Call forth Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠂⠀⠀⠀⠲⢮⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⢀⢢⠉⠀⢀⣤⣽⣿⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣶⣤⣤⣤⣛⣻⢿⣿⣿⣿⣿⣏⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⡿⠟⢏⡝⢿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢞⢿⢝⣿⣿⣿⣾⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠈⠀⠀⣀⡔⠈⢙⢋⡏⣹⠑⠁⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠴⡞⠏⠁⣐⣳⢞⡿⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠉⠉⠙⠻⣧⣤⡉⡀⠀⢋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢿⡾⠛⡁⠄⣸⣧⠈⠽⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⣴⣾⡷⠘⣷⣾⡋⠀⣿⡇⢰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣻⣿⣟⣿⢿⠑⠀⠸⣯⣤⠼⠋⡇⣸⢿⣷⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣥⣜⣾⣍⣟⡁⠆⠀⠀⠀⠹⣿⣗⠀⢁⣿⣤⢉⣷⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠪⠅⠹⣿⣿⣦⣄⠀⠀⠀⠹⣿⣄⠈⣽⢿⣿⠟⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠙⠳⣄⣻⡁⠟⢿⣷⡄⠀⠀⢻⡯⠀⢫⣿⠿⣚⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠠⣴⣶⣦⣤⣙⣧⠴⣢⣬⣽⢦⣄⠀⢣⠀⠸⠓⠞⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠋⠑⠋⠑⠂⠈
⠀⠀⠀⠀⠈⠸⠹⠈⠀⠰⠞⠇⠟⠷⠀⠽⠴⠘⠀⠨⠼⠃⠀⠚⠛⠆⡽⠻⠀⠘⠃⠯⠦⠿⠥⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠛⠛⠛⠛⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢀⣠⠴⠶⠖⠒⠒⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⠃⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣆⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡟⠀⠊⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠻⣆⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⡇⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠹⣆⣠⢤⣄⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⢸⡏⣼⠏⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⡟⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢤⡤⠄⠀⠀⠀⠀⠀⠀⣼⢡⡿⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣾⠃⠀⠀⠀⢀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠻⣄⠀⠀⠘⣯⡉⠉⠉⠉⣹⠟⠀⠀⢀⡾⠉⠉⠛⠛⠛⠛⠉⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠃⠙⠉⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠚⢻⡟⠶⢤⣤⣤⣤⣤⠴⠞⣿⠳⠶⠶⠶⠶⠶⠶⠶⠒⣿⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⣶⣤⣄⣀⠀⠀⠀⢠⡏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠚⠛⠛⠛⠛⠛⠓⠲⠦⣄⡀⠀⠀⠀⠀⢀⡾⢱⡿⠀⠈⠙⣿⠛⣻⣟⡀⢠⣟⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡴⠞⣻⢃⣿⠃⠀⠀⠀⠈⠛⣿⠈⠿⠿⠬⠿⠶⠴⠏⢸⡇
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠏⣼⠇⠀⠀⠀⣰⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⢳⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠻⣴⣟⡀⠀⠀⣀⡿⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀⠀⠀⠸⡇
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣏⠉⠉⠉⢹⡇⠀⠀⣰⣿⣿⡾⠟⠛⠿⣿⣿⣷⡄⠀⠀⣿
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⣦⡀⠀⣸⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢸⡿⠿⡇⠀⠀⢸⡆
⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠲⡟⠀⠀⠀⢿⣿⣿⣦⣀⠀⣠⣾⠁⢰⠇⠀⠀⠈⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⠉⣻⡟⣿⠄⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠛⠁⠀⠀⠀⠀⢸
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿
⣿⣿⣏⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣶⠶⢶⡶⠶⠶⣶⢶⡆
⠀⣿⣟⠩⣭⢸⡻⡿⣻⡇⡄⠀⠀⠀⠀⠀⡀⡄⠀⠀⢠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡄⠀⠀⠀⢠⡀
⠐⣇⣿⠃⣼⠘⣷⠓⠻⡄⢏⡟⣾⠟⡇⢸⡉⣟⢾⡃⢿⣄⡸⢒⠢⡖⠚⢶⡱⡆⢾⡄⣿⠀⣞⢰⡆⡶⣶⣰⡆⡇⢸⣶⢸⡃⣶⢰⢸⣹⣗
⠠⠗⠈⢡⡼⠸⠉⢢⢰⡆⣀⠁⠀⠀⠀⠀⣀⠈⠀⠀⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠘⠒⠂⠑⠛⠛⠓⠃⠘⠃⠙⠈⠐⠛⠙⠈⠛⠜⠂⠲⠆⠸⠂⠛⠘⠺⠀⠖⠃⠸⠸⠒⠺⠁⠺⠚⠐⠓⠂⠀⠖⠀⠇⠰⠸⠐⠂⢷⡺⠥⠮

⠀⠒⠶⣦⡐⣶⣶⣦⣔⢲⠒⢶⣖⡲⢲⠂⠰⡶⠂⣀⢲⣶⡒⢒⣲⣖⡒⠒⢲⣶⣶⣶⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠂
⠀⠀⠀⠘⠶⠛⠓⢻⣦⣽⣶⠦⡽⠣⢈⣰⡖⠑⠞⠛⠂⠙⠛⠿⣿⡋⠛⠚⠛⠛⠛⠠⣵⠀⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⡀
⠀⢄⠶⠲⠀⠀⠘⠛⠻⣿⣾⣅⡛⢷⡞⠿⢿⣆⠀⠀⢉⠀⠀⠁⠈⢻⠃⠀⠔⠲⠤⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡁
⠠⠈⠀⠀⠀⠀⠀⣴⣶⣭⡼⣿⡿⠞⣃⠀⠠⣿⡖⠀⠸⠇⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⢀⣤⣴⣦⣤⣤⣴⣴⣶⡶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣾⣍⣠⠤⠔⢷⣤⠻⠗⠀⠘⢦⣐⡀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⠀⢿⡿⠟⠿⠿⠿⠿⠷⠿⡿⢿⡇
⠀⠀⠀⠀⠀⠰⠀⢠⡴⠋⠀⡉⠁⠀⠰⢲⣮⢱⡆⠰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠷⣆⠀⠀⠀⠀⠀⡇
⢀⠀⠀⠀⠀⠀⠐⠀⢀⣀⢀⠀⠀⠀⠘⣿⢹⡞⠃⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣀⣿⠀⠀⠀⠀⠀⠆
⠈⠁⠄⠀⠀⠄⠙⠁⠀⠹⠃⠙⠃⠀⢰⣷⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⢀⠐⠁⠤⠀⡴⠶⣆⡀⠢⡀⠀⠀⢠⣌⡕⠛⠆⡀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢠⠘⠀⠀⣏⠀⠀⢈⢡⢂⠁⠀⢰⣗⣿⣶⣤⣀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄
⠘⠀⠁⠀⠙⠀⠀⡸⠈⠁⠀⠀⠾⠉⣿⣿⣿⣯⠉⠉⠠⠀⡅⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠟⠙⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡀⠀⠀⠀⡄⠛⠁⡀⡈⠀⡄⠀⠀⢹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠁
⢨⢹⡄⢀⠘⠻⣶⣼⡇⢹⠀⢹⣦⠀⠀⢻⣿⣿⠿⣿⠉⡦⠀⠀⢀⠛⠃⣥⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠸⣿⠁⣾⠇⣤⡀
⢨⡔⠁⢠⡤⣤⣽⢩⡀⡜⠀⢸⣿⣷⣄⡆⠙⠧⠁⠈⠁⢹⠀⠀⠀⠈⠃⢸⢀⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢻⢄⡛⠟⠋⠀⣀⣀⣄⢀⣠⡀
⢘⣓⡆⢀⠚⠙⠃⣺⣷⠁⠀⣼⢿⣿⡿⠛⢶⣶⣤⢶⡄⠋⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣮⣿⣿⣽⣿⡁
⢘⢙⠇⣀⢈⣆⠈⠻⣿⠀⢀⣿⣄⣧⠀⠀⠀⠁⠀⠈⠀⠀⢠⡆⢀⠀⠠⠘⠂⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⢸⣿⣏⣈⡛⣧⢾⠋⡏⠀⠀⣿⣿⣿⡄⠀⠀⠃⠀⠀⢰⡆⠘⣥⠏⠀⣴⠃⠀⠀⠀⢀⠿⡁
⠘⠁⢿⢯⢥⣿⣟⣷⡇⠀⢠⣿⣿⣯⡷⠀⠀⠐⠀⠀⠀⣿⡀⠈⠀⠘⠇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠨⠘⢻⣿⣻⡇⠀⣾⠹⣿⣿⣷⡀⠀⠐⠋⠀⠀⣿⣿⠓⠒⢴⡄⢀⠀⠀⠀⢐⣷⡀
⠀⠀⠸⢏⣨⣽⣿⡿⠀⠀⣿⢸⣿⣩⣿⠃⠀⠀⠀⠀⢠⣿⡿⣄⣐⣈⠙⠉⠀⠀⠀⠀⠉⠃⠛⠓⠋⠘⠒⠒⠚⠒⠒⠂⠀⠀⠀⠀⠒⠲
⠀⠀⠀⢴⠉⠡⢬⠁⢠⣰⡟⢸⣿⠉⢹⠀⠀⠀⢰⣶⣾⣿⣇⢿⠛⠻⣟⡀⠀⣀
⠀⠀⠀⢸⣴⣦⡞⢀⡾⢻⡇⠘⣿⡄⠸⢢⡀⠀⠀⣏⣸⣿⡟⠀⠀⠀⠈⠁⠀⠀⠁
⠄⠀⢦⡾⠿⠟⢁⡾⠃⢸⡇⠀⠙⠗⠓⣺⣿⣶⡼⢟⡟⠻⡅⠀⠀⣠⡾⠖⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠄⡀⠀⠀⠀⠀⠈⡄⠀⠛⠃⠐⠀⠀⠀⠀⢉⠝⠉⠉⠀⢀⣴⣤⡾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣠⡀⠀⠀⠀⠀⠀⣀
⡇⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄⠀⠀⠠⠀⠄⠀⠀⣀⣀⡀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠭⠽⠽⠯
⠇⠀⠀⠀⡀⢠⠀⠀⠀⠀⡀⣀⠀⠀⠀⢀⠀⠚⠛⡛⠛⠛⢣⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⢀⠀⠀⢀⡀⡀⠀⢀⡶⠷⢟⢲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢂⠔⠚⠄⠤⡀⢀⠀⠀⡴⣂⠂⠰⣥⡠⡾⠄⡖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠄⠀⢀⡀⠃⠁⠠⣲⠠⣤⠖⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠂⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢀⣀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠁⠀⠀⠀⠀⠁⢸⡰⠆⢀⣤⣤⣉⠻⠘⠂⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⡀⠀⠀⣼⠀⡏⠘⠛⠉⢳⢠⢍⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⣀⠄⠀⠀⠠⠀⠀⠘⠆⢻⣄⢀⡶⠋⠠⢌⢈⢍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠉⠐⣤⠀⢀⠀⢹⣟⣿⠇⠀⠀⠈⠈⣸⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⣤⡇⠀⠈⠀⠀⠤⠁⠀⠀⠔⠀⠈⣛⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠃⠹⣿⡿⠃⠘⠀⠀⠀⠀⠀⢁⢀⠀⠀⠶⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠃⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⣤⡄⢠⣤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢢⢠⠠⠀⠀⠐⠀⠀⢀⣗⣻⣷⣄⠈⠁⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠋⠀⠀⠀⠁⠀⠀⠀⠉⣟⣛⣫⡅⠀⣠⡒⠓⣢⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣀⢀⣶⠀⠀⠀⠀⠀⣘⡉⢉⠁⠃⣿⣆⢠⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠐⠂⢾⡶⡿⠈⢂⡐⣿⣏⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⡿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣸⣷⣦⡌⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢘⣿⣿⡿⠿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢀⣀⢀⣠⣸⣿⣄⣀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠿⠛⣿⣿⡿⠃⠘⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠰⢰⣿⣿⣿⢷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡊⡉⢉⣿⠀⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⠀⠈⣻⣿⣫⣾⣿⣿⣏⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣿⣿⡄⢉⡙⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣛⡛⢲⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⢀⡄⢹⣿⣇⠀⡀⠛⢹⢿⡿⠉⠛⠛⠛⠿⠄⠀⠠⣿⣿⣎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠽⠏⠐⠃⠀⠀⠸⠀⠠⢂⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⡈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⡀⡀⢠⣾⣿⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣼⠇⢨⡓⠷⡴⠯⣝⣳⡦⢦⣄⠀⢀⣴⠃⡀⠶⠃⣠⡞⠈⠁⣽⣿⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠀⢿⣦⣸⡃⣀⡀⡐⠛⠧⠖⣩⡉⣹⠿⢇⠀⣃⠀⢀⣟⢀⡄⠀⣿⠻⣇⠈⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⢀⡟⠀⣸⣿⢻⣀⡿⠀⠘⠊⠰⠀⣻⣿⣟⠳⡾⣿⣿⣷⣾⣯⣾⣷⣶⣿⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡞⢀⣾⢁⣀⣿⠿⠋⠀⠀⠀⢦⣀⡃⠀⠉⢻⣿⡃⠀⢍⣿⠛⠛⠛⠻⣯⠹⡿⠷⡻⠱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠊⡀⡚⡋⠀⡉⠁⠀⠀⢀⡤⢀⣘⣻⡅⢰⠔⠛⣿⣿⣦⣄⡈⠀⠀⠀⠀⢻⠁⣿⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⣣⡁⠀⣿⡀⢠⣾⣿⡇⣸⣿⣿⡇⢸⡇⣶⢹⣿⣿⣿⠛⠶⣦⣄⣀⠞⣸⣿⣧⣸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠃⢃⣠⣿⡿⠟⠻⢫⣥⣿⣿⣿⣿⢠⡇⢹⣆⠀⠛⠿⣶⣷⣦⣼⣥⣤⣿⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠈⣶⣥⠴⠖⠚⠉⠀⣿⣿⡿⠿⠋⠙⠁⠀⡏⠑⠀⢦⡀⠹⣿⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⣷⢄⠀⢄⣻⣷⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣰⣿⠃⠀⠀⢨⡄⠁⠈⠻⣿⣿⡻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⢉⣻⣁⣀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠼⠏⣰⣿⡇⠀⠀⠀⠘⣿⡀⠀⠀⢸⣿⣿⣮⡙⢿⣦⠀⢀⠀⠀⠉⠀⠀⣀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣶⣿⣷⣶⣶⣶⣶⣾⡆⠀⣿⣿⣀⣾⡀⣿⠀⢻⣿⣦⣤⣼⣿⣿⣿⣷⣄⠈⠁⢠⣤⣤⣴⣾⣿⣿⣿⣿⣦⣤⣤⣼
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣻⣛⣛⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣬⣷⡙⢿⣿⣿⡿⠿⣿⣿⣷⣦⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣶⣌⠳⠈⣧⠀⢿⡾⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⢹⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠴⠀⣠⠤⠤⠏⠀⠀⠙⠉⠙⠲⠒⠳⠄⠀⠀⣰⢣⡾⡮⠩⠐⠰⠽⠾⠉⠯⠓⠸⠔⠙⠯
⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠤⢀⠀⠀⠀⠀⣠⣄⠀⡀⢀⣀⠈⠓⠀⠤⢰⡡⡟⣰⣢⣴⠛⡾⣿⠿⠥
⠀⠀⠀⠀⠀⠀⠣⣀⠤⣀⣀⢀⠄⠀⠀⠄⠀⡇⠀⠀⠘⠍⣋⠠⠄⠀⠘⠒⠋⠩⠄⠁⠁⠀⠲⡃⠰⠆
⠀⠀⠀⠀⠀⠀⠀⡆⠰⠠⡌⠇⠀⠀⠀⠀⠋⣈⡤⢤⡄⠠⠄⠄⠂⠀⠀⠀⣀⣷⣾⠊⣄⠁⠀⠁
⠀⠀⠀⠀⠀⠸⠆⠀⡂⠤⠐⡈⠂⡄⠀⢚⠀⠀⠀⣸⡀⠸⠖⠈⢁⣠⣤⣤⣉⡙⠿⠀⠯⠀⢐⣲⡆
⠀⠀⠀⠀⠀⠄⠀⠰⠀⠣⠦⠄⠀⡀⠐⠎⠠⠐⠀⠈⠉⡗⠀⣰⠟⢉⣿⣿⡛⢻⣆⠀⢐⠲⠤⢙⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠘⠀⠀⢀⠀⠃⡌⢀⠂⠀⠸⣧⠀⣿⣘⠀⢈⠉⠀⠆⡿⢠⢿⢰⢦⣼⡾⢱
⠀⠀⠀⠀⢀⣤⡄⠄⠀⣀⠀⡤⠄⠠⠀⠂⢠⠈⠀⠠⠀⠸⡧⠘⣿⡅⠀⠀⣷⡾⢣⡆⣤⡬⠄⣡⣉⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠁⠙⠋⠁⢈⡤⠒⠒⠤⡀⠀⢂⠥⠒⢂⠐⣿⣿⣿⣿⣿⠄⠄⠍⡀⠺⠱⡇⢮⡞⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⡞⠄⠀⠀⠐⢹⠗⡎⢀⣖⠖⢄⢻⣧⣤⡿⠃⢈⠄⢄⠈⢢⠁⣤⣬⢃⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣀⣀⣠⣾⡆⠱⢀⠙⣊⠜⡸⢠⡄⠆⠀⡃⢎⣾⠂⢰⠀⠛⣟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁⢿⡜⢿⣿⣿⣿⣿⠀⢐⠤⠀⠁⠜⠁⠀⠠⠜⣆⠑⠠⠤⠔⠁⣀⡐⢿⠗
⠀⠀⠀⠀⠀⠀⢀⠀⠛⠋⠁⠀⠀⠈⠛⠿⠿⢁⠀⠈⠀⠀⡌⠀⠀⠀⠀⠈⠈⣆⠰⠆⣰⠈⠛⣣⣄⢆
⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⢐⡬⠄⠁⠁⢸⠸⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⢸⡄⠶⣿⠀⣦⠉⠉⠈⢁⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⡶⢀⠀⠀⠘⡌⠊⠸⠀⠃⢀⡇⠀⠀⠀⠀⠀⠐⠂⠀⡇⠀⠀⢂⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠃⠖⠎⠁⠀⠀⡃⡀⠀⠀⠀⠀⠀⣼⣿⡧⠠⣿⣿⠄⠀⠶⠀⢠⡄
⠀⠀⠀⠀⠀⢀⣴⣾⡶⣆⢠⣀⣠⠀⠀⢠⠐⠇⠀⠀⠀⢸⣿⠻⣿⣿⢧⠀⠈⠙⠁⠀⠉⠉⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠘⣦⡀⢠⡞⠘⠁⠀⠀⢼⠦⠤⢲⠀⠀⠰⠾⣿⣿⣿⣯⠿⡻⠀⠇⠀⣀⣤⣄⣄⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠠⠄⠀⠀⠀⠈⢀⠀⠀⠘⣷⣬⣥⣴⣿⠁⡸⢀⣞⡥⠀⠠⣌⣧⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠈⢠⣷⡄⠀⠉⠠⠈⡀⠀⠀⢺⠋⠉⠁⢈⠀⠇⢼⣯⣥⠀⣤⣭⣿⡄⠘⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠛⠙⢻⣿⡇⠀⢀⠀⡀⣅⢠⣖⣋⣙⣦⣐⣟⡤⢶⣾⣿⣿⠶⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠓⠊⠀⢉⡉⠘⠟⣿⠿⠟⠀⠈⣠⡄⢹⣿⣿⣤⣼⣿⣿⠵⠆

⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠹⣄⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⣰⠏
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣜⠉⠉⠻⣿⣟⡠⠂⢀⠀⠀⢀⠜⠻⣿⡿⠋⠁⠀⢨⠆⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⣷⣤⣀⣈⡿⣁⠀⠸⠇⠀⠀⠀⠀⢻⠃⣠⣤⣖⠁⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠐⠂⣰⣿⣿⢻⣟⣿⣿⠿⠖⠂⠀⠀⠀⢀⢻⣣⣿⣿⣿⠤⣉⢑⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢻⡹⡿⢿⠙⣿⣿⢆⠠⠀⠀⠀⡠⢂⣿⣿⣿⣿⣿⣭⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡳⣿⣮⣵⢧⣽⣄⠙⠂⡰⠊⡠⠊⢹⣾⣿⣟⠪⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠔⠋⠉⠛⢧⣇⠈⠑⢄⢀⠞⠀⠀⢸⡿⠿⡎⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠄⠀⠀⢠⢻⡆⠀⠀⠀⠀⠀⠀⡘⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣢⣎⡆⣇⠀⢰⠀⠀⠀⠀⣣⣲⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡄⢸⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⡿⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠃⠀⡰⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣋⣀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⣍⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣄⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣷⣻⣿⡆⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⡏⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⡿⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢿⣿⣿⣿⣿⣿⣯⠀⠀⠀⣰⣿⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣿⣿⣿⣿⠗⣀⣴⠟⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⠻⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠙⢳⠟⠻⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⣦⣀⣈⠿⣿⣿⣿⣿⡓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⣰⣦⣤⣤⣿⡟⠁⠀⠛⠛⣿⣿⠇⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠹⣿⡿⢿⣿⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠁⠀⠀⠉⠻⠳⢦⣤⣤⣤⣴⡃⠤⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠐⣦⡐⢤⣄⡀⠀⠀⣴⠀⣀⠉⢿⣶⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢾⣍⡒⠺⠿⢴⣿⣀⣾⣿⡆⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠉⠻⠧⣦⡀⠀⣉⡻⣿⡏⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠞⠿⠳⣮⠀⠲⢶⣿⣇⢸⣿⠁⣿⣿⣶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⠀⠀⣀⣀⣠⣦⠸⣆⣉⠀⣹⣾⡿⠉⢁⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⣀⣶⣶⣾⣿⢿⣿⣿⣿⣷⡬⣩⠴⠛⠉⠁⠀⣇⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⠿⠛⡉⣿⣶⣿⣉⠻⣿⣄⠀⠀⠀⠀⣠⣄⠹⣷⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠄⠸⣿⣿⣿⣆⠸⣿⡇⢀⠀⡘⢻⣿⣧⡙⠂⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⠐⠀⠀⣴⣿⣿⣿⣿⠀⣿⠇⠠⠈⣴⣿⣿⡟⠁⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⠀⠀⣿⣿⣿⣿⡟⢰⡿⢀⡀⠈⠱⣿⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⡀⠀⠉⣿⠿⠋⣠⣿⢧⣿⠁⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣤⣬⣤⣶⣿⠟⠋⣩⠿⣆⠀⠀⠀⠈⢿⣿⣷⡄⢰⣦⡀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣛⣛⣛⣛⣉⣠⠴⠚⠁⠀⠘⣆⠀⠀⠀⠈⢿⣿⣿⣮⣿⣿⣦⡀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠿⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢻⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⣻⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⠻⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠠⠀⡰⠂⠀⠉⣠⣦⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢈⣉⣀⡞⠉⠈⠙⠶⠒⠦⣤⡤⢦⡀⠀⠀⠀⢠⡞⣡⣤⠋⡇⠘⣫⣟⣦⡞⢻⡞⡆⢸⣏⠟⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠺⣌⡋⠙⠀⢠⠄⣠⠉⡀⠀⣨⢥⣠⣤⣥⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠁⠒⠒⠁⠁⠀⠀⡼⠒⠂⠀⣠⡴⠶⠀⠀⠀⠈⣁⠻⣬⢧⠼⠴⠿⠟⢘⠃⣿⠈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠖⠦⣤⣄⡀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠈⡄⠀⠒⠀⠀⠈⠋⠁⠐⠂⠀⠀⠀⠀⠙⡁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠂⢠⠌⠁⠀⠀⠀⠀⢈⠁⣠⠤⠴⡦⠀⠰⠀⠆⠁⠀⠀⠀⠂⣠⣷⣾⠟⢀⡀
⠀⠀⠀⠀⠀⠀⠸⠆⠀⢐⠠⡆⠀⡀⠀⣀⠀⠀⠘⠀⠀⠀⢰⡇⠀⢿⠖⠀⠁⣀⣠⣤⣈⣉⠙⢿⠆⠸⠯⠀⠐⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠠⡄⠁⠂⠈⠀⣤⠉⡀⠀⠀⠈⠙⢳⡆⠀⢠⡾⠛⣻⣿⣿⠿⢷⡄⠐⠀⠦⣄⠈⠉⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠤⠀⠀⠘⠀⠀⠀⠘⠀⠄⠐⠀⣼⡃⠀⣟⢠⠂⠙⠛⠛⠀⠈⣿⠀⣰⣶⢠⡀⢸⢲⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡈⠃⠐⠀⠀⠀⠘⣷⠀⢿⣶⡆⠀⠂⠀⣈⣡⡟⣀⠋⠋⠞⠱⠟⢿⠔⠁
⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠤⠆⠄⠛⢉⠀⠂⠀⠀⠲⠀⠀⢀⣀⠀⢙⢃⠈⣻⣧⣀⣀⣰⣿⠟⠀⠇⠸⠞⢾⠃⣾⢣⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⡴⠊⠉⠙⠲⣀⡀⠈⡐⠈⠈⠠⠈⣿⣿⠿⢿⣿⡗⢂⠒⠒⠤⣈⠊⡘⠣⠛⡞⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠀⠀⠀⠂⢸⡿⠘⠀⣰⡌⠌⠠⠹⣿⣦⣾⠿⠃⠈⠁⠄⡀⠈⢂⠀⣤⣤⡍⠘⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡀⠀⢀⣠⣾⡇⠘⠄⠐⠓⢀⠨⠠⢁⣄⠐⡀⠀⠅⠄⢰⡃⠀⢐⠀⠻⠿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡰⢠⣶⡘⣿⣿⣿⣿⣿⣿⠃⠠⢆⠁⠀⠀⡠⠃⠀⠉⠀⢳⡀⠡⢌⠁⢀⡐⠁⠀⠠⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡴⠶⠀⠙⠃⠈⢿⣿⣿⣿⠯⠀⠀⠂⠁⠀⡐⠁⠀⠀⠀⠈⠐⠩⠀⢠⣄⠀⡀⠸⣿⡇⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠂⠀⠀⠀⠀⣀⠠⠀⠀⠈⠉⠁⠈⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠁⡰⡃⠀⠀⢾⡷⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠴⠞⠃⠐⠀⠀⢀⢀⠁⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠸⠇⠘⠋⠇⠘⠿⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠸⣿⠇⠀⠀⠀⠀⠰⠉⠃⢸⡀⠙⠀⢐⡀⠀⠀⠀⠀⠀⠀⠘⢁⠀⡼⠀⠀⠀⡀⠀⠠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠒⠏⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣽⣿⣧⠀⢼⣿⣷⠀⠀⠶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⢦⡀⢀⡀⢀⠀⠀⠀⡀⠠⡇⠀⠀⠀⠀⣻⣷⠶⣿⣷⣶⡄⠀⠈⠹⠁⠀⠈⠉⠉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣉⠋⠉⣠⡇⠘⠂⠉⠀⢐⢆⠁⢀⠦⠀⠀⠀⣠⣾⣿⣦⣾⣿⢷⡿⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⠟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⡀⠀⠀⠀⢨⡛⠻⠿⠛⣉⣴⠃⠐⠀⣠⠖⠛⠙⠛⠳⡄⠀⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠈⢁⠀⠀⠀⢀⠀⠰⠄⠀⠀⠈⣿⣿⠿⠿⠿⡧⠀⡌⢠⣯⢾⠁⠀⠘⣶⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⣀⡀⣰⣿⣷⠀⠀⠀⠐⠀⢡⠀⠀⠀⣋⡀⠀⠀⢀⡄⠘⠃⢸⣿⣾⡆⠠⢲⣾⣿⡇⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠉⠁⢹⣿⣿⠀⠀⢀⠀⡄⣤⠀⣼⢛⣉⣉⣷⡀⣻⡐⢒⣀⣿⣿⣿⡷⠶⢿⣿⣿⡯⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠃⠊⠐⠈⣉⠀⠻⣛⣿⣟⠿⠃⠀⢉⣤⡀⢸⣿⣿⣧⣤⣼⣿⣿⡷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠘⠿⠃⠘⠛⠛⢿⠿⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣄⠀⣠⠦⠈⢁⣴⡤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣩⣥⡞⠁⠈⠑⠚⠓⢦⠴⠦⡀⠀⡄⢠⣟⣵⣾⣻⠀⡯⣿⣶⠛⣿⠷⠸⣏⠟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⢀⣀⢀⣠⠀⠀⢀⣀⠀⠀⠀⠀⠘⠷⠃⣩⢠⡖⡶⢢⢄⣠⡞⣶⣿⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣺⠉⠀⠐⡚⠟⢀⣀⠀⠐⢮⠽⢛⡘⠉⠉⠁⢧⠞⠠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠠⢉⡙⡆⠠⠀⠀⠀⠖⠋⣀⣀⡄⠀⡉⣀⠢⠀⠈⢠⠈⣤⣴⡴⠠⠀⠈⠋⠢⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠁⢌⢠⠡⡘⠄⣀⢀⠤⠃⠉⠁⢸⠃⢰⣦⠀⠃⣀⣀⣀⡚⠛⣿⡄⣿⡂⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡅⠎⣤⣈⠒⠉⠠⡜⢁⠀⡀⠈⠛⣶⠀⢠⡾⠛⣿⣿⡿⢷⣄⠀⠰⢤⣀⡛⠅⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠂⠰⠀⠑⠘⠀⠆⣕⠂⠦⠀⠀⣿⠀⣿⠰⠀⠉⠛⠁⡄⣿⢀⣾⣇⣦⢸⣷⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠄⠀⠀⠀⣀⡀⢈⡀⠰⢉⡈⠀⢈⠂⢽⣦⠸⣿⡇⠈⠁⣶⣶⠯⡬⣀⣝⣉⢛⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⠀⠒⡛⠖⠃⠐⣀⠤⠤⣌⡁⠂⠤⣂⠤⢌⠤⣿⣿⣶⣶⣿⣁⣈⣃⡉⢿⢣⡟⣿⡷⢓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣀⣼⣁⠀⠀⢠⢹⣶⡴⠁⣤⣪⢅⢿⣿⣉⣿⡟⠸⣂⡀⠉⠢⠈⢉⣘⡇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢿⣀⠀⠀⢀⣼⡇⠱⡀⠻⠜⡸⢀⢋⡉⣏⠀⡎⡑⢺⡇⢀⠇⠻⠿⠏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠆⣶⡼⣿⣿⣿⣿⣿⠇⠰⢌⠁⠊⡡⠃⠈⠁⣸⣆⠱⢌⣉⣠⠊⠁⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠾⠃⡈⠁⠘⡿⣿⣿⠿⠀⠌⠂⠀⡰⠁⠀⠀⠀⠐⠹⣆⢠⡄⢀⠄⠿⢟⣈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠁⠀⠀⠀⢰⣐⡄⠃⢀⢸⠀⡄⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠹⡄⣤⣻⡇⣠⡘⠟⠈⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣦⠈⠉⠀⠘⡜⢴⢲⡄⠤⠀⡆⠀⠀⠀⠀⠀⠀⠦⠈⠁⠀⠀⠢⠉⠀⡈⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠉⠀⢂⠀⢀⠧⣤⣾⠃⠀⠈⠏⠀⠀⠀⠀⠀⠐⢸⣶⣧⢀⣶⣶⡄⠀⢥⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⠀⡀⢀⠀⠀⡄⡀⡆⠀⠀⠀⢘⣷⡶⣶⣶⣶⠀⠈⢻⠋⠀⠙⠛⠁⠀⠀⠀⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡉⠁⣼⠰⠋⠉⠄⣐⣌⢁⡴⠀⠀⢠⣴⣿⣧⣿⣿⣾⠷⠀⡌⠀⠀⣀⣀⢀⠀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⣄⠀⢀⡄⢨⠀⠀⠘⢄⠀⠀⠐⣯⣙⣻⣩⣴⡏⢠⠃⡴⢋⡉⢉⡙⣦⠀⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⡇⠐⢀⣴⣸⠀⠐⢀⠈⠃⠀⠀⢸⠟⠛⠛⠙⠃⣿⣐⣿⣳⡀⢀⣞⣿⡇⢰⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠀⠷⠶⢺⣿⣿⠀⠀⠀⠀⢘⠀⣤⠾⠶⣤⠠⣾⣢⡀⣼⣿⣿⣭⣭⣿⣿⣧⠰⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠫⠯⠰⠼⠢⠛⠛⠘⣷⣿⣿⣿⠇⠙⢢⡍⠽⣿⣿⣆⣠⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⡉⠉⠀⠀⠀⠻⠟⠐⠿⠿⣿⢿⠿⠿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠀⠒⠢⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣶⣒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠤⠤⢄⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣑⣲⣶⣤⣤
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠿⠟⠛⠛⠛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠟⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢶⣤⣶⣿⣿⠟⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠉⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⢶⣒⣶⡚⠋
⠀⠀⠀⠀⠀⠀⢀⢼⣓⠢⢄⡀⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⣿⣧⣀⣀⣀⣀⣀⣤⡤⠶⠖⣚⣋⣭⣭⣶⡾⠶⠶⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⠂⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠓⠺⢿⣿⣿⣿⣵⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠐⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠒⠤⣄⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⣖⣛⣻⡟
⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡉⠓⠤⣀⠀⢸⣿⡤⠀⠀⠀⢀⣀⣠⣤⠴⣖⣾⣿⣭⣷⠶⠿⠛⠛⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠛⠿⠿⣛⣛⣯⣿⡿⠿⠾⠛⠛⠉⠉⠁⠀⠀⡀⠠⠄⠐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣶⣿⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⢸⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣸⣿⣿⡅⣀⣀⣤⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠉⠲⣄⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠸⠵⠋⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠔⠁⠒⠉⠁⠀⠀⠀⠀⠀⠉⠙⠂⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠿⠇⠀⠀⠀⠳⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⠃⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡄⠀⠀⠀⠀⣷⣬⣭⣭⣥⣼⠇⠀⠀⠀⠀⣇⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣄⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⢀⡜⣏⠙⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠤⣀⡀⠀⠀⠀⠀⠀⣀⡠⠖⠉⠉⠁⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡏⠉⢹⣉⠀⢀⡤⠦⣠⠤⠒⢚⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⠀⢀⡠⠔⠊⠉⠻⠤⠤⠤⠟⠉⠉⠀⠀⢳⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢨⠇⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢇⡎⢨⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⣠⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀⣀⡠⠎⡅⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡈⠂⠀⠀⠀⠠⠤⢄⣀⠀⠀⠀⠀⠊⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⠁⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⣉⣉⣑⡂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡴⠛⠀⠀⣠⠖⡉⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡤⢄⡀⠀⢸⠀⠀⠀⠀⠁⠠⠄⠀⠀⢠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠶⣝⣆⡹⠀⠀⠀⠀⠀⠤⠆⠀⠀⣃⠞⡠⠋⡴⢋⠜⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠳⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⣥⡚⣀⠮⢀⡆⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢤⣤⣹⣿⣿⣿⠛⠁⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢠⠀⠀⠀⠀⠉⠣⣐⠒⠤⢾⣿⣿⣿⣿⣿⠀⢀⢠⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢇⠀⠀⠀⠀⠀⠈⠉⠁⣖⠶⠈⠉⢙⣛⠀⠀⢸⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠇⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⣁⠴⠚⠉⠉⠉⠓⠠⠤⠤⠤⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⣀⠀⠀⠀⠈⠒⢤⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⣀⠝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⡀⢄⠈⠑⠢⣄⡀⠀⠰⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠃⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠦⢙⡈⢧⠀⠀⠀⣉⢣⣠⠀⢀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⢀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠀⠀⠙⡌⠀⠀⣴⠉⠁⣿⠀⡖⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠊⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠸⠀⠀⠀⠀⠘⡄⡰⠣⣝⠦⡈⢦⢣⠀⡍⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠑⢌⠢⣈⠀⢀⢱⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡁⢰⠀⢇⠀⠀⠀⠉⠒⠤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣸⢠⠎⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠘⡄⣀⢄⠤⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡴⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⠞⡇
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⡒⠤⣀⠀⠊⢸
⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣤⣢⣔⣁⣎⣀⣡⣀⣀⣀⣀⣀⣴⣠⣸⡅⠀⠈⠑⠢⣀⣄⣀⣀⣀⣀⣀⣀⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠎⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⠀⠀⣀⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣾⣿⣧⡙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠛⠋⠁



⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣧⣤⣴⣶⣶⠶⣿⣿⣶⢰⣶⣶⣶⣶⣤⣤⣤⣄⠈⣿⡆
⠀⠀⠀⢰⣾⣿⡏⣽⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠻⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⣤⣬⣤⠀⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⣡⣝⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷
⠀⠀⠀⠿⣦⣧⠀⠀⠀⠀⠈⠩⣗⣀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⢦⣘⣼⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢹⣿
⠀⠀⠀⣾⠿⣟⣄⠀⠀⠑⢀⡼⢯⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠤⡮⣸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣽⣿⡇⢸⣿
⠀⠀⠀⣿⠀⡋⣏⠀⠁⠠⡽⡷⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣶⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⢰⠈⠀⠀⠸⡇⢂⠀⠀⠨⣽⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣴⣶⣶⣶⣶⢰⣶⣶⣶⣶⣶⣶⣤⣬⣭⣅⠸⣿
⠀⠀⢸⣇⠀⠀⠀⣿⡀⢶⡖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢦⣿⣼⣿⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⡼⠿⣦⠀⠘⣿⣷⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣤⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⣷⠘⠿⠠⠊⢉⣥⣤⣤⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿
⠀⠀⣿⣤⣶⠤⠄⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣏⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹
⠀⢰⣟⣉⣐⣶⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠧⣼⣿⡛⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠇⡿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠠⠐⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⡄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠂⠀⠀⠀⣴⡾⠛⠛⠛⠛⠻⣦⡀⠀⠀⠈⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣶⡄⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⠀⠀⠘⠛⠛⠁⠀⠀⠀⠙⠛⠛⠀⠀⢀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⡶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢺⢻⢩⠀⣰⠃⠀⠀⠀⠀⡼⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠰⠸⢸⠀⠃⠀⠀⠀⠀⡼⠁⢀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⢀⡠⠴⠚⠓⠒⠦⢤⣤⠚⠀⠀⠁⢀⠜⡽⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠦⠌⠳⡶⢶⠾⠗⠋⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠐⠁⠐⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠴⠖⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡀⠘⡧⠀⣻⣦⠀⠀⠀⡚⢉⣶⠆⠀⠀⠸⢩⣍⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠞⠃⠑⠛⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⢐⡈⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⢴⣇⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡴⢧⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠠⠖⢲⠀⠣⡀⠀⢿⣅⡲⠀⠀⠀⠀⠀⠀⡠⠛⠛⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣩⠽⢷⡞⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡼⠁⠀⠀⢱⡀⢠⠉⠓⠲⠤⡄⠀⠀⠀⠀⠴⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠐⠦⡼⠁⠀⠲⢤⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠔⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣀⡀⢰⠃⠀⠀⠀⠀⡝⠀⠀⠀⣠⠴⣎⠀⠀⣀⡴⠛⠢⣔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⣠⠞⠁⣠⣴⡞⠁⠀⣈⣿⡋⠁⠀⣀⡴⢋⣵⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠀⣀⡤⠞⠁⣠⣾⣿⣿⣿⣄⠚⠓⣾⠀⠐⠞⣩⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠛⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣆⠂⡏⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⡉⠙⠛⠛⡿⠿⢿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠙⠪⠭⠅⣀⣀⣰⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⢷⣦⣀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡀⠀⠈⠻⣷⡄⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠈⢿⣆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⣿⡄⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢰⣿⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⢿⠿⠿⠋⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠊⡵⠐⠊⢁⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠒⠠⠤⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⣠⠤⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠢⢤⡀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠖⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⢀⢀⠎⠀⠁⠰⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⡉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠰⠉⠃⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⢀⠋⢀⠀⠀⠀⠀⢀⣉⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡆⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⣠⠊⠀⣾⡆⠀⠀⠀⠈⣇⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⢀⣾⣿⣿⠀⠀⠀⠀⣸⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠉⢀⣀⣴⣿⣿⠋⢰⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢇⠠⠊⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠢
⠀⠀⡠⠀⢄⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠇⠀⢀⡠⠐⢦
⠀⠀⠲⢄⡀⠑⠄⢀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⢤⣀⣀⡴⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠰⣏⡠⠔⠋⢀⡠⠞
⡠⠐⠠⠤⠌⡵⠀⠋⠀⠀⠑⠤⢴⠠⠆⠠⠄⠤⠤⠤⠄⠚⠁⡞⠀⠀⠀⠀⠀⠀⠠⡄⠈⠐⠀⠀⠠⠂⠐⠀⡄⠰⠀⠀⠀⠈⡇⠀⢩⠥⠤⠤
⠣⡤⠤⠤⢤⡄⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠃⠀⣸⣄⣀⣀
⠀⣀⠤⠀⠀⣠⠀⠀⠀⠀⣠⠃⢐⠀⡀⠀⠀⠀⠀⠀⢀⣠⡾⠀⠀⠂⠀⠀⠀⠀⠂⠘⠶⣄⠐⠀⠀⠀⠀⠀⡆⠘⢆⠀⠀⠀⠀⣀⠛⠠⣌⠉
⠀⠑⠤⠒⠉⡌⠀⡟⠒⠉⠀⠀⢸⣠⣇⣀⣀⣀⠦⠔⠋⢁⡆⠤⢀⡀⠀⠀⢀⣀⡠⠔⣧⠀⠉⠒⠲⠤⠤⢠⠇⠀⠀⠉⠒⠚⡀⢋⠢⣄⡸⠃
⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠂⠤⠶⠤⠤⠤⠀⠐⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡟⢱⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡇⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⠒⠋⠀⠀⠀⣿⠀⠀⢸⠀⠀⠐⠶⠋⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠚⠓⠢⠤⠤⠤⠤⡴⢞⠁⠀⠀⠘⣦⣤⣤⣠⠤⠤⠶⠶⠯⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠁⠢⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⡆⠀⠀⠎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢀⣰⣀⣀⠀⠤⠤⠄⣛⡉⠼⠀⠀⠀⡻⠄⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⢻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣓⣤⣭⣭⣭⣤⣤⣄⣒⣒⣊⣉⣉⣀⣀⣀⣀⣀⣀⣈⣁⣒⣂⣤⣤⣤⣤⣤⣄⣀⣒⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣄⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⠼⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠛⠛⠛⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⡀⠀⠂⠀⠀⠠⠶⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠤⡄⠉⠻⣿⣷⣾⣀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠶⢴⠞⠁⢰⠾⡇⠀⠀⢤⡄⠀⠤⢼⠟⠛⠛⠓⠀⢀⡀⠀⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠰⠃⢻⣤⣾⣧⠆⠈⠁⠀⡀⣠⣴⡆⠈⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⡀⠐⠀⢿⣶⡦⡤⠀⠖⠀⣶⢀⡄⠈⣿⠿⠋⠷⠖⠀⠀⡙⠛⠛⠀⠀⠐⠳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠉⠁⠚⠁⢄⡀⠀⢀⣤⣼⣿⡿⡘⢀⢠⣀⣶⣤⠀⣤⠉⠀⠀⢠⠤⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⡀⠀⠀⠀⠀⡀⠉⠩⠉⠓⢷⣿⣦⣿⣟⣛⣿⣿⣾⣆⡄⠐⠀⠐⠀⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠖⠀⠋⠉⠂⠠⣄⢀⣼⠟⠉⠁⠉⢻⣿⣿⣿⡿⣓⠂⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠸⣿⡇⠀⠀⠀⠀⠘⡿⢉⣿⣿⡋⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⢦⠄⠀⠞⠋⡀⠚⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠉⠉⠉⠉⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠈⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢡⠀⠀⠀⠈⡄⠀⠀⠂⠀⠀⠀⢀⣀⠀⠀⢀⣀⡐⠁⠀⠀⠀⠀⢰⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢱
⠀⠀⠀⠈⠀⠀⠀⡠⠠⠤⡀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡔⠀⢀⡠⢤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠈⠀⢰⠈⠀⠀⠀⠈⠀⠦⡀⠀⠀⡇⢸⠀⠀⠀⠀⠉⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢘⠀⠀⠀⠀⠀⡠⢀⡤⠴⢎⠑⢄⠀⢸⢸
⠀⡠⠄⠀⠀⠈⠳⣖⡒⠂⢐⠚⠀⡄⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⠒⠂⠐⠒⠒⠒⠒⠒⠂⠀⠀⢘⣠⠼⠀⠀⠀⠀⢎⠎⠁⠀⠀⠀⠱⡸⠀⢸⢸
⠀⡄⠀⠐⢀⣀⢸⡁⠀⡀⡂⢠⣴⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⡘⢰⠠⠀⣀⣀⠀⠗⠤⠰⠆⠢⠀⢀⠟⠀⢸⢸
⠀⢠⠔⠚⢋⠨⠿⡉⢉⣉⠝⠁⡏⠀⠀⠀⡇⢘⡠⣤⠀⠀⠀⠀⠀⠑⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⡴⠚⢋⡡⣿⠀⠀⠀⠀⠀⣶⡟⠀⠀⢸⢸⣀⢤
⠀⡃⠀⢀⣴⡪⠄⠐⣆⠀⢀⡂⠠⠤⣀⡀⡇⢘⢰⢠⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠤⠤⠀⠐⠂⠀⠤⠀⠵⠎⠀⠀⡠⠊⠀⢇⠋⠀⢁⡠⠊⠀⠀⠀⢸⢸⠁⡂
⢀⠆⠒⠙⣉⢹⠆⠀⠈⠋⠋⠀⠀⠀⠀⠀⡇⢸⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠘⣉⠩⣷⣤⠟⠉⠉⠙⡦⠠⢀⠀⢸⢸⠀⢸
⠈⠀⠀⣀⣤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣤⣎⡁⠇⠀⠀⠀⠀⡀⠀⠀⠑⢾⢸⠀⠈
⠀⢀⠞⠛⣍⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⠛⢋⡈⡿⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⢸
⡜⠁⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⡠⠐⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⢸⢸⠀⢰
⠇⠀⢀⢾⡄⠀⠀⣀⠤⠒⢁⡠⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠼⠀⢠⠳⠀⢀⣀⣀⡀⠀⢀⠁⠄⢸⢸⠀⢸
⠘⡄⢸⠘⠃⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠒⠃⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠒⠃⠐⠂⠀⠀⠀⠀⠀⠐⠂⠚⠂⢸
⠀⠐⣼⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⡄
⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢀⠠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡂⢄
⠀⠠⠁⢔⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠰⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠜⠈
⠀⢠⠀⠀⠈⠑⠂⠠⠤⠤⠤⠄⠐⠈⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠁⠒⠠⠤⠤⠤⠄⠐⠂⠉⠀⠀⠰
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰
⣿⣿⡟⠛⠛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣠⣺⣷⢀⡄⡀⠸⠘⠘⠘⠀⠀⠀⠀⣸⣿⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣼⣟⣻⣇⠀⠌⢊⡀⠁⠁⠁⠁⠀⠀⠀⣰⢻⣿⡟⣦⡗⠀⣀⣀⢀⣠⣤⣀⡄⠀⠀⠀⠀⣾⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣻⠟⣿⠻⠳⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡇⠀⠀⠀⠈⢉⣬⡉⠀⠹⠻⠁⠀⠀⠼⢿⣿⣿⣿⣿⣿⣝⣛⣛⣛⣛⣛⣛⣛⡛⠛⠂
⠀⠀⠀⠰⠏⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣻⣧⣤⣬⣀⣴⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣿⣿⣿⣿⡏⣿⣽⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢻⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡶⣾⣷⠿⠿⠿⠿⠷⣿⣿⢿⣿⡧
⠀⠀⠀⠀⢀⣿⣄⣀⡀⠀⠀⠀⢀⣀⣠⣮⡿⠇⢰⡟⠀⠀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⡇⠀⠀⠀⠀⠀⣿⠁⠀⢹⡇
⠀⠀⠀⠀⣾⣿⠤⠶⠿⠿⠀⠀⠸⣿⣿⣏⠧⢦⠈⠁⠀⠀⠀⠀⠀⣠⣶⣏⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⣻⠟⣦⠀⠀⠀⠀⠀⠀⠈⡏⠈⠤⣶⣷⣶⡤⠀⠀⠀⢾⣿⣟⢬⣹⣿⡆⢰⣦⣤⡀
⠀⠀⠀⠰⠃⠀⠹⠄⠀⠀⠀⠀⠀⠈⠃⠀⠰⠟⠛⠻⠿⠀⠀⠀⠚⠛⠛⠃⠚⠛⠓⠘⠛⠛⠃⠀⢀⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠉⢹⣿⡍⠀⠀⢤⣤⣤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣷⡖⣶⣿⣧⢸⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣆⠀⠀⠀⠀⣼⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡀⢸⣿⣿⡇⣿⣿⡇⠈⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠆⠀⠀⠀⠙⣻⣿⠏⢴⣦⣄⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣷⣶⣿⣷⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣦⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠙⠷⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣆⠀⠐⡟⢊⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠟
⠀⠀⢀⣀⡀⠀⠀⢸⣿⣏⠸⠶⡜⠄⠤⠥⠀⠀⠀⣀⡀⠀⡐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣓⡛
⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢠⡶⠻⣦⠀⠀⠀⠀⠀⠀⠀⢾⣶⣦⣤⣤⣤⣼⣿⣿⣬⣤⣤⣤⣤⣤⣶⡿⠟⠂
⠀⠀⠀⠀⠈⣿⠁⣠⡟⠉⢹⡍⠉⠉⠉⠉⠁⠈⣿⠀⠀⠀⢼⣧⣤⣿⡦⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣩⡟⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣰⡆⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣤⣀⣀⣠⣤⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣽⠛⠲⢦⣤⣤⣤⡤⠴⠚⠩⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⣹⡏⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⣦⣤⡀⠀⠀⠀⠀⠀⣿⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠶⠶⠶⠟⠻⠿⠷⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣸⢁⣾⡏⠈⠙⢷⡤⠶⢤⣴⡃⠀⣰⠋
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⣀⡤⢾⠋⣼⡟⠀⠀⠀⠀⠻⠤⢾⠋⢻⣴⣋⡑⣢⣀⢀⠀⠀⣶
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠋⠀⠏⢰⡿⠀⠀⠀⠀⢀⣤⣤⠼⣧⡦⠤⠤⠀⠤⠤⠤⡤⠴⠏
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⠀⡿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⢶⠦⠤⠤⠴⢾⡇⠀⠀⠀⢀⣺⡟⢺⣏⣵⣏⣹⣷⣦⡀
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⣾⠃⠀⠀⢠⣿⣉⡿⠋⠉⠉⠉⢻⣶⢼⣇
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⢦⣀⢠⡟⠀⠀⠀⢸⣏⣹⠀⠀⠀⠀⠀⢸⡿⠂⣿⠀⠀⠀⢰
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⢸⣿⣉⣧⣄⠀⠀⢠⠞⠁⢠⠏⠀⠀⠀⠘
⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠙⠿⣧⣶⡟⡂⢀⣸⣟⣿⠄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠟⠉
⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⢻⡷⣤⡤⣦⡤⡤⠤⠤⠤⡤⠤⠤⠤⠤⠤⠤⠤⣤⡤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣠⣀⠀⡀⣀
⣿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡠⠄⠒⠀⠀⠈⠁⠀⠀⣠⣔⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠖⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⠜⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠃⣰⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣉⣀⢀⠀⠀⡀⡀⠐⠨⡢⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣼⠇⢰⠃
⠀⠀⠀⠀⠀⠀⢠⡲⢿⣿⣿⣿⣟⠳⢖⣒⡚⠵⡲⣦⣄⢱⡀⠀⠀⠀⡠⠊⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⣿⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠿⢿⣶⡾⢿⡟⠛⠛⠂⠉⠫⣷⣿⣆⣠⢊⣠⣴⠶⠛⠉⠀⠀⠀⠀⠠⠤⢀⣠⠞⠁⢀⣼⣿⠏⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠒⢁⡴⠊⠙⠒⠦⢄⣒⠤⣄⣈⣿⣿⣿⣿⣭⣤⣤⣤⣀⣀⠀⠀⠠⠄⠠⢤⣀⣀⢠⣾⣿⡏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⢀⡙⢿⣿⣿⡿⣿⣿⣯⣭⣭⡭⠭⠍⠉⠒⠂⠒⠈⠉⠹⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠋⠉⠙⠛⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⠀⠀⢽⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⢦⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠋⠁⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠐⢿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⣸⠛⠿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣞⣉⠙⠛⠚⠛⢉⣉⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣾⣿⡷⣶⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⣇⢸⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠒⠒⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠴⠇⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⠟⠀⠀⠀⠻⡟⠛⠿⠟⠁⠀⢻⠉⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⡿⠛⠉⡇⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠓⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢑⡶⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡞⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣞⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢤⣀⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡉⠈⠛⠲⡄⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣴⣃⣀⣀⣀⣀⣀⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠂⠉⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠠⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠁⢂
⠀⠀⠀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢄⡐⢀⠄⢀⠗⠁⠀⠀⠀⢀⠁⠀⠀⠈⢄⡀⠀⢣
⠀⠀⠀⠮⠣⠠⠜⠀⠀⠀⠀⠀⠀⠀⣡⠄⠁⠀⢤⣆⠀⠀⠀⠀⠀⠂⣀⣇⠥⣀⡨⢍⣦⡀⢁
⠀⠀⠀⠐⡈⠢⠀⠂⠀⠀⢀⠤⠐⠘⣿⣷⣄⣴⣿⣿⡃⡂⠄⠀⠀⠢⠐⠀⠰⣍⡅⠀⡘⠰
⠀⠀⠀⠀⠈⢰⣶⣿⡄⢠⣿⠀⠀⠀⠈⠻⣿⣿⠟⡹⢋⠃⣀⣿⡀⠀⠈⠒⡒⠂⠒⡖⠈
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣿⡄⠀⠀⠀⠀⠈⠁⠀⠀⢠⣀⣼⣿⡇⣠⣾⡟⠊⠴⠌⠐⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠃⠀⠀⠀⠀⠛⠉⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠿⣯⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⢿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠈⢷⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣬⣷⣶⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⡿⠿⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡊⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠈⢄⠀⢀⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⢢⡀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠛⠲⠶⠦⠤⠤⠤⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠯⠭⣭⣉⡉⠉⠉⠁⠀⠀⢸⡇⣸⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⢘⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⢀⢆⡟⠀⠀⠀⠀⠀⠀⠀⠀⡰⣰⠇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠌⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠌⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠢⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠒⠒
⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠀⠙⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠶⠶⠶⠶⠶⠶⠶⢄⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⣠⠛⢥⠐⢄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡀⠊⠀⠀⠀⣀⡼⠃⠀⠀⢣⡈⠢⡄⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡔⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠙⢦⠈⠑⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣀⡄⠒⣩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠶⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠳⢦⣄⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣃⡸⠃⠀⠀⠀⢀⣤⡄⠀⠀⠐⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⢫⣀⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠉⢻⠁⠀⠀⠈⠀⠀⠋⠀⠀⠠⠟⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⢻⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢼⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⣆⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⢀⡾⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠐⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⣀⡀⠀⢀⣠⣄⡀⠀⠀⣤⣴⡾⠉⠱⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠉⠀⢹⣛⠋⠉⠉⠉⠉⢉⠁⠊⣴⡇⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢹⡇⠀⠀⠀⠈⣟⠳⢦⣬⣭⣥⡤⠖⠋⣹⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⢸⡇⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡞⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠄⠀⣌⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⢆⡇⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⢧⡰⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⣜⠁⠀⠀⠀⠐⣄⣠⡾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⣯⠀⠀⠀⠀⠰⡿⠁⣧⠀⠀⠀⠀⠉⢙⠋⠀⠀⠐⠂⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢣⡀⠀⠀⠀⣤⡇⠀⡇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣠⠖⠒⠲⣾⣧⠄⠀⠀⢀⡜
⠀⠀⠀⠀⠈⢳⣖⡩⠓⠚⠿⣀⡇⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⡄⠂⠀⠈⣴⠆⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣼⢡⣄⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣄⠚⣏⣴⠖⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡟⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢡⡾⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡤⠈⠀⠀⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣠⡔⢋⣀⣠⠶⠖⠛⠉⠁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⡟⠛⠓⠒⠲⠶⠶⠶⠒⠖⠛⠛⢙⠉⠉⠀⠀⢀⣀⡤⠔⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⡤⣤⣀⣀⣀⣀⣀⣀⣠⣴⠖⢭⡤⠤⡶⠚⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⠀⠀⠀⠸⡀⠀⠀⠈⠉⠉⢹⡏⠉⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡇⠐⠒⠒⠒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⣶⣄⣀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⡄⠈⠉⠳⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣀⠤⠔⠂⢈⣁⠨⣧⣀⢠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠉⠑⠠⢀
⢀⢴⡚⠉⣁⡤⣔⣲⠭⠕⠒⢉⣀⡄⠀⠀⠀⠀⢀⠄⢀⣀⣀⠤⠤⠒⠊⠁⠀⠑⢅⠀⠀⠀⠀⠀⠉⠒⢄⡀
⡇⠃⢹⡩⠾⣍⢁⡠⠴⠒⠋⠉⢚⢤⠀⠀⡀⠐⠁⡞⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀
⡇⠀⣼⠃⠀⢸⠀⠀⠀⢀⠔⣋⣑⣤⣤⠎⠀⠀⠀⠳⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢈⡵⠂
⡇⠀⣿⡄⠀⣾⠀⠀⢐⣤⠚⣀⢨⣯⣕⠐⠢⠁⠀⠱⡘⢆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⣠⠴⠊⡡⢧⡄
⣇⣆⡇⡇⠀⡇⠀⢶⡉⠙⠈⠀⠀⠉⠉⠙⠲⢄⡢⢹⡀⢈⣳⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣶⠉⣀⢔⡪⠚⠉⠓⢦⣀
⠀⠈⠷⣇⣀⣧⠶⣟⢍⡓⠦⣄⡀⠀⠀⠀⠀⢀⡹⢦⠿⣉⠀⠘⢄⠀⠳⣄⣀⣠⠤⠴⠖⠒⢋⣉⢿⣔⡪⠖⠉⠀⠀⠀⣠⢴⡟
⠀⠀⠀⠁⠸⣿⣗⠦⣕⡪⡂⢄⠉⠒⢄⡠⠖⠉⣠⣜⣄⡈⠓⢤⡈⠳⣠⣏⡂⢀⠤⢄⣒⠭⠕⠚⠉⢈⣱⠆⣀⡤⠖⠋⣁⠜⣳
⠀⠀⠀⠀⠀⠈⠪⢑⢄⡉⠛⢷⣬⣀⢠⣧⠴⣫⡞⠣⣀⠉⠢⣀⠉⠲⣌⠓⠒⠒⠉⠁⠀⠀⣀⡤⠞⣉⢿⣞⠉⢀⠴⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣐⠀⢈⣯⣀⠴⠊⠀⠣⣀⠈⠳⢄⡈⠑⢤⡀⠙⠢⣄⢀⡠⠖⠋⢁⠤⣪⢕⣩⡽⢚⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁⠀⠀⠀⠀⠈⠑⢤⡀⠙⠲⣄⡈⠳⣄⣠⣏⠂⠀⣊⣵⡯⠖⢋⡨⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⠢⣄⠉⢓⣲⠿⠋⠁⣠⢔⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣹⠫⠒⣀⠔⣪⠔⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣻⣆⣤⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣠⡤⣤⡤
⠀⢿⣿⡟⠛⠦⠀⠀⠀⣀⣠⠀⢸⣿⣦⠀⠀⠶⣾⣷⣿⣿⣦⣭⣂⠀⠀⠀⢠⣰⡦⠀⠀⢀⠀⣴⣄⠀⢀⢀⣤⣄⡀⣤⣄⡀
⠀⠈⢿⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠋⠁⠀⠀⢀⣼⡟⠀⢉⠉⠻⣿⠇⠀⠀⠘⠿⠁⠀⢀⣾⣴⣿⡿⠷⣦⣨⡉⣛⠿⢿⠟
⠀⠀⡈⠀⠀⠀⠀⢠⣾⠿⢿⡆⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⣀⣀⠈⠀⣠⣴⡿⠛⠀⠀⠛⢻⣿⠁⣀⠀⠈⠙⢿⣿⣿⡷⠁
⠀⠀⢿⣆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⡟⠙⠛⣳⣶⣶⡶⠀⠀⠙⢿⣿⡇⠀⢻⣿⠋⠁⠰⣶⣿
⠀⠀⠈⠛⣆⠀⠀⠀⠀⠀⠀⠰⣶⣤⣄⠀⢠⣾⣧⡀⠀⡀⠀⡼⣿⠃⠀⠠⣿⣟⣁⣤⠴⠶⠖⠈⠻⢿⡀⠈⡏⠀⢀⣴⣿⣿
⠀⠀⢠⣶⣾⠿⠷⠂⠀⠀⣤⡀⠀⠟⠁⢠⣿⢿⡿⣷⣾⡿⠄⠀⣯⠀⠀⠐⠉⠻⣿⠉⠀⠀⠀⠀⠀⠈⠁⠀⣤⡾⠛⠙⠉
⠀⡄⢈⣟⠁⠀⡀⠀⠀⠀⠙⢷⠀⣰⠀⣼⠃⠈⠀⠘⠿⠁⠀⠀⣿⣧⣤⣄⡰⢶⣷⣄⣴⠛⣷⣦⡄⠀⠀⣰⠏⠁⢀⣀⣠⣤⡤
⠀⣿⣾⡟⠀⢸⣿⣤⠀⡀⠀⠸⣷⣿⠠⠏⠀⠀⠀⣀⠀⠀⠀⠀⠁⠋⠻⠏⠀⠀⠀⢾⡃⠀⠹⡟⠗⠢⠀⠐⠀⠀⠀⠙⠛⠉
⠀⠋⠛⠙⠀⢸⣿⣿⣾⡄⠐⠞⠿⢷⡎⠀⢀⣠⣤⡽⠿⠛⠶⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠈⠀⠀⠀⠀⢳⣦⣤⣾⣧
⠀⠀⠀⠀⠀⢸⣿⡿⣿⡇⠀⠀⡀⢰⠀⢀⡾⠟⠉⡀⠀⠀⠀⠀⠀⠀⢉⣿⣷⠶⣾⣿⣿⡷⠄⠀⢀⡀⠀⠀⠀⢻⠙⠿⠉
⠀⠀⠀⣶⣤⢾⠋⠀⢻⠁⠀⣾⣧⡿⠀⠀⠀⠀⣰⣿⣦⣸⣷⠀⠀⠀⠀⠀⣀⠀⠋⠁⠀⣠⣤⠞⠿⠿⢿⣀⣀
⠀⠀⠘⠉⠀⠀⠀⠀⣠⣦⡄⠘⣿⠇⠀⠀⢰⣾⡏⠃⠉⢉⣻⠀⠀⠀⠐⣾⡟⠀⠀⠐⢞⣿⠇⠀⠀⠀⠀⠀⠹⣿⠟⠛⢇⡀
⠀⠀⠀⠀⠀⣀⠘⢶⡟⠻⣿⣶⠞⠀⠀⠀⣀⣿⠀⠐⣺⡟⠁⠀⠀⠀⠀⠼⣧⣄⠀⠀⠘⠋⠀⠀⠀⢰⣦⡀⠀⠏⠀⠀⢸⣿⣿
⠀⠀⠀⢀⣿⠿⠛⠛⠻⢦⠀⠀⠀⠀⢀⣠⣼⣿⣧⡀⣠⣦⣦⣤⣄⠀⠀⠀⠀⢉⡇⠀⣀⣀⠀⠀⠠⣬⣿⣷⣄⠀⠀⠀⠛⠋⠉
⢀⣤⣶⣷⡋⠀⠀⠠⣀⡀⢠⣿⡀⠀⠀⠈⠉⢉⣿⡿⠋⠉⠁⠀⠀⠀⢀⣀⣠⣾⠁⠀⠀⢙⣿⣓⠀⡄⠀⠉⢉⣁⡀
⠘⣿⣿⣿⣟⠀⠀⢸⣿⣿⠋⢛⣻⣶⡾⠾⠟⣿⡿⠀⣠⡀⠀⠀⠀⠀⣿⠋⠀⠃⠀⠀⠀⠼⠛⠛⣻⡏⢀⠀⠈⠙⢻⣍
⠀⠟⠛⠻⡏⠢⠀⢸⣿⠁⠀⣿⡟⠁⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⣀⣶⣿⡄⢰⣦⡀⠀⠀⡀⣠⠠⣿⣿⠃⠀⠀⠀⠀⠙⠆
⠀⠀⢀⡀⠀⠀⣀⣼⡃⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⢈⠁⠀⠐⣴⣿⣿⣿⣿⡄⠉⠉⠁⠺⣿⠋⠀⡿⠃⠀⢀⣶⣆
⠀⠀⠐⠛⠛⣹⣟⠀⢿⣆⠀⠀⠀⢼⣷⡀⠀⠀⣤⣶⠀⠀⢀⠴⠿⠛⠉⣿⠃⠀⠀⠀⠂⢻⡿⠟⠁⠀⠀⣾⣿⣿⣄
⠀⠀⢤⣤⢶⣿⡧⠄⠈⠹⠁⠀⠀⠉⢻⣷⣶⡶⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣄⣤⣶⣶⣦⣤⠤⠀⠈⠛⠉⠉
⠀⠀⡽⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡀
⠀⣾⣿⡶⠆⠀⠀⢀⣠⣴⠤⠄⠀⠀⢿⣶⠀⠀⣀⢠⣤⡔⠀⠀⠐⠻⠿⠆⠀⠀⠀⠀⣾⣷⡶⠀⠀⠀⣰⣶⣿⣿⠷⠃⠀⣴⡇
⠀⠀⠈⢐⠄⡠⠾⠛⠋⠁⠀⠀⠀⠀⠈⢿⠀⠀⢹⣿⣿⠃⠀⠀⠀⡀⢠⡆⠀⠠⣤⣽⣏⠉⠀⠀⣀⣿⠿⢛⣯⠀⠀⣠⠒⢹⠁
⠀⠀⠀⢤⣼⣿⡷⣶⣤⢶⣷⣄⣴⡆⠀⠀⣠⣄⣹⡿⠟⠀⠀⣼⣦⣵⣾⣿⡄⠀⠈⠙⠻⢤⣶⡿⠏⠉⠀⠸⣿⣔⡾⠁⠀⠌
⠀⠀⠀⠀⣿⣿⢀⡔⠀⠀⠉⣿⣿⡟⠀⢈⣹⡯⠉⠀⠀⠀⠀⠚⢿⡿⠀⠀⠀⠰⣤⠀⠀⠀⠈⢁⣀⣀⣀⣀⣘⣿
⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⢿⣏⣤⣾⣟⠉⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⢀⣀⣹⣿⠋⠀⠰⡾⣿⡟⠈⠉⠙⠻⣥⣀
⠀⠀⢠⣶⣦⠿⠛⠛⠓⠶⢷⡿⠉⠛⣿⡿⠷⣶⡤⠀⠀⣿⣤⡆⠀⠀⠀⠈⠉⠉⠉⢻⣄⠀⠀⠀⠛⠀⠀⢀⡀⠀⠨⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠈⠁⠀⠈⣋⠀⠀⠈⠙⢆⡀⣿⡿⠁⠀⣠⣀⡀⡀⠀⢀⠀⠀⠀⠀⢠⣦⠀⠀⣾⣷⠀⠠⢞⠉
⠀⠀⠀⠀⠀⠀⢠⣴⠿⠿⠛⠀⠀⣴⣿⡧⠄⠀⣠⣤⣤⡿⠁⠀⠒⠛⠻⠿⣿⣶⣾⠀⠀⠀⠀⠬⣿⣷⣤⣬⣿⠴⠛⠉⠀⠀⡀
⠀⠀⠀⢀⣠⠶⠹⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡁⠀⣠⣶⣿⣤⠀⠈⠙⣿⠀⠀⠀⠀⠴⠟⢛⣿⠁⠀⠀⠀⠀⣠⣦⡇
⠀⡶⠀⠀⠀⠀⡀⢻⢆⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⠂⠈⠉⠉⠛⠶⣶⣤⣽⡀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⢠⣀⠈⠛⠷
⢸⠇⠀⠀⠀⠾⣿⣇⠈⢻⡇⠀⠀⠀⢰⣾⣿⣶⣄⣀⣿⠃⠀⢀⣠⣦⡀⠀⠈⠙⠋⠑⠠⣤⣶⠃⠀⠀⠀⠀⠀⠀⠀⣙⣷⣦⣦

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⢀⣀⠀⠀⣀⣀⣀⠀⠀⢰⢴⠾⢿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠁⢢⠄⠀⠀⠀⠈⡿⢉⣻⠁⠤⠀⠒⢒⣦⡤⠟⠀⠀⠀⠢⣄⣀⢀⡀⣀⣀⣂⣀⢀⣀⡀⠄⠀⠀⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡶⠲⠤⠜⣉⡐⠶⢶⣯⣤⢴⣶⣶⠖⠉⢉⠠⠀⠀⠀⠀⣀⠎⠈⠋⠉⠩⠥⠄⠀⣙⣟⠁⠒⢂⣠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠉⠉⠑⠚⠳⡞⠉⠤⡧⢂⣠⠴⠓⠀⠀⠀⠙⠛⠓⠒⠒⠒⠒⠒⠚⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⢄⣀⡀⠀⠀⠀⡏⣀⡤⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠁⠀⠀⠀⠀⠀⠀⠀⣴⠶⡂⠀⠐⠌⠉⠁⠈⠉⠏⠓⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⡾⠷⠒⠚⠛⠿⠿⡶⠦⣄⣈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⢣⣤⣤⣤⣤⣤⣴⣧⣄⣀⠈⠙⠲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠭⠁⠀⠰⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠠⠶⠖⠶⠶⠶⠶⠶⢿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⡀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣧⡴⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠛⠛⠛⠛⠙⡿⣟⡛⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⣶⣇⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⣄⡀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠆⠀⡁⠀⢀⣤⡼⠔⡈⠀⠺⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠓⣦⣤⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠊⡍⠁⡀⠀⡿⣚⢉⡀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠑⠠⢀⠀⠀⠀⠈⠘⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆⠀⠀⠀⡇⠀⢇⠀⠃⠀⠸⡇⠀⡁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠢⢄⣠⣶⣿⣷⡀⠐⡀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠀⡇⠀⢻⠀⠁⠒⠒⠁⠀⡷⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⢸⠇⣹⠿⠋⢁⠰⢅⢀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⢀⠀⢸⠀⠀⠄⠤⣆⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠠⠈⠱⣄⣴⠾⠋⠀⠀⠀⠀⠀⣦⣼⠀⢸⠀⣍⠀⠀⣿⠁⡄⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠉⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠯⠁⠀⢿⡇⠀⠀⠀⠀⠀⢄⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡀⠠⠠⢊⠀⠀⠀⠀⡀⠀⠨⠷⠣⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⠀⢠⡀⠀⠐⠀⣿⡀⠀⡄⠀⠀⣾⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡇⠀⠀⠀⠀⢸⣷⣀⡿⣤⢿⡇⠀⠀⠀⢰⡟⢷⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠈⣿⣿⣷⠉⠉⠀⠀⠁⠀⢸⡇⠀⣿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡼⠋⡁⠈⠛⠛⠛⠛⠻⢿⠁⣿⣧⢀⠀⠀⣠⠀⠘⣇⣴⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡆⠀⠀⠀⠤⠶⠶⠶⠖⣊⡄⣹⣿⠉⢳⣶⣷⣶⣶⡏⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⢻⣿⡿⢻⣿⣿⡄⣿⣿⣶⣾⣿⣿⣿⣿⡇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠁⠀⠀⣸⠀⢹⣿⣿⡏⢻⣿⣿⣿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⣼⣿⡄⠀⢀⡏⠀⢸⣿⣿⠃⢸⣿⣿⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⢹⡇⣇⠀⢸⡇⠀⢸⣿⣿⠀⠈⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣾⡇⢿⡄⣾⡇⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢿⡇⠀⠀⣿⠀⠀⣸⣿⣿⠀⠀⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⣀⣿⡇⣀⠀⡘⠀⠀⠾⢿⣿⠀⠐⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠐⠒⠋⠀⠳⠶⠃⠀⠺⠶⠚⠋⠀⠀⠤⠾⠃⠀⢨

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣤⠀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢉⣼⣿⣿⣿⡏⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣏⣶⡿⠁⠀⠀⣠⣠⠀⠀⠉⠀⠀⣠⡿⠏⠀⠀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣯⣿⣿⣿⡿⠟⠃⠀⠀⣠⣿⠗⠀⠀⠀⠀⠀⠉⠀⠰⠟⠷⣵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠟⢛⣁⣵⠿⢯⣠⣀⣴⣽⠋⡄⢨⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣈⠈⣀⠙⣼⣿⣿⣶⣿⣿⠋⣉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⡇⢰⡴⣡⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⡟⠉⠀⠀⠀⠀⠀⡙⠦⠐⠊⠀⣠⣤⢀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢛⣷⠿⠀⠀⠀⠀⠀⢠⡀⠀⠲⡮⠳⠆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⡂⠀⠂⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠀⠀⠀⠀⣴⡖⢠⠁⠀⠀⠀⠀⠀⠀⠢⠄⠀⡀⣼⠁⠀⠀⠰⡄⠐⠂⠀⠀⢁⠀⠐⠲⠆
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠰⠂⠀⠀⣀⣔⠛⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠁⠆⠀⠀⠀⠘⢳
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠐⣠⠞⣽⠛⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⣀⠒⠂⠀⠾⠉⠓⣤⡅⠀⠀⠀⠀⣀⠉⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠨⠉⢦⠘⢗⡀⠀⠀⢋⠀⠀⠀⠀⠠⣄⠀⣲⠇⠀⠀⠘⢋⠀⠀⠀⢄⠐⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠀⣠⡿⢃⣦⣤⠀⠀⠀⠒⠀⠀⢀⡀⢠⣀⡈⠉⠀⠀⠀⠀⠀⠈⠒⠟⠛
⠀⠀⠀⠀⠀⢰⠀⠀⣠⠌⢀⡘⣿⣷⣜⠿⣿⠟⠀⠀⠀⠄⠂⠁⠀⢀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⢺⠀⢠⠟⠀⡈⡵⢊⡿⠋⠁⠀⠀⡀⢀⠌⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡻⡷⠋⠀⠀⠀⠀⠀⡀⠙⡆⠀⠀⠀⣴⠂⠀⠉⢠⣄⠀⠀⠀⠤⠶⠄⠀⠠
⠀⠀⠀⠀⢀⢀⠀⠈⠀⠀⠙⠁⠀⠀⠀⠀⠈⠉⠠⣠⠞⡲⣀⠀⠛⠡⠀⠀⠀⠀⠀⠡⢦⣀⣶⣖⣌⣃⠃
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡤⣺⣿⣿⣶⡃⢷⡄⠀⠀⠀⠀⠀⢀⠀⠊⡯⠋⣁⠈⠻⣧⡴⠲⣀⣀
⠀⠀⠀⠀⢀⡛⠷⠀⠠⠈⠀⠀⠀⢀⢀⣬⣯⣼⣿⡟⠋⠉⢳⡀⠙⢆⠀⠀⠀⠀⠀⠀⢀⡇⢠⡟⠛⡢⠀⣴⠄⠘⢻⡇⡀
⠀⠀⠀⠀⠘⠋⠀⡀⠀⠤⠄⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠦⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠸⠿⠆⠀⠈⠉⠉
⠀⠀⠀⠠⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠂⢸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⡀⠈⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠘⢷⣤⡤⣤⣤⢤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠀⠀⠀⠀⠘⠆⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠀⠉⠉⠀⠒⠂⠤⣀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⣿⠏⠀⠀⠈⠉⠉⠉⠉⠉⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠇⠈⢁⣀⠀⠀⠀⠀⠀⣀⡈⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠠⢻⠀⠀⠙⠛⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠉⠙⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⢀⣀⣀⣀⠀⠀⠀⠻⣄⠀⠀⢻⠀⠀⠀⢠⠤⠤⠀⠤⠤⡀⠀⠀⠀⠀⠀⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠬⠷⠀⠘⡄⠀⠀⠘⣿⣶⣶⣶⡿⠃⠀⠀⠀⠀⢠⢧⣄⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠔⠁⠀⠘⡉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⡤⠤⢤⠒⠊⠁⠀⠀⠀⢀⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⠃⢀⣼⣿⣷⣾⣿⡖⢉⡡⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢠⣄⠀⠀⠀⢀⠎⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠬⣐⡒⠢⠎⡠⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⠉⡩⢏⠉⠉⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡜⠁⠀⠳⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠘⢄⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠄⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠃⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠘⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠘⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠂⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⡇⠀⠀⠀⢀⠂⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠉⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣶⠀⠀⠀⢰⣾⣿⡀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠠⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠿⠿⠿⠿⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣆⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣼⣿⣿⣎⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡌⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣷⣜⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣦⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⡜⠿⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠶⠶⠿⠛⠛⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣷⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⣧⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣡⣶⣿⣿⡇⠀⣴⠟⢉⣤⣤⡉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⣴⣾⣿⣇⠀⣰⠟⠉⣀⣄⡈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡛⠛⠋⢸⡇⠀⠈⢡⡿⠇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⠉⢠⡏⠀⠀⢁⡼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠘⣧⡀⠀⢨⡅⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⢷⡀⠀⠨⡅⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣄⠀⠈⠻⢶⣤⣤⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠈⠛⠦⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠉⠰⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠈⢆⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡀
⠀⠲⠀⠀⠀⢠⠁⠀⠄⠉⣴⣏⡆⠀⠀⠀⠀⢼⣟⡅⠀⠀⠀⠀⠂⢰⠀⠀⠸⡇⠀⠀⡀⠀⠀⢠⠃⠈⠃
⠀⠀⠁⠀⠀⠘⠀⠐⠒⠒⠈⠛⠁⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⢷⠀⠀⠇⠀⡰⠃
⠀⠀⡄⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⡆⠀⠐⠁⠀⠀⠀⠒⠂
⠀⠀⢧⠀⠸⡆⠀⠀⠀⠀⠳⠶⠶⠂⠀⠀⠀⢀⠆⢡⣆⠀⠀⠀⠀⠀⠇⠀⠀⠸⠂
⠀⠀⠀⠀⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⣶⡿⡏⠀⠀⠀⠀⠀⡀⠀⠒⠀⢡⡄⠀⡈⠁⠀⠒
⠀⡈⠀⠘⡆⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠓⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡄⠀⠈⠀⠀⠀⠀⠉⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡄⢸⡆⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠄⠀⠀⠀⠀⠀⠀⢈⠙⠶⢦⣤⣤⣀⣀⣀⣠⣤⠴⠶⠋⠁⠀⡀⠀⠘⡅⠀⡇⠀⢡⡀⠀⠀⠰⠂⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⠋⠈⠯⠀⠀⠀⠀⠀⡎⠀⡌⠡⠀⠀⠁⠀⢘⡇⠀⠀⡈⠀⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠾⠀⠠⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡇⠀⡇
⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠇⠀⠃
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣠⠞⣥⢠⣄⠀⠀⠀⠠
⠀⠀⠈⠳⣄⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡰⠁⠀⡇
⠀⠀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⡘⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⣟⠐⠒⠒⠓
⠀⠀⠀⠀⠀⠨⣄⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠦⣌⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠸⠀⢸⠀⠐⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠚⠉⠠⢤⣤⣄⡀⠠
⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠠⢤⣤⣤⠆⠀⢇⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠁⠀⠀⢠⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣬⠀⠀⠀⠀⠀⠘⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⠀⠀⠀⠀⠠⠤⣤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⡏⠀⠘⠤⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠒⠒⠊⠉⠆
⠀⠀⠀⠱⢄⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠡⠄⣀⠀⠀⠀⠀⢀⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣄⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣿⠁⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⣀⣀⡭⠤⡄⠀⠀⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠈⠑⠲⡶⠃⠀⠀⢶⣤⢤
⠀⠀⠀⠀⢻⣢⡀⠀⠊⠉⠀⠛⢠⣈⡙⠿⡏⠀⠀⢀⡞⠀⠀⣤⠀⠀⣹⣷⡀⠀⣤⣤⣀
⠀⠀⠀⠀⠘⠃⠀⠐⠐⠠⣶⣦⣄⠀⠈⠻⠖⣀⣤⡟⠁⠀⠀⡆⠀⢸⣿⣿⡇⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠈⠿⢿⠷⡄⠀⠈⠱⣾⣷⣆⡀⠼⠁⢠⣾⣿⣿⣿⠀⠘⠉⠉⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡿⠋⣀⠀⢂⡀⠀⠠⠆⠀⠀⠀⡿⢹⡌⣵⣄⠀⣾⣿⣿⣿⣿⠁⠀⠀⠸⣿⣿⣅
⠀⠀⠀⠀⠀⣿⠇⠔⣡⣶⡿⠿⠆⠃⠄⢠⣄⠄⢁⣾⡿⣿⢃⣤⣿⣿⣿⣿⡿⠪⠀⠀⠀⣄⠿⠛⠂
⠀⠀⠀⠀⢸⣿⠀⠈⠁⣀⣀⡠⣄⠠⡄⢸⣿⣆⢀⣿⣇⠨⣾⢿⣭⣬⣭⡋⠁⠐⡀⢂⣀⡟⢀⠸⡖⣦
⠀⠀⠀⠀⢸⡿⡌⠻⢣⠍⣿⠁⠈⢻⡆⣌⢻⠟⠯⣿⣅⢀⡇⠈⢹⣿⡇⠀⢀⡘⠇⣻⡿⣶⠘⠀⡚⢹⠀⠠
⠀⠀⠀⠀⠀⠓⢻⣳⠇⠀⠉⠀⠀⠌⠁⠈⠚⠀⠤⠟⢺⣾⣿⣶⡄⠉⡇⠀⣿⡇⠀⠈⠱⠉⠘⠀⠆⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠰⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⡟


⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⡏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠉⠑⠶⠶⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢦⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⡞⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠈⠻⡄
⠀⠀⠀⣰⠋⠀⠀⢀⠶⢄⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣸⠇⠀⠀⠔⠁⠀⠀⠛⢦⡀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⣠⠟⠀⠀⠈⠀⠀⠔⠁⠀⠀⠙⢳⣄⡀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⡿⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⡴⠋⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠂⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠙⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠘
⣼⠁⠀⠀⠀⠈⠀⠀⠀⢰⠏⠀⠀⣴⠾⠆⠀⠀⠀⠀⠀⠈⠙⠛⠦⢤⣀⣀⡀⠰⠿⢦⡀⠀⣀⣴⡏
⣧⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠚⠳⠛⠉⠀⡇
⣟⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⡇
⣯⠀⢀⡴⠛⠛⠛⢶⡏⠀⠀⠀⣿⠁⢀⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢀⡀⠙⠀⠀⣼⡁
⡿⠀⣾⠁⡜⠃⠀⠀⠀⠀⠀⠀⠈⢦⠌⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠈⡴⠃⠀⢠⡟⢳⡄⢀
⢹⡄⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠁⢸
⠸⣇⠀⠉⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣸
⠀⢺⢁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣝
⠀⠀⢿⡠⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣒⣢⣀⣀⣤⡴⠋⠀⠀⠀⠀⠀⠀⢰⡟⣻⢿⣽
⠀⠀⠘⢧⡄⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣶⡿⠒⢸
⠀⠀⠀⠈⢿⡤⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣬⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣧⣴⣦⣭⠟
⠀⠀⠀⠀⠀⠻⣧⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⡴⠁⣴⠃
⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠈⠙⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠋⢀⡞⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡐⠀⠀⠀⠀⠈⠉⠉⣉⣱⠂⠈⠀⣈⠉⣉⣉⣀⣀⠀⣠⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⠀⠀⢀⣴⠞⠛⡉⠓⠀⠚⡉⠋⠉⠉⢹⡏⠉⠁⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠟⠁⠀⠀⠀⠈⠀⠁⠀⠀⣄⠀⣬⣷⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⣴⣄⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢠⠛⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠟⠛⠛⠛⠛⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣧⠀⣼⣦⣄⣀⣀⣠⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡤⡈⣶⡏⠀⠀⠉⠋⠙⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣧⠙⠛⣇⠀⠀⠀⢀⠈⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⣏⠀⠀⠠⣾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢹⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡟⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣤⣇⠀⣿⣤⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣟⣉⣉⣙⣀⣿⠀⣟⣐⣋⣉⣙⣷⣦⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⡆⡄
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣻⢸⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⡇⣟⣤⢀
⠀⠀⠀⠀⠀⢀⣴⠇⡇⡟⣸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣇⢻⢸⠸⣧⡀
⠀⠀⠀⢠⣤⣾⣿⠀⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⣿⣷⣤⡄
⠀⠀⠀⢸⣿⡿⢿⠀⡇⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⢸⠀⡿⢿⣿⡇
⠀⠀⠀⣿⡇⢧⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⡾⢹⣿
⠀⠀⠀⣿⡇⢸⠘⢸⢡⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡌⡇⠃⡇⠸⣿
⠀⠀⢠⣿⠀⠀⡇⢸⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⡇⢸⠀⠀⣿⡄
⠀⠀⢸⣿⠀⠀⡇⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢸⠀⠀⣿⡇
⠀⠀⢸⡏⠀⠀⠃⡌⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢡⠘⠀⠀⢹⡇
⠀⠀⣾⡇⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⢸⣷
⠀⠀⣿⡇⠀⠀⠀⠃⠀⠌⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣂⠠⠀⠸⠀⠀⠀⢸⣿
⠀⠀⣿⠁⠀⠀⠀⠂⠠⠤⣀⣉⠓⠺⣵⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣮⠕⢚⣉⣀⠠⠤⠀⠀⠀⠀⠘⣿
⠀⢀⣿⠀⠀⠔⠀⠀⠒⠒⠀⠤⠭⠕⣲⣭⣛⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣛⣭⣖⠫⠭⠤⡀⠐⠒⠀⠀⠢⠀⠀⣿
⠀⢸⣿⠀⠂⢀⡀⠤⠄⣀⣒⠢⠬⣍⡒⠮⢭⣛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣛⡭⠕⣒⣩⠥⠔⣒⣀⠤⠄⢀⡀⠐⠀⣿⡆
⠀⢸⣿⣍⣐⡒⠤⢭⣑⡒⠤⢍⣓⠒⠬⣙⡒⠬⣙⠢⢝⠻⣷⡀⠀⠀⢠⣾⠟⡩⠔⣋⠥⢒⣋⠥⠖⣚⡭⠴⢒⣊⡭⠤⢒⣒⣨⣿⡇
⠀⠈⠉⠛⠻⠿⢿⣶⣦⣭⣉⣒⠂⠍⠑⠂⠈⠁⠀⠉⠀⠁⠀⠙⢆⡴⠋⠀⠈⠀⠉⠀⠈⠉⠐⠊⠩⠐⣒⣉⣭⣴⣶⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣤⣤⣀⡀⠀⠀⠀⣀⣴⣾⣷⣦⣀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⡿⠿⠿⠟⠿⢿⣶⡶⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠒⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠐⠒⠋⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣄⡀
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡄⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⢦⣄⣀⡀⠀⠀⢀⣀⣀⣀⠀⢀⣀⡀⠴⠤⠤⠤⠶⣤⣄⣀⡤⠄⠈⠉⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⣀⡀
⠀⠀⠀⠀⡤⠤⠤⠦⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠉⠉⠉⢹
⠀⠀⠀⠀⣧⣤⣤⣤⣤⠄⣴⠀⠀⠀⠀⠀⣤⣤⣤⢤⣤⡼⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣟⣸⣁⣏⣸⣁⣿⠀⠀⠀⠀⠀⣿⣾⣷⣶⣶⣶⣶⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣼
⠀⠀⠀⠀⣿⣸⣂⣗⣸⡆⣿⠀⠀⠀⠀⠀⣿⣼⣿⣤⣤⣤⣤⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⢸⣇⢸⡇⢸
⠀⠀⠀⠀⣟⣸⣂⣓⣸⣄⣽⠀⠀⠀⠀⠀⣿⣹⣿⣉⣉⣉⣉⣉⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⣼⣯⣼⣧⣼
⠀⠀⠀⠀⠈⠉⠉⠁⠁⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠠⣉⡐⢬⣳⣄⠀⠀⣴⣫⣴⣾⡟⠍⠉⢉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠉⠛⠿⣷⣌⢻⣆⣼⣿⣟⠫⠭⠭⠻⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⠘⠁⠀⠀⠀⠀⠀⠉⠛⢿⠿⠿⡋⠄⠀⠀⠀⠄⠈⠉⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡤⠤⠄⠀⠒⠂⠁⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠧⣀⡀⠴⠂⠒⠒⠒⠒⠒⠒⠒⠚⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠲⠒⢲⠖⢉⣰⣄
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠸⠒⢇⣠⡤⠵⠶⠒⠛⠉⠁
⠀⠀⠀⣼⣇⣀⣀⣀⠀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⢈⣧⠀⠀⠈⢦⡀⠀⠀⠀⠀⣀⠔⠉⣀⡴⠗⠒⠋
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠾⠿⠡⠹⠛⠛⠶⠚⠛⠿⠿⠾⠿⠿⠿⠭⠀⠀⠀⠀⠱⡄⢤⣴⡾⠵⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠉⠀⠐⠂⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡔⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⢟⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠳⠄⠀⠀⠀⠈⠁⠒⠀⢤⠄⢒⢍⣢⡷⠖⠛⡁⠐⡖⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠉⠙⠛⠒⠂⠠⠼⠶⠿⠋⣁⠄⠂⣁⣤⠖⣇⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡜⠙⠆⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⡽⣾⠚⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢃⡀⠀⠀⠀⠈⠉⠁⠂⠤⢈⡉⠀⠀⡀⠶⠂⠛⡋⡕⣈⡴⠄⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠠⠴⠒⠲⠤⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠈⠁⠒⠐⠠⠄⣀⠀⠀⠀⠀⠙⡁⠀⣀⣶⠷⠟⠋⠁⠀⠀⠀⠀⠀⠸⠥⠾⠤⠶⠶⠶⠶⠈⣉⣉⣑⡖⠀⠈⢢⣄⣀⣀⣠⣤⣀⣀⣈⣁⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠴⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⠟⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠛⣫⡭⠄⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠁⣠⣄⠀⠀⠀⠀⠀⠀⣠⣄⠁⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠈⠻⠛⠀⠀⠀⠀⠙⡾⠶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠀⣿⣤⣭⣉⣩⣥⣼⡇⠀⠀⠀⠀⢀⣇⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠟⠀⠀⠀⠀⢀⣾⡋⠛⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠛⠃⠀⢷⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣤⠤⠤⢤⠴⠒⠋⠁⠀⠀⠀⠀⢀⣸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠀⠀⠸⣦⣤⣠⣶⣿⣷⡖⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢱⡀⠀⣠⣶⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡠⡝⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⣴⠃⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⣰⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡸⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⠃⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢰⠁⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⣀⣜⠀⠀⠀⢸⣤⣤⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠛⠀⠀⠀⢿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠂⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠧⠀⠀⡶⠲⡌⠳⠗⠈⠁⠲⣴⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⢹⠀⠈⠁⠀⣠⡀⠀⠀⠀⡄⠀⠘⠁⠀⠀⠀⢩⠶⠋⡥⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣼⠀⠀⢩⣿⠿⠄⠀⠀⠀⠞⢿⡿⠂⠀⠀⠀⢸⠀⠈⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠉⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⠀⠉⢢⡀⠀⢠⣤⡤⠀⣤⣠⠄⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠄⠀⠄⠃⠀⠀⠀⠈⣁⠀⢊⡟⠢⡀⠙⠓⠒⠛⠁⠀⣀⠠⠀⣀⠠⠀⠀⠀⠀⠀⠠⠀⠈⡆
⠀⠀⠀⢠⠈⠙⡏⠀⠀⠀⠀⠐⠂⠈⢦⠀⠘⢓⠙⠣⠄⣀⣀⠠⠀⠀⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁
⠀⠀⠐⠀⠶⡀⠈⠀⠀⠀⠀⠛⠀⠀⠀⢃⠀⢡⠂⣀⠀⣤⢄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢧⠈⠀⠈⠠⢀⡀⢀⡀⠜⠀⠀⠀⠀⠂⠀⠉⠁⠀⠙⠄⠙⠄⠀⠀⠀⣘⡡⠋⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀⡎⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠁⠂⠀⣤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⡴⠼⢷⠂⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢣⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠉⠲⠤⠀⠴⠟⠁⠀⢠⠆⠀⢀⠉⠢⡀⠀⢰⠇⣤⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠉⠀⠀⠁⠀⠛⢿⣷⠧⢰⣁⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠈⢂⠀⠀⠀⠀⠀⠈⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠀⠀⣠⣀⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠦⠀⠀⠀⠀⣸⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡠⠋⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠸⣀⠋⠀⠉⠲⠄⠀⠑⢄⠀⠀⠀⠀⡄⠁⡀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠐⢦⠀⠀⣀⠂⠀⠀⣾⣆⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢠⡄⠀⡎⠀⠀⠀⠀⠀⠀⠀⠠⠚⠃⠀⠸⣾⣩⣿⡧⣤⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠸⢿⣽⣶⣿⣦⣤⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠀⠰⣀⠀⢃⠀⠠⠤⠀⠀⠀⠀⠀⣠⣤⣼⣍⠛⠸⠗⣿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠤⠄⠐⠒⡚⠇⠀⣾⠋⣠⠅⠀⣀⠀⠀⠀⠀⣴⠿⣿⣷⣼⣿⣦⣄⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⡤⠒⠆⠀⠛⡒⢈⡀⠀⠠⢶⣤⠀⠀⠓⠤⢾⡏⠀⠙⠋⠛⠿⠿⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⡀⠸⣄⠐⠴⠀⠘⠉⡈⢁⠀⠀⠀⣠⠀⣀⡀⠀⠀⣄⡀⢀⠀⠀⠀⠀⠈⢻⡁⠀⠀⠄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⣀⠁⠰⠶⠂⠀⠀⠀⠀⠀⠉⠀⠈⠶⠈⠁⠀⠉⠀⣃⠘⣢⡀⠀⠀⠀⢻⡄⠀⠀⠛⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⣀⣤⣤⣄⠠⠶⠶⠶⠶⠚⠉⠛⠛⠉⠻⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠈⡛⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠈⠓⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠐⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⣀⣀⠤⠖⠋

⠋⠁⢀⡞⢄⠢⡑⢿⣿⣿⣶⣿⠿⡿⠀⠀⠀⠀⢠⢞⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⢰⣿⠿⡏⠀⠀⠀⠀⢠⢞⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠐
⣄⣰⠋⠀⣤⡀⠑⠌⠛⠛⠆⠈⠀⠀⠀⠀⠀⠀⠀⠼⠲⠆⠀⢰⣶⡿⠿⢷⣦⡄⠆⠁⠀⠀⠀⠀⠀⠀⠀⠾⠲⠆⠀⠰⠿⠄⠀⠀⣠⡀
⣟⢻⡄⠆⠁⢩⣰⢀⠑⠘⠃⠀⠀⠀⠀⠀⠸⠖⠀⢀⠔⠄⢠⣾⣿⣷⣶⣿⣿⣟⠃⠀⢀⠀⠀⢀⣸⠖⠀⠀⠔⠄⠀⠀⠀⠀⠀⠺⠿⠅
⣿⣿⣯⠅⠔⢢⡄⡘⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⣼⣿⣿⣿⣽⣿⣿⡿⢊⣐⣇⠀⠫⠁⠀⠀⠀⠙⠒⢆⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢀
⠋⠀⠀⠰⣿⠏⣅⣐⣤⣶⣿⣿⣶⣶⣆⠀⠀⠀⠄⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⡾⠛⠿⣷⣿⣷⣶⣦⣄⡀⠀⠀⠐⠃⠀⠀⡈⠀⠀⠀⠨⠃⠁⠀⡀
⣀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡰⣦⡴⢂⡆⠀⢹⣿⣿⣿⣿⣿⣤⣁⡀⠀⠀⠀⠉⠛⠿⠛⢿⣦⡀⠀⠀⡀⠠⠄⠀⢀⡄⠀⠀⠐⣀⣀⣀⡀
⡿⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠂⣸⠑⠂⠈⣿⣿⣿⣿⣿⠿⠿⠿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠉⠀⣟⠁⠀⣾⣿⣿⠐⡀⢀⣞⣿⣿⣿⢧
⠁⠰⠀⢻⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣤⣷⠒⠀⣠⣿⡏⣿⠿⢷⣤⣀⣤⠍⠛⠛⠻⢿⣿⢶⣤⣄⣀⣠⡃⠀⢂⠙⣩⣤⣦⠀⢈⡻⢿⣿⣿⢟
⠠⠄⠉⠀⠈⠛⣻⣿⣿⣿⡿⠿⡿⠛⠿⢿⣿⠿⠿⡿⠇⣺⣾⣿⣿⣿⠇⠀⡠⠀⠋⠹⣀⠀⠀⠀⠀⠁⠀⠹⣿⣿⠏⠿⠒⠈⢦⡈⢻⡏⡄⣼⣿⣮⡀⠀⠈
⠀⢨⡄⢠⣾⣿⡿⠟⣻⣿⡧⠀⠷⠂⠐⣂⣤⣄⣶⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠓⠻⠷⠆⠀⠘⠛⠶⢴⡿⠃⣸⡆⡓⢄⣘⣻⣦⣽⣿⣛⣩⡄⠀⠀⠉
⣶⠀⣀⠘⠛⠋⠀⡈⠀⠉⠀⣀⣀⣀⣾⢫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠠⠀⠀⡀⣰⡿⠗⣠⣴⣾⣿⢟⣽⢿⣿⣮⠥⠥⠄⠉
⠃⠀⠩⠀⠀⢀⠁⣀⠀⣰⡿⠫⢍⠹⠟⢸⡿⢿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⢸⣿⣿⢂⣾⣿⠿⠛⡱⢋⡿⠌⢿⣿⣧⡀⢠⣤
⣀⠠⠄⠉⠀⠙⣀⣹⣿⣿⣧⡀⢀⣷⡰⠚⠀⠀⠀⢢⠂⠙⣿⣿⣿⠏⠉⡍⠙⠻⠿⢿⣿⣿⢿⣿⣿⣷⣶⣿⣿⣿⣦⣴⡄⢀⣈⠀⣼⣿⣦⣆⠻⣿⣧⠸⣿
⠿⠆⠘⠀⡄⠰⠈⢿⣿⣿⡿⠛⣉⣀⣁⡈⠛⠀⠀⠀⠀⠀⢠⣿⣿⣶⣄⣀⡀⠀⠀⠀⠉⠁⢰⣿⣄⣼⣿⣿⣿⣿⣿⡀⠑⠀⢱⡾⢿⣿⡿⢃⣄⠤⢍⣀⣻
⠀⠀⠀⠸⣿⡄⠀⢠⠉⣿⣴⣿⣿⡧⠈⠀⢠⡤⠀⢠⡔⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠞⠄⠢⠐⣽⣾⣿⠤⣺⢿⡏
⠀⠀⠀⠀⢻⢇⠀⠀⠁⠀⢻⣿⠀⠀⣀⣤⠘⠂⠠⠞⠉⢲⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣰⠃⠀⣭⠀⠑⠌⠛⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠴⠶⠶⠦⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⢸⡇⠀⢠⣶⣶⣶⣶⣤⣤⣴⣿⣿⣟⣁⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣷⣄⣀⣀⣀⣤⣀⣀⣤⣤⣤⠄⠀⠈⣧
⠀⠀⠀⢀⡾⠁⠀⠀⡅⠀⠀⠀⠈⠉⡁⠤⠜⠛⠛⠻⠛⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠉⠉⠉⠉⠉⠉⣿⠁⠀⠀⠀⢹⡇
⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⡼⢿⡖⢺⣷⣶⣶⠔⢿⡀⠀⠀⠀⠀⠀⢠⣄⠀⣴⣶⣶⣶⡄⠚⢷⡄⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢸⡇⠀⠀⠀⢿⠆⠀⠀⠀⠀⡞⠻⢿⣿⡿⠟⠀⣬⠁⠀⠀⠀⠀⠀⢸⣿⣄⣼⣿⣿⡿⠀⠀⠘⠁⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⣴⠛⢦⣄⠀⠀⠀⣈⠉⠋⠉⠀⢀⡴⠂⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠙⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠞⠋⣀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠀⠀⠀⠀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠛⢷⡾⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠏⠉⠙⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⢀⡾⠛⠳⣶⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⡴⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠑⣷⠀⠀⠀⠀⠈⢻⠟⠉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⣾⠀⠀⠀⡀⢰⣾⡀⠀⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢀⠛⠀⠀⢀⣰⢾⣅⠸⣷⣀⡸⠿⢆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠰⠎⣽⠟⠓⠚⢿⣄⠀⠀⠁⠀⠀⠀⠀⠀⣀⠋⠁⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⠀⠀⠀⠀⠀⠙⠛⠃⢠⡤⠀⠚⠛⠀⠀⠀⠀⢀⡀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡿⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢸⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠆⢾⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠀⠀⠀⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣶⣶⣶⣶⣦⣄⣀⣀⣀⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠶⠾⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠿⠧⣤⣤⣤⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠻⢦⣴⣦⣤⣀⣀⣤⣤⣤⣤⣶⣤⡶⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⡟⠉⠉⠉⠉⠁⢈⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢸⡇⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⡇⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠀⢸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠛⠁⠀⠀⢸⡇⠺⠂⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠁⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⡤⠶⠞⠛⢿⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⠴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠒⣒⣒⣒⣒⣒⣲⣶⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⣭⣴⣶⡦⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣽⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣵⡿⢿⣿⣿⡟⠙⢿⣏⠉⠉⠉⠉⠉⠁⠀⠛⢿⡿⠟⠛⠷⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠀⢘⣿⣿⢿⣷⣆⠀⢹⣿⣿⣿⣿⣿⠀⢀⣾⡳⢶⣦⣶⣶⣾⣾⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⢐⢋⣐⣻⣿⣾⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣇⠘⢿⣿⣷⣿⣭⣟⣛⣛⡅
⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠈⢻⣿⣷⠽⠟⠋⠁⠀⠀⡿⢻⣯⡉⡽⣿⢛⡄⠀⠉⠛⠻⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢁⡼⠂⠑⠧⣿⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⠱⠀⠀⠀⣠⣤⢠⣤⢀⡏⠀⠀⠀⠀⠈⣧⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⢠⣿⠃⠉⣀⡸⡇⠀⠀⠀⠀⠀⡟⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⢀⣿⠇⠀⣜⠧⢡⣽⡲⠦⠤⢶⣂⠳⠀⣠⣿⡟⣇
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⡾⠉⠀⣴⣿⢦⣝⠖⠰⣏⠛⠻⢍⣠⡬⢿⣿⣿⢹⡄
⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⣸⠃⠀⠸⠛⠁⠀⠈⠛⠛⠿⠿⠛⠛⠉⠀⠀⠙⠿⠘⣇
⠀⠀⠀⠀⣴⣿⠛⠁⠀⠀⠀⠀⠀⣷⣤⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⡄
⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣩⣤⣶⣶⣶⣶⣦⣄⣀⠀⡀
⠀⠀⢸⣭⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠿⠛⠋⠉⠉⠉⠉⠙⠛⠛⢿⣧⡥⡀
⠀⠀⠀⢻⣧⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⢿⣿⣯⣕⣦⣤⣤⣤⣐⣢⣴⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠻⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⢠⡄⢠⡾⣿⠀⠀⣸⣿⡟⢆⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⠀⠀⠀⠀⠀⢸⡩⣿⣿⣿⣿⣿⢃⣾⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠉⠹⣿⣷⣀⠀⠐⠃⠸⣆⣈⣻⣿⣋⣡⣼⠏⡀⠐⠂⠀⠀⢠⣾⡏⠀⠁⢤⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠉⠛⠷⢤⡀⠀⢈⢙⡏⣍⢹⡛⢧⠐⠀⠀⠀⠀⠤⠚⠋⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢨⣿⢷⣿⡾⢷⣾⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⡀⠀⠰⠆
⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⣸⣦⣤⣴⣶⠶⠶⠀⠲⣄⣠⠜⠿⠿⣿⡦⠀⠀⠒⠐⠛⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠰⠀⣷⠀⠀⣶⣠⣾⣿⠟⠁⣼⣿⣿⣦⣀⣀⠉⢁⣀⠀⠀⠈⠿⣃⣴⡟⠓⡀⠀⣈⣴⣾⣿⣶⣶⣿⣤⣤⣄⣤⣶⡶
⠀⠀⠀⠀⠀⣸⣷⣄⣈⣽⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣧⣿⣿⡇⠀⠀⢸⣿⡿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠘⣿⣿⣿⠹⡏⠀⠀⠀⡀⢈⣤⣾⣿⣿⠏⢀⡀⠙⢿⣿⣿⠻⢠⣄⣴⣿
⠀⠀⠀⠀⣼⠁⣿⣿⣿⣿⣿⡟⠁⣠⣶⣿⣧⡀⠈⠛⢿⣿⣿⡿⠀⠀⠠⣾⣿⣿⠟⠁⣰⣿⣿⣦⡀⠹⣧⣤⣿⣿⣿⢻⣴⡄
⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣷⣾⡿⠋⠉⠻⣿⣶⣄⠀⠁⠻⠞⠁⠀⠀⠈⠁⢠⣴⣿⡿⠋⠀⠹⣿⢶⣿⣿⣿⣿⣧⠘⠹⡇
⠀⠀⠀⠀⠸⣾⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⡦⢈⡉⠐⠀⠀⠀⠀⠀⠀⡀⣠⣴⣦⣬⣤⣴⣾⣷⣴⣦⢸⣿⣿⣿⣿⡟⠰
⠀⠀⠀⠀⠀⢻⣿⣟⢿⣿⣿⣿⣿⡿⠛⢋⣾⣿⡶⣗⡛⣛⣿⣿⣿⣾⣿⣿⣿⣿⣮⣽⣛⣀⡙⠀⢻⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⡄⠀⡙⠿⣱⣿⣿⣬⣝⡻⣿⡻⣿⠛⢻⣫⠟⣡⣿⣿⣿⡇⠀⣽⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⡷⠟⠁⢻⣿⣿⣿⠃⠒⣼⢸⣿⣿⣿⣿⣽⣿⣷⣯⡗⠀⠀⠵⣻⣥⣿⡙⠟⠁⣼⣿⣿⣿⡿⠧⢹⡇⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠂⠘⢹⣿⢿⠀⠀⠛⠻⣿⣿⠋⠻⣿⣿⡿⣿⡇⠀⠀⢩⣍⣿⣿⠇⠀⠀⣿⣿⣿⣯⣆⠀⠈⠃⠙⡎
⠀⠀⠀⠀⠳⠄⢠⣿⠀⠀⠿⣿⠀⠀⠀⣰⣀⣎⠃⠐⣀⠈⢷⣿⣿⣧⡀⠀⣼⣿⣿⣿⣦⣁⠀⡹⠙⢿⣿⣿⡀⢳⡀⠀⠁
⠀⠀⠀⣀⠄⢀⣾⡯⠀⠀⣠⡿⠄⠀⠀⢹⠉⠛⠢⠄⠠⠆⠘⣿⣿⣿⣷⣾⣟⠻⣏⠻⡿⠛⣠⣅⠀⠈⣿⠻⡗⠈⣿⣧⠀⠀⡄
⠀⠀⢦⡠⠀⡾⠟⠛⠦⠿⠯⠀⢀⣀⣠⣤⣦⠀⢱⣴⡿⢹⣷⣿⣿⣿⣿⣷⣟⢀⡀⠀⠙⣿⣿⣿⡆⠘⡛⠀⠙⠄⠘⠿⠀⠀⢳
⠀⠀⠀⠓⠀⠀⠀⠀⠒⠀⠀⠀⠉⠀⡄⠀⠀⡀⠈⣷⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⣿⣿⣏⡀⣈⡉⠀⠀⠀⠠⠄⠀⠠⠤
⣤⣤⣄⣀⡀⠤⢂⣤⠴⢸⠛⠁⠀⡄⣟⣠⣿⣿⣿⡿⣷⣿⣿⡇⠙⣿⣿⡏⣿⣿⣛⣛⠛⠿⠿⠿⠿⢿⣛⣋⠁⠠⠀⡠⢀
⢀⣀⡀⣸⡇⠀⠀⠀⢰⡏⠀⢀⣈⣉⣽⣿⣹⣿⡟⣰⣿⣿⣿⡇⠀⠘⠋⠀⠸⣿⣿⣿⡿⢿⣷⠶⠃⠉⠉⠉⠛⠲⠒⠒
⣟⠋⠁⠿⣇⠀⢀⡀⠘⠋⣀⣴⣿⡿⠛⠛⠋⠙⢁⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢻⣿⡶⠄⢀⣀⣀⣀⣀⣀⠀⠀⠁⠠⠤⠤⠶⠶
⣿⣦⣤⣤⣤⣶⣿⣿⣿⣿⣛⣉⣁⣀⣀⣀⠀⠀⠘⠋⠛⠿⠇⠀⠀⠀⢀⠀⠀⠈⢉⣡⣤⣠⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠶⠶⠦⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⠛⠛⢷⡄⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠉⠲⠿⠃⠀⠀⠰⢆⣠⣾⠏⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⣤⣀⣿⣄⣀⡀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠚⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠙⢻⣿⡁⠀⠀⠙⢿⣦⡄
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠀⠀⠀⠈⠙⠻⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠷⠶⠶⠾⠟⠻⡿⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⠾⠋
⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠟⠻⠿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⣾⣏⣤⣄⣤⡶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠻⠆⠀⠀⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠄⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡷⠦⣤⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣀⣤⣶⣦⠤⠤⠤⠄⢺⣷⣄⣀⣤⣾⡿⠛⠀⠀⠀⠀⠠⠄⠀⠀⠁⠈⠁⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⣼⡏⠙⡏⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⢀⣠⣴⠟⠁
⠀⠀⢸⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠟⠋
⠀⠀⢸⠃⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣰⡶⠛⠉
⠀⠀⣠⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠁
⠀⠀⢻⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⣾⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⢰⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⡇⠀⢸⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢸⡇⠀⠈⠓⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⡇⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠁⠀⢿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣦⠀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠉⠀⣿⡟⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠖⠛⠟⠛⣿
⢀⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⣿⣴⣆⣠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣿⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠛⠛⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠰⣿⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠠⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢠⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠏⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⣿⡆⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣶
⠀⠀⠀⣀⠀⠠⠆⠈⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⡄
⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⢰⣿⠁⠀⠀⠀⠀⠁
⠀⠐⢿⣿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠁⠀⠀⠸⢿⣧⣠⣄⣀⣰⣷
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠟⠻⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠐⠀⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢠⠄⠀⠄⠠⠄⠀⠄⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡴⣟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣥⠾⢋⡄
⠀⠀⢀⣀⣀⣀⣀⣀⡄⡠⠚⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣡⠾⣛⠴
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣄⣀⣤⣤⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡥⢚⣥⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠋⠥⠴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠒⠂⠈⠉⠉⠁⠒⠒⠒⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⡠⠤⣖⡒⠂⠉⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠓⠒⠲⠶⠤⢤⣤⣀⣀
⠀⠀⠀⢸⢡⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣄⣀⣀⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⡟⠁
⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠸⡇
⠀⠀⠀⣀⣑⣲⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠷⠓⠤⢤⣄⡀
⠀⠀⠀⠙⡟⠲⠤⣄⣀⠀⠈⠉⠙⠒⠲⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠚⢷⣦⣀⣀⣤⡶⠿⠿⠗⠒⠒⠊⠉⠉⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠈⠉⠉⠛⠓⣶⣦⣤⣄⣀⣈⣇⣠⣤⡶⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠶⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⣈⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⢶⣿⣶
⣰⡿⠛⠳⠷⣦⣭⣛⣿⣶⣤⣀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠔⠒⢒⣋⣩⣭⡥⠶⠶⠛⠛⠋⢹⠃
⣿⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀⣠⠏⡄⢀⣀⣀⣤⣤⠶⠶⠒⣚⣋⣩⣭⡤⠶⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣭⣴⡶⠶⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣦
⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣖⣷⣀⣀⣤⣤⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⠈⢉⡉⠭⠍⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠲⡀⢺⠉⡁⠀⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣷⡀⢤⠒⠾⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣇⠈⠢⡀⣵⢦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣐⠊⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⢤⠎⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⢄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⢄⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠖⠂⢽⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠦⡀⠀⠀⢠⠋⠁⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡀⠀⠀⠀⠠⡑⢄⣠⠜⢣⣴⣇⠀⠀⠀⠀⠀⠀⠀⠒⠀⢲⣶⠖⠂⠐
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠀⢀⣠⡴⠎⡀⠀⠀⠈⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥
⠀⠀⠀⠀⠀⠀⠘⣄⢀⠀⠀⠀⠉⠙⠠⠄⣀⠈⠀⠀⠐⢖⠷⠄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⢖⠂⠤⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠒⠀⢤⣤⣄⡀⠀⠀⠉⠙⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠂⠀⢤⠄⠟⢿⠛⣄⠀⠀⠀⣠⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⢸⠀⠀⠈⠉⣈⠯⡃⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆⠇⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣘⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣾⠟⠁⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⢀⣴⠞⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣴⡿⠃⠀⠀⢠⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣿⡿⠁⠀⠀⠀⢿⣸⣠⣤⣴⣾⣿⣶⣖⡦⡀⣀⣴⡾⠟⠋⠁⠀⠀⣄⠈⢻⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣴⣿⣿⣿⡿⠷⠶⠺⢿⣿⣿⡟⠉⢁⣴⣿⠿⠛⣿⠾⣿⠍⠀⠀⠀⠀⠀⠀⢹⣿⢸⢸⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢁⣾⣿⣿⣿⣿⢠⡿⠿⠿⠦⣿⠛⣆⠀⠀⠿⣅⠀⢈⡽⡆⢸⠀⠀⠀⠀⠀⠀⠀⢸⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣠⠄⣿⠀⠈⠦⣀⠀⠀⠀⠀⠀⣸⠜⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠀⠀⠀⠀⣰⠟⠀⠀⠀⠈⠛⠻⣶⣆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⠤⠶⠖⣿⠀⠀⠀⢀⡀⠒⠂⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⣠⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢈⠓⠲⠆⠞⠀⠀⠀⠀⠀⣤⣼⣿⠆⠀⠀⠀⠀⣿⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⡏⢀⣠⣶⡖⣶⣛⣭⠤⠒⠋⠀⢿⠀⢳⡄⠀⠀⣿⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⡙⠛⠏⠙⠛⠋⠉⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⢀⡟⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢠⠎⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⢲⣠⣤⠄⣰⣾⣯⡄⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⡁⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⢀⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣄⠀⠀⢠⠀⡇⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠛⠿⠛⠿⠟⠛⠿⠛⠿⠟⠏⠛⠛⠷⠆⠀⠓⠳⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⡀⠄⠄⠒⠚⠉⠉⠩⠤⠀⠐⠓⠛⢀⠛⣾⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⡀⠀⠀⡀⠀⠀⠘⠄⣿⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠔⠦⣼⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⢴⣡⣄⣤⡼⡇⠀⠀⠀⢻⠀⠀⠀⠀⢀⠂⠀⠀⠹⣆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠈⠀⠈⠡⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡢⢀⠀⣀⣄
⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠸⢄⣀⣔⠓⠢⠴⢥⣀⢸⠀⠀⠀⠀⠀⠨⢀⠤⡀⠘⠻⣧⠆⠼⢿⡁⠠⡀
⠀⠀⠀⠀⠸⠀⠀⢤⣂⠐⠂⠤⠄⣳⣶⡶⠮⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡠⠓⡀⠠⠕⠀⠀⢬⣇⠄⠰⠄
⠀⠀⠀⠀⠈⡆⠀⠸⠁⠀⠔⣨⠵⣞⠿⡇⠏⠊⢅⠠⠀⣊⣁⠀⠐⡒⠰⠂⠀⠀⠈⠀⠀⠀⡀⠙⣄⡐⠁
⠀⠀⠀⠀⠘⢷⡀⢚⣀⠔⢛⢴⡾⣻⠃⠀⠈⢚⠅⡂⢔⠛⣫⣴⡒⡽⠁⠀⠀⠈⢱⡀⠂⠀⣀⡤⢋⠂
⠀⠀⠀⠀⠀⠀⠉⢁⠚⠖⢋⠕⢈⠳⠀⡄⠀⠰⠫⢈⣤⠞⠋⣀⣴⠇⠀⠀⠠⠦⠴⡄⢤⣶⠏⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣄⠀⠘⠳⣶⡁⠀⠀⠀⠀⠀⠚⠿⡟⠀⠀⡀⣍⣁⠉⠉⣙⣉⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠛⣢⣀⡒⠂⠀⠀⠈⠂⠀⠀⠀⢠⠂⠀⠀⠀⢲⣮⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠩⠟⠛⠓⠲⠤⣀⡀⢀⠀⠂⠀⡀⠀⠀⣟⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⠀⠀⠀⠀⠀⢰⠩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⠀⠘⠃⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⡿⠋⠈⠁⠉⠉⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢁⣀⡀⠀⠀⢀⡤⣄⠀⠀⠀⣹⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠚⠀⠉⣼⠀⠈⠀⠈⠀⠀⣸⣿⡿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠛⣀⣀⠀⠀⠀⢠⣿⣯⣿⡏⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠐⢿⡉⠉⣹⠃⠀⢀⣿⣛⠳⡌⠧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠙⠚⠁⠀⢀⣼⣷⡌⡇⠛⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠿⡖⠒⠚⠛⠉⠙⣿⠇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠻⣄⡀⠀⣀⡼⢻⠀⠀⠀⠀⢸⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⢷⠶⠒⠛⠙⣇⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠹⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠺⠿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡢⠤⠞⢋⣩⣿⡇
⠀⠀⠀⢀⣴⠀⠀⣄⠀⣄⠀⡴⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠏⠀⠀⠀⣠⣿⢟⠛⠃
⠀⢢⡀⠈⠛⣶⣀⢀⣁⣿⣾⠃⠲⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠞⢉⡁⠀⢀⣠⡾⠛⠛⢉⣀⡄
⠀⠀⠁⠀⠈⣾⡿⣿⣿⣦⡀⢩⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁⠀⠛⣯⣤⣤⣴⣿⣿⡇
⠀⠀⠀⠀⢿⡿⠶⠙⢿⣿⣧⣀⠉⠳⠈⠃⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣀⣄⣠⣴⢟⣥⣶⣤⣴⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠛⡄⢿⡾⠟⠉⠁⠀⠀⠶⠾⠿⠋⠀⠉⠀⠀⠀⣰⣿⣿⣿⡿⠉⠀⢼⣿⣾⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠐⠀⠀⠀⠀⢿⡜⠃⠀⠀⠀⠀⣄⠀⠀⡀⠀⢀⣤⠀⣀⣴⣿⣿⣿⣿⡶⠚⣡⣿⣿⣿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⣼⠟⠀⠀⢇⣠⡾⠿⠿⠟⠛⢋⣡⣾⣯⣤⣾⣿⡿⠋⠁⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⣰⣿⣀⣤⣾⣿⣿⣿⠃⠀⠀⠘⠉⠁⢀⢀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄
⠀⠀⡀⣼⣿⡏⠉⣿⣿⣿⡟⠁⠀⢀⣀⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣴⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠃
⢀⢀⣾⣿⣿⣷⣤⣿⡿⠋⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡻⠿⢶⣒⡲⠄
⠀⣮⣾⣿⣿⢻⣿⣿⡀⠀⣠⣼⣿⢻⣿⣿⣿⣿⣅⣉⠁⠀⣰⡇⠈⠀⠙⠻⣿⣿⣿⣿⢿⣿⣿⣭⣛⠻⠿⢷⣶⣤⡅
⠀⣿⣿⣿⣿⠘⣿⣿⣿⠿⣏⡉⣀⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣦⣤⣼⣷⣬⣿⣿⣍⠳⣮⣝⡿⣿⣷⣦⣀⠀⠉⠁
⠀⣿⣿⣿⣿⣄⢻⣇⠻⢷⣮⣝⣻⠆⠀⠻⣿⣿⣿⣿⣧⠀⠻⣿⣿⢿⡿⣿⣿⣿⣿⣿⣷⣌⠙⢿⣦⣉⠛⢿⣿⣦
⠀⣿⣿⣿⣿⣿⣮⣿⡀⠀⠈⠙⢷⣿⣦⣄⡘⢿⣿⣿⡧⠀⠀⠈⠻⠀⠀⠈⢫⠘⠁⠘⣿⣿⣷⣤⡙⢿⣷⣄⠈⠙⠂
⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⢻⣿⣿⣿⣦⣍⣛⠶⣤⣄⣀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣦⡛⠻⣷⣦
⠀⣿⣿⣿⣿⡜⣿⣿⣿⡆⠀⠀⠀⠀⠀⠠⣉⠛⠿⣿⣿⣷⣶⣭⣙⡷⢦⣤⣾⡿⠀⠀⠈⠉⠉⠻⣿⣟⢿⣦⡀⠉
⠀⣿⣿⠘⣿⣷⡘⣿⣿⣿⣄⠀⢷⣦⡀⠀⠀⠉⠒⠐⢿⡛⠿⣿⣿⣿⣷⣶⣭⣿⣶⣴⣄⡐⠀⠀⠈⠻⣿⡉⠻⣦⡀
⠀⢹⣿⡇⢹⣿⣷⡘⣿⣿⣿⣦⡀⠻⣿⣷⡀⠀⠀⠀⠀⠙⣦⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠙⢿⣦⠀⠉
⠀⠸⣿⡇⠀⢿⣿⣿⣾⣿⣿⣿⣷⣄⠙⢿⣿⣦⡀⠀⠀⠀⠁⠀⠻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣝⡛⠶⣿⣷⣦
⠀⠀⣿⣿⠈⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠹⣿⣿⣷⣤⡀⠀⠀⠀⠈⠻⡟⢿⣿⣿⣿⣿⣭⡛⠻⢿⣿⣿⣷⣦⣭⡉
⠀⠀⢸⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠛⢿⣿⣿⣷⣦⣄⡀⡀⠈⢂⡜⠛⢿⣿⣿⣿⣷⣦⣀⠉⠛⠻⠟⠁
⠀⠀⠀⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⠻⣿⣿⣿⣿⣦⣄⡈⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠸⢿⣧⠀⠀⢿⣯⢿⣿⣍⢿⣿⣿⣿⣿⣿⣦⣄⠀⠈⠻⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣇⠀⠈⢿⣷⣿⣿⣦⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠈⠛⠿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠙⠻⢿⣻⣧⡀⠙⠻⠿⠟⠛⠻⠿⠷⠀⠀⠀⠈⠛⠋⠉⠛⠙⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⡟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠈⠉⠀⠀⠀⠙⠁⠀⠈⠻⣿⣆⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹⣿⣆⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢧⡀⢀⣄⠀⠀⠀⠀⠀⠰⣌⢠⣿⡄⠀⠀⠀⢿⡿⠆⠀⣼⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠈⠱⢝⠷⣦⣀⣤⣶⣦⣜⠃⠙⠳⠀⠀⠀⠀⠀⠀⠀⠺⡏⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠈⠛⢿⣿⣿⣶⡄⠀⠀⠀⠈⠙⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠉⠉⠉⠙⠛⠛⠿⠿⠷⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠀⢶⣾⡷⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡦⠀⠘⠛⢋⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠀⣠⣤⠴⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠂⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣍⣉⣽⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠻⢷⡜⣇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⡅⢀⣠⣤⣥⣄⠀⣀⡀⠸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⢛⣉⡙⠿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣠⣶⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣥⣤⡀⠀⠀⠀⢠⡄⠀⠐⣩⠶⢿⡻⡿⠇⠈⠙⠛⠁⠀⢹⡇
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣌⠻⣿⣿⣦⡈⢻⣿⡶⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⠀⠐⣒⣢⡖⠀⠘⠻⣿⣷⡀⠀⠈⣧⠀⠀⠉⠀⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡿⠿⣿⠟⠋⠙⣷⡘⢿⣿⠿⠀⠻⣷⣶⣿⣿⣿⣿⣿⣆⢀⣤⠀⣠⣶⣿⣻⠏⠈⠉⠶⢀⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⣷⣤⣾⣷⣴⣾⣿⣿⡄⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣧⣼⣿⠎⠁⠈⣉⣉⣠⣤⠀⠀⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢠⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣷⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣯⣽⠟⡋⠀⠀⠈⠛⠛⠉⠁⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⣿⣿⣁⣀⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸
⠀⣠⠟⠛⠋⠉⠉⠉⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣽⣧⣀⠀⠀⠈⠁⠀⠀⠀⠙⢿⣿⣦⠀⡇⠘
⠀⣿⡆⠀⠀⠀⠀⠀⠀⣠⠀⣠⣴⠿⠋⠁⠀⠙⢿⣏⠙⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⣿⡇⠀⠀⠐⢢⣤⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠙⢷⡈⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⡗⠀⠀⠈⠉⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⢿
⠀⣿⡇⠀⠀⢀⣤⡿⠟⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠒⠉⠻⠋⠀⠀⠀⠀⠸⡀
⠀⣿⣇⣠⣴⠿⠋⠀⢀⣀⣤⣤⡶⠾⠛⠉⠉⠀⠉⠀⠉⠳⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣀⣤⡶⠶⠞⠛⠉⠉⢁⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⡿⠋⣡⡴⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠉⠻⣿⣿⣿⣿⣿⡉⢿⣏⠉⠀⠀⠀⠚⠛⠀⠀⠠⠀⠶⠶⠟⠛⠋⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⣿⡀⢾⣇⠀⠀⠀⠀⠀⠰⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠀⠘⢦⣘⣿⣿⣿⣿⣧⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣠⣤⣴⣶⣿⠇⠀⠀⠀⠀⠀⢠⣿
⢹⣿⣇⣸⣿⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢻⣿⣿⣿⣿⣧⡈⢻⣆⠀⠀⠀⣠⡶⠚⠉⠁⠀⠀⠉⠛⢿⣿⣉⠁⠀⠀⠀⠀⠀⢀⣿⠃
⠀⢻⣿⣿⣿⡆⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣤⣞⠁⠀⠀⠀⢹⣿⣿⣿⣿⣷⡄⠹⠇⢀⣾⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⢠⡍⠙⢦⡀⠀⠀⠀⣸⡟
⠀⠀⢿⣿⣿⣿⣶⡄⢻⣇⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⠛⣧⠀⠈⣿⠻⣿⣿⣿⣿⣦⣤⣿⣷⠀⠀⠀⠈⠻⣆⠀⢠⣴⣾⣿⣿⡄⢀⣹⣦⠀⣠⡟
⠀⠀⠘⣿⣿⣿⣿⣇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⠈⢷⣄⣿⠀⢀⣉⣿⠿⣿⣿⠛⠁⠀⠀⠀⠀⣀⠙⣧⠈⢿⣿⣿⣿⣷⡼⣿⣿⡆⠉
⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠙⠛⠛⠋⠉⣀⣀⡈⠙⢷⣄⠀⢀⣠⣾⣿⣿⣿⣧⣾⣿⣿⣿⣿⣇⢿⣿⣷
⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⣿⣿⣿⣿⣷⣦⣀⣴⣿⣿⣿⣿⣦⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⠇
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣷⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠈⠛⠂⢀⡀⠀⠀⢀⣴⡿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⣻⣿⣿⣿⣿⠿⠛⠻⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠘⢛⣧⣤⣴⠿⠋⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⢀⡾⠃⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠈⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⢒⣚⣛⣛⣛⣛⣛⣓⣶⣶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⣭⣶⣾⣧⣹⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣷⣯⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡵⠿⣟⣿⣿⣿⠉⠩⢯⣁⣀⣀⣀⣀⠀⠀⢉⠻⠟⠛⠙⠛⠟⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢾⡇⢀⣼⡿⣿⣷⣿⠀⢸⣿⣿⣿⣿⣿⡄⢠⣿⣝⡷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢌⣥⣬⠹⣿⣿⡿⠋⠀⣸⣿⣿⣿⣿⣿⣧⠀⠻⢿⣿⣿⣶⣿⣭⣵
⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠈⠛⠟⠓⠋⠁⠀⠀⠀⠟⠩⠳⡆⢞⠳⣨⠄⢄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠓⢿⣷⣿⣤⡤⠀⠀⠀⠀⠀⠀⠸⠉⣣⠞⠁⠀⠙⢻⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣄⠀⠀⢠⣿⢰⠛⠆⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⢠⣿⠃⠀⡴⠦⠻⣄⡀⠀⣀⡀⠆⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⢀⡿⠏⢀⣵⣍⢱⣛⠃⡭⢭⢽⠖⠈⠠⣾⣿⡏⣷
⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠀⠀⠀⡼⠀⠀⣼⠟⠉⠓⠦⣤⣵⣤⣬⡵⠞⠋⠙⢿⣿⢸⡆
⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣷
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣉⣭⣍⣭⣭⣍⠁
⠀⠀⢹⣹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠛⠛⠋⠙⠛⠛⠻⠿⣾⣄⡀
⠀⠀⠈⢿⣎⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠈⠻⣿⣾⣕⣠⡄⠤⢀⣀⣀⣐⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡴⠖⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠄⠐⠀⡀⠀⢀⡀⠀⣀⣀⣀⣀⠀⠀⡤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛
⠀⠀⠀⠀⠀⠀⢡⣬⣭⠥⠤⠤⠤⠄⠀⠀⠐⠒⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⠻⡄⠀⢀⢀⡴⠛⠉⠀⠀⡀
⠀⠀⢀⠀⣈⠀⠀⠀⠀⣄⣙⣘⠀⢀⡠⠤⠀⠀⠀⢠⣐⣲⠶⣾⠀⠀⠘⢿⡵⠋⠀⠀⡠⠀⠀⠀⠀⠂
⠀⠀⡁⠀⡇⠀⠰⣆⣀⡏⢹⢨⠖⢰⣧⣤⣄⠀⠀⠜⠿⠋⠀⠸⠲⡒⠒⣦⠀⠀⠠⠪⣀⠀⠀⠀⠀⠀⠀⠈⠐⡤
⠀⠀⠃⠈⠁⠀⠀⠉⠀⠁⠈⠉⠀⠈⠀⠀⠐⡋⠀⡇⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⢈⣲⣤⡁⠂⢀⠀⠀⢀⠔⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠂⠀⠀⠀⠈⠈⠈⠀⠀⠀⡊⠳⢤⣉⡓⠦⣌⣒⣤⡞⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⢀⡀⡧⠄⡀⠀⠀⢐⠙⠳⢄⡈⠉⠓⠢⢍⣑⡴⠃⠀⠂
⠀⠐⡷⢾⡄⠠⢤⣀⣀⣾⡇⢰⣶⣶⣧⣤⡇⠀⠀⡄⡀⢹⣻⠇⠇⠉⠁⡀⠀⠀⠑⢄⠀⢝⠲⢄⡀⠰⢈⡴⠚⡠
⠀⠀⠀⠐⠃⠄⠜⠀⠐⠋⠓⠘⠂⠚⠃⠂⠓⠐⠒⠋⠀⠋⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠑⠀⠑⠠⡉⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠕
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠤⠀⠈⠁⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⡴⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢸⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠇⠀⠀⠿⠴⠶⠶⠾⠿⠛⠓⠶⠶⠶⠖⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠈⠁⠈⠉⠁⠉⠉⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠒⠂⢹⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢴⡖⠛⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣀⣴⣿⡇⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠰⣦⣤⣶⣶⠿⠿⢿⡏⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠘⠿⠏⠀⠀⠰⠂⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⣀⡀⠀⠀⠀⠀⠀⢀⣀⣴⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠶⠶⢶⡶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠊⠀⠀⠀⠀⠰⠶⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠈⠉⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⢁⠈⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠱⣦⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡀⠋⠀⣴⠤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠶⣀⣀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⡙⠂⠀⠀⣀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠤⠴⠞⠁⠀⠀⡴⠉⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠃⠀⠀⠀⡆⠀⡄⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⢻⡦⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠁⡀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⡇⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢦⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠘⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣠⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠐⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠘⠛⠛⠛⠛⠲⣶⡖⠀⠐⠒⠚⠛⠛⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠇
⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇
⠀⢠⣿⠀⢠⣤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⢿⡿⠟⠀⠀⠈⠻⠖⢶⡶⠶⠾⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠓⠀⠀⢺⡇⠀⠀⠀⠀⠀⢸⡇⠀⠲⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠛⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠘⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠈⠁⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⠁⠀⠀⠀⠀⠀⠀⠈⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠁⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠉⠁⢶⣦⡀⠀⢰⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⡆⠀⣀⣴⣦⠤⠤⠼⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠛⢃⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⡾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣶⣦⠄⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⣤⣶⣶⣴⣶⣤⣤⣶⣦⣶⣦⣤⣤⣤⣤⣶⣶⣶⣦⣤⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣶⣦⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣶⣦⣤⣤⣴⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣴⣶⣦⣴⣴⣶⣶⣦⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠿⠿⠿⠿⠿⠿⢿⠿⠿⣿⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣶⣄⣤
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢀⡤⠛⠉⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠼⣿⣧⣤⣤⣤⣴⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣦⣤⡴⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡆⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣠⣿⣇⠀⠀⠀⠀⢀⣴⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢀⣴⠿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠈⢹⣿⣿⣦⣴⣶⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⣰⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⢀⣤⣴⠿⠋
⠀⠀⠀⠀⣿⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠿⠋⠉
⠀⠀⠀⣸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠉
⠀⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡾⠛⠉
⠀⠀⠀⣸⡀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡾⠟⠁
⠀⠀⠀⣿⡟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⣿⠇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⢘⡏⠀⠀⠀⣸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⣾⡇⠀⠀⠀⣿⡇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡇⠀⠀⢠⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⣿⡇⠀⠀⢸⡟⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡇⠀⠀⢸⡇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠁⠀⠀⢸⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣤⣦⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣉
⠀⣴⠀⠀⠀⠀⢰⡄⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢠⣿⠇⣠⣄⠀⢸⠁⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⢿⣿⣄⣿⣿⡆⣸⡇⠀⢸⣷
⠸⣿⣏⣻⣿⣿⡿⠁⠀⢸⡏
⠀⠻⠿⣿⠟⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⠀⢀⣀⡀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠙⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠿⠄
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠃⢺⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⡿⠟⢻⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠀⠀⢭⣝⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣼⣧⠍⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠏⠀⠈⠉⠉⠀⠛⠓⢦⡘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⢈⠀⠀⠠⠤⠀⠀⠀⣻⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣿⠁⠉⠀⠀⠀⠶⠀⠀⠀⣿⡀⢹⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⡿⠀⣺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠘⢿⣿⣿⡟⠶⠤⠒⠀⠀⠀⢺⣾⣷⣿⠇⢬⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢳⡿⠿⣧⣄⠀⠀⠀⠀⠀⠸⡶⢾⣭⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠙⢿⣿⣛⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡔⠀⠀⠀⢰⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡤⠄⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⠴⣿⣶⣤⡀⠀⠀⠈⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⢌⠻⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠐⠂⠀⠀⠀⣾⡒⠀⠘⠐⣘⣻⠄⠒⣷⡀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢿⢿⣿⠋⠀⡿⢿⡷⢄⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣾⣶⣮⣚⣦⡈⠳⣈⢛⢫
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⣟⣻⢻⣯⣽⡽⣧⢨⡴⠚⢷⠦⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣛⣻⣛⣿⣿⣿⣮⡿⠀⠀⠈⡇⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣷⠀⠀⠀⢻⣄⣀⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠋⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠉⠒⠤⢀⣀⠤⠒⠉⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠔⠒⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠒⠤⠤⢄⣀⣀
⠀⠀⠀⣠⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⡌
⠙⠢⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⠿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⠿⢷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⠍⢙⡛⠻⠿⢷⣶⣦⣤⣤⣤⣤⣤⣀
⣴⣾⣟⠿⠿⣿⣿⣟⣣⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠖⠛⠉⠡⠶⠚⠉⢁⣠⣴⣶⣿⠿⠿⠟⠛⠛⠛⠛⢿⣦⡀
⠈⠛⠿⣿⣶⣦⣬⣉⠙⠛⠿⠿⣷⣦⣶⣿⣁⣤⣀⣤⡤⠖⠀⠀⣀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠛⠉⢉⣀⣤⣤⡶⠶⠶⢶⣦⣄⠀⢹⣷
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣶⣤⣄⡈⠉⠛⠻⠿⣿⣿⣶⣿⣶⣋⣤⣤⣤⠶⠞⠛⠉⢁⣀⣤⣴⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⢻⣿⡀⠈⣿⡇
⠀⠀⠀⢀⣸⣿⣿⣿⣷⣦⣉⠛⠛⠿⢿⣶⣦⣤⣀⡈⠉⢻⣿⠋⠉⢁⣀⣤⣴⣶⣿⠿⠟⠛⣉⣠⣤⣶⣶⣾⡁⠀⠀⠄⠀⣼⣿⠀⢰⣿⠇
⣠⠶⠞⢻⣿⣿⣿⡿⢭⣒⠩⠙⠲⠤⣄⡀⠉⠙⠛⠿⢿⣿⣿⣶⣿⡿⠿⠛⣋⣩⣤⣶⣾⠿⢟⡛⢯⡱⣶⡬⡙⠳⣤⣴⡿⠟⠁⠀⣾⡿
⣿⣦⣄⡀⠈⠛⠿⣿⣶⣬⣍⡒⠦⣄⣀⠉⠙⠲⠦⣤⣠⣤⣽⣿⣷⡀⠀⠉⠉⣀⠀⠈⠻⣷⡈⠙⠆⠙⡌⢳⡌⠂⠙⣿⣀⣀⣠⣾⡿⠁
⠀⠉⠛⠻⢷⣦⣄⣀⠉⠛⠻⢿⣷⣦⣌⣙⠒⠦⢤⣀⣈⠙⠛⣿⣿⡷⠀⠀⠈⠁⢀⣀⣤⡼⢿⣦⡀⠀⠀⠀⢷⠀⠀⢹⣿⠻⠿⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⢈⣿⡿⣷⣦⣤⣀⠈⠙⠛⠿⣿⣶⣤⣈⠉⠛⣿⣿⣿⣃⣤⣴⡶⠿⠛⠉⠁⣀⣠⣽⣷⡀⠀⠀⠈⠀⠀⠈⣿⣦⢀⣀⠀⠈⠉⠙⢳⣦⡄
⠀⠀⢸⣿⡛⠛⣿⡇⢁⠏⣿⣿⣷⣦⣤⣀⠀⠉⠙⠻⢿⣾⣿⠿⠟⠛⠉⣁⣠⣤⣶⡶⠿⠛⠛⠉⠙⣷⠀⠀⠀⠀⠀⢀⣿⣿⣭⣤⣶⣾⣿⡿⠿⠟⠃
⠀⠀⠈⢿⣷⡀⢿⣇⠀⠀⠋⢴⠋⡿⢿⣿⣿⣦⣤⣀⠀⢀⣤⣴⣶⣾⣿⠿⠛⠉⠁⠐⠉⠀⠀⣀⣰⣿⡆⠀⠀⢀⣠⣼⣿⡿⣿⣿⣿⣏
⠀⠀⠀⠈⢿⣿⡌⠙⢷⣦⣄⠀⠈⠁⠾⢡⡟⣿⣿⣿⣷⣿⣿⣿⠟⡋⢀⠀⠀⢀⣠⣤⣶⣾⡿⠿⣻⣿⣷⣶⡿⠿⠻⢟⣑⣬⣿⣿⣿⡙⠛⠷⢶⣤⣄
⠀⠀⣀⣠⣼⣿⣿⣆⠀⠙⠻⣿⣦⣄⠀⢸⠁⡟⣼⡏⢸⣿⠟⣿⣿⣶⢋⣶⣿⠿⢛⣫⠥⢐⣚⡥⠿⠛⠉⠁⠀⠲⠾⠟⠛⢋⣩⣴⣿⠿⠃⠀⣀⣩⣿
⠀⢿⣏⠀⣶⣟⢻⣿⣧⡀⠀⠈⠙⠿⣿⣦⣀⢡⣿⠀⣿⡇⠀⢿⣿⡇⢸⡟⠁⠀⠀⠐⠊⣉⣤⣴⡶⠞⠉⢀⣀⣤⣴⠶⠿⠛⣋⣁⣤⣴⣶⡿⠿⠛⠉
⠀⠈⢿⣆⠘⣿⣆⠫⣿⣷⡄⠀⠀⠀⠈⠙⠿⣿⣿⣤⣿⡇⠀⢸⣿⡁⢸⣷⣄⣤⡴⠾⢛⣋⣩⣤⠶⠾⠛⢋⣉⣥⣴⣶⣾⠿⢿⣿⡟⠋⠁
⠀⠀⠘⣿⣆⠹⣿⣧⠈⠻⣿⣆⠀⢤⡀⠀⢀⠀⠨⠉⢿⣧⣄⡀⢿⣇⠀⠻⠿⢷⡶⠿⠛⢉⣡⣤⣶⣾⣿⠿⠟⢛⡉⠁⠀⠀⣶⣍⠻⣷⣄
⠀⠀⠀⠘⣿⡆⠹⣿⣷⡀⠹⣿⣷⠈⠃⠀⠀⠁⠀⠀⠀⣉⡙⠻⢿⣿⣷⣦⣤⣤⣴⣶⡾⠿⠟⠛⠉⠁⠐⠊⢉⣁⣀⣀⣀⣠⣭⣤⡀⠙⣿⣧
⠀⠀⠀⠀⠘⣿⡄⠘⣿⣇⢀⠙⣿⣷⡀⠀⠒⠀⠀⠀⣈⣉⣉⣀⣀⣈⣭⣭⣭⣭⣤⣶⣶⣶⣶⣶⣿⡿⠿⠿⠿⠿⠿⢛⡛⠛⠛⢿⣿⡄⢸⣿⡄
⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⣎⠃⠈⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⣛⣛⣛⣋⣭⣭⣭⣭⣴⡶⠶⠶⠒⠚⠛⠋⠉⠭⠤⠴⠶⠂⢀⣿⡇⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⢿⣆⠀⠈⣿⣿⣿⣦⣒⢂⡀⠈⠉⠉⠉⠉⠉⠋⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣭⣭⣤⣾⣿⠁⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠈⢿⣦⣼⣿⣿⣿⣯⣍⣀⣐⣒⣒⣠⣭⣭⣭⣥⣤⣶⣶⣶⣶⣶⣾⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢮⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁