿⣿⣿
⠀⠀⠀⠀⣠⡾⠛⠁⠀⣾⣿⣿⠋⣩⡿⣿⣿⣷⠀⠈⠙⢷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣠⣾⣿⡇⠀⠀⣰⡾⢂⡀⠀⠀⣿⣿⣦⣄⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠙⢷⣦⡀⠀⢿⣿⣿⣾⣋⣠⣿⣿⡿⠀⢀⣤⡾⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠙⢿⣿⣇⠀⠀⠉⣰⡿⠃⠀⠀⣿⣿⠟⢁⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠻⢷⣮⣿⣿⣿⣿⣿⣿⣿⣷⡾⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣁⣀⡈⠙⠀⠀⠀⠀⠀⠀⠀⠚⣋⣡⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣛⣛⣻⣟⣓⣒⣒⣒⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⢨⣿⣭⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡆⠀⠰⣿⡷⠀⠀⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠁⠀⢘⣿⣟⡀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⠟⢻⣿⣶⡸⣿⣿⣿⣿⠾⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⠋⠁⠀⠀⠈⠻⡇⠈⠻⢿⣿⣿⢩⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣶⣄⠈⠻⢣⣾⣿⢽⣽⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠘⠟⣤⣼⣭⣿⡇
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣷⡄⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣾⣿⣧⣿⣿⣿⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢏⣾⣆⡈⠻⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣏⣾⣷⣿⣿⣦⡘⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⡆
⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⢼⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣽⣾⣿⠇
⠀⠀⠀⠀⠹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠠⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠁⠁⠀⠀⠀⢀⠞⠁⠀⠀⡤⠼⠓⡄
⠀⠀⢸⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⠀⠀⠀⠘⢀⠰⠇⢀
⠀⢀⠈⠾⣆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⣳⠶⡌
⠀⠈⠑⠦⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠣⣤⣤⠔⠊⣁⠔⠁
⠀⠀⠑⢆⠈⠺⣄⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⠴⠖⠒⠉⠀⠀⠀⠉⡉⠉⠁
⠀⠀⠀⠀⠈⣙⡿⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠒⠿⠺⡖⡄⠀⠢⣴⣠⠔⠊⠀⠉⠉⠁⢲⡒
⠀⠀⠋⣿⡐⠟⠃⠀⠀⠀⠘⡇⣀⣀⣀
⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠉
⠀⠀⠃⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡰⣷⠒⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⢤⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠑⠒⠒⠊⠁⠀⠀⠀⠀⢀⡤⠄⢀⡤⠛⢧⠞⠋⠉⠉⠀⠀⠀⠀⠀⠈⢄⠀⠀⢀⠇⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⡁⠈⠎⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠆⣤⣦⣴⡋⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡫⠄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣀⢞⣀⠀⢀⣀⣤⡞⠋⠁⠈⠉⠓⠳⠧⢀⡀⣀⠀⠀⢀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⠦⣤⢠⣭⡤⣤⢔⡿⠛⠉⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠠⠀⣀
⠀⡀⢾⣡⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⣀⣀⠀⢀⠀⠀⠀⠀⣠⠖⢉⠙⠋⠓⢄⣸⠁
⢠⣆⡶⠁⠀⠀⠈⠙⡦⠤⠤⠤⢊⡀⠀⠀⠙⠳⢄⣀⣀⡜⣁⡤⠆⠀⠀⠀⠈⠇
⠀⢯⣄⠀⠀⠀⠀⠠⠇⠀⢀⣰⣃⠀⠀⠀⠀⠀⠀⢣⠀⢛⠋⢻⣄⠀⠀⠀⠀⣹⣄⣀⣀⣀⣀⡀
⠀⠈⠀⣉⣀⠀⠠⠧⢴⡖⠉⣁⠉⣆⠀⠀⠀⠀⢀⣾⠛⠿⡕⠾⣯⣀⣠⣤⣾⠃⠸⣿⣭⡁⠀⠀⠉⠐
⠀⠀⠘⠉⠁⠀⠀⠀⠀⣄⠀⠉⡔⢫⠀⣀⠀⠀⢠⡟⠀⠀⠙⠒⠠⣤⠐⠋⠁⠀⠀⣿⠀⠉⢶⡄⠀⠀⠀⠀⠄
⠀⠀⢠⡀⠀⠀⠀⠀⢀⡅⠀⠀⠈⠛⠒⠀⠂⣤⠞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⠇
⠀⠀⠈⠑⠂⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⡀⠀⠀⠀⠀⡻⠦⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠒⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡠⣤⣄⡀⠀⠀⣠⣿⣀⣁⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⢰⡶⠭⣭⣤⣓⣒⠒⠂⠀⠀⠀⠀⠀⡀⢒⣛⣋⣭⠭⠥⠖⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⣶⡇⠀⡈⢀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠲⣦⢀⢸⣿⠇⡀⠁⢸⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⣤⣶⣿⣷⣤⡀⢠⠂⠀⠀⣶⣤⣿⢸⡟⣿⢶⠶⣄⣸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢿⣧⡉⠉⠉⠙⠁⠀⠀⠀⢠⣿⣿⡟⢸⡇⣟⢈⣿⠈⣿⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡿⣿⣶⡤⠀⠀⠀⢀⠀⢼⡿⠀⡇⠈⠙⠋⣑⣶⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡷⠟⠋⢀⠠⠀⠀⢀⣼⡶⠣⠀⠆⠘⠿⢿⣿⠿⡟⠇⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⡁⠁⣀⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⣀⡐⠐⠿⠿⠿⠿⠷⠆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢬⣭⣙⣛⣛⠛⠳⠰⠶⠟⠛⢛⣋⣡⣤⡄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣟⣿⣿⢿⡟⣿⣿⣿⣿⢹⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢘⣿⣿⣿⣿⢹⣿⣿⣹⠟⣹⣿⣿⠇⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣷⣟⣤⣶⠀⠙⣿⡇⢠⣷⣼⣷⡂⣶⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣶⢿⠀⠹⠟⠹⠷⠴⠨⣿⠇⠠⣿⣸⣿⡇⠛⣸⡷⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣿⣿⣼⣿⡇⣿⣮⣤⣜⠷⢦⣤⣀⣀⡀⠛⠂⢰⣞⣻⣽⣦⢔⣩⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣷⣶⣮⣭⣭⣭⣭⣭⣭⣭⣷⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋