Bots Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⡇⣿⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⣿⣿⠃⢠⣆⣹⣷⣄⠈⣿⣿⣿⣿⣿⠃⢀⣆⣹⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⣿⣿⡄⠘⢿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡄⠈⢿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣄⡀⠉⠁⣀⣼⣿⣿⣿⣿⣿⣷⣄⡀⠉⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣀⣀⡀⢸⣿⢸⣿⡏⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⢻⣿⣿⣿⠀⣀⣠⣄⡀
⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣾⣿⡿⢯⣻⣆
⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣿⢿⣿⣷⣾⣿⡟
⠀⠀⠀⠀⢸⣿⣿⣿⠟⢹⣿⣿⣿⡇⠀⠀⣀⠀⠀⠀⡀⠀⠀⣀⡀⠀⣀⣀⠀⢀⣀⠀⢸⣿⣿⣿⠀⠹⣿⣿⣿⡆
⠀⠀⠀⢀⣿⣿⣿⡏⠀⢸⣿⣿⣿⡇⠀⠸⣽⡷⠀⢾⣿⠀⠸⣯⡿⠰⣯⣿⠃⢿⣽⠇⢸⣿⣿⣿⠀⠀⢹⣿⣿⣿⡀
⠀⠀⠀⣸⣿⣿⡿⠀⠀⢸⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⠀⢿⣿⣿⣇
⠀⠀⢠⣿⣿⣿⠃⠀⠀⠘⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⡄
⢀⣴⣾⡧⣼⣧⣄⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⣠⣿⣷⣶⣿⣤
⣾⣿⣿⣤⣼⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣿⣿
⠻⣿⠁⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢸⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣦⣶⣾⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣶⣾⣧⣀⠀⠀⠀⠀⠀⠀⣀⣼⣶⣶⣶⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣫⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡾⣻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⡆
⠀⠀⠀⠀⠀⢶⣿⢿⡶⠶⠀⠀⣶⠅⠀⠀⠀⠀⢀⣀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡟⠃
⠀⠀⡀⠀⠀⣀⣐⣛⣀⡀⠀⠀⣿⠀⠀⢸⡟⠛⣏⣛⣹⣙⣻⠛⢻⡟⠀⠀⠀⣠⣶⠶⢦⣾⣿⣷⡴⡶⣦⣄
⠀⢸⡏⠉⠉⣿⣏⣛⣿⡏⠉⠉⠛⠀⠀⢸⣇⠀⣿⣿⣿⣿⣿⠀⢸⣧⠀⠀⠚⠯⠀⠀⣾⣹⣨⣽⣷⠀⠀⠿⠓
⠀⢸⡇⠀⠀⢻⡿⠿⢿⠃⠀⠀⠀⠀⠀⠈⠃⠀⠻⡾⠧⢿⡼⠀⠈⠃⠀⠀⠀⠀⣤⠷⠛⠛⠛⠛⠛⠾⣤
⠀⠀⠈⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⣸⣇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣾

⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣀⣶⣀⡀⢀⠄
⠀⠀⠀⠀⠀⠻⠿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⣠⣄⣄⣴⣷⣦⣀⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣤⡀⡀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣤⣿⣤⣄⠀⢀⡀
⠀⠀⠀⢸⡅⠈⣿⣿⣿⠀⢹⠀⠀⠀⠀⠀⠈⣿⢻⣇⣀⣿⠟⣟⠀⠀⠀⠀⠀⢸⡏⠉⢫⣯⣿⣽⡝⠉⢹⡇
⠀⠀⠀⠚⠃⠀⣿⣿⣿⠀⠚⠂⠀⠀⠀⠀⠀⣿⠘⣿⣩⣿⢠⣧⠀⠀⠀⠀⠀⢼⡇⠀⣼⣿⣿⣿⣿⠀⢼⡧
⠀⠀⠀⠀⠀⠀⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⣿⠋⠈⠙⠀⠀⠀⠀⠀⠀⠁⠀⢳⡟⠒⢻⡞⠀⠀⠁
⠀⠀⠀⠀⠸⠇⠿⠸⠷⠾⠆⠀⠀⠀⠀⠀⠀⢺⡇⢾⠹⡧⢾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡄
⠀⠀⠀⠀⢠⣦⣶⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⣤⣤⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠃
⠀⢻⡟⠀⠀⠛⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠁⠀⠀⠀⠀⢀⣤⣠⣴⣶⡿⣿⢷⣶⣦⣄⣤⡀
⠀⢨⣧⣠⣶⣾⢛⣛⢻⣶⣆⣄⣤⠀⠀⠀⠀⣴⣶⡶⢿⠿⢶⣶⡆⠀⠀⠀⢠⣯⠀⠙⣿⣏⣋⣹⣿⠋⠀⣽⡀
⠀⠈⠁⠀⠛⣿⠨⠋⢸⡿⠃⠀⣿⠀⠀⣶⣾⠋⣻⡯⢥⠨⢿⡏⠹⣷⡆⠀⠸⠛⠀⠀⣨⣟⠛⣻⡅⠀⠐⠟⠃
⠀⠀⠀⠀⠀⢹⡟⠛⣿⡅⠀⠰⠿⠆⠀⠈⠠⡜⠛⠻⠿⠿⠛⠟⢦⠁⠁⠀⠀⠀⠀⠀⢾⣿⠀⣿⡇
⠀⠀⠀⠀⢠⣼⡇⠀⣿⣥⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠀⢸⡯⠀⣽⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠊⠄⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢉⡐⢄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠄⠳⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⡄⠀⠀⠀⠀⠀⢀⡮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣞⣄⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡉⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣷⡆⠀⠀⠀⣉⡁⠐⡄⠀⠀⢠⠈⠀⠠⢉⣢⠀⠀⢰⣞⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠃⠀⠰⢀⠙⠁⣀⠂⠀⠀⢨⡈⠑⠟⠃⡀⠁⠀⠛⠿⠅⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢈⡋⡂
⠀⠀⠀⠄⠤⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣠⢞⡽⢲⡾
⠀⠀⠀⠢⣭⣴⠻⠷⣔⡢⢄⡀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠈⠐⠀⠤⠤⠀⠂⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀⢊⡴⠋⠀⣠⠇
⠀⠀⠀⠀⠈⢧⡳⡀⠀⠈⠉⠓⢯⣖⠤⣀⠀⠀⠀⠀⠀⠉⠉⠐⠒⣶⣶⣶⣶⡖⠒⠐⠒⠒⠊⠀⠀⠀⢀⠄⣪⠔⠋⠀⢀⠊⡞
⠀⠀⠀⠀⠀⠀⠱⣌⢦⡀⠀⠀⠀⠈⠉⠛⢿⡖⢀⣀⠈⢁⣈⣈⡋⠉⠙⠁⠉⠙⠉⠉⠉⠋⠉⠰⡤⢴⡵⠊⠀⠀⠀⠠⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⢄⠀⠀⠀⠀⠀⡞⠀⠑⠝⠀⢸⠣⠄⠀⠀⠀⠀⠉⠉⠀⠀⣠⡤⢤⣤⠈⠺⠄⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠃⠀⢮⡵⠀⢸⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⢤⣄⣀⠶⠀⠀⡀⠀⣀⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⣄⠀⠀⠀⠐⡂⠄⢸⠀⠀⡠⠄⠆⣄⣪⣀⡀⠀⠶⣒⣐⠶⠀⠀⡏⠣⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡩⢶⠀⢈⢍⠀⠈⠒⠒⠒⠒⠒⠋⠀⠐⠃⠀⠛⠬⠤⠓⠀⢰⡇⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⣸⠀⠐⠒⠁⠀⠰⢶⢶⣶⠲⡶⠀⠀⠀⠀⣙⣩⠭⣛⠀⠘⢃⡼⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣧⠸⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡇⠀⠀⠀⠀⠉⠐⠒⠋⠀⠰⠦⣈⣉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⢡⡐⠄⠀⠀⠀⠀⠀⠛⠖⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢦⣴⣾⠶⡏⠉⠋⠀⠈⠛⣒⠒⡒⠒⠐⠒⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⠀⠀⠀⠀⠀⠉⠁⠀⠉⠙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⡀⡑⡄⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⢤⡀⠄⡒⠄⣼⡀⠀⡰⠈⠘⣃⢡⡏⠀⡊⠄⡂⢽⣄⡀⠠⠠⠀⠀⠀⠙⣧⠂⡠⠀⢄⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠉⠙⠋⠙⠉⠊⠁⠂⠁⠐⠁⠂⠡⠊⠙⠋⠵⠒⠖⠲⠊⠁⠁⠀⠔⠱⠊⠊⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣀⣬⣬⣴⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠹⡟⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⡇⣠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠱⢙⠛⡁⠀⠀⠀⠐⠟⠁⠁⠀⠀⣷⡧⣾⡇
⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠠⣙⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠀⡇⠛⡟⠁
⠀⠀⠀⠀⠀⠀⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠈⠁⠀⠀⠀⠀⠁⠀⣯⢰⡇
⠀⠀⠀⠀⠀⡀⠴⡄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣭⣭⣥⠀⠀⠀⠐⠓⠃
⠀⠀⠀⠀⠀⠆⠐⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠘⣿⣿⣀
⠀⠀⠀⠀⠀⠈⢤⣬⣇⡆⠀⠀⠀⠀⠀⠀⢀⠀⠄⠉⠆⣀⠀⢀⣠⣀⣸⣭⣥⣀⡢⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢫⠀⢳⡀⠀⠀⠀⡠⠂⠁⠠⠂⣸⠋⠀⠀⠀⡲⠂⠨⣿⠤⡷⣅⠀⠀⠀⠀⠀⠉⠐⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠳⣀⠔⠁⢀⡠⠚⡃⠀⢻⠀⠀⠠⠊⠐⠀⠄⣿⠀⡷⠈⢹⠉⠙⠒⠲⠤⢤⡵⢧⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡜⠁⣰⠔⠉⠀⠀⠃⠐⣘⠀⢠⣶⣥⣤⣤⣴⣿⠆⡟⠄⠾⠀⠀⠀⠀⠀⢠⠃⢄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠴⠀⠁⢠⣿⣶⡇⠀⠀⠀⠀⢠⡇⣰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⡿⠁⠀⠀⠀⢀⡏⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠄⡀⠀⠉⣭⢉⡙⠛⠋⠁⠀⠀⠀⠀⠘⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠻⢿⠁⠀⠛⠻⠿⡇⠀⠀⠀⢀⠤⠒⢷⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠄⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⢹⠀⠀⠀⠀⢜⡃⠀⠀⠀⠀⢀⠔⡠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣮⡆⠀⢀⢀⡩⡇⠀⠀⠀⠀⠈⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣀⣀⣀⡄⢰⡁⠀⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠂⠠⡐⢇⣠⣤⣡⣤⡿⣇⣀⣉⣉⣉⡷⠒⢠⠆⠀⠀⠀⠀⠠⠀⠄⠀⠤⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠁

⣿⠿⠋⠁⠀⠀⠀⠀⠀⢠⡏⠀⡾⠆⠄⢲⣦⡀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠟⠋⠁⣠⣶⡖⢄⠰⡧⠀⢻⠀⠈⠙⠿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⠁⢀⣄⠈⢰⣯⣹⢿⠀⠀⠀⠈⠻⣟⣿⠛⠁⠀⠀⠠⣿⣏⣷⠀⡆⣤⠀⠸⡇⠀⠀⠀⠈⠛⢿⣿⣿
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠈⠁⠈⢄⠉⣁⠄⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠢⣈⢉⠀⠀⠉⠀⢀⣿⣦⣄⠀⠀⠀⠀⠙⢿
⢀⣴⣿⣿⣿⣿⡿⠟⠉⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣼⠛⠠⢀⠀⠀⠀⠀⢤⡤⠀⠀⠈⠁⡀⠀⣤⠄⠀⠀⠀⠀⡀⠌⠛⡆⠀⠙⠿⣿⣿⣿⣷⣄
⣿⣿⡿⠁⠀⠀⠀⠀⣠⣶⣿⡿⠁⠄⠀⠀⠀⠀⠐⠈⠁⠀⠀⠠⠂⠀⠀⠈⠀⠂⠀⠀⠀⠀⠐⠘⢧⡀⠀⠀⠈⠻⣿⣿⣿⣧⡀
⡿⠋⠀⠀⠀⢀⣴⣾⣿⣿⡟⠁⠀⠀⠴⣦⣀⣈⡁⠚⠂⠀⠀⢠⡀⠀⠀⠀⠚⣊⣀⣠⣴⠦⠀⠀⠘⢿⣦⡀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⣠⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠉⣻⣿⠀⠀⠀⠊⣼⢻⣷⠀⠀⢨⣿⡟⠁⠀⠀⠀⠀⠀⠀⠻⠿⣆⠀⠀⠈⢻⣿⣿
⠀⠀⢀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣆⠀⠈⢄⠙⢋⠈⠁⠀⣼⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢻⣿
⠀⢀⣾⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠉⣷⣄⠀⠸⠆⠀⣤⡾⠉⠀⢀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⢿
⢀⣾⣿⣿⠏⠀⠻⣤⣀⣀⣀⣀⣤⣤⡤⠶⣶⠖⢣⡀⢋⠉⡀⢀⠀⠈⢍⠵⣀⠒⠶⡶⠶⣤⣤⣄⣀⣀⣀⣀⣴⣿⣿⡄⠀⠀⠈
⠾⠿⠿⠿⠤⠤⣤⡴⠿⠿⠿⠿⠿⠯⠤⠴⠤⠶⠶⠦⠤⠤⠤⠤⠤⠤⠬⠥⠶⠦⠴⠤⠦⠬⠿⠿⠿⠿⠿⠿⠾⠿⠿⠷⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⣶⣶⣶⠄⠀⠀⠠⣶⣶⣶⠄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣶⣤⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣤⣶⡿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣧
⠀⠀⠀⢀⣤⣾⣿⣇⡀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸⣿⣷⣤⡀
⠀⠀⠀⣿⡟⠉⠉⢿⣷⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡿⠉⠉⢻⣿
⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠈⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⠛⠛⢛⣛⣛⣛⣛⣛⣛⡛⠛⠛⣿⡟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠂⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣄⢀⣓⣞⣘⣀⣀⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢼⣵⢺⡲⡖⡾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣷⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡟⠀⠀⣠⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠀⠀⢻⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠁⢸⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⣧⠀⢻⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⢰⠒⢒⠆⠀⠀⠀⠀⠰⡒⠒⡦⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⡄⠘⡄⠀⠀⠀⠀⠤⠒⠒⠒⠒⠤⡀⠀⠀⠀⢠⠇⢀⣧⣤⠾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⠈⠉⠓⠒⠒⠒⠂⠒⠒⠚⠋⠁⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⢛⡛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠙⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠄⠀⠀⠀⠈⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠐⠀⠀⠀⠀⠀⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠐⠋⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣤⣄⣠⣤⣀⣠⣄⣠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢸⣿⢈⡁⡇⢐⣾⠸⣿⠸⣿⠰⠆⡧⣼⣿⠀⣿⠀⢀⣤⢶⢶⣶⡶⣤⡀⠀⣿
⠀⠀⠀⠀⠈⠛⢾⣷⣷⣶⣾⣶⣾⣶⣾⣷⣾⡷⠟⠁⢀⣿⡴⢋⣘⣘⣈⣉⣁⣃⠙⢦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⠉⠉⠁⠀⠀⠀⡞⢹⣷⡫⣿⡍⠉⠉⢩⣿⢽⣾⡏⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠙⢾⡇⠛⠛⠒⠒⠒⠒⠛⠛⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡐⢿⣛⣛⡿⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡶⠶⠶⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞





⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣴⡏⠉⠉⢲⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣾⣻⣧⣤⣠⣸⣷⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢹⡀⡀⠈⠁⣠⢦⣤⣄⢀⡀⠤⢤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠾⠶⡁⢀⠀⣀⢤⡀⣿⢿⡅⢧⣨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⢰⠃⠀⠙⠶⠞⠁⠆⠙⠖⠊⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣮⣾⡧⠀⠀⢠⡤⠤⠤⠤⠤⠤⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢡⠇⠀⠀⢘⣟⣛⣛⣟⠿⠿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⠞⣶⣤⣶⣤⣤⣤⣤⠶⣖⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣶⣶⣖⠒⠓⣄⢻⡿⠿⠿⠿⠿⠶⠞⠩⠤⠅⠙⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⣄⠀⠄⠀⠡⠀⢸⠀⠀⢐⠒⠄⠀⠀⠀⠀⠀⠀⠐⢄⠟⠀⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄⢈⠁⠀⠀⢠⠀⢸⠀⠀⢈⡀⣠⣤⣤⣴⣤⣤⣴⡶⣖⡦⠀⠹⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢷⣀⢁⣠⣾⣿⡏⣴⢸⠀⠀⣷⠉⣉⣩⠟⠟⠙⠰⠶⣤⠸⡇⠀⣠⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢙⣿⣿⣿⣿⢇⡿⢸⠀⠀⣿⠀⢏⠀⡠⠆⠀⠀⠠⠎⢸⡇⠀⢰⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⣄⠀⢀⣾⣿⣿⣿⣿⣸⣇⢻⠀⠀⣿⢀⡀⠁⠀⠀⠀⠀⠐⢒⢸⡇⠀⣾⡄⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⢤⠈⣧⣾⣟⣾⣿⣿⡇⣿⡟⢸⠀⠰⣇⠳⠄⠂⠤⠀⢛⠒⡚⢨⢸⡇⠀⣿⣿⠒⠀⠉⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠠⠀⡏⠉⣹⣿⣿⣿⢷⣿⠇⠰⠀⠀⡾⠀⠂⡘⢀⣁⣬⣀⣠⠶⠦⠃⠀⣿⣿⣤⣤⠤⠄⢁⠶⠺⣏⠉⠁⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣼⡏⠀⢱⣼⣿⣿⣾⡟⢸⢈⠀⠀⠙⠒⠛⠻⠛⠛⠋⠉⠡⠀⠀⠀⠀⠻⠛⠫⠬⢀⡀⠎⢄⡱⢸⣷⡶⠠⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⠀⠀⠀⠘⣿⣿⣿⣿⣁⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢶⠖⠛⠋⠉⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣆⠀⠀⠀⣿⣿⣿⣏⣸⡇⢸⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⠔⠀⠀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠹⣆⡀⠀⣘⠛⠻⢿⣿⣅⡸⢃⡀⣀⣀⣀⣤⣬⣽⣶⣶⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢢⠙⠿⡍⠑⢄⠸⣿⣿⣿⠿⠟⠛⠛⠉⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣉⢼⡆⠈⠀⠨⠀⣿⣿⡿⠀⠀⠀⠀⠀⣿⠁⠸⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠃⠀⣷⢀⣰⠶⠀⠉⢽⡇⠀⠀⠀⠀⠀⢸⡀⢈⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠋⠉⠁⠀⠀⠀⢰⣻⣀⣀⠀⠀⠀⠀⢈⡉⢾⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣀⠈⢻⠦⠐⠚⠈⠃⢸⣴⣶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢸⠀⢡⠀⣦⣤⣶⠶⣧⠎⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠁⠁⢠⡏⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣿⢿⡟⠁⠀⠀⠀⣿⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⡀⠀⣀⣠⣤⣶⣤⣤⣤⣀⣀⣀⣀⣬⠀⡟⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⢀
⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠇⠏⠀⠀⠀⠀⠀⣥⣤⠶⠞⠋⠈⠉⠑⠒⣤
⠀⠐⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣭⠐⢀⡀⠀⠀⠀⣀⣿⣿⣶⣦⣀⣤⡴⠖⣛⠿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣬⣅⣛⠳⠚⠋⠉⠀⠀⠈⢉⢻⠒⣊⠭⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⣿⣿⣿⣽⣛⡿⢿⣿⣿⣿⣷⣦⣀⡴⠖⢚⠉⢽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣶⣭⣝⣛⢿⡟⢂⣨⡤⠖⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⢿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣦⣤⣄⠀⣴⣿⣿⣿⣿⣷⡿⢺⣿⣿⣵⣽⡇⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣇⣹⣿⢰⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⡃⡀⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣅⣉⡛⠃⠈⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣟⣿⡇⠀⠐⣽⣯⣾⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⡟⢻⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠃⠸⣿⣴⣟⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣯⣿⣿⣏⠠⠈⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣤⣌⣻⣿⣿⣽⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣨⣿⣿⣿⡿⣋⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⣛⠛⠛⠁⠘⠿⣼⣿⣿⣹⣿⡇⠀⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢻⡿⢿⣷⠀⡀⢾⣓⣮⢹⣿⡟⠀⣽⣾⠉⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠻⣾⡟⠀⣍⠀⣿⣷⣿⣿⡟⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠷⣶⣤⡀⠀⠂⠀⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⠙⣷⠀⣍⢰⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣾⠟⠀⠷⢨⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠇⢀⣀⣤⣤⣀⡀⠸⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣼⡿⠿⠟⠻⠿⢿⣧⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢉⣿⣿⡉⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡶⠀⠀⢶⠆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠂⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣿⣿⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠖⠲⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠆⠀⠀⠰
⠀⠀⠀⠀⠀⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠈⠻⢷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠠⡚⠳⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⢀⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⠾⠉⠙⠷⠦⠤⠤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠇⠀⠂⠒⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⢤⠒⠈⠹⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠃⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠀⠲⢦⢄⠀⠀⠀⠀⢀⠠⠤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠂⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢔⡢⠀⡀⠀⠀⠀⠀⠃⢠⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⡀⢀⡴⠔⠀⠀⠀⠀⠢⢀⣠⡖⠔⠀⠀⠀⠀⠀⠀⣀⢐⣠⠞⠀⠀⠀⠀⢫⠀⠈⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣤⠀⢀⠀⠀⣀⠀⢠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣵⣶⣆⣸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⢸⠀⠀⠟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠚⣆⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠀⠘⠀⠒⠛⠀⠚⠒⠒⠒⠀⠀⠀⠀⠀⠖⠈⠀⠀⠀⠀⢹⠀⠀⣠⠞⠣⢄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡴⠊⠡⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠓⠉⡃⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡅⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⢀⣀⣀⠤⠒⠋
⠀⠀⡏⠀⠀⠀⠀⢀⣀⡴⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⢸⠀⠀⠀⠀⡴⠋⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣈⠷⣀⣶⡼⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡎⠀⡠⢄⠀⢳⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠙⠛⠁⠘⠓⠊⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠈⠐⠒⠒⠒⠒⠂⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⠉⠀⠀⣰⠖⠒⠒⠶⠤⠤⠤⡤⠤⠤⠔⠋⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⡏⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠦⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠂⢐⣧⡀⠀⠀⠀⠀⣰⡇⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠈⠀⢹⠀⠀⠀⡞⠁⠀⠈⠁⠉⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡙⠋⣾⣿⡟⣻⣿⣿⡛⣿⣿⡎⠛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⡹⣿⣿⣿⠿⢿⣿⣿⡿⣱⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣛⣛⣛⡋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣩⣭⣤⣭⣭⣭⣥⣬⣭⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣦⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣥⣶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣽⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣥⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠈⡄⢡⣤⣤⢠⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣤⣥⡜⣿⡟⣤⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉




⣿⢿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠼⠤⠤⠤⠤⠤⠤⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠤⢀⠀⠀⠀⢀⠠⠤⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⡂⠀⠄⡂⠁⠀⠀⠁⡊⠄⠀⡆⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠐⠠⠩⠔⠁⠀⠀⠑⠬⠡⠐⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠂⠘⠈⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠈⠀⠂⠃⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠐⠆⠀⠀⠖⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢾⣯⣾⣿⣧⡤⣤⣿⣿⣿⣿⣿⣿⣿⡤⢤⣼⣿⣿⣞⡿⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⣿⣿⢯⣶⣻⣻⣮⣻⣿⣿⣿⡿⣵⣞⣟⣶⣝⣿⣿⡇⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⢹⣿⣿⣿⣺⢿⣿⣿⣿⡇⣿⡿⣿⣗⡿⣼⣿⡇⠀⠀⠀⣿⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢸⡿⣿⣷⣝⣛⣛⣯⣿⣿⣿⣿⣿⣮⣛⣛⣫⣽⣿⣿⡇⠀⢀⣼⠏⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠻⣿⣿⣿⣿⣏⡍⡉⡍⠉⡏⠙⣿⣿⣿⣿⣿⣿⡿⡿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣭⣭⣯⣯⣯⣿⣿⣿⣿⣿⣿⣟⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣺⣵⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣅⣨⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣽⣿⢾⠇⠍⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣳⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⢾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣻⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣿⣿⣇
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠜⠓⠒⠚⡙⣾⠻⡛⡛⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢗⠄⠪⠊⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⢸⠑⢕⠕⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢘⡢⡠⡢⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑⠂⠀⠒⠁⠀⢨⠆⠔⣀⢨⡄
⠀⠀⢀⡀⢎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⣤⢤⣤⣼⠗⠒⠛⠋
⠀⠀⠈⠻⢬⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠉⠀⠉⠉⠉⠉⠉⠩⢹⡛⠉⡭⠭⡍⢽
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⢬⡇⠀⠠⡶⢈⡶⠮⠶⠀⠀⠀⢸⡦⢞⣣⣶⣘⢼
⠀⠀⠀⠀⠀⠀⠀⠑⢜⢄⠀⠀⠀⢀⠠⠴⢸⣦⠵⢊⡉⡃⠀⠀⣤⠶⠶⠶⠖⡺⠗⣢⢥⡖⠋⠝⡡⡉⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡕⣲⡬⠕⠋⠈⠁⠈⠉⠓⠢⢭⣤⣆⣿⢤⠔⣎⡥⠖⣿⠁⠀⡣⡢⡪⡢⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⠛⢁⡴⡁⣿⠀⠠⢆⢔⡺⡢⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠐⠀⠉⠀⠂⢻⠁⢸⢗⢕⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠁⠀⢸⢗⢕⠋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣂⣀⡀⠀⠀⠀⠀⠀⠠⣸⣗⣇⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠏⣽⠉⠋⠉⡿⠉⠉⠉⢈⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠊⡛⠓⡞⢣⠀⠀⠑⠕⢑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀⣢⣨⡇⢸⡀⠀⠀⣢⣢⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠉⠉⡉⡁⡽⠁⠉⠉⠉⠫⡉⠁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣀⡀⣀⣬⣮⣇⠄⢀⠀⠀⠈⡴⣋⣰⡇
⠀⠀⠀⠀⠐⠂⠐⠀⠐⠀⠂⠀⠒⠒⠂⠒⠀⠒⠒⠂⠄⠞⠲⠞⠒⠛⠖⠚⠈⠛⠫⠟⠫⠛⠭⠿⠥⠐⠄⠠⠀⠄⠠⠤⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡆⠊⠱⡆⡇⣸⠀⡀⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⣟⡬⣻⢍⡭⣝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣯⣧⣽⣬⣧⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣉⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⢀⡠⠖⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⢤⡀⠀⠀⢻⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢸⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢸⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⡇⠀⠀⢠⠖⠲⡀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⢸⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⡇⠀⠀⠘⠢⠞⠁⠀⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⢸⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣧⠀⠘⣄⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⢠⠏⠀⣼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⠀⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⢀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠐⠑⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠖⠠⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢱⠀⢀⡀⠤⠐⠒⠒⠒⠒⠠⠄⣀⣠⠞⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⡤⣤⡀⠀⢰⠛⠁⠀⣀⠀⠉⠓⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⡴⠌⣹⠀⡏⡀⠀⢾⣿⠇⠀⠀⡂⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠛⠉⠀⠈⠢⣤⣤⣤⣰⡶⠞⠃⠐⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠐⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠂⠠⠤⢀⣀⣈⣉⣀⣉⣉⡈⣁⣀⣀⣀⣠⡠⠶⠚⢡⠤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣮⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⠀⠐⣐⣨⠇⡀⡞⠉⠑⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠈⢀⣄⣽⣵⡄⠀⠈⠋⠁⠀⠐⠀⠀⠀⠄⠖⠖⠒⠒⠊⠉⠁⠔⢠⣺⡇⠀⠀⠀⣈⢧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⣏⣮⣁⣩⠗⢶⣎⣑⣀⡤⠀⠄⠒⠃⣉⣫⡿⢯⣉⣙⣶⡁⣐⣙⣷⡂⠀⠀⠚⢏⡂
⠀⠀⠀⠀⠀⠈⠀⠀⣐⡠⣏⣽⣭⣽⠀⠔⠛⠉⠀⠀⣀⣤⣔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠙⠄⠠⢀⠀⠤⠰⠀⠃
⠀⠀⠀⠀⠀⡄⠈⢉⠈⠉⢩⠙⡋⠀⣀⣠⢤⠶⠛⠉⠉⠆⠩⢉⢿⠿⠷⠿⠿⠾⣾⣧⣷⠷⢚⡶⣶⣾⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠬⠀⠤⠄⠃⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠖⠠⠐⠴⠴⠨⠷⠾⠿⠟⢿⠛⠁
⠀⠀⠀⠀⠀⠑⠄⣐⢀⢀⣢⠎⠠⠈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⢞⠖⢒⢲⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠨⠉⠉⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁⠀⠹⠋⠉⠙⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⣴⣠⣤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠤⠬⢬⣅⡉⣀⡀⠂⠈⠀⠶⠾⡆⡶⣿⠿⠿⠾⠛⠛⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼⠁⠀⢸⡄⠀⠀⠀⠀⠀⡫⠈⠸⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢪⣤⣬⣁⣁⣁⣴⠆⠀⠀⠀⠀⠀⢪⣰⣀⣂⣊⣂⡱⠀⠀⠀⢀⢀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠙⠉⠀⠁⠈⠀⠀⠀⠁⠉⠋⠁⠉⠉⠉⠀⠀⠁⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠠⠡⠂⠩⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡐⠶⠿⠿⠿⠯⠭⠄⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣏⠀⠩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠔⢁⣴⣌⠀⠀⠀⠀⢀⢀⣄⢄⠀⠀⠀⢓⢀⡾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⠀⠈⢄⠻⠷⠋⡀⠀⠀⠀⠸⠾⢋⠀⠀⠀⠀⢸⢼⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢸⠀⠀⠀⠉⠀⠁⠀⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⢸⠹⢶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠊⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠢⣄⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⢀⠔⠀⣸⠔⣨⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠒⠤⠄⢀⣀⣀⣀⠠⠐⠁⠀⠄⢸⢁⢸⠁
⠀⠀⠀⠀⠀⠀⡄⠐⢣⡌⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣶⣲⣶⣶⡆⠀⠒⠐⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠓⠐⠚⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣒⢒⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠈⣍⣉⣹⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢧⠀⡀⠀⠀⢀⠀⡈⣏⣛⣍⠐⠢⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢙⢧⠀⠀⠀⠀⠀⠀⡀⠔⠉⠰⡀⢹⢲⠚⠀⠈⢉⢉⡍⠉⣹⣿⣈⠯⡞⠀⢈⠀⠁⠈⠁⠒⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠘⢆⠀⠀⢀⠔⠉⠀⢄⣠⡇⠀⢽⠀⠀⠀⠀⢢⢊⠈⡐⣸⣿⠀⢟⡹⠷⡤⢤⣠⣁⢀⠀⠀⠀⣄⡜⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⣧⠜⠁⠀⣬⠔⠉⢠⠀⡀⢸⠀⠀⠀⡰⠁⠣⠄⠜⢸⡏⠀⢘⠀⡈⡇⠀⠀⠈⠉⠉⠑⠒⡞⢛⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣞⠈⢢⡴⠎⠀⠀⠀⠈⡀⠉⠏⠀⠀⢶⣶⣶⣶⣶⣾⣾⡟⠂⣾⢀⡨⡇⠀⠀⠀⠀⠀⠀⡼⠁⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠐⠂⠀⠀⠀⠰⠆⠈⠁⢠⣿⣷⣿⡇⠀⠀⠀⠀⠀⣰⠇⣰⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠙⠩⠋⠀⠀⠀⠀⠀⢰⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⣀⡀⠠⠤⠤⠷⠾⣿⣾⡶⠎⠀⠀⠀⠀⠀⢀⡟⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣤⣤⠀⠀⢸⣤⣴⣥⡥⠀⠀⠀⠀⠀⠀⢀⣀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠛⣟⠀⠀⠘⠉⠻⢛⡇⠀⠀⠀⠀⠰⠋⣉⠙⠟⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣷⠀⠀⠀⠀⢐⢰⡇⠀⠀⠀⠀⠐⠊⠀⠁⡀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣿⠀⠀⢠⠀⢠⢙⠃⠀⠀⠀⠀⠀⠀⡠⢃⠇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣥⣺⠀⠀⢸⠀⠰⢹⡆⠀⠀⠀⠀⠀⠘⠖⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠉⡋⡆⠀⡜⠓⠘⠷⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⡓⠶⠶⠖⣇⡜⠒⠤⠤⡤⡴⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠠⢀⠄⡠⠔⠠⠀⠀⠉⠄⠂⠜⠧⠰⢴⠧⡦⢶⠟⣧⢤⣤⣠⣡⣡⣝⡫⠀⠔⠋⠀⠈⠐⠄⠂⠜⠡⠃⠕⠠⠒⠐⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠯⠵⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⡁⡖⡗⡄⡔⡟⡆⡱⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠧⢧⠭⠯⠤⠬⠾⢥⠥⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⣉⣉⣉⣉⣉⣉⣉⡉⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠫⡃⠀⠥⠧⠭⠼⠬⠥⠧⠄⠀⣫⢳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠒⠚⢯⡣⡱⠇⠀⠀⠀⠀⢎⠆⠀⠀⠀⠀⠷⣡⢪⠟⠒⡐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠈⠋⠀⠀⠠⠔⢦⢤⠔⢤⢤⠖⠤⠀⠀⠈⢛⠰⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠃⠛⠀⠃⠃⠀⠃⠃⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⣤⠀⢠⡄⠀⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣒⣿⠀⢸⣗⡒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⣿⠀⢸⣗⣒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡒⣿⠀⢸⡗⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠛⡆⡞⠓⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠃⠓⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣍⠉⣉⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠈⠉⠁⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠠⠖⠂⠀⠤⠀⣀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣈⠉⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠈⢣⣀⠐⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠀⢈⠁⠀⠀⠌⣀⡑⠀⠀⠀⠀⠀⡘⣁⡈⡆⠀⠀⠀⠇⠸⢙⡏⢢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠆⠀⢨⠀⠈⠀⠀⠀⡄⠻⢁⠀⠀⠀⠀⠀⢇⠻⠃⡇⠀⠀⠀⠀⠀⠸⠁⡃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢸⡠⢅⣀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⠀⠀⠠⠼⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠤⠀⠀⠀⠀⢀⡁⢀⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢄⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠉⠁⠀⠀⡎⠡⠀⡠⠔⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⠘⢌⠂⠀⠠⡄⠀⢀⠾⠊⠐⠤⢀⡀⠀⠀⠀⠀⠀⢀⠠⢔⡌⣪⡀⢀⠀⣀⠀⠀⠀⠁⢐⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠓⠚⠂⢌⠁⡄⠐⢀⠈⠀⠐⡑⠂⠀⠤⠄⠀⠀⡀⠀⠀⠤⠀⢬⠀⠉⠉⠃⠐⠁⠀⠀⠀⢀⢠⠄⠬⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡆⠀⠐⠉⠀⢰⣬⠇⠀⠀⡠⠒⠂⢄⠀⠎⡱⠀⠀⠘⢠⣤⡤⠧⡘⣤⠖⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⡤⠤⠠⡤⠀⠃⠀⠀⠀⠀⠸⠓⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠦⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠂⠐⠣⠀⠀⠀⠀⠀⠀⠀⠍⠐⠀⠀⠱⠦⠤⠥⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢫⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⢄⠀⡀⠀⠀⠀⠀⠀⠈⠲⡤⣤⣀⣀⡀⠀⠀⠀⣀⣀⢠⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡀⢘⠉⠀⠀⠀⠠⠀⠀⠴⠧⠤⠤⠵⠄⠀⠀⡨⣀⣀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠐⡄⠈⠄⢘⠁⠀⠀⠸⡀⠀⠀⠐⠂⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠛⠒⣤⠀⠀⠀⠀⠈⠓⠦⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠐⠚⠛⠒⢋⠒⠲⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡋⠀⣠⣄⠀⢻⡄⠀⠀⠀⠀⠀⣼⠋⢀⡄⣿⣠⣤⣤⣤⣤⣀⡀⣰⠟⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣾⠁⢠⣿⡏⠀⣾⣁⣀⣀⠀⠀⣼⠃⢀⡾⣱⡿⢁⣤⣶⣶⣤⣬⣭⡉⠀⣐⣛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠿⠋⣠⣾⠟⠉⢨⡍⢷⣼⠃⠀⠨⢾⣿⣷⡿⠟⠋⣛⢿⣿⡿⠀⢰⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠰⣿⣿⠃⢀⣾⣷⡍⣨⡅⠀⣴⣿⡆⣩⡅⠀⣠⣶⣝⢁⣩⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢠⡏⠀⣼⣧⠀⢻⣿⠀⢸⣿⠟⣴⣿⠀⢸⣿⠟⣱⣿⠀⢠⣿⣿⢋⣾⡇⢀⣿⡾⣻⡇
⠀⠀⠀⠀⠀⠀⣿⣠⡴⠋⠘⢧⡀⠻⣷⣶⣶⡛⠁⠈⠳⠦⠴⠞⠃⠈⠷⣤⣤⠴⠎⠜⠷⢬⠽⠾⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣭⣭⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁
⠀⠀⠀⠀⡠⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠻⠬⢅⡀⠀⠀⠀⠀⠉⠁⠒⠤⢀
⠀⠀⠀⠀⢆⣁⠌⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠤⠄⣒⣒⡂⠀⢀⠹⡄⠀⠀⠀⠀⣤⣒⡬⣥⣍⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⠙⢢⡀⠀⠞⡡⠞⢉⣉⡉⠉⠳⣄⠙⣿⡀⠀⠀⠀⣯⣶⣶⣤⡈⢷⡌⢆
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⡄⠀⢟⠎⡜⢡⢾⣿⠟⠛⢷⡀⠘⣧⠘⡇⠀⠀⠀⣿⣿⠋⠉⣷⠈⣿⠸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⡇⠀⢸⣴⡇⣿⣿⣿⣀⣀⣼⣷⠀⣻⠆⣷⠀⠀⠀⣿⣿⣷⣶⡿⠀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⣧⠀⢸⣿⣇⢻⣿⣿⣿⣿⣿⡏⠀⣿⢀⡇⠀⠀⠀⣟⠻⠿⠛⠁⣴⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠯⣴⠟⠡⡹⣦⣙⠿⠛⠛⠊⣀⣼⢇⠜⠀⠀⠀⠘⠻⣟⡲⠶⢛⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⡪⢝⣻⣿⣿⣛⡋⠕⠋⠀⠀⠀⠀⠀⢀⡠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣲⣦⣤⣤⣤⣤⠤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣴⣿⠛⠋⠉⠻⠃⠈⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠔⣿⣯⡀⠀⠀⠀⢀⠀⢸⠓⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣿⡛⠍⠀⠐⠐⣛⠀⢸⠀⠀⠙⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠀⢰⠋⠙⣇⠀⠈⡡⣁⢴⠋⢰⠀⠀⠸⡄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖⠘⠀⠀⠈⠷⠊⠈⠓⠖⠋⠀⠀⣻⠁⠀⠀⠀⠻⢮⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⡯⠿⠏⠀⢀⣔⡣⠄⡀⠀⠀⠀⢀⣶⠵⠒⠒⢢⠀⠉⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠅⠆⠁⣴⣿⡯⠀⠀⠹⡀⠀⢀⣿⣻⠄⠀⢀⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣣⣴⡶⠶⢧⣀⣼⣗⡛⣠⣞⣿⠉⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣿⣿⣇⣀⡈⠟⠛⠿⠿⠿⠿⠿⠤⠄⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣄⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⡠⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢸⣿⣿⡇⠀⠀⠀⡇⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⡇⠀⠀⠈⢸⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡁⠀⠀⣺⣿⣿⣇⠀⠀⠀⡇⠀⠈⠀⢸⡇⠀⠀⠀⠀⡏⠀⡷⠀⣄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⠀⣿⣿⣿⣿⣤⣤⣤⡇⠀⠀⠈⢸⣇⠀⠀⠀⠀⡇⠀⣿⠃⣿⣆
⠀⠀⠀⠀⣴⡇⠀⠀⡀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠⣿⡇⠋⠀⣿⣿⣿⣿⠀⠋⠀⡇⠀⠀⠀⣸⡇⠀⠀⠀⠀⡇⠀⣿⢠⣿⣿
⠠⠀⠀⠀⢸⡇⠀⡄⡷⡇⢨⠠⠈⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⢀⣿⣿⣿⣿⠲⠲⠶⡇⠀⢀⣀⢸⡇⢸⡀⠀⠀⡇⠀⣿⡈⣿⣿
⠀⠀⠀⠀⢸⡇⣆⡇⠇⡇⣤⣄⠠⠄⠤⠀⣠⣤⣤⣶⣿⣧⣤⣶⣿⣿⣿⣿⣴⣶⣦⣧⣴⣻⣭⢿⡏⠀⠁⣀⠀⡇⢄⣿⣫⣿⣿⠀⣶
⠀⠘⡆⢀⢸⡇⠁⠃⠀⣇⣭⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣓⡇⠑⣿⣷⣿⣿⣠⡤
⠀⠐⣿⠙⠋⣇⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⣿⡿⣀
⠎⠐⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣫
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⢯⢿⣿⣽⣿⣸⣿⢺⣻⣿⣻⣿⣿⣺⣿⣿⣿⣿⣿⣾⣿⣟⣿⣿⣿⢻⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣻⣿⣻⣟⣻⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠋⣶⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣒⣒⣒⣒⡶⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⢞⣫⣶⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠱⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢶⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣦⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⣿⣿⣿⡇⡀⣠⠾⣟⡳⣆⠀⠀⠀⢀⡴⢿⡛⢶⡀⠀⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠇⢿⣶⠿⢇⣿⠀⠀⠀⠸⣷⡾⠿⣸⡇⡆⣿⢸⣿⠟⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠉⠛⠛⠋⠀⡇⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⡀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⢀⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢶⡶⠶⠦⠶⠶⢶⠶⠶⠟⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⡶⠶⠿⠿⠶⢦⣤⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⢠⣾⣿⣿⢻⡟⣿⢛⣻⢻⣿⢻⣿⢋⡙⣿⣶⣄
⠀⠀⢀⡴⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢦⠀⠀⣿⣿⣿⣿⢨⡅⣿⢨⣽⠸⣿⠸⣿⢸⡇⣸⣿⣿⣧
⣀⣠⡞⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢳⣈⡛⢿⣿⣿⣾⣷⣿⣶⣾⣶⣾⣶⣾⣶⣶⣿⣿⣿⣿
⡏⢸⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⠁⢹⢸⡇⠿⢸⡇⣿⢸⠀⢻⠁⢸⢃⢸⡇⠸⠀⣿⣿
⡇⢸⡇⠀⠀⠀⠀⠀⠘⠿⣿⠿⠃⠀⠀⠀⠀⠀⢸⠀⢸⢸⣇⣿⣸⣇⣛⣸⣀⣄⣆⣛⣨⣄⣇⣧⣀⣿⠟
⡇⢸⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⠀⢸⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⡇⢸⡇⠀⠀⢰⣳⣒⡎⡆⠀⢸⢱⣲⢽⡆⠀⠀⢸⠀⢸⠀⠀⠉⠙⠛⠛⡿⠛⠛⠛⠋⠉⠉
⠛⠚⢧⡀⠀⠀⠓⠻⠚⠁⠀⠈⠓⠗⠋⠀⠀⢀⡜⠒⠋⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠙⢦⡀⠀⠀⣤⢤⣤⣤⣄⠀⠀⢀⡴⠋⠀⠀⠀⣀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠙⣄⠀⠙⠛⠛⠛⠋⠀⣰⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⠤⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣀⡀⡇
⠀⠀⠀⢀⣠⠔⠚⠋⠉⠉⠉⠉⠉⠙⠒⠢⣄⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⣶⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⡀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⠀⠀⠒⢸⠐⠂⢈⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠑⠢⠒⠁⠉⠊⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠈⢦
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⢄⡀⠈⠙⠓⠒⠒⠛⠉⠀⣀⠜⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡔⠉⠈⠑⡄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉⠓⠓⠂⠰⠄⠒⠓⠊⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡔⠉⠈⠑
⠙⢄⣈⣤⡃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣣⣌⣀⠔
⠀⠀⠈⠣⡑⢄⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠉⠢⢝⣒⢒⡇⢸⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⢸⠀⣗⢒⣚⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠒⣶⣶⣶⣶⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠿⣻⠃⠀⠀⠀⠀⠀⠀⢻⡛⠟⡿⠋
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣦⣤⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⣻⣿⣿⣿⠟⢛⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿⣿⣿⣿⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣷⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣽⣯⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⠛⠛⢻⠛⠛⠛⢿⠛⠛⠛⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣾⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⣰⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣤⣤⣤⣼⣤⣤⣤⣾⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⠉⠈⠙⠋⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⣾⠿⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠾⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠛⠛⠛⢻⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠉⠀⠈⢹⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠋⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⡀⣰⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⡿
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣿⣾⣷⣶⣶⣿⣾⣷⣿⣾⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠇⣠⡆⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠐⠂⠁⠀⠀⠀⠰⢋⠿⠇⣠⠆⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠂⠁⠀⠀⠀⠰⢊
⠼⡱⠀⠀⠀⡠⠔⠒⠂⠉⠀⠀⠀⢂⡤⠀⠀⠀⠀⠀⠹⢗⠺⡱⠀⠀⠀⡠⠔⠒⠂⠉⠀⠀⠀⢂⡤⠀⠀⠀⠀⠀⠹
⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠂⠀⡄⠘⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⡄⠘⠀⠐⠂
⠐⢶⣕⠀⠐⣄⠀⢀⢎⠄⣀⣄⠁⠀⠀⠈⠉⠀⠀⢀⢪⠶⠀⢾⣕⠄⠐⣄⠀⢀⢎⠄⣀⣄⠁⠀⠀⠈⠉⠀⠀⢀⢪
⡄⠀⡈⠁⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡄⠀⠈⠁⠀⠀⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⢰⠚⠀⢠⡠⢆⡀⠀⠀⠀⢠⡸⠀⢙⡃⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⢠⡠⢆⡀⠀⠀⠀⢠⡸⠀⢙⠃
⠐⠐⠸⠀⠀⠀⢆⠚⢠⢃⠀⠐⠀⠀⠤⠴⠀⠀⠀⠀⠀⠰⠐⠐⠘⠀⠀⠀⢆⠚⢠⢃⠀⠐⠀⠀⠤⠴
⣀⠀⠀⠀⠀⠉⢦⠛⢂⡌⠀⠀⠰⠶⠀⠬⢴⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⠉⢦⠛⢂⡌⠀⠀⠰⠶⠀⠬⢴⠀⠀⠀⣄
⢉⡆⠂⠀⠱⠓⡄⠖⢀⠰⠀⠀⢀⢖⡤⠄⠘⠲⠀⠐⠇⢸⢉⡆⠂⠀⠳⠑⡄⠖⢀⠰⠀⠀⢀⢖⡤⠄⠘⠲⠀⠐⠇
⠀⠁⠀⠀⠃⠴⡬⡄⠋⣠⠀⠀⠄⢸⣇⠂⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠃⠴⡬⡄⠋⣡⠀⠀⠄⢸⣇⠂
⠀⠀⠀⠀⢰⣆⠏⣺⣆⡆⠀⠈⢌⠢⠉⠡⠢⠃⠀⠀⢤⡀⠀⠀⠀⠀⢰⣆⠏⣺⣆⡆⠀⠈⠌⠢⠉⠡⠢⠃⠀⠀⢤
⠁⠨⠀⠀⠀⠙⠁⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢏⠀⠁⠨⠀⠀⠀⠘⠁⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⡀⠠⠄⠀⣶⠒⠂⠀⢠⣈⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⣶⠒⠂⠀⢠⣈⠐⠀⠀⠀⢰⠠
⡙⢏⠉⠀⠀⠴⣐⠐⢒⠚⠀⠀⠀⠀⣹⠆⠀⠀⢀⣠⠀⣁⡙⢋⠉⠀⠀⠴⣐⠐⢒⠚⠀⠀⠀⠀⣹⠆⠀⠀⢀⣡
⣾⡾⢷⣶⣶⣶⣿⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⣶⣾⣷⣾⣿⣾⣾⣷⣶⣶⣶⣿⣶⣾⣷⣶⣶⣶⣿⣿⣶⣶⣶⣾⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠁⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⢣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡜⠋⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣤⣤⣤⣄⢀⣤⣤⣤⣤⡀⣠⣤⣤⣤⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠺⢏⣩⡽⠗⠘⠿⣍⣍⠿⠃⠹⢯⣉⡹⠟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣈⣉⣁⡀⢀⣀⣉⣉⣀⠀⢀⣈⣉⣁⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣏⣉⣹⡟⠸⣿⣉⣉⣿⠇⢺⣏⣉⠹⠿⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠛⠃⠀⠀⠀⠛⠛⠀⠀⠀⠘⢛⠂⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾⡟⠛⢻⡷⢸⣿⠛⠛⣿⡇⢸⡟⠛⢻⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠛⠃⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠃⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡿⠿⢿⣗⢰⣿⠿⠿⣿⡆⣸⡿⠿⢿⣗⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣇⡈⠱⠶⠎⠁⠀⠉⠶⠶⠉⠀⠈⠱⠶⠎⢁⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⢐⣒⣒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣩⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣮⣗⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣷⣮⡳⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⠉⠀⠉⠉⠉⠉⡛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⣤⣤⣀⠀⠐⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⢟⠝⠒⢄⠨⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠋⠁⠈⣝⢷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣿⠀⣆⠀⠀⠀⢠⣤⣼⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡿⢟⠂⢄⠀⠀⠀⣸⡇⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢚⡷⠃⣑⣥⣤⡶⢻⣀⠘⠗⠷⠶⠶⠶⠶⠶⠲⠶⠶⠶⠶⠒⣶⠶⠶⢖⣶⠖⠲⠞⢓⡾⢂⡐⠂⠐⣈⠟⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⡤⡖⠒⠯⣠⠊⠁⠀⢠⠏⠃⠀⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠁⠀⠈⣧⢔⠯⠟⣣⡴⠋⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡏⡀⠀⡀⣙⠀⠀⠀⠸⣷⢀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠰⢙⣿⣃⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⢳⣬⠸⣃⡕⠀⠀⠀⠀⠙⢮⡌⠱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠉⢮⣳⣻⠛⣣⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣄⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠃⢸⡇⢹⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣈⢀⡇⠀⠀⠀⠀⠈⢷⠤⠰⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠘⣦⠼⠇⠀⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⢙⣒⣒⣒⡒⢦⣿⣀⣀⣮⣤⣦⣤⣌⡙⢦⠀⠀⠀⠀⠀⠀⢠⢷⣤⣴⣿⣿⣟⠀⠀⣙⣛⣒⣒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣯⣷⣶⣿⣿⣿⣿⣿⣦⣸⡿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⣫⣶⣶⣿⣿⣿⣿⣿⣧⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠙⠖⠊⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠂⡂⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣧⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡽⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠰⠀⠀⠀⢠⣶⣒⠀⠀⠀⠀⣶⣖⣆⡆⠀⠀⠀⢰⠃⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⢀⣀⣀⣀⣙⣉⣀⣀⣀⣀⣙⠛⣩⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⣀⡸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠙⠀⠀⠉⢱⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠀⡀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⠂⠭⠌⣩⣉⣉⣭⣉⣤⡀⠤⠚⠀⠀⠀⠈⣇⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠤⠈⠳⠤⠞⠀⠁⠘⠀⠀⠀⠀⠀⢹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣠⠇
⠀⠀⠀⠀⢸⡆⢄⠀⠸⢀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⡶⠒⠢⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠼⠖⠋⠀⠀⠀⠀⠄⡠⠂⠀⡂
⠀⠀⠀⠀⠀⢄⠈⢢⣀⠂⣜⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⠤⠤⠧⠤⠤⠤⠤⠤⠀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⡈⠃
⠀⠀⠀⠀⠀⠈⠑⠶⠞⠛⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢠⠶⠒⢂⠁⠈⠉⠠⢄⡀⠀⠀⠀⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠠⣟⠀⠀⠀⡆⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡏⠀⠀⠀⡀⠙⢦⡤⠊⠀⠈⠢⣤⠶⠃⠀⠀⠀⡄⢀⣠⣄⣀⣀⣀⣀⡀⡀⢘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠙⠒⠒⣾⠀⠀⠀⢨⠱⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠤⢄⡀⠌⡉⠉⡟⢤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠒⢉⡝⡆⠀⠀⠀⡏⠀⠀⠀⢸⠀⠱⡀⠀⠀⠘⠇⠀⣀⣤⠀⠀⠀⠀⡇⠀⠀⠡⠠⠾⣐⡠⢇⣀⡸⠀⠣⢘⡃⢠⠔
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠑⠯⠤⢧⣀⠤⣼⠃⠀⠀⠀⠘⡄⢀⣕⠀⠀⠠⠀⣉⠀⡆⠀⠀⠀⠀⡇⠈⠳⠒⠲⣤⠤⡤⢄⣀⣀⣸⣀⡼⠀⠀⠙
⠀⠰⠅⢈⣐⠔⢋⠁⡖⠲⢤⣄⣀⡀⠀⠀⠀⣼⠀⠀⠀⠀⢼⣤⠀⢠⣋⡀⢠⠆⠉⡶⠀⠀⠀⠀⠀⡇⠀⠰⢀⠜⠀⠀⠀⠀⠀⠏⠉
⠀⠀⠀⠈⢭⠀⣅⣿⠀⠀⠀⠀⠉⠉⠛⠛⠓⠏⠀⠀⠀⠀⠀⢣⠀⠈⢩⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡶⠁⠀⠀⠀⠀⠀⠀⣂
⠀⠳⠤⠐⢚⠷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠠⠮⢤⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢀⢄⠶⠉⠗⢠
⠀⠀⠀⠀⣄⡄⠀⠙⠚⠒⠦⠤⠴⠴⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠉⢉⡉⠳⠒⠒⠒⠒⢶⠶⠶⠶⢶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⠀⠈⠳⡀⠀⠀⠀⣨⠄⠀⠠⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢘⠀⢙⡄⠀⢳⠀⠀⡼⠁⢀⠄⢀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠓⠋⠀⢀⠇⠀⠀⣅⠀⠐⣀⡤⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⣢⡤⠄⠲⠁⠀⠀⠀⠙⢦⣀⣀⣀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢀⡀⢀⠉⠳⡀⠀⠀⠀⠀⡰⠇⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠀⢀⠆⠀⠃⠀⠀⠀⠞⠀⡠⠒⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠉⠁⠀⣠⠇⠀⠀⠀⢆⠀⠐⠒⠂⢀⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠙⠓⢛⡟⠳⡀⠀⠀⠀⠈⠳⠤⠤⠶⠋⠀⡇⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⢀⡴⠋⠀⠀⠱⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣥⣤⣤⣤⣄⣴⣯⣤⣤⠤⠤⠴⠿⠀⠀⠀⠺⠤⠤⠤⠤⠤⠷⠤⠤⠦⠶⠶⠚⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⢠⡴⠏⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠁⠀⠀⣿⣿⣿⡄⠈⣹⠾⠃⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠋⠉⠉⠉⠠⠔⠀⠀⠀⠀⣀⢤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⡀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣍⡳⠂⠀⠀⠠⣄⠠⠀⣀⣊⠁⠀⠀⠀⠀⠀⠁⢣⢈⣄⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢠⠀⠁⠀⠀⠀⢀⠺⢦⡴⠀⠉⠸⠠⠒⠂⠀⠀⠀⠈⠩⡁⠈⠂⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠀⢸⠀⠀⠀⣠⣀⠋⠀⢻⠐⠄⠆⠀⠀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡀⠂⠸⠀⠀⠀⢣⣤⣄⡀⠈⠀⠀⣣⣴⡴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⡛⠉⢙⠗⠾⠶⠛⣿⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢤⣠⠶⠋⠀⠀⠀⠀⠘⠛⢤⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠁⠀⣠⡾⡁
⠀⠀⠀⠀⠀⠐⠃⠀⢀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠂⡁
⠀⠀⠀⠀⠀⢷⡀⠀⠈⣁⡐⠁⠀⠀⠀⢠⢀⡘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢍⡜⠈⢋⠀⠀⠀⠀⠘⠻⠞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡰⣤⣤⠤⡀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠠⠃⠀⠤⡀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣧⣄⣠⡄⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠄⠀⠀⢻⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠚⠁⠛⠲⠗⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠩⠂
⠀⠀⠀⣀⠀⠀⣇⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠒
⠀⠀⠈⠁⠀⠀⠀⣤⡀⠘⡀⠀⠄⠒⠂⠀⠀⠅⠀⠠⣄⠀⣀⠤⠀⠀⠠⠀⡀⠒⠤⡀⠀⠀⠀⡠⠒⠀⠀⠑⠢⠐⠢⡀⠀⡠⠀⠒⠀⠀⠈⠀⠠⣄
⠀⠀⠀⠀⠀⢠⠶⠛⠒⠚⠁⡘⠀⠀⠀⢀⣀⡀⠀⠀⠀⢳⠅⠀⠨⠶⠀⡀⠀⠀⠀⠘⡆⡠⠋⠀⠀⠀⣀⢄⡀⠀⠀⠘⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠱
⠀⠀⠐⠀⠀⠘⠂⠁⢱⠀⠀⠆⠀⠀⣰⠁⠀⠉⡃⠀⠀⢈⡆⠀⠀⡞⠈⠀⢁⠀⠀⠀⢸⠀⠀⠀⡴⠋⠀⠀⠈⡆⠀⠀⡝⠀⠀⢠⠏⠉⠉⠳
⠀⠀⡆⠀⠀⠀⠀⣀⣀⠇⠰⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⣼⠀⠀⠀⠓⠀⠀⠁⠀⠀⢠⠇⠀⠀⡞⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠁⠀⠀⣰
⠀⠀⡇⠀⠀⢰⠋⠁⠀⢀⡀⠀⠀⢰⢆⠀⠀⠀⠠⡴⠊⠈⠀⠀⠀⣄⠀⠀⠀⠀⢖⢹⠀⠀⠀⡇⠀⠀⠀⡠⠂⠀⠀⠰⠀⠀⠀⣦⡀⠀⠀⠀⣤⠖⠁
⠀⠀⡇⠀⠀⠈⠁⠀⠀⠸⡀⠀⠀⡞⠀⠱⣄⠀⠀⠙⢆⡀⠀⠀⢰⠀⠑⢄⡀⠀⠀⠙⢦⡀⠀⠈⠂⠐⠀⠀⠀⠀⣰⠇⠀⠀⢰⠀⠑⣄⠀⠀⠈⢢⡀
⠀⠀⡇⠀⠀⣀⣀⣀⡤⠴⠧⢤⣰⠃⠀⠀⠈⢣⡀⠀⣠⡗⠀⠀⢸⠀⠀⠀⠙⢦⡀⢀⠞⠣⣄⡀⠀⠀⢀⣠⠴⠋⠰⠤⣄⣀⡇⠀⠀⠈⠳⡀⠀⢀⡵
⣤⣤⣽⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣯⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣾⣥⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⢤⣤⣼⣿⡷⠆⣀⣤⣿⡋⠵⠶⣶⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠶⠲⣿⢿⣿⣿⡀⠈⣿⣿⡧⠖⠠⢴⠾⢿⣿⣶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠋⢘⣳⠈⢸⣿⠛⣀⣀⣀⣸⡗⢉⢑⠘⠂⢉⠋⠛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⢈⠀⢸⣯⣹⣿⡄⣿⣿⡑⠤⠥⢾⡗⠓⠒⠐⣛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠛⡉⣣⠘⠧⠿⠿⠿⡿⠅⠉⣿⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⢀⣵⠀⠁⠀⠀⠀⠀⠀⣏⠀⡤⠸⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣬⡄⠀⠀⠀⠀⠀⠀⢰⡆⣧⠄⠱⠤⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠄⡤⠤⢄⠀⠀⠀⡠⠤⠤⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⡎⠉⠱⠃⠀⠘⠔⠉⠑⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠱⢑⣀⠪⠆⠀⠐⢔⣠⢚⠌⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⠉⠿⠉⠉⠉⠉⠉⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⣀⠀⠀⠀⢸⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢃⠀⠀⢀⣿⠀⠀⠀⠈⠳⠶⠶⠶⢶⡶⠶⢶⡶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣒⡴⣯⠚⠀⠀⠀⠀⣀⣀⣀⣀⣸⣇⣀⣸⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡈⢻⡁⠀⠀⠀⠀⢰⡏⠉⣩⣭⣍⡉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢠⠒⢛⠦⢤⠤⣤⢼⣇⡊⠂⣀⠊⢙⡄⠀⠀⠀⠀⠄⢸⡧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣇⠀⡁⠸⡀⢸⡇⠁⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⢈⠆⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⢹⡇⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⢸⡏⠑⠒⢬⡀⠜⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⠤⠔⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡼⠦⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⢀⢞⠝⠉⠙⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠇⠀⠀⣿⠀⠀⠀⣸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⠂⢲⠃⠀⠀⠀⠀⢸⡂⠐⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠤⠥⡏⠀⠀⠀⠀⠀⢸⡢⠭⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢍⣒⣸⠁⠀⠀⠀⠀⠀⢸⠧⠉⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠦⠤⢾⡀⠀⠀⠀⠀⢰⠟⠛⠒⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⠀⠀⠀⠀⠀⠘⠒⠤⠤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡶⠿⠿⠿⠿⠿⠿⢶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀
⣶⡾⠿⠟⢻⣿⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⡟⠛⠋⠙⠙⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠈⣿⠟⠻⠿⢷
⣿⡇⠀⠀⢸⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇⠀⣿⡏⠀⢴⣶⣶⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⠇⠀⣿⣏⠀⠸⠿⢿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⠋⠀⣈⠙⢿⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⣿⣿⠀⢀⣀⣤⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⡇⠀⣾⣿⣧⠈⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⡇⢰⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⡿⠻⣿⡟⠛⣿⡟⠙⣿⡇⠀⣿⣿⡇⠀⣿⣿⠀⠘⠉⠙⣿⠇⠀⠙⠛⣿⠀⠘⠛⠛⣷⠀⠙⠿⠋⢀⣾⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣷⣴⣿⣷⣴⣿⣷⣴⣿⣷⣶⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣿⣶⣶⣶⣶⣿⣷⣦⣤⣴⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⠟⠛⣿⣿⡟⢛⣿⡟⠛⣿⣿⡟⠛⣿⡟⠛⢻⣿⣿⡿⠛⠛⣿⣿⣿⠛⠛⣿⣿⣿⠛⠛⣿⣿⠛⢻⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⡁⠐⣿⣿⠇⢸⣿⠃⠀⣿⣿⡇⠀⣿⡇⠀⠈⣿⣿⡇⠀⠀⣿⣿⡇⢀⡄⢸⣿⣿⠀⠀⠙⣿⠀⢸⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⡇⢀⣄⣀⠀⢸⣿⠀⢠⣿⣿⠁⠀⣿⡇⠀⠀⢹⣿⠁⡄⠀⣿⣿⠇⢸⣇⠈⢿⣿⡀⢰⡄⠹⠀⢸⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣇⢸⣿⣿⠀⢸⣿⠀⢰⣿⣿⠀⠀⣿⡇⠀⣆⠘⡿⠀⡇⠀⣿⣿⠀⢸⡿⠀⠸⣿⡇⢸⣿⡄⠀⣸⣿⣿⣿⠀⠀⣿⠀⠀⠀⢸
⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⡀⠘⣿⡿⠀⢰⣿⡇⠀⣿⠀⠁⢸⡇⠀⢻⡟⠀⣠⣤⡄⠀⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⠀⠀⠀⢸
⠿⢷⣶⣦⣼⣿⡀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣠⣾⣿⠃⠀⣿⡆⠀⣼⡇⠀⢸⡇⢀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⢠⣿⣤⣴⣶⡾
⠀⠀⠀⠈⠉⠙⢿⣦⣄⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⣠⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⢴⡦⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⢀⣀⠁⣴⣄⠈⣀⡀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠙⠛⠀⠈⠁⠀⠛⠋⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡀⣈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⠂⠀⠀⠀⠀⠘⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⣄⠘⠗⣀⠖⢀⠐⠗⢀⡼⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠋⠉⣁⠀⠀⢉⡉⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠯⣥⢤⣭⠟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⣤⣤⣤⣤⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠒⠚⠒⠚⠚⠛⠒⠛⠛⠚⠒⠒⠒⠓⠚⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠠⡆⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡾⠀⢨⠀⣷⠀⢠⡞⠁⢀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⣽⣄⠈⠋⣿⠀⠀⢧⡀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣟⣁⣴⡋⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡴⠋⠀⠉⠛⠶⠾⡆⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠙⢦⡀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠶⠉⠉⠃⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠀⢸⡇⠀⠀⠀⠀⠙⢷⡎⠁⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠘⣀⠐⠗⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣾⠀⠀⠀⠀⠀⠀⠐⣆⡈⢉⣠⡆
⠀⠀⠀⠀⠀⠀⢀⡟⢳⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠁⡄⠀⠀⠀⠀⠀⠀⡰⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⣏⠹⣇
⠀⠀⠀⠀⠀⢀⡞⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⠢⠄⠀⠀⠤⠚⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸
⠀⠀⠀⠀⠀⣾⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧
⠀⠀⠀⠀⡸⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⡿⠶⠶⠶⠶⢶⡟⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡄
⠀⢀⡴⠞⢳⠔⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡾⠁⠀⠀⠀⠀⠀⢻⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⣟⠙⠓⢄
⢰⠏⢀⡴⠛⠶⢄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡎⠉⠙⢦⠀⢇
⣇⠀⡞⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⣘⠀⡈
⡟⠀⢻⡄⠀⢀⡼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠿⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣹⣤⣀⠀⠀⠀⠀⠀⠈⢳⡀⠙⡇⠀⠙⠶⠁
⠹⣆⢀⡿⠀⠀⠳⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠛⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⢾⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣤⣄⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣉⣉⠁⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡀⣼⡀⣤⣠⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢠⡄⣸⣠⡇⢀⠀⣀⣀⣀⡀
⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠳⣄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠿⠀⠤⢤⠤⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣦⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⠀⠀⠀⠀⠰⠃⠈⡙⠢⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⢠⠀⠄⠀⠀⠀⡅⠀⠀⢷⠀⢁⡠⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⢠⡇⠀⣀⣀⣻⢀⢀⠀⠀⣄⢰⣀⣌⢺⣢⢎⣔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡈⠠⢌⣁⠖⢀⠀⠀⠀⠈⣻⣭⣤⣴⣆⣾⣇⡷⣣⡼⡵⠃⡜⠀⣤⠿⣳⣽⢻⣾⢣⢎⣠⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢈⠩⢦⡄⠈⢂⡴⠋⠐⠂⠙⠶⡶⢿⢿⡟⣯⣾⣡⠞⣁⡤⢶⣟⣁⣨⣿⠁⣐⠟⣃⣿⡤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⣙⣦⡏⠠⠀⠠⠂⢀⠀⡄⠈⡀⠛⠟⣿⡟⡿⠋⠰⠈⣉⠉⠿⢻⣀⣿⣷⣿⣕⣾⣫⣀⢄⠨⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⣀⣁⣿⠀⢀⠀⠀⠀⠁⠀⠀⠈⠀⠀⡀⢨⣿⠁⢀⠀⡀⠁⡆⢰⠈⠉⢿⣿⣿⣿⣿⢿⣫⡛⠿⡲⠂⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣨⣿⡟⠆⠘⠀⠊⠀⠂⠀⠂⠀⣠⡀⣵⣯⣿⠀⠈⠐⠃⠀⠀⡀⠀⣰⣢⣿⣿⣿⣿⠓⠂⠨⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⢀⠀⠆⠀⡀⢀⣆⣀⣜⣥⣾⣟⣿⣿⠴⠀⡆⠐⠃⠀⢃⣤⣿⣟⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢀⠀⢈⣁⣴⣿⣿⣿⣿⣿⣿⣿⠻⡇⢀⠀⡄⠀⣧⣼⣿⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⣘⣤⣾⣾⣿⣿⣟⣿⣿⣾⡟⠑⠆⣷⠉⣀⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣻⣿⣿⣿⡿⠿⣟⡙⠙⢣⡄⡁⢸⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⠀⠈⠉⠒⠨⣱⠂⠀⠙⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡱⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣰⠋⢀⣀⣀⠀⠀⣀⣀⣀⠀⠈⣇⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣀⣠⣿⣈⣀⣛⣿⡷⠋⠘⠿⠧⠿⡆⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡞⠁⢠⡇⠀⠀⠀⠀⠉⠉⠁⠈⠀⠈⠁⡴⢰⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡆⠀⢸⡇⢠⠤⠤⠤⠤⠤⡤⣀⣀⣀⠀⡃⣿⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⡇⠸⣆⠀⠀⠀⠀⠀⠀⢨⡏⢰⡇⢿⠀⠀⠀⣰⠟
⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠓⠦⢤⣤⡤⠴⠛⠀⢸⠇⠈⠛⠛⣿⠉
⠀⢀⡴⢦⠸⡅⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⠿⠟⠛⠳⠶⠶⠶⠶⣤⣤⣾⣤⢤⡴⠶⠟
⢀⡈⢧⡈⢣⡹⣄⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣉⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⢦
⢯⡙⢦⠙⣤⠟⠉⠀⠈⡗⢉⣹⢰⠉⡟⠙⠛⠛⣿⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡤⠼⠧⠤⠶⢶⣄⡀
⠀⠙⢦⡟⠁⠀⠀⠀⠀⡟⠉⠀⢸⠀⣇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢻⠸⣇⠀⠀⠀⠀⢰⡏⢹
⠀⠀⠘⢦⣀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡏⠉⠛⣿⠀⠀⠀⠀⠀⡁⠀⠀⠊⠀⢡⠘⡇⠀⠀⢸⡆⠉⠛⠛⡟⠛⢻⡇⢸
⠀⠀⠀⠀⠈⠛⠶⠤⠞⠣⡀⠀⢸⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⢤⣀⡞⣠⠇⠀⠀⢸⡇⠀⠀⠀⣿⠀⢸⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢆⡠⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⢀⢁⡀⠀⠀⢸⠃⠀⠀⠀⣿⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡿⢾⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⣭⣀⢉⠆⠀⣸⠀⠀⠀⠀⣿⠀⣾⠀⣿⠀⠀⠀⠀⠀⠈⡅⢸⣀⣤⣄⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠁⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⣤⣇⣀⣀⣀⣀⣠⡤⠞⠋⠉⠈⡷⠀⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⢧⣤⠯⢭⢭⢥⠼⣇⡀⠀⠀⠀⠀⣟⢉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣠⡷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣳⠶⠶⠤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠋⠉⢹⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⣻⠛⢚⠳⢚⣛⣻⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡶⠦⠤⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠋⠀⠀⠈⠀⢹⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡇⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠘⢧⣤⡤⢤⣤⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠾⣇⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠦⠶⠾⠶⠒⠛⠉⠁⠀⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢺⣀⡟⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⠤⢄⣈⠉⠀⢀⡠⠔⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠰⡇⠸⡦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣉⠀⠉⠒⠤⣀⡀⣸⢀⡠⠖⠉⠀⣈⡿⠂⠀⠀⠀⠀⠀⠀⠀⣾⠙⠒⠤⣄⣀⠛⠚⠀⣀⡠⠼⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠉⢹⡅⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⠏⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠓⠦⢤⣀⠀⠀⣱⠀⠀⣀⠤⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠉⠓⢻⠊⠉⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣼⣧⠶⠛⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣄⠎⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠠⡔⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠆⠈⡄⢀
⠀⠀⠀⠀⠀⠀⠈⢘⠱⠁⠰⣇⣠⠒⠀⠀⠀⠀⠀⠀⠀⢹⢰⣼⢸⠘⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠃⠀⠏⠉⡎⢠
⠀⠀⠀⠀⠀⠀⠀⢘⢰⠀⢀⠈⠘⢇⡜⠀⠀⠀⠀⠀⠌⢸⠈⠡⢸⠀⠘⠃⡆⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⢀⢘⢸⠙⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣈⢰⠀⣰⡀⠀⠁⡇⠀⠀⠀⠀⠀⠀⢀⠑⡸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡄⠀⣰⣼⣼⠀⢸⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⠀⠸⠅⡟⠛⣇⠈⠀⠀⠀⠀⡀⠬⠰⠀⡤⢠⣤⣤⡷⢾
⠀⠀⠀⠀⢁⣀⣾⠉⠈⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⡀⠆⠀⠀⢰⣀⣿⠀⣀⢤⢠⢄⠲⣤⠈
⠀⠀⠀⠀⢸⠉⠉⠀⠘⠻⣦⡘⡆⠈⢸⠀⠀⠀⠀⠀⡆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢩⢐⢙⣾⣷⠌⠆⠛⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠐⠀⣿⡇⠀⡄⠸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠰⠀⢀⡇⡀⢰⠀⠀⠀⠀⠁⢻⡔⢬⣉⣯⠔⢁⠰⠢
⠀⠀⠀⠀⠀⢀⠰⡀⠦⠾⠟⠀⡰⢃⠆⠀⠀⠀⠀⠀⠸⡄⡙⢷⣦⣴⠞⡁⠀⡆⠀⠀⠀⠀⢡⡀⠰⣫⡖⡴⢰⣼⢋⠇
⠀⠀⠀⠀⠀⠈⢆⡑⠤⠀⠠⠞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠑⢌⡃⠳⡌⢓⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡑⡠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢶⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣤⠠
⠀⠀⠀⠀⠀⢠⣲⡂⠸⡆⠉⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⣴⠟⢇⢦⠄⠀⠀⠀⠀⠠⠒⡆⠸⠇⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠁⢸⡇⠀⠗⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢹⢶⠉⠀⡾⠃⣶⠀⠀⠀⠀⠀⠸⣇⣀⡀⠇⡅
⠀⠀⠀⠀⠀⠆⡈⠇⠀⡆⠀⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⢩⣾⡀⠟⢷⠀⡅⠉⠀⠀⠀⠀⠀⠊⠏⠉⠇⠂⡇⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠰⡆⠀⡇⡀⢀⡀⠀⠀⠀⠀⢐⡀⣸⡀⢱⠛⠀⠀⠖⠛⠀⠀⠀⠀⠀⠀⡀⠀⠁⠋⠛⡇⠀⣤⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⠁⠁⣾⠀⠀⠀⠀⠀⠂⢱⣦⠉⣜⠀⠏⠆⠀⡀⡿⠀⠀⠀⠀⠀⠁⠁⠀⣄⢄⠉⠀⣽
⠀⠀⠀⠀⠀⡀⠀⠀⡀⡄⡄⡀⡀⠚⠶⡇⠀⠀⠀⢠⢦⠘⣄⠙⠆⠀⢰⠿⡇⢸⠀⠀⠀⠀⡇⡔⢈⡸⠖⡤⡈⠰⡀⡀⠆
⠀⠀⠀⠀⠀⢁⠀⠋⠈⠰⠆⠁⠙⠀⠀⠀⠀⠀⠀⠈⡄⠘⠋⣿⠸⠳⡀⠀⡴⠛⠀⠀⠀⠀⢛⡐⠁⠀⠐⡀⠈⠑⢮⢸
⠀⠀⠀⠀⠀⠰⡲⣠⢀⠒⡒⣀⣤⠴⠢⠀⠀⠀⠀⠀⠀⣷⠀⣮⠃⠚⠿⠀⣤⠆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠱⣽⠻⠇⠐⠯⡛⡵⠁⠀⠀⠀⠀⠀⠘⢎⢶⡤⠐⠚⣯⢾⠏⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣧⣭⣭⣬⣥⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣽⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣯⣿⣽⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⣼⡀⠀⠀⢆⣠
⠀⠀⠀⠀⠀⠀⠙⠻⡦⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣾⣿⠀⠀⡾⠁
⠀⠀⠀⣰⠜⣿⣿⣿⣿⣿⣿⡟⢴⠀⠀⠀⠀⣠⠝⣿⣿⣛⣛⢻⣿⡟⢵⡄⠀⠀⠀⢰⡾⢷⣶⣶⣿⣶⣶⡾⢷⡆⠀⠀⠀⢀⡞⢹⣿⣿⣿⣽⣿⠗⠚⠁
⠀⠀⠀⣇⠀⣿⣿⣿⣿⣿⣿⡇⢸⡄⠀⠀⠀⣯⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⢸⡁⢸⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⠅
⠀⠀⠀⢩⠧⠙⢛⡟⠛⣿⠛⠷⢯⠀⠀⠀⠀⢫⢥⡙⢻⡟⠛⣿⠛⠧⢽⠃⠀⠀⠀⠞⠃⢸⣿⣿⣿⣿⣿⠁⠛⠇⠀⠀⠀⠃⠘⢀⠬⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⢸⡕⠀⣽⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡗⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⠙⢷⡀⠀⠀⠀⠀⢀⣠⣞⠉⠀⠀⠀⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠴⠷⠰⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠰⠿⠇⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠘⠉⠉


⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶⣶⢶⣆
⠀⠀⠀⠀⠑⢢⣶⣿⣀⣿⣦⠒⠁⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⣛⠿⠟⠁
⠀⠀⠀⠀⣠⢼⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⢀⡿⠻⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣮⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠿⣶⣶⣤⣤
⠀⠀⢀⡼⡅⢸⣿⣿⣿⣿⣿⠀⡽⣄⠀⠀⠀⣾⣇⠀⢸⣿⣿⣿⡏⠀⢸⣷⡀⠀⠀⠀⠀⠻⠟⠛⡟⠛⠿⠃⠀⠀⠀⠀⢠⠋⠀⣿⣿⠆⠀⠈⣿⡇⠈⢣
⠀⠀⠈⣷⠋⢸⣿⣿⣿⣿⣿⠀⢳⡏⠀⢠⠶⢿⡟⠀⢀⣯⠀⢸⡄⠀⠸⣿⠷⣄⠀⠀⡄⣆⣿⣿⣿⣿⣷⡦⡀⠀⠀⠀⢸⣆⠀⠿⢿⡿⠶⢾⡿⠇⢀⣾
⠀⠀⠀⠟⢃⡴⠟⠿⠿⠿⠿⣄⠚⠇⠀⠈⠀⠈⠀⠀⣾⣿⠀⣾⣿⡄⠀⠁⠀⠉⢀⣼⡻⠁⢻⠿⠿⠿⠃⠻⣳⡀⠀⠀⠘⠉⠀⠀⣿⠀⠀⠈⣷⠀⠋⠘
⠀⠀⠀⣴⣾⠇⠀⠀⠀⠀⠀⢼⣶⡄⠀⠀⠀⠀⠠⠶⠿⠿⠀⠻⠿⠶⠄⠀⠀⠀⠈⠉⠀⠀⠼⠅⠀⠿⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⣿⣄⠀⣴⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⣶⣶⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⣄⣤⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠭⣿⣿⡷⠥⠿⠀⠀⠀⠀⠀⠀⠀⠈⢠⣠⣤⣄⡌⠁
⠀⠀⠀⠀⠀⢾⣥⣽⣿⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⠿⠿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⢫⣿⡜⣼⡆
⠀⠀⠀⣾⣷⣿⠉⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⣿⣾⣿⡿⠛⠻⣿⣷⣿⡇⠀⠀⠀⣾⡇⣿⣿⡟⠛⠛⣿⢺⣿⠀⠀⠀⠀⣰⣶⣾⣛⡟⠛⠛⢻⣷⣶⡄
⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣇⣀⣠⣿⣿⣿⡇⠀⠀⠀⣿⡇⣿⣀⣀⣀⣀⣿⣺⣿⠀⠀⠀⠀⣷⡇⢸⣿⣿⣿⣿⣿⡇⢻⣻
⠀⠀⢠⡿⠇⢿⣿⣿⣿⣿⣿⠸⢿⡄⠀⠀⠀⢠⡿⠴⣿⣿⣿⣿⣿⡷⢿⡇⠀⠀⠀⣿⠇⢹⡏⠉⠉⢽⡏⠼⣿⠀⠀⠀⠈⣷⠆⢸⣿⣿⠿⣿⣿⡇⠼⣿
⠀⠀⠀⠁⣠⠞⠛⠿⠿⠛⠻⣤⠈⠀⠀⠀⠀⠀⠉⠀⣿⠿⠿⠿⣿⠁⠉⠀⠀⠀⠀⠈⠁⢻⡟⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠈⠀⠀⠙⢰⣶⡆⠋⠀⠀⠁
⠀⠀⢠⣶⣿⠀⠀⠀⠀⠀⠀⢼⣷⡆⠀⠀⠀⠀⠀⢰⣿⡦⠀⢴⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇


⣿⣿⡿⠿⣿⠿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠁⠀⠀⠙⣆⠀⠀⢀⠀⠀⠀⠈⣷⠀⠀⢰⠏⠀⠀⠀⢢⠀⠀⢠⡟⠁⠚⠙⢳⡀⠀⠀⣾⠁⠐⠒⢦⡀
⠀⠀⠀⠀⠱⣄⣀⣀⣼⠇⠀⠀⠘⢦⣀⣀⣠⡟⠀⠀⠘⢧⣀⣀⣠⡾⠀⠀⠘⣧⡀⠀⢀⡼⠁⠀⠀⣿⡀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⣄⣰⠃⠀⠐⣆⢠⡞⠀⠀⢰⣄⣼⠋⠀⠰⣆⢀⡼⠁⠀⠠⣄⢠⠎⠀⠀⠰⣄⣴⠋⠀⠀⢶⣀⡞⠁
⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠘⠏⠀⠀⠀⠀⠻⠃⠀⠀⠀⠹⠟⠁⠀⠀⠀⠹⠏⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢰⠒⠒⠟⠀⢸⠀⠀⠸⡆⠀⠸⢦⣤⠖⠀⠐⢦⣤⡾⠃⠐⢶⣤⡶⠂⠀⠻⣦⡴⠂
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠸⠀⠀⠀⠀⢾⠃⠀⠀⠜⠀⠠⠞⠙⠦⠀⠠⡾⠛⠧⠀⠀⠰⠋⠓⠀⠀⣴⠏⠛⠆

⠀⠀⠀⠀⢸⠛⠛⠛⢻⢸⠀⠀⡞⠛⠛⠛⢛⣤⠀⠀⡟⠉⠉⠙⣢⢞⠀⢸⠛⠛⠛⠛⡻⣂⠀⢶⠒⠒⠒⢠⠄
⠀⠀⠀⠀⢸⠐⣆⣰⠋⢸⠀⠀⡇⠀⣄⣠⠏⢸⠀⠀⡇⠲⣤⡞⠁⣸⠀⢸⠀⢶⣀⡾⠁⣿⠀⢸⢰⡄⣰⠋⣀
⠀⠀⠀⠀⢸⣀⣘⣃⣀⣸⠀⠀⢧⣀⣼⣯⣤⣼⠀⠀⣧⣀⣍⣀⣀⡽⠀⢸⣤⣬⣿⣥⣤⠟⠀⢸⣀⣻⣥⣄⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⢛⠁⠀⡴⠋⡀⠀⡴⢁⠀⠀⣠⠊⠀⢠⢞⠁⠀⣠⠛⠂⠀⠀⡜⣁⠀⢠⠟⠀⠀⢀⣖⡀⠀⢀⠞⠁
⠀⠀⠀⠀⢻⡀⠱⡼⠁⠀⡇⠀⢇⠘⣆⡴⠃⢀⠀⢸⠈⢣⡰⠃⢰⠀⠀⠸⡆⠘⣦⠏⢀⡇⠀⢸⡀⢻⣤⠏⢀⠇
⠀⠀⠀⠀⠀⠑⠢⠥⠴⠊⠀⠀⠈⠓⠼⠧⠐⠁⠀⠀⠑⠤⠥⠤⠊⠀⠀⠀⠙⠲⠾⠶⠞⠁⠀⠈⠳⠦⠤⠶⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⠿⠿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣿⣿⡿⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣟⠀⠀⣸⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⠶⠶⠶⠶⠶⠶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣶⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣠⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⣀⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠁⠉⠉⢠⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⠀⠉⠉⠈⢿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⣠⣿⣿⣄⣠⣿⣿⣄⣀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠚⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠞⠋⠁⠀⠸⠿⢦⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⣾⠀⠀⠀⢰⣹⡆⠀⠰⣽⡷⠀⠀⣾⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣦⣀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣼⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⡿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠦⣤⣉⣙⡋⠀⠀⠀⠀⠀⠙⠛⠉⢁⡰⠏⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠠⠄⠀⣀⠉⠉⠉⠛⠛⠓⠂⠉⠉⠉⡀⠀⠤⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⡂⠈⠀⠃⡄⠀⠀⠀⠀⠀⡇⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣏⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⣠⡄⠠⠀⠀⠀⠀⣀⠀⡇⠀⢀⣇⠀⣹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠻⣄⡼⠋⠙⢦⡀⡸⠀⠀⠀⠀⠀⠀⠀⠐⢇⠘⠋⠀⠐⠀⠀⠀⠀⠁⠀⣇⣠⠞⠉⠳⣇⢨⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⣀⡟⠀⠀⠀⠀⠉⡛⠀⠀⠒⡀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⣇⢈⣇
⠀⠀⠀⠀⢀⠴⠚⠓⠾⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠙⠀⠀⠂⠀⠐⠀⠈⠁⠈⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⣹⠤⠿⢤⡀
⠀⠀⣀⣤⡏⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⡄
⣴⠋⢁⡴⠛⣄⣀⣀⣠⠛⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡤⠻⣄⠀⠀⢀⡼⣍⠓⢦
⠈⢳⣏⠀⡼⠀⢸⠇⠘⢦⣠⠷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⢾⣀⡴⠋⢹⡏⠙⣆⠈⢳⣠
⠀⠀⠈⠙⠧⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢶⠀⠀⠀⠀⠀⠈⠀⠀⢸⣇⣀⣽⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠉⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣈⣀⡀⡸⣄⠀⠀⠀⠀⠀⠀⡧⣐⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠙⢳⠾⠿⠗⠂⠉⠑⢤⢀⡴⠊⣙⣳⠶⠤⠬⠒⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠎⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⢹⡔⠉⠀⠀⠑⢄⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⡀⠀⠈⢢⣀⣀⣀⠠⢤⠂⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠁⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⢠⠃⠈⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡀⡀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⢀⡌⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢀⠜⠁⠀⡀⠄⠊⠘⢄⠀⠀⠀⢀⡴⠁⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄⢀⣤⣖⣒⣒⣒⡒⣒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠠⠤⡀⠀⣈⡉⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢒⡆⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠂⠔⠁⠸⠄⠝⡇⠀⠘⡇⠀⠀⠀⠀⠀⣎⠐⣅⡠⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠤⣄⣀⡠⡤⡀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣼⠲⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠒⠧⠦⠓⠃⢸⢸⣠⠟⠀⠀⠀⠀⡼⠣⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⡟⢻⠛⠓⠛⠋⠀⠀⠀⠀⢀⡞⢤⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠛⠧⠾⠛⠓⠒⠒⢲⣤⣄⡴⠫⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡅⠈⠉⠉⠉⠉⠓⠒⠒⡖⠉⣯⢺⠹⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢤⡇⠀⠒⡂⡒⠒⠒⢲⠀⡇⠀⣙⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢤⣷⠀⠟⠲⠃⠀⠀⢸⡗⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣸⢹⠀⢰⣞⡳⠀⠀⢸⣷⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⡼⢸⡄⠘⠒⠒⠒⠒⠚⠁⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣤⡇⠈⡇⠀⠀⣤⣤⣤⣤⣤⣿⠔⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣤⠗⡄⠓⣒⢲⡖⠒⠒⠒⠲⡿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⢇⣸⠀⣍⣹⠀⠀⠀⠀⠀⡷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠘⠉⠀⠆⢾⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡟⠀⠀⠀⠀⠀⢹⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣇⠀⠀⠀⠀⢠⢾⠚⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠘⠒⠃⡇⠀⠀⠀⢻⣈⡉⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠒⣒⣒⡭⠃⠀⠀⠀⠈⠒⠬⠭⡩⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠉⠛⠀⠀⠐⣏⣱⠀⠀⢀⠓⠀⠛⠀⠀⠀⠹⠤⠃⠀⠘⠛⠈⠙⠳⠀⠀⠸⣁⡵⠀⠀⣐⠓⠀⠋⠀⠀⠀⠳⠔⠁⠀⠘⠃
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠊⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠰⠗⠈⠂⠄⠀⠀⢀⣤⡀⠀⢀⡔⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠈⠀⠀⢠⣄⠀⠀⠀⠞⠁⣴⡡⠄
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠤⠇⠀⢨⡀⠿⠆⢠⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⡀⠀⠀⠀⠡⠴⠃⠀⠀⣘⣈⡉⠀⠟
⣀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⡼⠂⠀⠀⢀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡁⠀⠀⠰⢅⠽⠀⠀⠀⠀⠎⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡴⠀⠀⠸
⠀⠀⠀⠀⠘⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⢃⠀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠖
⠢⢤⠀⠀⢀⡤⢄⠀⠀⠀⣤⠉⢤⠀⠀⠀⢴⠙⡆⠀⢠⡴⠀⠤⢤⠀⠀⢠⠤⢄⠀⠀⢀⣄⠁⡤⠀⠀⠀⣞⢙⠄⠀⢠⡆
⠀⠀⠀⠀⠀⠓⠚⠀⠀⠒⠀⢠⡠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠒⠋⠀⠀⠒⠀⢤⠂⠀⠀⠀⠈⠁
⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢃⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠈⠉⢀⠉⠁⠀⠀⢐⠛⡆⠀⠀⠎⢡⣝⣡⠀⠀⠀⠀⠀⠀⠀⠠⠜⠃⠘⠢⠀⠀⠞⢓⡄⠀⠀⠘⢡⣍⡠
⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣚⣐⠉⠀⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⢓⡙⠋⠰⠃
⢲⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⢀⠔⢦⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠
⠋⠀⠀⠀⠀⡀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠈⢠⢀⡁⠀⠀⠀⠑⠉⠀⠀⠀⢀⠠⠄⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈

⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⡂⠀⠐⠀⠀⢀⠤⡀⠀⠀⣈⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⢤⠀⠀⢀⣁
⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣻⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠒⠒⠛⠉⢀⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⡇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠠⠚⠟⠀⠀⠀⠰⠟⠀⠀⠀⣿⣥⣿
⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⡇⢻⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⣷⡸
⠀⠀⠀⠀⢀⠈⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢲⣷⣷
⠀⠀⠀⠀⠐⣂⣻⢀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⣉⢉⣖⣒⠀⢄⡀
⠀⠀⠀⠀⠀⠈⠏⠙⡄⠀⠀⢀⠀⠀⠀⠄⡷⠋⠁⠀⠭⠍⣽⡧⢿⣇⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠐⡀⠸⡄⠀⠀⡠⠐⠁⠀⡇⠀⠀⠈⠀⠀⢺⠃⢰⠈⡏⠉⠑⠒⠂⢤⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠶⠊⠀⠀⠀⠀⠁⢀⠺⠷⠶⠶⠿⠃⣸⢀⡇⠀⠀⠀⠀⡌⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣿⣿⠃⠀⠀⠀⡼⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠐⡖⠚⠿⠿⠋⠀⠀⠀⠰⠁⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢿⠇⠀⠟⠿⠿⠀⠀⠀⠠⠔⠦⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠐⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡀⠀⡇⠀⠀⠀⠀⡄⠔⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠛⠃⢀⠂⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣈⣘⣓⣿⡎⠐⠒⠒⢧⠄⢀⠀⠀⠀⠀⡀⡀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣉⡉⣉⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠁⠑⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣈⣉⣉⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⣿⣇⣀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡤⣼⠀⡀⢀⠤⢄⠀⡀⢸⡤⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⠹⣀⣇⣎⣀⣈⣆⣇⡸⠉⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡚⠀⠀⠀⠀⢨⣿⣯⠀⠀⠀⠀⠘⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡇⠀⡞⠉⡞⠁⠀⠀⢙⡎⠙⡆⠀⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢟⣿⠀⢇⢀⡯⠝⠽⠿⠯⣇⢀⠇⢰⣟⢇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣩⠁⠀⠀⠀⠀⢩⣹⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠵⠀⠀⠀⠰⠥⠬⠇




⣏⣈⣏⣯⣟⣯⣿⣹⣽⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣦⣄⣀⣀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡈⠛⠛⢻⣿⣿⣿⠛⠛⢛⣿⣿⣿⣿⡿⠻⢿⣿⣿⡅⢻⣿⣿⠀⢀⣸⣿⣿⣿⣿⣿⡟⠹⠿⣿⣿⣧⣿⣿⣿⡏⠛⠛⠛⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣧⣤⣶⣿⣿⣿⣿⡿⠁⠀⠘⣿⣿⣷⣾⣿⣿⣾⣿⣿⣿⡟⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣍⡀⠀⣿⣿⣧⠂⠀⢠⣿⣿⡿⣿⣿⣿⠉⠉⠉⠉⢹⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠛⢿⣿⣿⣿⣷⣿⣿⣿⣧⣤⣾⣿⣿⣧⣿⣿⣿⣀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠈⠙⢿⣿⣿⣿⢿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠛⠿⡿⠿⠟⠁⠀⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⢰⣿⡆⢸⣿⠀⣾⣷⠀⣿⡇⢰⣿⡆⠀⢰⣿⡆⢸⣿⠀⣾⣷⠀⣿⡇⢰⣿⡆⠀⢰⣿⡆⢸⣿⠀⣾⣷⠀⣿⡇⢸⣿⡆
⠀⠀⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⠀⣿⣿⡀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⡀⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⠀⣿⣿⡀⣿⡇⢸⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⣾⡿⢻⣿⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣾⡿⢻⣿⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣾⡿⢿⣿⣿⡇⢸⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⣿⠃⠀⢿⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣿⠁⠀⢿⣿⡇⢸⣿⡇⠀⢸⣿⡇⣼⣿⣿⠁⠀⢿⣿⡇⢸⣿⡇
⠀⠀⠀⣼⣿⣧⣼⣿⣿⣄⣀⣼⣿⣧⣼⣿⣧⡀⣼⣿⣧⣼⣿⣿⣄⣠⣾⣿⣧⣼⣿⣧⡀⣼⣿⣧⣽⣿⣿⣄⣠⣾⣿⣧⣼⣿⣧
⠀⠀⠀⢿⣿⡙⠛⠻⢿⣿⣿⣿⠟⠛⢋⣿⡿⠀⢿⣿⡛⠛⠻⣿⣿⣿⡿⠟⠛⢋⣿⡿⠀⢿⣿⡛⠛⠻⣿⣿⣿⡿⠟⠛⢛⣿⡿
⠀⠀⠀⠈⢿⣷⣄⠀⣸⣿⣿⣇⠀⣠⣾⡿⠁⠀⠈⢿⣷⣄⠀⣸⣿⣿⣇⠀⣠⣾⡿⠁⠀⠈⢿⣷⣄⠀⣸⣿⣿⣇⠀⣠⣾⡿⠁
⠀⠀⠀⣀⣀⣙⣿⡿⠿⢿⣿⠿⢿⣿⣋⣀⣀⠀⣀⣀⣙⣿⡿⠿⣿⣿⠿⢿⣿⣋⣀⣀⠀⣀⣀⣙⣿⡿⠿⣿⣿⠿⢿⣿⣋⣀⣀
⠀⠀⠀⣿⣿⠿⠿⢿⣷⣾⣿⣾⡿⠿⠿⣿⣿⠁⣿⣿⠿⠿⢿⣷⣾⣿⣾⡿⠿⠿⣿⣿⠈⣿⣿⠿⠿⢿⣷⣿⣿⣾⡿⠿⠿⣿⣿
⠀⠀⠀⠘⣿⣧⡀⠀⢸⣿⣿⡏⠀⢀⣼⣿⠃⠀⠘⣿⣧⡀⠀⢸⣿⣿⡇⠀⢀⣼⣿⠃⠀⠘⣿⣧⡀⠀⢹⣿⣿⡇⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣶⣿⣿⣿⣿⣶⣿⠟⠁⠀⠀⠀⠈⠻⣿⣶⣿⣿⣿⣿⣶⣿⠟⠁⠀⠀⠀⠈⠻⣿⣶⣿⣿⣿⣿⣶⣿⠟⠁
⠀⠀⠀⣿⣿⢿⣿⣷⣦⣼⣿⣴⣾⣿⡿⣿⣿⠂⣿⣿⢿⣿⣷⣦⣿⣿⣴⣾⣿⡿⣿⣿⠐⣿⣿⢿⣿⣷⣦⣿⣿⣴⣾⣿⠿⣿⣿
⠀⠀⠀⠹⣿⣆⠀⠈⢹⣿⣿⡟⠁⠀⣰⣿⠏⠀⠹⣿⣆⠀⠈⢹⣿⣿⡏⠁⠀⣰⣿⠏⠀⠹⣿⣆⠀⠈⢹⣿⣿⡏⠁⠀⣰⣿⠏
⠀⠀⠀⠀⠙⢿⣷⣦⣾⣿⣿⣷⣦⣾⡿⠋⠀⠀⠀⠙⢿⣷⣦⣾⣿⣿⣷⣤⣾⡿⠋⠀⠀⠀⠙⢿⣷⣴⣾⣿⣿⣷⣴⣾⡿⠋
⠀⠀⠀⣶⣶⣶⣿⣯⣍⣹⣿⣩⣽⣿⣶⣶⣶⠤⣶⣶⣶⣿⣯⣅⣻⣿⣩⣽⣿⣶⣶⣶⠠⣶⣶⣶⣿⣯⣅⣿⣿⣩⣽⣿⣶⣶⣶
⠀⠀⠀⢻⣿⡌⠉⠙⢻⣿⣿⡟⠋⠉⢡⣿⡟⠀⢻⣿⡌⠉⠙⢻⣿⣿⡟⠋⠉⢡⣿⡟⠀⢻⣿⡌⠉⠙⢻⣿⣿⡟⠋⠉⢡⣿⡟
⠀⠀⠀⠀⠻⣿⣦⣤⣼⣿⣿⣧⣤⣴⣿⠟⠀⠀⠀⠻⣿⣦⣤⣼⣿⣿⣧⣤⣴⣿⠟⠀⠀⠀⠻⣿⣦⣠⣼⣿⣿⣧⣠⣴⣿⠟
⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠛⠋⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀
⠀⢀⣤⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷⣤⡀
⢠⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⠶⠶⠾⠷⠶⢶⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⢠⡶⢶⣄⢠⣿⡟⠁⣴⠶⢦⡀⢻⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⠀⠀⠹⣯⣼⠏⢸⣿⣇⠀⢿⣬⡾⠃⣸⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣤⣤⣶⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⡁⠀⠀⣀⡀⠀⠀⠀⠈⣹⣿⢿⣿⣟⢸⡟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇⠀⠀⠈⠻⠶⣶⡶⠟⠉⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣠⣟⠀⠀⠀⣠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡏⠉⣹⣦⠀⢴⣿⣿⡆⢹⣷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣷⣾⠁⠘⣷⡀⠉⢹⣇⠸⣿⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⢉⣹⡟⠁⠘⢷⣄⠈⣷⣤⣤⣿⡶⠟⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⠀⠀⣠⡟⠉⢷⣄⠀⠀⣹⣿⣯⣤⣤⣤⣄⠀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⣿⣠⡾⠛⢷⡀⠀⠙⣷⡾⠋⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣴⠟⣷⣴⠟⠹⣧⡀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⣀⣨⣿⣄
⠀⠀⠀⠀⢸⣿⠾⢿⡄⠀⠀⠈⢳⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣟⠉⠁⠀⠹⣆
⠀⠀⠀⢀⣿⠃⠀⠈⢻⣄⢀⣤⠾⣯⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⣿⢻⣆⠀⠀⠀⠹⣦
⠀⠀⠀⠈⣿⡀⠀⣀⣠⣿⠟⠁⠀⠈⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠀⣿⡶⢶⣦⣶⣿⡆
⠀⠀⠀⠀⠈⠛⠛⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣤⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⠾⠁⠀⠘⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡿⠀⠀⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⡿⣧⡀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢠⣞⠋⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣹⡇⠀⣿⡀⠈⠻⣶⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⣿⢿⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣶⣆⠀⠀⣤⣤⣤⣤⣤⣤⡴⠄⠀⠀⣿⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣤⣬⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⢶⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠁⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠲⠞⠓⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠶⠾⠓⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣿⣍⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠞⠛⠛⠛⠻⣦⣄⠀⠀⠀⠀⠀⢀⡿⠋⠉⠁⠈⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣿⣦⣶⣶⣶⢶⡶⣶⣶⢶⣶⣾⣇⠀⠀⢸⣟⣿⣿⣿⡟⣿⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠂⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠠⠑⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣲⠇⠀⠀⠀⠀⠀⠀⠀⠐⣄⢂⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡣⡀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠘⠛⠀⠀⠀⠰⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠐⠒⠠⡀⠀⠀⠀⠀⠀⢐⠢⡀⠐⠈⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠇⠀⠄⠰⠷⠀⡈⠀⠀⢘⠒⢦⡴⠒⠹⠀⠠⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⠲⠠⠄⠐⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠒⠄
⠀⠀⠀⠄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡔⡷⢬⡴
⠀⠀⠐⢄⣂⡾⠶⣦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠁⠂⠀⢀⠀⠠⠐⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡵⠋⠀⣾⠇
⠀⠀⠀⠀⠙⣜⢆⠀⠉⠘⠛⠿⣖⠤⡀⠀⠀⠀⠈⠈⠉⠐⠂⢲⣶⣶⣶⡶⠒⠐⠒⠒⠈⠀⠀⠀⡠⢒⠴⠋⠀⢀⠊⡎
⠀⠀⠀⠀⠀⠈⠢⡑⣄⠀⠀⠀⠀⠉⠛⢿⠞⣁⡀⠀⣀⣉⡁⠁⠉⠁⠉⠉⠉⠉⠉⠉⠀⢢⠤⡾⠂⠁⠀⠀⠠⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡠⡀⠀⠀⠀⠀⡏⠀⢑⠃⠀⡗⠒⠂⠀⠀⠀⠀⠈⠀⠠⣤⠤⢤⠄⠙⠇⠀⠀⠀⡀⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡢⡀⠀⠀⠃⠀⠣⠃⠀⡇⠀⠀⠀⢀⠄⢔⠀⠀⠐⠲⠬⠴⠂⠀⢀⣀⠤⣬⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣢⣄⣀⠀⢨⠅⠀⣇⠀⠐⠒⠓⣢⠕⠆⡀⠈⣛⠥⢜⡃⠀⣼⠈⢦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠜⣹⠀⢐⢢⠀⠀⣈⣉⣈⣁⣀⠀⠈⠀⠀⣭⡲⢭⡅⠀⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢺⠀⠀⠁⠀⠀⠉⠋⢿⠉⡟⠀⠀⠀⠀⠶⣚⣳⠆⠀⡁⠚⠵⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡚⠿⢸⢀⠀⠀⠀⠀⠀⢠⣌⣀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠒⠀⢤⠀⣠⣄⡤⡠⠤⠤⠤⢠⣤⡤⣤⡄⠤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠁⠈⠑⠒⠒⠒⠓⠛⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠇⠧⢄⠀⠀⠀⠀⠀⠀⠀⡀⠴⢴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡄⢠⢠⢱⠀⠀⠀⠀⢀⠊⡀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠀⠌⡁⠳⡤⣤⣭⣤⣼⠤⠚⠀⠀⠀⠀⠙⠢⢤⣬⣁⣐⡀⣀⢼⠋⠄⠀⠀⠀⠠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠤⣄⠀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠀⢸⡇⠀⢸⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⣶⠛⢀⡀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⣤⡤⠤⠤⠤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⠟⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢸⣀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣄⣿⡀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⠀⠀⠀⢀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⡶⠒⢳⡟⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡞⠛⠛⠋⠉⠉⣉⠛⠛⠓⠒⠲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢯⣡⠖⣿⠀⠀⠀⠀⠀⢿⣬⠇⠀⠀⠀⠀⢹⣍⣙⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢧⣠⠞⠁⠀⡇⠀⠀⠀⠀⠀⣖⣛⡆⠀⠀⠀⠀⢸⠀⠙⢦⣀⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠦⣴⠋⠀⠀⠀⣷⠀⠀⠀⠀⠀⣨⠭⡀⠀⠀⠀⠀⣼⠀⠀⠈⢻⣄⡬⠿⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠤⢀⠀⢸⠀⠀⠀⠀⠸⣦⣄⣀⣀⣀⣙⣲⣃⣀⣀⣀⣰⠏⠀⠀⠀⢸⠃⢀⡤⣄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⢸⣇⣸⠃⠀⠀⠀⠀⢀⣏⣀⣯⠈⠉⠉⠉⠉⣽⣁⣹⡀⠀⠀⠀⢸⡀⣸⡀⠈⠳⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⣀⠁⣿⠀⠀⠀⠀⠀⣿⠈⣈⣧⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣏⡉⠉⢹⣆⠀⠀⢀⣰⡏⠉⢉⣹⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠉⠛⠉⠉⢳⣰⠋⠈⠉⠛⠋⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣛⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣙⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣼⣷⢸⣿⣿⡿⠋⠉⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢀⠈⠉⢻⣿⣿⣿⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⠁⢠⣆⣨⣷⣦⡀⠸⣿⣿⣿⣿⣿⡏⠀⣰⣀⣹⣷⣆⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⠀⠸⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣧⡀⠈⠉⠋⠁⢀⣾⣿⣿⣿⣿⣿⣿⣄⠀⠈⠙⠉⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⢸⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⣿⠀⣿⣿⠁⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⢹⣿⣿⣿⡇⠀⣀⣤⣤⣀
⠀⠀⠀⠀⢠⣾⣿⡿⢿⣿⣿⣿⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣾⣿⡿⢯⡙⣷⡀
⠀⠀⠀⠀⠸⣿⣿⣦⣾⣿⣿⣿⢸⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣧⣿⠁
⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⠛⣿⢸⣿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣿⣿⣿⡇⠘⢿⣿⣿⣿⣇
⠀⠀⠀⠀⢠⡏⣿⣿⡿⠁⠀⣿⢸⣿⣿⠀⠀⢠⣒⣶⠀⢀⣶⣦⠀⢰⣚⣷⡄⣴⣛⣷⢠⣞⣻⡆⢸⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣹⣿⣿⠃⠀⠀⣿⢸⣿⣿⡀⠀⠈⠛⠋⠀⠀⠛⠋⠀⠈⠛⠛⠁⠙⠛⠋⠀⠛⠛⠁⢸⣿⣿⣿⡇⠀⠀⠘⣿⢿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⣿⠘⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣇
⠀⠀⢀⡿⠿⣿⡟⠀⠀⠀⠀⣿⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⠟⢿⡄
⢀⣴⣿⡿⠶⣿⣷⣄⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⣠⣾⣿⠶⢿⣿⣦
⣾⣿⣿⣦⣤⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⢷⣤⣴⢿⣿
⠻⣿⡇⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿
⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣥⣤⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣠⣶⢿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣼⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣄⠰⠞⣠⢀⣀⢠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣨⣥⠟⣠⢮⣘⢂⣸⢶⣝⣲⠖⣲⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠀⡠⢮⣩⣤⢥⣼⠁⡮⡭⣘⣩⢔⡓⠪⢼⠇⣰⡏⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣄⡛⠠⠆⣭⢛⣷⠈⡦⣬⣿⣧⠞⠉⡬⣥⠀⢈⡿⠾⢌⡅⣴⣘⡄⡎⠴
⠀⠀⠀⠀⠀⠀⠠⠀⡠⠤⡞⣙⣻⣣⠔⠚⣵⣴⡗⠒⢻⣧⠀⢮⠳⣯⠶⠛⠉⣍⠩⢍⠟⡗⠳⡀⢭⠲⡂
⠀⠀⠀⠀⢠⡔⠖⢒⡔⢚⣳⡾⢥⠷⣞⣤⢀⡾⠛⣴⠀⢻⣆⢰⡾⠁⠀⠀⣘⣦⡍⣬⣸⢉⠿⠍⠀⢬⣋⠴⢀
⠀⠀⠀⠀⠰⢥⢏⣀⢚⣦⡤⢷⣄⠛⣭⡷⣼⡇⠛⣰⡟⣌⢿⣿⢁⣘⣀⠀⠤⣧⢮⠓⢸⣿⡒⠲⢈⣓⠶⣉⠭
⠀⠀⠀⢠⣓⣒⠂⠾⣞⡗⠽⢮⣽⣷⣄⠀⢹⡇⠌⠉⣬⠧⣜⣿⣆⠳⣸⢳⡖⠛⣆⠀⢸⡏⠁⠬⢠⣹⣉⢮⠹⢸⠃
⠀⠀⡀⠰⠽⣭⠆⢍⣿⡂⠀⣅⠀⠈⠙⢷⣮⣿⣍⠒⠸⠀⠟⠹⣿⡆⠠⠜⡄⠿⠻⣆⣸⡇⣌⣞⠂⡜⣏⣑⣞⡃
⠀⢀⡉⠆⡙⠽⣛⣶⣏⡡⠰⠽⠂⠆⣦⣰⡬⢿⠿⣷⣦⣄⠀⠀⣿⣿⠀⠈⠁⠀⠀⣿⡿⠑⡾⠆⡸⠐⠾⣾⣚⠀⣀
⢀⡀⠒⣛⣶⡰⠉⣼⠏⠛⠀⢛⣶⡡⠁⠃⠣⠤⠀⠀⠉⠻⣷⡄⣿⣿⠀⠀⠀⣠⣾⠟⠁⠀⣯⡟⠍⠀⢀⣟⠡⠦⠄
⢨⢁⡄⠐⠶⢻⣴⡟⠼⡡⢀⠀⡚⣷⡁⢄⣷⣍⢠⠰⣙⠄⢹⣿⣿⣏⣠⣶⣿⣿⣷⣤⣄⣀⣙⣦⣀⣠⣾⣯⠉⡄
⠀⢫⡾⣰⠠⡜⣿⡇⠈⠽⢺⣴⢉⡿⠛⢙⠘⠃⢃⢃⠈⢀⣼⣿⣿⡿⠛⡅⢆⠐⡀⡌⢝⣻⠟⠛⠋⢥⡾⢤⡼⠁
⠀⠀⠘⠋⠽⠓⢿⣷⡀⠀⠀⠈⣿⠁⠛⠨⢘⣺⢥⠂⢀⣾⣿⡟⠁⣄⠷⣭⠶⣖⠳⢹⠋⠁⠀⣀⣻⣾⣿⣋⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⡀⢿⡀⠀⠀⣵⣟⠃⠀⣼⣿⡏⠀⡐⣾⡾⠏⢂⣣⣽⣶⣶⣿⠿⠟⢻⡟⣲⢳⡏⢦⡑
⠀⠀⠀⠀⠶⢠⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⣤⣧⣀⡀⣿⣿⡇⠀⠈⣿⣽⣾⣿⣿⡟⠋⠁⢀⠠⣆⠆⡞⣤⠦
⠀⠀⠀⠐⠚⣨⠼⣂⣆⣲⣀⣯⣭⣿⠿⠿⠿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⡿⢷⣾⢶⢧⣯⣜⣏⢫⢡⠤⠐⡐⠃
⠀⠀⠀⠀⠀⠀⠘⠃⠐⢣⡐⢦⡀⠗⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⡟⠁⠀⠈⢛⡽⢮⢐⠙⠉⠏⠏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠺⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠱⠺⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⠾⠆⡈⠚⠀⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠰⣄⣠⢂⡁⠀⠀⠀⠀⠈⠀⠿⢢⠀⠀⡀⠀⠀⠀⢀⡀⠒⠀⠀⡈⢙⠒⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⣿⣻⣇⠀⠀⠀⢠⠄⡀⢰⠀⢀⠟⡹⠚⠓⠛⡟⠂⣰⡶⠒⡷⠾⠓⠖⠾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡾⠁⢿⡏⠀⠀⠀⠞⠠⠿⢻⠀⠈⠁⣄⣴⠟⠀⠛⢿⡇⢰⡀⠐⡀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠳⠀⠘⠃⢂⣴⡶⠀⡐⠚⣜⠀⣸⠀⠾⢯⣛⣶⣶⣿⡇⢸⣇⠀⢃⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠄⢀⡾⠾⠿⣧⣤⡾⠋⣹⡋⣠⠿⣄⡰⢿⡄⠀⣰⣿⠃⠀⢿⡆⢸⡄⠀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣱⣷⡾⡿⠲⢦⣈⠉⣷⣾⣟⡷⠃⠀⠉⠀⡘⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣦⢙⣷⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷⠄⣿⠒⢤⡈⢛⣾⡿⠛⠀⠀⠀⠀⠀⠱⠘⠛⢻⠃⠀⠀⠀⠀⢈⣿⣁⠀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⢹⡄⢿⠠⢤⣉⣻⣿⣧⠀⠀⠀⣈⠀⠀⠀⠐⢠⠂⠀⠀⠀⠀⠀⣼⡯⢀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⣹⠁⠘⠆⢤⣽⣿⣿⣿⡀⠀⠘⢿⠃⠀⠀⠀⠸⠀⠠⠀⠀⠀⣰⢻⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠾⣿⣇⡿⢦⣄⠀⢼⡟⠸⠿⣿⣿⡤⢴⠿⠿⠿⢿⡷⠿⣶⣶⣦⣤⣴⣿⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⢀⣿⣿⠓⠀⠸⠁⠘⠀⠀⣀⣼⣧⠈⢉⠓⠒⠂⠴⠲⠶⣿⣿⣿⣿⣿⣝⠣⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⡏⠈⢻⠁⠀⠀⠀⠀⣧⡞⢉⠜⠉⡛⠋⠀⠐⠒⠂⠚⠛⠿⡿⠿⣿⣧⠈⠳⣄⡁⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣸⠀⠈⠒⠉⠠⠀⠀⣿⠉⠙⠲⢶⠁⠀⡜⠀⠀⠀⢠⡆⠀⢸⠗⢾⣟⠃⢄⠈⠙⢾⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠁⠀⡇⠠⠀⠀⠀⠀⣹⠀⠀⠀⠀⡀⢀⣧⠀⠀⠀⠘⣷⡆⡆⠀⠀⢿⡏⠀⠑⠀⢠⣟⣴⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⢹⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⢡⣤⣿⣧⠀⠀⣾⣿⣷⠀⠀⠀⠘⣿⡀⠀⠀⣾⢪⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⣠⣶⣤⣾⠀⠀⠀⠀⢸⣿⣿⠟⠐⣴⣿⣿⣇⠄⠀⠀⠀⢿⣿⣔⣾⣿⡉⡷⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⢋⣴⣾⣿⣿⣿⡏⠀⢈⠀⠀⢸⣿⠇⠀⠀⠘⢿⣿⣏⠀⡄⠀⠀⠘⣿⣿⣿⣟⠁⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣰⡿⡡⣿⣽⣿⣻⡟⢻⡇⠀⠘⠀⠀⣹⣿⠀⠀⠀⠀⠈⢻⡿⣦⠐⠤⠀⠀⠈⢿⠛⠁⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣾⣿⠉⠙⡿⠋⠀⠘⣗⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⣤⣧⣽⣇⠂⡀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢈⡇⠀⠀⠀⡿⠻⠶⣶⣶⡿⣿⡇⠀⠀⢠⢸⣿⡟⠛⢿⡟⠛⠛⢋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠇⠀⠀⠀⣿⡶⢦⣽⣿⢘⣿⠇⢀⣀⣼⡠⠏⣠⠇⢈⢿⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢹⠁⣼⡿⣿⢻⣿⣀⠘⣿⢽⣧⣶⠇⠀⣚⠋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣤⣶⣿⣧⠈⠁⣸⣿⣿⣿⣿⣷⡦⠥⢾⣤⡶⣿⣿⣉⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣤⣴⣾⣿⣿⣿⣿⣿⣆⣴⣏⣼⣧⣿⣿⣿⡆⠀⠀⠈⠁⠘⣿⣿⣿⣿⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠿⠋⠀⠀⠘⣿⢿⣿⣷⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⣼⣿⣿⣶⣄⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠈⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣠⣶⣿⣿⣿⣿⣿⣋⠉⠁⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠻⠿⠿⠿⣿⡿⠿⠟⠛⠉⠉⠁
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⡀⠂⠀⠐⠀⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠠⠀⠀⠀⠒⠒⠐⠀⠂⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⠄⠀⠀⣠⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⣓⠄⠀⠀⠈⠩⠊⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠀⠠⠀⠀⠐⠀⠀⠂⠀⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣢⢄⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠬⠤⡤⠤⠤⠤⠤⠤⠦⠤⠶⠤⠒⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⡰⣅⡠⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡠⢊⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡠⠊⡠⠊⠀⢀⠀⡀⠀⠀⠀⠀⡠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⢀⠔⠈⠑⣎⠀⠀⠠⠀⠐⠈⠒⠒⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠴⠧⡀⠀⠀⠀⠐⠀⠈⣄⠀⠀⡊⠀⠑⠄⠀⠀⡎⠀⠀⡃
⠘⣄⠠⢂⡴⠀⠒⠚⠣⠄⠔⠋⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠈⠀⠀⠀⠀⡄⣠⠗⠒⠒⠳⠤⠞⠁⠉⠉⠙⠖⠒⠁
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠓⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⡀⡀⠀⠈⠈⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣾⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢹⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⢼⡀⠀⠀⠀⠀⠀⡧⠄⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡼⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⣳⠀⠀⠀⠀⢞⠀⠐⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠊⠧⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⠃⠀⠀⠀⠀⠈⡗⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢼⡀⠀⠀⠀⠀⠀⡧⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠺⣀⣠⡇⠀⠀⠀⠀⠀⢇⢀⠜⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠒⠒⠒⠒⠒⠚⠓⠳⠀⠀⠀⠀⠋⠀⠀⠐⠒⠒⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢰⣿⣦⠀⠀⠀⠀⣴⣿⡆⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣀⣉⣁⣀⣀⣀⣀⣈⣉⣀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⣤⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⡏⠈⠉⠉⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⠉⠉⠁⢹⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠟⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠻⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠅⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠰⣶⣄⣀⢴⣶⢦⡀⢱⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣿⠒⠀⠉⠙⠥⠤⠽⠿⠿⠘⢀⡤⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡁⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⠐⢸⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡇⠸⣆⠀⠀⠀⠀⠉⣹⠃⡆⢸⠀⠀⣀⡾
⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣈⣓⠢⠤⠦⠖⠋⠀⡇⠈⠉⢹⡏
⠀⢴⠛⢎⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠉⠉⠙⠛⠛⢷⡚⠓⠒⢟⠋
⡔⢦⠳⡌⣮⠗⠋⢳⠤⢦⢠⢦⠤⠤⠤⣴⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⣄⠀⢀⣳⣀⣀⣀
⠈⠢⣱⠎⠀⠀⠀⢸⠚⠉⢸⢸⠀⠀⢰⡇⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠈⡟⡇
⠀⠀⠳⢄⠀⠀⠀⢸⠀⠀⡃⢸⠉⠛⣿⠀⠀⠀⠀⡀⠀⢠⠁⢹⠙⡄⠀⠘⡇⠙⠛⢻⠛⢺⠇⡇
⠀⠀⠀⠀⠉⠒⠲⡛⢆⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠄⠀⠘⢤⣎⡰⠁⠀⢀⡇⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡱⣆⡛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢐⠠⠀⢠⠀⢸⠇⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⠀⠴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠗⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠐⠚⠒⠊⠀⢸⠀⠀⠀⢸⠀⢸⢠⡇⠀⠀⠀⢀⣸⡤⠗⢻⣗⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠠⢸⠀⠀⠀⢸⠀⢯⣹⣏⣉⣉⣹⣏⠀⠀⠀⠈⣗⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣸⣇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠬⢽⠓⠒⠲⠶⠤⠤⢤⣄⣀⣀⣀⣀⣿⣀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢇⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡩⠛⠻⠳⠛⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠉⠉⠙⢄⠀⠀⠀⠀⠀⠀⠀⣰⠁⢧⣀⣀⣀⣠⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⢿⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠴⠥⠔⠒⠉⠀⠀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡽
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣂⠉⠳⠞⠈⠉⣲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠒⠲⠖⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⠀⡧⣀
⠀⠀⠀⠀⠀⠠⢞⡉⠙⠒⠤⣀⢸⣄⡠⠖⠉⢈⡵⠂⠀⠀⠀⠀⠀⠀⡟⠓⠤⣄⣈⠒⢃⣀⠨⠕⢦
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⠉⣇⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠈⢹⠉⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣈⠉⠒⠦⢄⣸⣀⡤⠔⠊⢉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⣈⣇⣠⡤⠞⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡻⢾⣿⣿⣿⣿⡿⡏⣿⣿⡟⢹⡇⡖⣷⣾⢉⡇⢸⡏⣿⢹⣿⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡙⠛⠛⠛⠃⠈⠉⠻⠿⠻⠿⠳⠜⠻⠛⢸⡇⡇⣿⣿⠨⡇⢸⡇⣿⢸⣿⣿⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣻⡟⣿⡿⣿⣿⡟⢻⣿⣸⣿⡆⣤⣤⣤⢸⠀⢠⣀⣀⣄⣠⣧⣀⣀⣸⡇⡇⣿⣿⠀⠀⠀⡇⣿⢸⣿⣿⣿⣿⣿⢻⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣨⡛⢬⣶⣘⠒⣿⣿⡇⠀⠀⠈⣿⡇⠂⠀⠀⣸⠛⠛⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⣿⠀⠀⠀⡇⣿⠚⠛⠛⠛⠛⢿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⢿⣟⣙⣩⣕⣮⢻⠁⠀⠈⠅⠀⠀⢰⣿⠃⡀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⢸⡟⠻⢿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣏⠙⢯⠀⠀⠀⠀⢰⣷⣿⣷⠀⠀⠅⠀⠀⠀⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⡀⡀⢐⡇⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣾⣝⣭⣭⠼⢦⣤⠀⠀⢘⣿⣿⣿⠀⠐⠃⠀⢀⣤⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣄⠀⢸⡇⢸⠀⠀⠀⠀⠀⢸⡇⠐⠞⣿⠛⣿⣿
⣿⣿⣯⣿⣙⣫⣞⢳⡇⢹⡀⠀⠈⠛⠛⠛⠀⠀⡆⣴⢻⡇⡇⢀⣤⣦⡀⠀⠀⠀⠀⠀⢠⣴⣤⠀⣿⢹⢳⠸⣇⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠐⣿⣿
⣿⣿⡿⢋⠿⢃⣔⢺⣇⣠⣾⣿⣿⣿⣿⣿⣿⠠⠃⣇⢸⡇⣇⠈⠻⠿⠃⠀⠀⠀⠀⠀⠘⠿⠟⠀⡿⢸⢸⢀⣯⣽⣤⣤⣤⣤⣤⣼⡇⠀⠀⣗⢀⣿⣿
⣿⣿⡿⣾⣟⣩⡁⢥⢽⣿⠁⠀⣿⢉⡉⠉⠉⣸⠀⠈⠘⠧⣿⠓⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠶⠚⣿⠞⠈⠀⡧⢹⡏⢩⣭⣭⣿⣿⣿⣿⣷⣿⣤⣯⣿
⣿⣿⣿⣶⢍⣍⡁⡆⢸⣿⠀⠀⠿⠹⠀⠀⢸⠐⢶⣶⣶⣆⣿⠒⠒⢒⣒⣒⣒⣒⣒⣒⣒⠒⠒⢚⣿⠀⠀⡄⠇⢹⣿⣏⠛⠋⠛⢻⣿⣽⢿⣿⣿⣿⣿
⣿⣛⣿⣗⡊⠀⢰⡧⢸⣿⠀⡆⠀⠀⠀⠀⠸⣀⣄⣈⠉⠉⢹⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡄⠀⢸⣿⣿⠀⠀⢀⣘⣿⠿⠿⢿⣛⣛⠛
⣿⣯⡿⣭⠁⠀⢸⡇⢸⣿⠀⠇⠀⠀⢀⣠⣤⣤⣦⣤⣤⣤⣟⠓⠀⠘⠛⠛⠛⠛⠛⠛⡛⠃⠐⠚⣿⣄⣀⣤⢤⣼⣿⣿⣀⠀⠀⠀⢸⡇⢏⣿⣿
⣿⣷⣿⣙⠿⠿⣻⡧⢼⣿⠀⣡⡴⢞⣩⠅⠀⠀⠀⣧⣤⢸⣷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⣿⢠⣤⡇⠀⠀⠀⠈⢭⣛⢶⣤⣼⢳⡿⣿⣿
⣿⣿⡿⠭⠯⢤⢽⡇⠀⣀⣼⢏⣴⠏⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠈⢷⣝⢿⣼⡟⢻⣿
⣿⣯⣿⣿⡷⢞⣨⢏⠹⣯⡟⣼⡏⠀⠀⢀⣤⣄⠀⣿⣿⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢸⣿⡇⠀⢀⣤⣄⠀⠀⠀⢿⡎⣿⠛⠛⣿⠈
⣿⣿⣿⣿⣿⢼⠁⣠⣤⣤⡇⣿⡇⠀⠀⠻⣿⡿⠀⣿⣿⣼⣿⣿⣿⣿⣷⣤⣄⣀⣤⣶⣿⣿⣿⣿⣿⣼⣿⡇⠀⠸⣿⣿⠃⠀⠀⢸⣧⣿⠶⠦⠾⢠⣤
⣿⣿⣛⣾⠾⢴⠀⣹⡇⠀⣇⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⣿⣿⠀⠀⠋⠉⣿
⣿⣶⣿⡃⣁⣺⠀⡿⠃⠀⣿⣿⡇⠀⠀⠀⢻⣇⠀⠀⠀⢸⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⠀⠀⠀⠀⢀⣿⣷⠀⠐⠀⢸⣿⣿⠂⠂⢦⢰⢻
⣿⣿⣿⣭⣾⣿⠀⡇⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡆⠀⠀⢸⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⠀⠀⠀⢠⡿⢹⣿⠀⠀⠀⢸⣿⡟⠀⠀⢻⢿⣶
⣿⣿⢿⡥⠴⢿⡦⣇⢀⠀⢸⣿⡇⠀⠀⠀⢸⡟⣷⠀⠀⢸⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⠀⠀⠀⣾⠁⢸⣿⠀⠀⠀⢸⣿⣇⢀⡀⣸⣿⣿
⣿⣿⡯⢶⣼⠞⡁⣿⠀⠀⠈⢹⡇⠀⠀⠀⢸⡿⢸⡆⠀⢸⣿⣿⣿⣿⣷⣝⣛⣟⣛⣵⣿⣿⣿⣿⣿⠀⠀⣸⡷⢶⢸⣿⠀⠀⠀⢸⡏⠁⠀⠀⣿⡟⣿
⣿⣿⣟⣛⡷⣟⠃⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⢸⢿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⢻⣾⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠛⠃⣨⢋⣿⣀⢀⣀⣼⡇⠀⠀⠀⢸⣇⠀⣿⣷⠈⠙⠛⠛⠛⠛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠉⠀⣼⡧⠚⣾⢻⣿⠀⠀⠀⢸⣇⣀⣀⢀⣿⣿⣿
⣿⣿⣶⣷⢣⣯⠀⣹⣿⡿⠛⣿⡇⠀⠀⠀⢸⡟⠀⣨⣹⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡿⠀⣀⡟⢸⣿⠀⠀⠀⢸⣿⢹⣿⡿⢿⣿⣿
⣿⣟⣿⣟⣿⣽⠟⢻⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⣧⡀⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣼⡇⢐⣿⡇⢸⣿⠀⠀⠀⢸⣷⢈⣙⢩⣾⣿⣿
⣿⣟⣿⣛⣟⢋⣶⣿⡇⠀⠀⢸⡷⠀⠀⠀⢾⡇⠸⣯⡄⢸⡆⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⢠⡟⢰⣾⣿⡇⢸⣿⠀⠀⠀⢼⣿⣀⡸⢀⣿⣿⣿
⣿⣿⣿⣿⡿⣻⣷⣾⡇⠀⠀⢸⣷⠀⠀⠀⢸⣷⠀⠙⡀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣿⣻⣿⡇⢸⣿⠀⠀⠀⣾⣿⠀⢀⢐⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⢄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠋⢰⡟⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠈⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠳⠦⠤⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⢶⣀⡇⠀⠀⣼⠀⠀⣀⣀⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢿⡀⠀⠀⣸⢿⡇⠀⠀⡿⢠⡞⠁⠀⠈⢷⠀⢿⠀⠀⠀⠀⠀⠀⠘⢷⠦⣤⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣈⠛⠲⠚⠉⢸⠀⠀⠀⡇⠈⢧⣄⣀⣠⠟⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠶⠦⢬⣭⣥⣤⣄⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⡶⠴⠤⠤⠄⠀⣠⣤⣀⣀⣀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⡀⢀⣀⣀⣀⣧⣠⣠⣤⣄⣤⣤⣤⣤⣄⣀⣀⣉⣉⣉⣹⣇⣈⣁⣀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢸⣧⡁⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢸⠤⢴⠇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡏⠀⡆⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⣸⡇⢸⠀⡾⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠓⠒⡂⠈⣿⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣿⠃⢸⣺⡃⠀⣀⡴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠁⠀⢹⣷⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⡏⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣼⠟⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⢰⠇⠀⠀⠀⠃⠀⠀⠀⠙⢿⣦⡀⠈⠳⣄⣀⠀⠀⠀⣀⣤⠞⢁⣠⡿⠋⠀⠀⠀⢸
⠀⢀⣤⣾⣁⡀⠀⠀⡿⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣉⣛⣛⣋⣉⣤⡶⠟⠉⠀⠀⠀⠀⠀⢸
⡴⠋⠀⠀⠈⠉⠙⠛⢳⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⡇⠀⣿⠙⠒⣶⠄⢰⠏⠀⠀⠀⠓⠒⡔⠐⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢲⡶⠢⠞
⠀⢻⣀⣻⡄⣰⠋⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠉⠉⠉⠛⠲⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣦⣀⣀⣿⢦⣄⣠⣤⣿⣀⣀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠤⣤⣤⣬⡉⠉⡏⠀⠀⠀⠀⡏⠉⣽⠧⠶⠶⢦⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠾⠃⠀⠀⠀⠈⠷⢶⡇⠀⠀⠀⠀⣷⠞⠃⠀⠀⠀⠈⠓⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠛⠋⠉⠙⠛⠶⣄⣿⠀⠀⠀⢸⣠⠶⠛⠋⠉⠉⠛⠲⣤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢶⣦⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⢆⢡⣤⡌⠁⠐⠠⠀⠀⠀⠀⠀⠀⠄⡀⠤⡤⠞⠳⠔⢮⠥⣋
⠀⠀⠀⠀⠀⣡⠤⡀⡄⠀⠀⢿⣷⣀⡄⣀⠜⠀⠀⠀⠀⠀⠸⠎⠙⠅⠠⠀⠀⠀⢆⠀⠀⠀⠀⠐⠷⠀⡃⠉⠅⠆⠈⠀⠈⠽⢶⡷⢀
⠀⠀⠀⠀⠒⠙⠋⢁⠏⠀⠡⣾⣿⠝⢏⣈⠢⡀⠀⠀⠀⠠⢴⠆⠀⠂⠄⠘⠀⢁⢌⠂⡄⠀⠐⡆⠀⠚⢯⣿⣦⠀⠀⠀⣀⣶⡾⠶⡄⢱
⠀⠀⠀⠀⠀⠂⠧⣈⣀⣠⠼⠟⠁⠀⠂⠀⠈⠁⠀⠀⠀⠀⠱⠄⣀⣀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⠠⣤⣶⡿⠟⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠀⠄⠀⠀⠀⢀⠀⡀⠬⠙⠸⠲⢄⣧⠀⠀⠀⠀⠀⠔⠂⢛⡄⠀⠀⠘⣿⡅⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠄⠉⠀⠠⠀⢀⠰
⠀⠀⠀⢀⠀⢂⣴⠞⠉⠀⠀⠀⠀⠀⢀⣾⣤⠀⠀⠀⠀⣠⡀⣸⡵⠀⠀⢸⣇⡂⢧⠀⣤⠀⠀⠀⠀⢱⠑⣞⠀⠀⠀⣔⣠⡴⡺⡇
⠀⠀⠀⠈⠀⣿⣴⣴⣤⣤⣤⣤⡄⢠⣜⠿⡾⢤⠀⠀⠀⠀⠹⡄⢅⠐⠴⠔⣿⡇⠟⠀⠉⠀⠀⠀⠀⠬⠊⠙⡂⠀⢀⠈⢝⢷⣵⠇
⠀⠀⠀⠘⢀⠊⠈⠙⠿⣿⣿⣿⣷⣤⣾⣤⠐⠀⠀⠀⠀⠀⠐⠇⠀⡓⠄⢤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠀⠀⠰⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠀⠍⠍⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡞⡲⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣎⡉⠀⠄
⠀⠀⠀⠀⠀⢀⡠⠔⠛⠋⠡⡓⠯⣽⡀⠀⢀⠀⠀⠀⠀⢀⣶⡿⠿⠏⠏⠭⢟⣟⠁⠀⢀⣤⠀⠀⠀⠀⠀⠜⠉⠚⢿⣷⣄⠅⣠⠄
⠀⠀⠀⣰⣆⢸⣿⠀⠀⠀⢂⡠⠉⠌⠱⣲⡭⡇⠀⠀⢠⡶⣦⣤⣤⣶⣶⣶⣤⣽⣿⣶⣿⣯⠀⠀⠀⣠⠀⠀⠀⣀⠀⠈⢻⣦⣻
⠀⠀⠀⠘⢿⠃⢀⣴⠂⡀⠁⠒⡀⢄⠡⠉⠛⠁⠀⠀⠘⢷⠿⢟⣩⣭⢯⢽⣻⡿⠋⠀⠙⠺⠀⠀⠀⠘⡇⠀⠀⠟⠀⠀⡞⠻⡿
⠀⠀⠀⠀⠀⠁⠉⣉⠈⠀⢐⢀⢠⣼⠿⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⢾⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⣀⡜⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠖⣤⠀⡶⠀⢰⡂⣴⡄⡆⠀⢠⡇⠄⢰⢠⡤⠀⣶⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠚⠒⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠑⢄⡠⠂⠁⠀⠀⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⠀⠀⣀⡀⠀⠈⡅⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠘⡀⠀⠛⠃⠀⢀⠁⠹⠟⠀⠀⠠⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢘⠢⢄⠈⠂⠤⠄⠒⠁⠐⠠⠀⠠⠔⠁⢀⢰⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢈⠀⠨⠁⠢⠤⣀⡀⠀⠀⠀⣀⠠⠄⡖⠁⢀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⠀⢘⠀⠀⠀⠈⠁⠀⠈⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠢⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡶⠶⢤⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠑⠢⠄⣠⣀⡀⢀⣀⣠⠠⠄⠛⠉⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠘⣇⠀⢧⠀⣠⠃⠀⡸⠓⢤⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⣦⠚⠉⠉⠲⡞⠁⣠⠜⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣲⣤⣤⣤⣤⣤⣤⣖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣍⣀⠤⠜⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠂⢹⠋⠀⠋⠉⠁⠀⠉⠧⠈⢳⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄⣀⣀⡤⠋⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⡻⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠉⠳⣤⣤⣀⠀⠀⠀⠀⠀⣠⠞⠑⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢾⠃⠀⢰⠃⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠚⠊⠁⠀⠀⠀⠀⠀⢻⡀⠈⢣⡄⠤⠖⡋⠁⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠇⠀⣿⠀⠀⡟⠀⢀⣾⣿⣷⣶⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡇⠀⣀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⣀⡼⠋⠈⠧⣸⠃⠀⠘⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢇⣠⠋⠀⠀⠀⠀⡿⠀⠀⠀⠈⠛⢿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋
⠀⠀⠀⠀⠀⣠⠖⠛⠶⣏⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢀⣀⣼⠁⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣠⠞⠉⣠⠿⣄⠀⠀⠀⡰⠳⡀⠀⠀⠙⠦⣄⡐⠉⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⣸
⠘⢦⡞⠁⢰⠏⠉⡟⠉⢣⡀⢙⡦⠀⠀⠀⠘⠈⠉⠒⠢⠤⠤⣀⣀⣉⣀⣀⣀⣀⣀⣀⡠⠤⠤⠲⠊⠉⠁
⠀⠈⠙⠲⣿⣀⣀⡇⠀⠀⠙⠉⠀⠀⠀⠀⠀⡷⠋⠉⠓⢦⡀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠲⣄⠀⡘
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⠁⠢⡀⢹⡄⠀⠀⠀⣰⠃⣠⠖⠀⢦⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢠⡿⠾⠧⠀⠀⠀⡸⢦⣧⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠒⢾⡀⠀⠀⠀⠀⠀⠀⠀⢨⠶⠒⠋⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠉⢵⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠙⠛⠋⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠙⢛⠒⠲⠦⢬⣧⠀⠀⠀⠀⠀⠀⢀⣞⡤⠴⠖⢶⠛⠲⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠤⠀⠀⣀⠀⠀⠈⠳⣄⠀⠀⠀⡴⠛⠁⠀⢀⡀⠀⣁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡤⠤⢤⡤⠤⢤⡤⠤⢤⡤⠤⡤⢼⡆⠀⢸⣅⣄⣠⣠⣄⡠⣤⣤⣤⣤⣤⢤⣤⢤⣵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⣠⡴⠶⢾⡷⠶⣦⣄⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡯⠀⠀⠸⠃⠀⠹⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣶⣤⡀⠀⠀⠀⠈⣩⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠛⠛⠛⡏⢻⠛⠛⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⢿⣴⣊⠀⠐⠊⠈⣷⣮⣿⣿⣷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠿⢯⣍⠙⣿⣦⣉⠳⣶⣬⣭⣭⣭⣭⣵⣾⠟⢋⣵⣿⠋⣹⡿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⢀⣿⣧⣿⣿⡟⠓⠚⠛⠛⠛⠛⠛⠛⠓⠚⢻⣿⣿⣸⡏⠀⠀⢈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣤⣤⣾⢧⣿⣿⣿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠘⣿⣿⣿⣦⣤⣤⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢻⣯⣭⣷⠛⠿⣿⣿⠖⢲⠒⠒⠒⠒⠒⠒⡖⠲⣿⡆⢹⣿⠿⠛⢿⣯⣿⡟⢻⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⡭⠛⠻⢿⡁⠀⠀⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣇⣾⣿⠀⠀⣈⡿⠛⣛⢯⣹⣿⠄
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠈⢳⠶⠿⣿⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⣵⣿⠿⠶⡞⠁⢀⣠⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠉⠻⣾⣿⣶⣶⣾⣷⣄⠀⣸⣯⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⡇⠀⣠⣿⣶⣿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠿⣷⣿⣿⡃⠀⠀⠀⢰⡆⠀⠀⠀⢹⣿⣿⣿⠿⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠿⠿⠶⠶⠿⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣤⣤⣤⡴⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢿⣿⣿⣷⣿⡿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⣿⣿⣿⣄⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣉⣉⣉⣉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⠉⣉⣹⣿⣿⣿⣿⣏⣉⠉⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⠀⠙⠛⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⠛⠋⠀⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⣴⣶⣿⣿⣿⣶⡆⢠⣶⣿⣿⣿⣶⣦⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡁⢘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⡄⢠⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠉⠁⠀⠁⠀⠀⠉⠉⠁⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠁⠞⠁⠀⡐⠂⢂⡐⠀⠄⠀⠀⠀⠀⠀⠀⠰⠂⢂⣀⠐⠄⠀⠈⠣⡈⢲⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠁⠎⠀⠀⠀⡇⣎⠙⠴⡇⡇⠀⠀⠀⠀⠀⠀⠐⠈⠉⠦⢷⠀⠀⠀⠀⠱⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣸⠀⠀⠀⠀⠣⠌⠵⠊⠠⠁⠀⠀⠀⠀⠀⠀⠰⠌⠱⠎⠡⠆⠀⠀⠀⠀⣇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠉⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡔⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠺⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⠗⠠⢄
⠀⠀⠀⠀⢀⡰⢄⠀⠀⠙⠢⠘⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡵⠀⠀⠀⠀⠀⠸⠃⠖⠋⠀⠀⡠⢆⡀
⠀⠀⢀⠔⠁⠀⠀⠑⠂⡄⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠸⠀⠐⢠⠐⠊⠀⠀⠈⠢⡀
⠀⠠⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈
⢆⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⡀
⢨⠓⠠⢀⣀⠀⡠⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣬⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠀⣀⡀⠄⠚
⢨⠒⠐⠒⠣⡀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠜⠒⠐⠒
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠂
⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢆⠀⠀⣠⠰⡎⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⡏⢙⠁⠀⠀⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢱⠆⣄⠀⠀⡰
⠀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠈⠁
⠀⠐⢀⠀⠈⢄⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡠⠁⠀⡀⠁
⠀⠀⠀⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⣀⡴⠂⠀⠀⠐⣄⡀⠀⠀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠁⡁⠀⠀⠀⠰⡌⠀⠀⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⠍⠈⠉⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⡈⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠠⠤⠀⠀⠠⢀⡀⠸⠀⠀⠀⢀⡠⠄⠀⠀⠤⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠑⣮⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⣀⣽⣤⣭⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣄⣽⣧⣈⠃⣿⣗⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣿⣿⣿⣿⣛⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣶⣾⣿⣿⣿⣿⣿⣷⡶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢸⣿⣷⣿⠀⣨⣟⠛⠛⠛⠋⡀⢸⣼⡟⢹⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⠛⣿⣄⣧⣿⡀⢰⣿⡆⠀⣼⢻⣇⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⢿⣿⣿⣿⣃⣀⣉⣀⣠⡟⠀⢻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠆⠀⠸⣿⣯⣴⣶⡶⣾⣭⣽⠇⠀⢺⣿⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⣀⠛⣛⡷⣷⡛⢛⡀⠀⠀⠀⠹⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣿⡿⠿⠷⠾⠿⣿⣽⡄⠀⠀⢀⣜⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠛⠆⠀⣿⡿⣿⠀⠀⠀⢸⣏⠉⠁⠀⠠⠟⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠁⠀⢿⣿⣿⡀⠀⠀⢸⣿⣷⠏⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢿⡿⠟⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡏⣸⠖⠀⠀⠻⣿⣩⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⠋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡴⠋⢳⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠻⣤⠞⠁⠘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠼⠦⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⠁⠀⠀⠀⠀⠀⠀⠀⠿⠜⠸⠇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⡿⢹⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣇⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⢸⣤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⢿⡄⠀⠀⠈⠉⠛⠛⠛⠿⠛⠛⠛⠉⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣄⣀⣀⣀⣀⣀⣤⡤⣤⠤⠤⠶⠶⠒⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒⠒⠒⠛⠒⠚⠲⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣯⠵⣾⠀⠀⠀⠀⠺⡭⠆⠀⠀⠀⢹⢍⣳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⡴⠁⠀⣿⠀⠀⠀⠀⢺⣽⠆⠀⠀⠀⢸⠀⠑⣤⡼⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⢿⠀⠀⠀⢹⡀⠀⠀⠀⢰⣒⡆⠀⠀⠀⡾⠀⠀⠘⡷⠚⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠏⢀⡆⢸⠀⠀⠀⠀⢛⡗⢲⠒⠒⠚⠒⠲⠖⢺⠁⠀⠀⠀⠁⢰⠉⢳⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⣸⡉⢹⡆⠀⠀⠀⢸⡋⢉⡇⠀⠀⠀⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⣏⣉⣉⣧⡀⠀⣀⣼⣉⣉⣽⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠉⠉⠀⢹⣼⠁⠀⠉⠉⠀⠀⠀⠀⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⡄⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⡈⠀⠀⠀⠁⠃⠆⠀⠈⠀⠀⠀⠀⠀⠀⠘⠂⢤⠀⠀⠀⠀⠤⢀⠀⠀⠀⠇⠀⠀⡀⠀⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠄⠐⡆⠄⢠⡀⠀⠀⠀⠆⠀⣀⣀⡇⠀⢱⠀⠀⠀⠀⠀⡄⢀⡄⠀⠀⠑⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⡀⠀⠀⠀⠀⢸⠀⠠⡄⠀⠁⠀⠀⠙⢢⡤⠀⠃⠈⠀⠀⠈⢀⢸⠀⠀⢰⢀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠠⠈⠁⠐⠀⢁⣀⡀⠀⠀⠀⠀⠀⠡⠰⠀⠀⠀⠠⢀⠀⠀⠀⠀⡇⠀⢨⡘⠂⠈⡍⠒⢄⠀⢰⠀⠘⢨⠁⠙⠀⠄⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢀⠁⠀⠀⠈⢑⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⠀⠀⢘⠀⢸⢘⡀⠀⠀⠀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣿⣿⣶⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣴⣿⡿⠿⢷⣦⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠿⣿⣿⡏⠉⠻⢿⣶⣦⣤⡀⢀⣴⣶⣤⣼⣿⣷⣶⣄⣼⣿⠉⠁⠀⠀⠀⠈⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠈⠑⠂⠀⢸⣿⣿⣿⣷⣿⠋⠉⠙⠻⣿⣿⡿⠟⠛⠿⡀⠀⠀⠀⠀⠀⡿⠛⡛⠻⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠛⠉⡍⠛⢿⡀⠀⠀⠀⠋⢁⠠⠄⡇⠀⢸⠀⠀⢀⠀⠀⡆⢀⣇⠀⠀⠈⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢀⠀⠀⠀⠀⠸⠀⠐⠄⠀⠇⠀⠀⠈⢰⠄⠀⠉⠀⠀⡀⠀⢡⢸⠀⠀⢸⢠⠀⠁⠀⠀⢡⠀⡀⢸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⣇⡀⢀⣀⠀⠀⠀⠀⡥⠒⠀⠀⠀⠨⢄⠀⠀⠀⠀⡇⠀⢸⠈⠂⠀⡏⠘⢄⠀⠈⠀⢸⠈⠀⠉⠙⠛⠿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢀⠀⡇⠀⠀⣁⠀⠀⡀⡀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⣰⠀⠀⠀⣇⠀⣐⠀⢸⡀⣸⢀⣄⣀⣀⣀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣦⣤⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⡟⠛⢿⣿⣦⣤⣀⠀⠀⣤⣦⣤⣸⣿⣧⣤⡀⣲⣿⠛⠉⠁⠀⠀⠙⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠈⠙⠓⠇⠀⢸⣿⣿⣿⣧⣾⠟⠛⠛⢿⣿⣿⣿⡿⠿⣿⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠿⠛⡛⠿⣿⠀⠀⠀⠀⠟⠋⢁⡀⡅⠀⢱⠀⠀⢀⠀⠀⡄⠀⡆⠀⠈⠙⢻⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⡇⠀⠀⠑⢢⣀⠀⠒⠈⠀⠀⠉⢂⢸⠀⠀⢸⢀⠀⠂⠈⠉⠒⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠈⡆⠀⠀⠀⠁⠀⠀⠀⣂⠤⠀⠀⠀⢨⡀⠀⠀⠀⠀⡇⠀⢸⠈⠂⠀⡟⠸⢄⠀⢀⠀⢸⠐⠁⠚⠛⠿⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⡇⠉⠉⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⡇⠀⠠⠀⢨⠀⢸⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠁⠀⠀⠉⠀⠀⠁⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠁⠀⠉⠈⠈⠁⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠶⠆⠿⢷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣼⣶⣶⡴⠤⣴⣶⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⠿⠿⠿⢿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣴⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⠃⠛⠤⢌⣁⣬⣥⣌⡡⢶⠛⠈⣿⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣿⣿⠀⠀⣦⣯⡾⠋⠙⢳⡟⣤⠀⠀⣿⣿⣤⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣽⣻⣿⣷⡆⣿⣼⣠⣤⣤⣀⣷⣿⢰⣾⣿⣟⣯⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⠉⣿⣿⡇⣿⣷⣿⣾⣷⣿⣿⣿⢸⣿⣿⠉⡿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣶⣿⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣶⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠇⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠸⣟⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠟⠀⠀⢠⣾⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡄⠀⠀⠻⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣤⠀⠀⣿⣾⣇⠀⠉⠛⠋⠉⠀⣸⣷⣿⠀⠀⣤⣾⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠟⠁⠀⢟⢿⣟⠀⠀⠀⠀⠀⠀⣻⡿⡻⠀⠈⠻⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⣭⣿⠷⠂⠀⠀⠐⠾⣿⣭⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠐⠒⠢⡄⢢⡄⠄⠀⠀⠎⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠘⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠀⠀⠚⠢⡄⠁⠀⢠⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠘⢀⡀⠓⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠠⢐⠠⠀⢆⠂
⠁⠀⠀⠀⠀⠀⠀⠰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠣⠁⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠞⠀⠨⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠀⠐⠀⠀⠀⠤⠀⠀⠬⠀⠀⠀⠀⠤⠀⠀⠁⠄⠀⠀⠄⢈⠀⢰⡀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠃⠈⠁⠂⠈⠃⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⣀
⢈⡁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠂⠀⠀⠀⢐⣉
⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢐⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀
⠒⠂⠀⠈⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠠⠒
⠀⣀⠀⠀⠀⠀⠀⠀⠈⠘⠄⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠀⠀⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠁⠀⠀⠀⠀⠀⠀⢁⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢀
⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⡀⠈⠀⠉⡀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠁⠈⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⠁⠀⠀⠀⡆⠀⠠⠀⠐⠂⠐⠖
⡄⠐⠐⢀⡀⠀⠀⠀⠀⠀⠀⢠⠤⠴⠰⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠅⠀⠀⠀⠀⠕⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰
⠤⠤⢀⢀⠀⠀⠀⠀⠀⠀⠠⠁⠠⠂⡄⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠃⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠠⠈⠀⠀⠀⠀⢀⣅⠀⠀⢲⠄⡁⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣈⠀⠀⠀⢤⠤⠀⡄⠐⠃
⠁⠀⠁⠀⠀⠀⠀⡤⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠈⠀⠀⠀⠐⠀⠸⠀⠀⠀⠤⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠀⠀⠀⠀⠀⠈⠠⠤⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡐⠈
⠀⠪⠀⠀⠀⠀⠀⠀⣤⡀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⡀⠂⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡁⠀⠐
⠐⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠁⠀⠀⠘⠀⠆⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⣀⠀⠀⢀⡐⠃
⠀⠀⠀⠀⠀⡀⠺⡀⠂⠠⠋⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠊⠂⠐⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀