Monarch butterfly Emojis & Symbols

⠀⠀⠀⢀⡔⠒⣶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⡶⠒⣤
⠀⠀⠀⢸⣿⣹⣿⡓⠒⠿⠛⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠛⠗⠒⣻⣿⣹⣿
⠀⠀⠀⠀⢿⡤⣇⣿⡀⠁⢠⣿⡇⠑⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠁⢿⣷⠀⠁⢸⡇⡧⣼⡇
⠀⠀⠀⠀⢸⡇⢧⠸⡆⠀⠪⠅⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⡄⠩⠂⠀⣼⢣⠇⢿
⠀⠀⠀⠀⢸⡇⢺⠈⣷⠀⠀⠀⠈⠉⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠈⠉⠀⠀⠀⢰⡏⢸⠃⣿
⠀⠀⠀⠀⠀⢿⠘⡆⢻⡆⠀⠀⠀⠀⠀⠈⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣾⡇⡾⢸⡏
⠀⠀⠀⠀⠀⠘⣎⢣⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣄⢲⣷⣆⣼⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢹⡟⢡⢋⡾
⠀⠀⠀⠀⠀⠀⠈⠂⠑⠙⠂⠀⢀⣤⣤⣀⣀⣀⣹⣿⣿⣿⣯⠈⣿⣿⣿⣿⣁⣀⣀⣠⣤⣤⠀⠀⠸⠁⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢸⡗⠀⠉⠉⠉⠉⠉⠉⣿⣿⣷⣶⣿⣿⡏⠉⠉⠉⠉⠉⠁⠀⣿⠑⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠹⡶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⢛⠳⠵⢛⡻⠃⠀⠀⠀⠀⠀⠀⠀⣿⠐⣾⠡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⣟⢹⡅⠀⠀⠀⠀⣀⠀⠀⠀⢨⡙⠚⣋⠇⠀⠀⢀⡀⠀⠀⠀⠀⢸⠙⣧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣧⢸⣇⠀⠀⠀⠀⠉⠀⠀⠀⠸⡆⠔⡺⠀⠀⠀⠀⠁⠀⠀⠀⢀⣼⢀⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠻⣆⠹⣦⣤⠀⠘⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠙⠀⢀⣤⡾⠃⡾⠃⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠛⣶⣄⠸⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢁⣴⡞⠓⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠗⠛⠻⣄⠙⠛⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⠛⢁⡼⠛⠓⢿⣆
⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠙⢧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⠀⠀⠀⠀⣼⣿⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠫⠀⠹⡆⠀⠀⠀⠋⢹⠀⠀⠉⢻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⢰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠴⢣⣄⠀⠀⡄⢸⠀⠀⠀⠀⢛⣭⠼⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠠⠟⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣧⡖⠛⠁⠀⠀⢸⡇⠀⠘⣼⡆⠀⠀⠀⠋⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⡆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⡇⢙⠶⠁⢀⣠⡾⠁⠀⠀⠈⢿⣤⣀⡀⠀⠀⠀⠀⢈⣧⠀⠀⠀⣼⣿⣦⣠⢾⣆⠀⠈⠁⠀⢻⣄⡀⢀⡼⠀⠁
⠀⠀⠀⠀⣰⡿⠓⠀⠀⡾⠁⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠐⠠⣦⡼⠃⠀⠀⢠⣿⡋⣼⡿⠋⠿⣆⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠿⠷⠶⠶⠖⢳⣄⣀⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢹⠏⠱⠀⠀⢹⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠸⠋⠛⠳⣦⡀⠀⠀⠀⠀⠀⣇⢸⠀⠀⢀⡴⠋⠘⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⠀⠀⠈⣿⣤⠞⠉⠁⠀⠤⠀⣾⠇⠀⢀⣶⠖⠺⡟⠓⠄
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡠⠒⠈⠻⣿⠀⠀⢰⡏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠻⡶⠀⠀⠈⠉⠃⠀⠉⠀⠠⡼⠏⠀⠀⣼⠃
⠀⠀⠸⣿⠇⠀⠈⠒⢤⣀⡴⠁⠀⠀⢸⡿⠀⠀⢰⠇⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠐⣡⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠹⣷⡀⠀⠀⠀⢹⡇⠀⠀⠤⠾⣅⠀⠀⣿⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⢟⣢⡀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⢈⡷⠋⠀⠀⠀⠇⠀⠀⠀⠀⢸⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠙⣡⣾⡄⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣀⡾⠆
⠀⠀⠀⠀⠘⣇⠀⠀⠀⡀⠀⠐⣄⢀⣰⡟⠀⠀⠀⠀⢠⡦⠀⢀⣴⠆⠀⠀⠀⠀⠀⣸⡿⠁⠀⠻⣿⣿⣄⠀⠀⠀⠀⠸⣧⡀⢀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠈⠻⡦⠴⠃⠀⠀⠉⠉⠉⠱⠀⠀⠀⢀⡾⠁⠈⢟⠁⠀⡆⠀⠀⠀⠈⢿⣇⢠⠖⢰⣿⣿⣿⣿⣶⠤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⡟⠁⠀⠀⠰⡄⢀⡄⠀⠀⠀⠀⠘⠛⠀⠀⠉⣴⠀⠘⠛⠡⣼⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠀⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⢠⣦⡀⠀⢀⣤⣤⣀⣤⣴⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⠋⢧⡀⠀⠠⠅⠤⠛⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⡰⡿⣿⣦⣶⣆⠀⠀⠙⢧⣄⡒⠀⠀⠀⣠⡶⠷⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⣿⡻⣾⡀
⠀⠀⠀⠀⠀⠀⢰⠛⣻⠿⢮⢼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⣜⢻⠛⠛⠇⠀⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠄⠀⠀⠀⠈⠳⣄⠀⠀⠀⣨⣿⢄⣀⠀⠀⢀⣠⣶⢺⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣄⠀⠘⠿⣿⣳⡀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣀⣠⣾⣿⡟⠁⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢹⣿⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⣀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⢠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⠁⠈⠛⠃⢸⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⠁⠀⠀⠀⠀⠀⠐⠒⠛⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⣏⡷⠞⠻⡶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢶⠟⠳⢾⣹⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣣⣿⣁⣩⣽⡗⣸⣿⣷⢦⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡴⣾⣿⣇⢺⣯⣉⣈⣿⣜⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣼⣯⣄⣨⣿⠷⠼⡿⠷⠾⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠷⠾⢿⠧⠾⣿⣅⣠⣽⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣅⣹⡇⠀⠀⠀⣀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣀⠀⠀⠀⢸⣏⣨⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠛⠀⠈⠉⠑⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠊⠉⠁⠀⠛⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢿⣄⠀⠀⠤⠤⠚⠓⠤⠀⠀⠈⠳⣄⣤⣤⣠⠞⠁⠀⡀⠤⠚⠓⠤⠤⠀⠀⣠⡿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⢈⣿⣿⡁⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣶⠾⠿⠿⢿⣿⠶⠶⡶⠶⣾⣿⣿⣷⠶⢶⠶⠶⣿⡿⠿⠿⠷⣶⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣛⣿⣁⣀⣀⡀⣠⠊⠀⠀⠀⢀⠴⢩⢻⡿⡝⠦⡀⠀⠀⠀⠁⣄⢀⣀⣀⣈⣿⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⡟⠉⠀⠀⡠⠀⢀⣀⡤⠚⠁⢀⠀⣾⣷⠈⠀⠈⠓⢤⣀⡀⠐⣄⠀⠀⠉⢻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡤⠂⠈⡡⠚⠉⠀⠀⠀⠀⢁⡌⢸⡇⢡⡘⠀⠀⠀⠀⠉⠓⢌⠁⠐⢤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⠃⠀⢻⡀⠀⠀⠀⠀⠀⠀⠑⣦⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⢤⣶⣅⣠⡎⢀⡞⠀⠀⠀⠀⠀⠀⢳⡀⢱⣄⣨⣶⡴⣾⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣿⣿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⣾⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⢀⡤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣻⡿⠿⣟⣫⣍⣲⣄⡀
⠀⠀⠀⠀⠀⠀⡽⢾⠯⠹⢯⣉⠂⢬⣣⣙⣦⡀
⠀⠀⠀⠀⠀⣾⣷⣻⣷⠶⠦⠤⣽⠶⣏⢹⡏⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣒⢶⣶⡶⣄
⠀⠀⠀⠀⢠⣿⣤⡿⠿⠒⠒⠤⢬⣻⡏⠙⠻⢌⣿⣆⠀⠀⠠⠊⢀⠄⠂⣠⡴⣾⣷⠌⠉⣻⣎⣿⣽⠇
⠀⠀⠀⠀⢸⢷⣾⣤⣀⣀⣀⣀⣀⣈⢳⡀⠀⠀⠹⣿⡄⠀⠁⡐⠁⣠⣾⡩⠾⣷⣻⠭⣍⣉⡴⡿⠁
⠀⠀⠀⠀⢸⣯⣀⣀⣀⣀⡀⠀⠀⠈⠉⠙⢦⡀⠀⢹⣷⣈⡐⣠⠞⠋⠀⢀⠴⠋⠠⠤⣼⡿⣿⠇
⠀⠀⠀⠀⠈⠓⠿⢤⣤⣴⣾⣿⣶⣦⣄⣀⠀⠙⢦⡀⣿⣿⡟⢁⡠⠴⠚⠓⠦⢄⣀⠀⣸⢷⠇
⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠭⢭⣍⣉⣛⣻⣿⣿⣷⣿⣿⣿⣮⣁⣀⣀⣀⡀⣀⣀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⡰⢿⠾⠤⠴⠒⡠⠖⠉⠀⢀⣀⠭⣿⣿⣿⢿⠛⠛⢭⣛⡻⢿⣶⣬⣿⠟⠁
⠀⠀⠀⠀⠀⣷⣇⡤⠔⠒⣉⠷⢖⡶⠛⢉⡤⣺⣿⡿⡿⡀⠳⣄⠈⠦⢈⠙⣿⣆
⠀⠀⠀⠀⢰⣯⡏⢀⡴⠊⢠⠞⠁⣠⠔⡫⣺⣿⡿⠁⡧⠸⡀⢻⠳⠓⢄⠑⡿⡏
⠀⠀⠀⠀⠀⣿⢿⣏⣀⡴⠃⣠⠞⢡⢊⡼⠁⠟⠁⠀⣇⠀⡇⠘⡆⢣⠀⣷⡿⠃
⠀⠀⠀⠀⠀⠈⣿⣙⡿⣧⡴⠃⣰⡷⠋⠀⠀⠀⠀⠀⢻⡆⣿⢠⣇⣸⣶⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣽⣯⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣯⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⠒⠒⣿⠩⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡇⣿⡓⢓⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣉⣉⣹⣿⣿⠿⢉⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣉⠝⣿⣿⣯⣉⣉⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣿⡋⠉⠉⠙⠻⠋⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⡟⠋⠁⠀⢈⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⠟⠛⠓⠒⠒⠳⢄⠀⠀⠙⠷⣄⠀⠀⠄⠀⠀⠠⠀⠀⣠⡾⠋⠀⠀⡠⠞⠒⠚⠛⠋⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⠤⠤⠤⠤⠽⢦⣀⠀⠈⢳⣄⠀⢂⣠⠁⣠⡾⠋⠀⣀⡴⠾⠤⠤⠤⠤⠤⠤⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡙⢷⣿⣿⣼⠋⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⣈⣿⣿⢹⣿⣥⣤⣠⣤⣴⣶⣶⣶⣶⣶⣾⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛⠃⠛⠛⣩⠭⠉⠉⠉⣈⣿⣿⣼⣿⠇⠈⠉⠉⠭⢥⠛⠛⠛⠛⠣⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡿⠄⠒⠀⢀⡼⠁⠀⢀⡠⠊⡀⡏⣿⣿⠟⠀⠑⠠⡀⠀⠀⠣⣈⠀⠒⠀⠾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⡠⠄⠀⠀⠉⢀⡠⡴⠋⠀⡐⠀⠀⣿⣿⡀⠀⠡⡀⠈⠲⡤⣀⠈⢉⠀⠂⠤⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⢀⠄⠊⢩⠋⢁⡴⠀⢀⠜⠀⠌⡔⢻⣿⠱⡐⡀⠑⡀⠀⢱⡀⠉⢏⠀⠀⣄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣇⠀⣠⠃⠀⡞⠁⢀⠎⠀⢌⠌⠀⢸⣿⠀⠑⡄⠀⠘⡄⠀⠹⡄⠈⢢⣀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣷⣤⣴⡀⢀⠎⠀⣴⠊⠀⠀⠀⠁⠀⠀⠈⢧⡀⠈⣆⢀⣷⣤⣼⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣯⣿⣷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣯⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⢰⣾⣿⣏⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣶⠒⣷⣶⣦
⢿⣿⣿⣾⣿⣯⣽⣿⣿⣿⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣴⡾⠟⣿⣿⣛⣛⣿⣟⣿⣿
⠘⣿⣿⣾⣿⣿⣍⣉⣻⣿⠀⢸⣿⣤⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⣷⣶⣿⡃⠈⣿⣛⣛⣻⣿⣭⣿⡿
⠀⠙⣿⣯⣻⢿⣛⠉⢉⣿⣷⣿⡿⠿⣌⣻⣭⣃⣠⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣿⣴⣿⣥⠾⠻⣿⣿⣏⠉⠙⣿⣟⣻⡿⡿⠁
⠀⠀⠈⢳⡿⣿⠛⣿⡿⠛⠛⠛⠳⢶⣿⡿⠿⠻⠿⣿⣷⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣤⣾⣿⣯⣿⣿⣿⣿⠷⠲⠟⠻⢿⣷⣾⠿⣿⣯⠟
⠀⠀⠀⠀⢹⣿⣿⣯⣽⣤⣀⣀⣤⣄⣬⣿⢦⡀⠀⠀⠙⠿⣿⣦⡈⠢⠀⠀⠀⠀⠀⠀⠀⡀⠐⢁⣴⣿⠿⠛⠁⠀⣀⢾⣿⣁⣤⣤⣤⣤⣾⣟⣿⡿⣿⠋
⠀⠀⠀⠀⠀⢿⣿⣍⣟⠁⠀⠀⠀⠀⠀⠉⠉⠛⣷⡦⣄⠀⠈⠻⣷⣄⠀⠰⣄⠀⣀⠄⠉⢀⣴⣿⠟⠃⣀⣤⣶⡿⠛⠉⠉⠀⠀⠀⠈⠉⢻⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣾⣷⠶⠶⠶⠶⠶⠶⠶⠚⠋⠉⠛⠛⠳⣦⣈⠙⣷⡀⢸⣿⣧⠀⣰⠛⢉⣥⡶⠿⠛⠋⠛⠛⠛⠻⠷⠶⠶⣶⣶⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⢙⣿⣿⣿⢞⣿⣟⣾⢃⣘⣛⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣿⡿⠿⠿⠟⢿⣿⢃⣾⣋⣿⣿⢉⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣩⣩⠏⠉⠉⠉⠉⢛⣛⣲⡶⠾⠛⠋⠉⡹⢿⣿⣿⣿⣿⣟⠉⠙⠲⢶⣶⡶⠾⠛⠛⠉⠉⠙⠻⣯⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⣿⣁⣤⡴⠶⠛⠛⢉⣿⡏⠀⠀⠀⣠⡜⢡⡾⣿⣿⢻⣿⡈⢳⡄⠀⠀⠈⠻⣿⡶⠶⢤⣄⣀⠀⠙⣾⣗⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⡿⠛⠉⠀⢀⣠⣴⠾⡿⠉⠀⠀⢀⡼⠋⢀⡟⣿⣿⣿⣿⣿⢳⡀⠙⢦⡀⠀⠀⠙⢷⢤⣄⡈⠉⠛⠻⢿⣿⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⢾⢿⣾⣇⣀⣤⠞⠋⠁⣠⣞⣀⣠⣴⣾⠏⠀⢀⡞⢠⠇⢸⣿⣿⠘⡌⢷⡀⠀⠻⣷⣤⣄⣈⣷⣌⠛⠶⣤⡀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠀⠀⣠⠞⠋⣿⠟⠉⣼⡏⠀⢰⡿⠁⡜⠀⢘⣿⡇⠱⢻⠈⢷⡈⠀⢹⣯⡉⠻⣿⠻⢷⣄⠈⠙⢳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣤⡞⠁⢀⣾⠃⠀⣸⡟⠀⢀⡿⠀⠐⠀⠀⢸⣿⣇⠀⢹⡇⠈⢷⡐⡄⢻⣧⢀⠹⣄⠀⠙⢷⣌⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣟⣿⣤⣾⡇⠀⢀⣿⠀⠀⣾⡇⢀⡾⠀⠀⠈⣿⣇⠀⠀⢷⠀⠘⣧⠘⡀⢻⡆⠆⠹⣄⣱⣼⣿⣿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠹⣿⢿⣿⣻⣷⣿⡿⣷⢿⣟⣿⣿⠁⠀⠀⠀⠙⠃⠀⠀⠈⣆⡀⣹⣦⣥⣼⣷⢶⣾⣿⣿⣿⡯⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣿⣿⣿⣿⠯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⣯⢿⣞⣿⣿⡿⡿⠿⠟⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠙⡄⠀⠀⡀⠀⢀⠟⠁⠀⡞⡠⠀⠀⢄⢳⠀⠀⠁⣟⠀⠀⠀⠀⠀⡷⠉⠘⣇⠀⠠⣀⡀⠀⢀⡏
⠀⠀⠀⣨⠁⢸⡇⠀⣭⠀⠀⠀⣃⡇⣿⣿⢘⣘⠀⠀⠀⢈⡀⠀⡿⠀⢀⠁⠀⠀⠸⡤⣀⣽⣅⢠⡜
⠀⠀⠀⠘⡦⠴⠧⢴⠋⠀⠀⠀⠹⣷⠈⠁⣾⠏⠀⠀⠀⠘⢷⣤⠀⣤⠾⠃⠀⠀⠀⢧⣻⠈⢼⣣⠃
⠀⠀⠀⠘⠁⠀⠀⠀⠓⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⡀⠀⢤⠀⠁⠀⠀⠁⢠
⠀⠈⢏⠀⠀⢀⡀⠀⠀⣹⠁⠸⣌⣵⢦⣀⡴⢮⢡⠇⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠈⣦⡄⢀⢀⣤⡎
⠀⠀⠘⣦⠀⢸⠇⠀⣴⠃⠀⠀⢻⢄⠐⠿⠂⡠⡟⠀⠀⠻⡄⠀⢸⠀⠀⡴⠃⠀⠀⡈⠀⠸⠀⠈⡀
⠀⠀⠀⠙⠷⠆⠰⠾⠋⠀⠀⠀⠀⠻⠭⠀⠭⠟⠁⠀⠀⠀⠑⠤⠀⠠⠴⠃⠀⠀⠀⠳⠦⠀⠠⠶⠁
⠀⠀⣆⠠⡀⠀⠀⢀⠄⣰⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⡀⡀⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠈⢦⠀⢰⡆⠀⡴⠃⠀⠠⡆⠀⠈⣿⠁⠀⢰⠆⠀⠀⠸⠅⢰⡆⠀⠇⠀⠀⠀⠀⢤⡿⡧⠄
⠀⠀⠀⠹⢤⠌⢡⡤⠏⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠘⢆⠀⠁⣠⠃⠀⠀⢀⡾⣓⣺⣒⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠈⠁⠉⠉⠉⠁⠉
⠀⠸⠏⣵⠂⠀⠀⠐⣦⢹⠇⠀⡟⠳⢢⣀⣠⡴⠞⡷⠀⢺⠛⠂⠀⠀⠘⢻⠇⠐⢶⡆⠀⠀⠀⠀⣶⠒
⠀⠀⠀⠹⣄⢸⡇⢠⠏⠀⠀⠀⠹⠄⡗⠸⠐⡧⠰⠃⠀⠸⣅⡀⢰⠃⢀⡸⠀⠀⠈⢧⠀⢸⠀⣤⠏
⠀⠀⠀⢲⣌⡋⢙⣡⡖⠀⠀⠀⠀⢧⣤⠀⠀⣤⡏⠀⠀⠀⢧⡤⠈⠠⣄⡇⠀⠀⠀⠸⣤⡞⣦⡼
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⠉⠋⠘⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⣴⣿⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⡿⠋⢈⡽⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣼⣠⠞⠉⣀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣰⡏⢀⣴⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣴⣿⣿⡣⠔⣋⠤⠝⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠋⠁⣿⣿⣿⣷⣷⣶⣶⣦⣤⣀⢠⣦⣽⣿⣿⠿⠛⠉⢁⣀⣴⣒⣛⠛⣻⡷
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠗⢀⣬⣿⣿⡿⠤⣤⣤⣬⡿⠿⣿⣿⣿⣿⣿⠓⢒⣛⡛⠛⠷⢶⣮⣍⣙⣿⡷
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣤⣿⡟⢁⣠⠖⠉⣀⣤⣶⣿⣿⠟⡝⡉⠻⣆⠀⠀⠩⢍⡓⠲⢤⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⠋⢀⣴⣿⡿⢛⡿⠏⠁⣼⢁⠃⠀⠘⢆⠀⠀⠀⠈⠑⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠟⢋⣿⠗⢋⠄⣠⢞⣿⢸⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⢁⡞⣹⢸⡏⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣀⣿⣈⣷⣼⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣻⠿⡿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⢿⣯⣿⠇⡇⠀⡇⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣶⣦⡀⠀⠀⣿⡏⠉⢹⡏⠀⠙⡿⠙⣷⣠⣿⣼⣠⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡧⠛⢏⠉⠹⡟⠻⣿⣷⣄⣿⡇⠀⠈⡇⠀⢠⡇⠀⡾⣿⣡⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⢧⡀⠈⢣⠀⢿⠀⢻⠘⣿⣿⠇⠀⠀⡇⠀⢀⣀⣴⡴⡏⣹⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣀⡀⠈⠢⣄⡶⠤⡄⢸⠀⣾⣿⠀⠀⠀⠁⣠⠞⠉⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠈⠙⠿⣶⣌⢣⠀⠈⠲⣴⢻⡟⠀⠀⠀⡾⠁⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡈⠙⠲⢤⣌⠉⠻⣆⠀⠀⢻⣿⡇⠀⢀⠞⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⠉⠛⢦⡈⠳⣄⢘⣿⡇⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣬⣾⣿⣧⣤⣴⠞⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠟⠿⠿⢯⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡤⠚⣩⢻⣿⡏⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠛⠉⣀⡴⠞⠁⣠⠞⠁⢸⣿⡇⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠴⠖⠋⢁⣤⡴⠾⠁⠀⠀⡇⣿⡇⠀⠈⢢⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣤⡤⠶⠛⢋⣴⠃⢀⡴⠛⢹⣿⡇⠀⠀⠈⢳⡀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣇⠀⣠⠔⠉⡸⠋⡙⠀⡇⠈⣿⣧⠀⠀⢐⠀⠙⣦⣀⡀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⠁⠀⡴⠁⢰⠇⢠⣇⣼⣿⣿⠀⠀⢸⠀⠀⢀⠈⢿⢻⣈⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣵⣿⣤⣤⣿⣶⣿⠿⠋⢸⣿⠀⠀⣸⠀⠀⢸⠀⠘⣿⣏⢹⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠛⠁⠀⠀⠸⣿⣤⣠⣿⣀⣀⣼⣦⣼⠋⢹⠟⡏⢣⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣿⣿⣷⣸⠀⢱⢸⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣻⣯⣷⣻⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⢚⣧⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣧⡴⠚⠋⣻⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡶⠛⣯⣤⠴⠚⢫⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣾⣿⣧⣤⣴⠿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣉⣯⣴⣇⣼⠟⠋⠉⢹⣿⣶⣟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⣿⠟⠁⠈⣿⠁⢀⣠⠴⠿⠿⣧⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣰⣴⠋⠀⠀⢰⡷⠞⠉⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣰⡿⠁⠀⠀⢠⡟⠀⢀⣀⣤⠴⠒⠋⠛⢷⣯⣼
⢀⣤⡶⠲⣾⠿⠿⢿⣟⣛⣶⡖⢲⣦⣤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣠⠿⠖⠛⠉⠁⠀⠀⠀⠀⠀⣸⣏⡽
⠸⣿⣛⣏⣠⡴⠖⠛⢻⣏⣙⣿⡿⢏⣉⠶⠿⠿⣿⣄⣦⣤⣀⠀⠀⠀⠁⠢⠀⠀⠀⠀⡾⠁⠀⢀⡼⠃⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⡿⠋
⠀⠙⣿⣉⣧⣤⡤⠞⠛⢻⣿⡿⠷⠶⣿⣄⠀⠀⠀⠀⠈⠉⠙⠻⠷⣤⣄⠀⢱⣴⣦⣸⠁⢀⡴⠋⠀⠀⠀⢀⣠⣶⣿⠿⠟⠛⢿⣏⣹⡷⣄⡀
⠀⠀⠈⠻⣙⣾⣶⣤⣶⣟⠁⠀⠀⢀⣀⣭⠷⢦⣄⣀⡀⠀⠀⠀⠀⠀⠉⠙⠺⣿⣿⣷⡴⢋⣠⣤⣤⣶⣿⡿⠛⠉⣀⣤⠤⠤⠖⠛⠻⢷⣿⣻⡆
⠀⠀⠀⠀⠈⠛⣽⣦⣼⣿⣷⠖⠋⠉⠀⠀⠀⢀⣠⠽⠋⠉⠛⠓⠒⠒⠒⠒⢒⣻⣿⣿⡞⠛⠛⠉⠉⠉⠀⠉⠛⢤⣤⣤⣤⡤⢤⣤⡤⠼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⢳⣿⣾⣿⡀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⢹⣿⣿⣖⠠⢄⣀⠀⠀⠀⠀⠀⠀⠉⢓⣄⠀⠀⠀⠀⠈⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⢿⡟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⠋⠀⠀⢠⠋⢿⣿⣿⠑⠦⣈⠉⠉⠓⠲⣶⠒⠚⠉⠉⠓⢤⣀⠀⣸⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣇⣀⣀⣀⣤⣶⣿⠿⠛⠋⣉⢴⡏⠀⠀⠀⣠⠏⣸⠈⣿⣿⣷⣄⠈⠓⢦⡀⠀⠈⢻⣶⡄⠀⠀⠀⠈⣿⣿⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢽⠿⠿⣿⢿⡏⠀⣀⡴⠊⢁⡼⠀⠀⠀⡴⠃⢀⡇⠀⡟⢿⣿⠈⠳⣄⠀⠉⠳⣄⠀⠉⠙⢦⡀⢀⣼⣷⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⠚⠁⢀⠴⣿⠀⠀⣠⣾⠁⠀⢸⠃⠀⡇⠈⢻⣧⠀⠈⠳⣄⠀⠈⠑⢤⡀⣠⡿⣻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡀⣠⠜⠁⠀⣷⠖⠋⠁⣿⠀⠀⣿⠀⠀⡇⠀⠘⣿⣧⠀⠀⠈⠻⣦⣤⢼⣟⣹⣿⡿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⡿⠁⠀⠀⠀⡇⠀⠀⢸⣿⠃⠀⡇⠀⢰⡇⠀⠀⢻⣿⡇⠀⠀⠀⠈⠛⠾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣿⣦⣀⠀⣸⠁⠀⠀⢰⡇⠀⢠⡇⠀⣼⡇⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣽⣿⣿⣿⣶⣤⣤⣼⡷⣤⡾⢿⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⢿⣿⣻⣿⡿⣷⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣶⣿⣿⠋⣠⣴⣶⣤⣄⣀
⠀⠀⢿⣿⣯⣼⣿⣿⣿⣿⣿⣿⡟⠛⢷⠀⢠⣀
⠀⠀⠸⣿⣯⣽⣿⣭⣍⣥⣽⣇⣀⣀⣿⣿⣿⠿⣟⠀⣤⡀
⠀⠀⠀⠘⢟⣯⣽⢿⣥⡀⣀⣽⣿⣿⣿⣉⠉⣦⣤⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠳⡿⣿⣉⣿⣿⠉⠉⠉⠉⠉⠙⠻⣟⠉⠀⠉⠙⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⣿⡷⠶⠶⠶⠒⠒⠒⠛⠷⣤⡀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣴⣿⡅⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣶⣤⣀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠖⠚⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠓⠦⣀⠀⠙⢿⣆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣈⡀⢀⣿⣷⣴⢿⣿⡇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣟⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣧⠀⣉⣩⠭⠿⠿⢿⣁⣈⣀⣀⣀⣠⠤⠤⠒⠂⠀⠀⣿⣿⣿⣿⠋⢻⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⡋⠁⠀⠀⣠⠖⠋⠉⢁⣤⠟⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢃⠀⢸⣿⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣿⣷⣄⠴⠋⠁⠀⣠⠶⠋⠀⠀⠀⣠⠔⠀⠀⢀⣴⣿⣿⡿⠁⡎⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣿⣿⣀⠀⢠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⣠⣿⠟⣵⡟⠁⣸⠁⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣷⢿⡀⠀⢀⠔⠁⠀⠀⠀⠀⣼⡿⣣⣾⡟⠀⠜⢹⠀⢀⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣽⢿⣽⡞⢃⠀⢀⡴⠗⠀⡼⠋⢘⣽⠟⠀⠀⠀⠚⡶⠟⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠻⠋⠒⠋⠀⠀⠀⠐⢡⣿⠏⠀⠀⠀⠀⢠⣅⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠀⢀⣾⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠔⢠⣿⣦⣴⣦⡴⣷⡴⠋⠹⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣌⣻⣿⢿⡴⡗⢉⣇⡄⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⢿⠏⠁⠀⠉⠃⣽⠟⢻⣿⡿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⡿⣧⢴⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⡱⠚⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣴⢁⠠⢶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣤⣏⢰⠟⢀⡠⠞⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠿⣫⠭⠍⠴⣦⡤⠌⣿⡶⢎⠁⠈⠢⡀⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣥⢂⡤⠂⢀⡴⠋⢇⢹⣦⢷⣌⣈⠉⢄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣾⠁⢶⠀⠆⢻⠈⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡻⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣧⡽⠟⠀⠀⡶⢶⣦⣄⡀⠀⠀⠀⠀⠀⣀⡴⣾⢟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢿⡤⠙⢢⡀⠀⢀⡜⠡⢰⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡯⠉⠂⠤⣻⣾⣟⠥⠀⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⢒⡟⠓⡏⡗⠛⠯⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣄⡠⢊⡾⢇⠒⢤⠤⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡤⠞⠁⠀⠀⠉⠓⠦⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠒⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣋⣤⠶⠚⢻⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡴⠿⣟⣁⣠⡴⢿⣹⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣿⣧⣴⣿⣏⣁⣠⣾⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡏⣹⠋⣹⡿⠟⠛⠻⣿⣥⣽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣰⣿⡿⠋⠹⣿⠋⠀⠀⣀⣼⣿⣟⣻⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣐⣹⠏⠀⠀⢰⣏⣠⠶⠛⠉⠁⠈⢻⣻⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣿⠃⠀⠀⢀⡾⠋⠀⠀⠀⢀⣀⡤⢿⣟⣿⢻⡀
⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⡿⠁⠀⠀⢀⣾⣁⣤⠴⠖⠚⠉⠀⠀⠀⠙⣿⢿⡇
⣴⣿⣯⡼⠟⠋⢉⣹⣷⣤⠿⣿⣩⣿⠿⣟⣰⣦⣤⣄⠀⠀⠀⠈⠁⠒⠠⢀⠀⠀⢰⠀⠀⢀⡿⠁⠀⠀⣠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠞
⠹⣯⣼⣷⡶⠛⠉⢉⣹⣧⣤⣿⣟⠛⣶⡞⠛⠛⠛⠳⠾⢿⣶⣦⣄⡀⠀⠀⠈⢄⠘⡀⠀⣾⠃⠀⢀⡼⠃⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣅
⠀⠙⢿⣴⣷⢶⠚⠉⠉⣹⣿⠿⠛⠛⠻⢷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⣰⣿⣿⣾⠃⢀⡴⠋⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠙⣷⣼⣿⣳⣦⡀
⠀⠀⠀⠙⢽⣿⣿⡶⢿⣿⡀⠀⢀⣀⡤⠴⠞⠛⠶⠦⣄⣀⣀⣀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡖⣉⣤⣴⣶⣾⣿⢿⣋⣁⠠⠤⠴⠖⠒⠚⠉⠉⠻⣿⣏⣳
⠀⠀⠀⠀⠀⠑⢼⡷⣾⣿⣿⠟⠋⠁⠀⠀⠀⢀⣠⠶⠊⠁⠈⠉⠉⠙⠛⠛⠛⣋⣩⣿⣿⣿⡉⠉⠁⠁⠀⠀⠀⠈⠙⠲⠤⣤⡴⠶⠶⠶⠶⠶⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠱⣿⣶⡿⣄⣀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠉⢨⣿⣿⣯⣐⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⠟⠋⠀⠀⢀⠏⡿⣿⣿⡏⠓⢦⡀⠉⠉⠙⠻⣶⡚⠋⠁⠀⠙⠢⣄⡀⣠⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣧⣀⣀⣀⣠⣴⣾⡿⠿⠛⠉⣁⢴⠏⠀⠀⠀⢠⠎⢰⠁⢹⣿⣿⣦⡀⠈⠳⢤⡀⠀⠈⢿⣶⣄⠀⠀⠀⠈⣻⣿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢭⣿⠿⣿⡿⣿⠃⠀⣠⠴⠋⢁⡼⠀⠀⠀⣰⠋⠀⡾⠀⢸⠻⣿⣇⠙⢦⡀⠀⠙⠶⡄⠀⠉⠙⢦⡀⠀⣰⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⠶⠊⠁⣀⠴⡿⠀⠀⢀⣴⠁⠀⢀⡃⠀⢸⠀⠹⣿⡄⠀⠙⢦⡀⠀⠈⠳⢄⡀⢀⡿⢿⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⡏⠀⣠⠞⠁⠀⣧⠴⠚⠁⣿⠀⠀⢸⠁⠀⣸⠀⠀⢻⣿⡄⠀⠀⠙⢷⣤⣀⣼⡿⢛⡷⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⡞⠁⠀⠀⢀⡇⠀⠀⢰⣿⠇⠀⣼⠀⠀⣿⠀⠀⠘⣿⣿⡀⠀⠀⠀⠙⠻⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢻⣧⡀⠀⠀⣸⠀⠀⠀⠀⡏⠀⠀⡇⠀⢠⣿⠀⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⣿⣗⣶⣿⣄⣀⣀⣰⣧⣀⣰⣷⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢽⣿⡾⢿⣽⣿⣭⣽⢯⣼⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⢰⡆⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠷⣄⠒⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢱⡍⢢⠀⠙⣆⠀⢀⠹⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣇⠐⠴⢚⠜⠀⠸⠈⣠⠇⢧⠀⠈⠃⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣡⣈⢕⠂⠀⠀⠀⠀⠃⢐⣰⢸⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡟⢚⠖⢸⠆⡆⠀⠀⠀⠀⠈⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⢸⣙⡇⢀⡉⠁⠊⠀⠀⠀⠀⠐⢄⣈⣄⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⣉⣹⣄⠀⢸⡼⡇⠗⣨⢆⠀⠀⠀⣀⣤⣶⡿⠿⠟⠋⠉⠀⢀⡴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⣿⠿⡿⡝⣆⠀⠳⣷⡄⢀⣠⠤⠚⠉⣾⠟⠋⠀⠀⠀⠀⠀⣴⣿⢇⡀⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡷⢛⣉⣺⣷⣵⡟⣿⣆⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡼⢻⡤⠔⢻⣷⡀
⠀⠀⠀⠀⠀⠀⣸⣟⣧⠻⣿⠿⣷⠘⣇⣸⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣶⣷⢋⣷⡆⢉⢾⣳⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠏⠲⣾⣇⡘⠣⣹⣿⣧⣧⠈⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣸⣹⡻⣿⣣⠖⢻⡤⠚⡽⣷
⠀⠀⠀⠀⠀⠀⠈⠻⣴⣿⣿⠛⠀⠀⢻⣿⢹⣿⠀⠀⠀⠀⠈⠀⠀⠀⢶⠀⠀⠀⠟⡟⣒⢼⣸⡵⠂⣉⣷⠊⡀⢻
⠀⠀⢀⣤⣶⣶⠶⢶⡮⣽⣧⣀⢁⡀⠀⠙⠂⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡏⠸⢪⣿⡞⣋⣀⣼⠏⠀⠊⠉⣷⡀
⠀⢠⣯⡽⠁⠚⢷⣸⣿⠴⠿⣽⣻⣾⡳⣄⠻⣾⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⢧⠀⠊⠿⢿⣉⣀⠴⢚⢷⡆⠀⢀⣹⡆
⠀⢸⣒⣧⠙⣽⠓⠳⢦⣤⣀⣪⣽⣿⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣁⠈⠀⠀⠀⢿⣹⢤⡄⠀⠚⢻⣋⡉⢀⣹⡆
⠀⠈⠳⣞⣶⣁⣾⣳⣾⠴⠟⠛⠛⠉⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⣀⣀⠛⣷⢻⡆⠀⠐⢿⠆⠀⣤⣤⠟⠒⠶⠦
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣯⠉⠉⠛⠛⠾⣍⣉⡻⢓⡤⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠉⠙⠒⠉⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⣷⣾⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡾⢻⡟⢻⣿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣾⣿⣶⣟⣻⣦⣤⣤⣄⡀⠀⠀⠀⣴⢃⣼⣠⣟⣠⠟⣹⣿⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⡿⣿⠈⡟⢻⣿⣤⣿⣿⣿⣦⣤⣴⣇⡾⢫⠟⢉⡷⠚⢁⣿⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇⣿⡶⣧⡎⠀⣿⣷⣿⣄⣼⣿⠟⡟⢠⠃⢠⠞⠁⣠⣾⣿⣶⡟⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⢻⠃⢸⠁⢰⡿⠛⢻⣿⠿⣿⢰⣷⡇⣰⠏⣠⠞⠉⠈⢹⣿⣷⣶⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣧⢸⡄⢸⠀⣸⠁⠀⡼⠁⣸⣿⣿⠋⢹⠟⢿⠃⠀⣠⠖⠋⠉⠉⢻⣦⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⣌⣧⣸⣧⣿⠀⢰⠇⠀⣿⠉⢹⣄⡾⣶⡟⢠⠞⠁⠀⠀⠀⣠⡾⠟⢷⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣹⣟⠙⣧⣸⣇⢸⠀⢰⣿⠶⢾⡟⠀⢸⣷⠃⠀⠀⠀⣠⠞⠁⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢦⣼⠉⠙⣿⡀⢸⣿⢀⡟⠀⠀⢸⡇⠀⢀⡴⠋⠀⠀⠀⣠⠴⢿⣯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡼⢿⡆⠀⠈⢷⣼⣿⡿⠀⠀⠀⢰⣧⠞⠁⠀⠀⢀⡴⠚⢁⣤⡾⠋⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡘⡇⠀⠀⠀⢻⣿⡇⠀⠀⠀⣾⠃⠀⠀⠀⡰⣋⣴⣾⣿⣿⣷⣟⣻⣿⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⢸⣿⡇⠀⠀⢰⠇⠀⠀⢀⣼⣿⣿⠟⠋⢁⠴⠊⠉⠉⣛⣿⣿⡉⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠘⣿⡄⠀⠀⡟⠀⠀⢀⣾⣿⠟⢁⣤⠞⠁⢀⡤⠖⠛⠉⠀⠀⢙⣯⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠦⣈⠻⣿⣄⠀⣿⡇⠀⣸⠁⢀⣴⣿⠟⣡⡴⢿⣥⣤⣞⣉⣀⣤⠴⠖⠒⠛⠛⢿⣏⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠂⠒⠒⠠⢌⡑⣌⣿⣧⣹⣇⢠⣇⣴⣿⣿⡯⠊⠁⠀⠀⠀⠀⢀⣽⣋⣀⣀⠀⠀⠀⠀⢀⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⣿⣿⡿⠋⠀⠀⣀⣠⣤⣶⣾⣛⣉⠉⠉⠁⠈⠉⠙⠻⣿⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣗⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⣤⣀⠀⣀⣽⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣯⣝⡒⠶⠶⠶⢦⣤⣤⣀⣀⠀⠀⠀⠙⣿⣿⣿⣀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣭⣓⠲⠦⢤⣀⡈⠉⠛⠻⢶⣤⣀⣼⡋⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠷⣦⣤⣉⣳⣤⣀⡀⣀⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣏⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢿⣣⠀⣠⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡿⢿⣻⣼⣾⣽⣯⡿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣠⣧⣼⣯⣿⣿⠟⣻⡗⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣿⡿⢹⣟⠛⣳⡊⢹⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⣼⣇⣿⣩⢿⣿⠟⣿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢩⣟⣷⣷⡞⠁⡠⠟⠻⣵⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⡟⠉⠈⡷⠊⠀⢀⡴⠟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠀⠀⣰⠁⡠⠖⠉⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⣴⠟⠁⠀⠀⣠⣴⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⡧⢚⠏⠀⢀⣤⣾⡏⢹⠿⠿⣞⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⡈⣿⠀⢊⣠⡶⠟⠋⣹⣞⣁⠤⠔⠚⠉⠙⣿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣏⡁⠀⠀⠊⠉⠉⠓⠶⣖⡂⠀⠈⠉⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣆⠀⠀⠤⠤⣤⣤⠤⠴⠏⠉⠉⠙⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⡢⣄⠀⠀⠈⠑⠀⡀⠀⠐⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣝⠢⢄⠀⠀⠈⣷⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⢿⣿⣶⣬⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠈⠛⠿⢿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢙⣷⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⣷⡾⣆⠀⠀⢠⣾⣿⡏⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡾⢯⠙⣏⢻⡟⣆⢀⣾⢷⣼⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⣿⠛⢌⣷⢿⣦⣷⡞⢻⣿⠿⣏⢹⡟⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣬⣿⣦⠀⢻⣴⣿⣿⠿⣾⠋⢣⣹⢾⣇⣸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣶⣾⣿⠟⠙⣿⣿⣯⣀⣿⣿⣿⣦⠀⢻⠀⢻⠹⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣾⠋⠉⠉⣳⠾⣿⣤⠿⠿⣿⡏⠉⠹⣆⠘⡇⠘⡀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠛⢧⡀⣠⣿⠗⠻⢿⡀⠀⠀⠙⣆⠀⢸⣆⣿⣠⣧⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣧⡀⠀⠀⢹⡟⣧⠀⠀⠈⢳⡀⠀⠀⠘⣆⢠⡏⢹⠋⢻⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣽⣦⣄⠀⣿⡏⢳⡄⠀⠀⠳⡄⠀⠀⢸⣼⡟⢻⣤⡞⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⡷⠾⣿⣟⣻⣿⣷⣜⣿⣄⠹⣦⠀⠀⠹⣆⠀⢰⡟⠀⠈⣿⢧⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣉⣿⣿⣻⡿⠚⠻⣿⡙⠙⠻⡟⠿⣿⣷⡜⣆⠀⠀⠘⢧⣼⠁⠀⠀⢿⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣽⣿⣧⣼⠿⠷⣤⣀⠀⠙⢦⡀⠘⢦⠈⢻⣿⣾⡄⠀⠀⠈⣿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⣿⣿⣼⡇⠀⠀⠀⠉⠓⢤⡀⠹⣄⠈⢷⡀⢻⣿⣧⠀⠀⠀⣿⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣯⣨⣿⠿⠓⠲⠶⠤⣤⣀⣹⠚⠚⠲⠤⣿⣄⢻⣿⡄⠀⠀⡏⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣯⣬⣿⡟⠻⣯⡀⠀⠀⢀⣀⣀⡈⠙⢷⣄⠀⠀⠀⠙⢆⢿⣷⡀⠀⡇⠀⢠⣿⠏⠀⡠⠔⠊⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣿⠛⢿⣷⣾⣿⣿⠟⠛⠉⠉⠉⠛⠛⠓⠿⢷⣤⡀⠀⠈⢻⣿⣷⡀⡇⢀⣾⣿⣶⡭⠄⠒⠂⠉⠁⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡿⢿⣧⣼⣧⣀⣠⣤⣤⣤⣀⣀⣀⠀⠀⠈⠙⠳⣤⣈⣻⣿⣿⣇⣾⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⣿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠲⠶⣦⣼⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣶⣾⣿⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⡿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⢰⣿⠿⠟⠛⠉

⠀⠀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣀
⠀⢸⣷⢼⣿⣟⣓⣬⠭⢷⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡾⠭⣥⣚⣻⣿⡧⣾⡇
⠀⠀⠻⣿⣷⣒⣿⣍⣁⣐⢺⡷⢿⠓⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠚⡿⢾⡗⣂⣈⣩⣿⣒⣾⣻⠟
⠀⠀⠀⣽⣹⣷⣤⡤⠤⠼⢯⣱⡶⠷⢮⣙⣷⣄⠀⠀⠢⡀⠀⠀⢀⠔⠀⠀⣠⣾⣋⡵⠾⢶⣎⡽⠥⠤⢤⣤⣾⣏⣯
⠀⠀⠀⠘⣿⠻⣿⡦⠴⠒⠒⠚⠷⣄⠀⠀⠉⠻⣷⣄⠀⠘⡀⢀⠃⠀⣠⣾⠟⠉⠀⠀⣠⠾⠓⠒⠒⠦⢴⣿⠟⣿⠃
⠀⠀⠀⠀⢿⡟⣿⣀⢀⣀⣀⣠⠤⠬⠷⠦⣄⡀⠈⠻⣦⡀⢀⡀⢀⣴⠟⠁⢀⣠⠴⠾⠥⠤⣄⣀⣀⡀⣀⣿⢻⡿
⠀⠀⠀⠀⠀⢻⣻⠋⠁⠀⠀⣀⣀⡀⠀⠀⠀⠉⠓⠦⣌⣳⣾⣷⣞⣡⠴⠚⠉⠀⠀⠀⢀⣀⣀⠀⠀⠈⠙⣟⡟
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣋⣩⣥⣶⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣶⣬⣍⣙⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⠯⠤⠒⠒⡻⠋⠉⠀⣠⠞⣿⣿⣿⣿⠳⣄⠀⠉⠙⢟⠒⠒⠤⠽⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣽⣃⠤⠒⢊⡟⢀⣠⣴⠛⢁⡜⣼⣿⣿⣇⢣⡈⠛⣦⣄⡀⢻⡑⠒⠤⣘⣯⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣏⡏⢀⡴⠊⢁⠜⢉⡞⠁⣠⠎⡼⡿⣿⣿⢯⢧⠱⣄⠈⢳⡉⠣⡈⠑⢦⡀⢹⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡏⠀⣠⠋⢀⡞⠀⣰⠏⡸⣰⠇⢹⡏⠸⣆⢇⠹⣆⠀⢳⡀⠙⣄⠀⢹⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣾⣇⣀⣼⡁⢰⡏⢀⣷⠏⠀⠀⠀⠀⠹⣾⡀⢹⡆⢈⣧⣀⣸⣷⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⣙⣿⣳⣿⣴⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⣦⣿⣞⣻⣋⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉

⠀⣠⣶⣶⣦⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⣴⣶⣶⣆
⠀⣿⣻⣧⣼⠛⠛⠛⣿⣻⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⣟⣿⠛⠛⠛⣧⣼⣟⣿
⠀⢹⡿⢤⣼⡛⠉⣉⣩⡿⠆⢸⣷⣽⣶⣄⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣤⣶⣭⣾⡇⠰⢿⣍⣉⡉⢛⣧⡤⢿⡏
⠀⠀⢻⣿⣉⣿⡏⠉⠉⣿⣷⣋⣙⣿⣷⣦⣙⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣋⣴⣾⣟⣋⣙⣾⣿⠉⠉⢹⣿⣉⣿⡟
⠀⠀⠀⠻⣟⣿⣾⠖⢾⣿⠋⠉⠉⠙⢄⠀⠉⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠉⠀⡠⠋⠉⠉⠙⣿⡷⠲⣷⣿⣻⠟
⠀⠀⠀⠀⠙⣿⣿⣤⣼⣿⡀⠀⠀⠀⢈⣄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣿⣧⣤⣿⣿⠋
⠀⠀⠀⠀⠀⢻⣿⣧⣼⣿⡿⠋⠀⠈⠉⠉⠙⠢⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠔⠊⠉⠉⠁⠀⠘⢿⣿⣧⣼⣿⡟
⠀⠀⠀⠀⠀⠈⣿⣽⣥⣼⣇⠀⠀⠀⠀⢀⣀⣠⣶⣄⡀⠀⠀⠀⠈⢷⡀⠀⠀⡀⠀⢠⡞⠁⠀⠀⠀⢀⣠⣶⣄⣀⡀⠀⠀⠀⠀⣸⣧⣬⣯⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣥⣾⠟⠀⠉⠉⠉⠀⠀⠀⠀⠈⠁⠂⢄⡀⠀⠙⢶⣿⣿⡶⠋⠀⢀⡠⠐⠈⠁⠀⠀⠀⠀⠉⠉⠉⠀⠻⣷⣬⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣟⢹⣿⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠒⠄⢨⣿⣿⡅⠠⠒⠉⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢠⣿⡋⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠷⠶⠶⠶⢾⣿⣿⣿⣿⡷⠶⠶⠶⠾⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⣿⠋⠁⠀⠀⠀⠀⡠⠋⠀⠀⠈⠁⠀⢀⠜⣹⢿⣿⣟⠣⡀⠈⠈⠁⠀⠀⠙⢄⠀⠀⠀⠀⠈⠙⣿⡿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⢾⣿⠶⠖⠒⠒⠚⠘⠁⠀⠀⠀⢀⡠⠖⠁⢠⠃⣾⣷⠘⡀⠈⠲⢄⡀⠀⠀⠀⠈⠃⠒⠒⠒⠲⠶⣿⡷⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠁⠀⠀⠀⢀⡴⠁⣀⣠⡴⠞⠉⠀⠀⣠⠀⠀⣿⣿⠀⠀⣄⠀⠀⠉⠳⢦⣄⣀⠘⢦⡀⠀⠀⠀⠈⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣶⡤⠒⠈⠉⡠⠔⠉⠀⠀⠀⠀⠀⡀⠁⣰⠃⢿⣿⠘⡄⠈⢀⠀⠀⠀⠀⠀⠉⠢⢄⠉⠁⠒⢤⣶⣿⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⣀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢸⠇⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣿⣿⣿⣧⣀⠀⢀⣰⠁⠀⢠⠂⠀⣸⠇⠀⠀⠀⠀⠀⠀⠸⣇⠀⠐⡄⠀⠈⣦⡀⠀⢀⣼⣿⣿⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣛⣾⣟⣻⡿⢿⣤⣴⡏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢹⣦⣤⡿⢿⣟⣻⣷⣟⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢿⣯⣿⣳⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣞⣿⣽⡿⠶⠛⠋⠁

⣞⢛⣿⠿⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⢿⣟⢛
⣽⣟⣻⣿⠿⠿⠾⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠿⠿⢿⣿⣛⣿
⣿⣽⡟⠀⠀⠀⠀⠈⢿⣯⠙⠛⠿⣦⣄⠈⠳⢦⡀⠀⠀⠀⠀⢀⠴⠋⢀⣠⣾⠿⠛⢉⣿⡟⠀⠀⠀⠀⠈⣿⣽
⣿⣿⠿⠿⠶⠶⠦⠤⣬⣿⣧⡀⠀⠈⠛⢷⣄⠀⠑⢄⠀⠀⡴⠁⠀⣴⠟⠋⠀⠀⣠⣿⣯⡤⠤⠶⠶⠾⠿⢿⣿
⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠙⢦⡀⠈⣆⡜⠀⢠⡞⠁⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢻⣽⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣭⣿⣦⣀⠀⠈⢳⣀⣿⣇⣰⠋⠀⢀⣠⣾⣏⣥⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣽
⢸⣟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣻⣿⣿⣇⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣻
⠈⢿⣽⣷⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⡶⠶⠶⠶⠾⢿⣿⣿⡿⠶⠶⠶⠶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣿⣽⠏
⠀⠀⠙⠳⠾⢿⣿⡿⠿⠿⠿⠭⣤⣤⠴⠶⠒⠚⣩⣽⣿⣿⣯⡙⠒⠲⠶⢤⣤⣬⠽⠿⠿⠿⣿⣿⡿⠶⠛⠁
⠀⠀⠀⢀⣶⣿⡧⠤⠤⠴⠶⣶⠟⠀⠀⣀⡴⠞⢡⠾⣿⡿⢦⠙⠲⣄⡀⠀⠘⢷⡶⠦⠤⠤⢤⣿⣿⣦
⠀⠀⠀⣾⡟⠀⣀⣠⠤⠖⢾⣥⣤⢶⡞⠉⠀⣰⠋⢀⣿⣇⠈⢳⡀⠈⠙⣷⠦⣤⣽⠗⠢⢤⣄⡀⠈⣿⡇
⠀⠀⠀⢻⣻⡟⠉⢀⣠⠖⠉⠀⣠⠏⠀⢀⡾⠁⢀⡞⣿⡏⢆⠀⠹⣆⠀⠈⢳⡀⠈⠑⢦⣀⠈⠙⣿⣻⠃
⠀⠀⠀⠀⢻⣷⣴⠟⠁⠀⠀⣼⠋⠀⣠⡟⠀⢠⠎⠀⣿⡇⠈⢣⡀⠘⢧⡀⠈⠻⡄⠀⠀⠙⢷⣴⣿⠏
⠀⠀⠀⠀⠀⠻⣾⣷⣄⢀⣼⠁⠀⣴⠏⢀⡔⠁⠀⠀⢸⡇⠀⠀⠑⢄⠈⢷⡄⠀⠹⣆⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⡿⣶⡾⣿⡿⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⠻⣿⡿⣶⡶⣿⣯⡿⠟⠁

⣰⣿⣿⣛⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣞⣻⣿⣷
⣿⣿⣾⣿⣥⣩⣿⣿⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⣿⣯⣡⣽⣿⣾⣿
⠹⣿⣾⢿⠤⡤⠤⣿⣁⣼⣷⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣿⣄⣹⡥⠤⡤⢼⢿⣾⡿
⠀⠙⢿⣺⡶⣶⣤⣾⡿⣿⣏⣻⣭⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣯⣽⣋⣽⡿⣿⣧⣴⣶⣾⣺⠟⠁
⠀⠀⠘⣽⢷⡾⣿⣿⠀⠀⠈⠙⢯⠉⠉⠛⠿⣦⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⡾⠟⠋⠉⢩⠟⠉⠀⠀⢸⣿⡿⣶⢿⡝⠁
⠀⠀⠀⠸⡿⣟⣻⡟⠉⠑⠒⠒⠘⠢⡀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡾⠋⠀⠀⠀⡠⠚⠐⠒⠒⠉⠙⣿⣛⡿⡿
⠀⠀⠀⠀⣟⣻⣿⣇⣀⣀⣀⣀⠀⠀⠬⠢⢄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠤⠨⠄⠀⠀⣀⣀⣀⣀⣿⣿⣛⡇
⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠙⢦⣠⣿⣧⣠⠞⠁⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣶⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣄⣦⣿⣶⡿⣿⣿⣷⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣷⡾⠁
⠀⠀⠀⠀⠀⣰⣯⣿⡿⠟⠛⠉⠉⠉⠉⣩⠟⠉⠉⠉⠉⢉⡹⣿⣿⣻⡿⣉⠉⠉⠉⠉⠙⢯⡉⠉⠉⠉⠙⠛⠿⣿⣯⣷⡀
⠀⠀⠀⠀⠀⣿⣾⣿⠤⠴⠒⠀⢀⠜⠁⠀⠀⠀⣀⠤⠐⠀⠀⢻⣿⣿⠃⠀⠐⠢⢄⡀⠀⠀⠀⠙⢄⠀⠐⠲⠤⢼⣿⣾⡇
⠀⠀⠀⠀⠀⣿⣿⣯⢀⠤⠂⠊⠙⠶⠀⠐⡲⠋⠀⠀⠀⠀⠀⢈⡏⢸⠀⠀⠀⠀⠀⠈⠓⡒⠀⠰⠞⠁⠂⠠⢄⢨⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣯⡀⢀⠐⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⢀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠐⠀⠀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⠀⣀⡜⠁⠀⠀⠊⠀⠀⠠⠀⠀⢸⣷⢼⡇⠀⠠⠀⠀⠈⠀⠀⠀⠙⣄⡀⢨⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠡⣿⣿⣿⣿⣟⣷⣄⡔⠀⠀⣠⠎⠀⠀⠀⠀⣷⣺⠁⠀⠀⠈⢦⡀⠀⠐⣄⣴⣟⣿⣿⣿⣻⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣟⣷⠿⣯⣉⠟⠁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠙⢏⣩⡿⢷⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠤⠤⠤⡄⡄⢠⠦⠠⠆⠄⢠⠄⡄⠤⢠

⠀⣤⣤⣤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⣤⣤
⠀⣿⡷⢾⣉⣉⣹⣿⠷⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠾⣿⣏⣉⣉⡷⢾⣿
⠀⢹⣵⠾⣥⡤⠴⣯⣁⠾⣷⣟⡲⣄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⢖⣻⣾⠷⣈⣽⠶⢤⣬⠷⣮⡏
⠀⠀⠻⡶⣿⣦⢤⣾⠟⠓⠺⠛⠛⠚⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠓⠛⠛⠗⠚⠻⣷⡤⣴⣿⢶⡟
⠀⠀⠀⠙⣿⣧⣤⣿⡀⠀⠀⣀⣀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣀⠀⠀⢀⣿⣤⣼⣿⠋
⠀⠀⠀⠀⢻⣿⡶⢿⡏⠁⠀⠀⠀⠑⢀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⠊⠀⠀⠀⠈⢹⡿⢶⣿⡟
⠀⠀⠀⠀⠀⣿⣷⡞⣧⡄⠠⠤⠄⠒⠛⠒⠠⠀⠀⠀⠱⣄⣤⣤⣠⠎⠀⠀⠀⠄⠚⠛⠒⠠⠤⠄⢠⣼⢳⣾⣿
⠀⠀⠀⠀⠀⢹⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⢈⣿⣿⡁⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⡶⠿⠿⠿⣿⠿⠶⠶⠶⠶⣾⣿⣿⣷⠶⠶⠶⠶⠿⣿⠿⠿⠿⢶⣶⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⡟⣻⣏⠀⠀⣀⡀⡀⠁⠀⠀⠁⢀⠴⢹⢻⡟⡍⠦⡀⠈⠀⠀⠈⢀⢀⣀⠀⠀⣹⣟⢻⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣽⠋⠉⠀⠀⡠⠀⠀⣀⡤⠚⠁⠀⠀⣾⣷⠀⠀⠈⠓⢤⣀⠀⠀⢄⠀⠀⠉⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠄⠐⠚⡡⠘⠉⠁⠀⠀⠀⠃⡄⢹⡏⢠⠘⠀⠀⠀⠈⠉⠃⢌⠓⠂⠠⣤⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⠇⠀⢷⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣷⣤⣤⣴⠀⢀⠎⠀⡼⠁⠀⠀⠀⠀⠈⢧⠀⠱⡀⠀⣦⣤⣤⣾⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣷⣿⣭⣿⣿⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣿⣿⣯⣿⣾⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⠙⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠁⠀⢰⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠤⠞⠁⢀⡼⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠄⠀⠀⠈⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠈⠉⣻⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠒⠛⠛⠛⠛⠿⠿⠿⠷⣶⣶⣤⣴⣿⣿⣿⣿⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡇
⠀⠀⢀⣴⢟⠋⠑⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠻⢿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡇
⠀⠀⢾⣷⣃⡔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⠃
⠀⠀⠈⠛⢿⣷⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠨⠋⠉⡿⣿⣿⣿⡙⠦⣄⠀⠀⠀⠀⠀⠀⣤⣉⡿
⠀⠀⠀⠀⠀⠈⠙⠳⣶⣶⠀⢀⣠⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⠹⣷⡄⠀⠉⠀⠀⠀⠠⠄⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⣁⢀⣤⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣇⣠⣇⠀⣰⡄⠀⠀⢠⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠚⠿⠟⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣶⣶⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢀⡼⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠐⠁⠀⠐⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡄⠀⠀⠀⠀⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡟⠀⠀⠀⠀⠀⠀⠠⠞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⠀⠀⡠⠀⠂⠀⠀⠀⠀⠘⢛⡆
⠀⠀⠀⢀⣀⡤⢤⡴⠶⠒⠋⠉⠉⠙⠛⠛⠛⣿⣶⣶⣼⣿⣿⣿⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⣾⡫⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡼
⠀⠀⠻⣶⣇⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⣿⣿⡷⣄⡀⠀⠀⠀⠀⠀⠐⢺⡇
⠀⠀⠀⠈⠙⠳⢤⣤⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣏⢿⡄⠉⠂⠀⠀⠀⠀⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣁⣠⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣯⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣴⠀⣤⠀⢀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠤⠾⡤⣤⡴⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡍⠀⠾⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡙⢢⡀⠀⠀⠀⠀⠀⣴⠋⡣⠔⢒⣐⣶⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠸⠟⢀⢳⠀⠀⠀⢠⠞⡠⠋⢀⠔⠉⣀⣼⠶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠒⡴⠲⡌⡈⣇⠀⢠⢋⡜⢁⡔⢁⠔⠉⢀⣼⠤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠁⢰⢳⢉⡸⣄⢏⠎⣠⢋⠔⠁⢀⠔⠉⠘⣆⣤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⡆⠘⡄⢤⠹⣌⡴⣡⠋⣠⠞⠁⠀⠀⣠⠽⣍⣠⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠇⠀⢸⠖⢤⡊⡳⣅⡼⠁⢀⣠⠖⠋⠀⢀⡬⢧⣠⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⠀⠀⠀⢸⠀⠀⢧⡐⠺⣄⠴⠋⠀⠀⢀⠔⠁⠀⡞⣈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⡇⠀⠀⡴⠉⣧⡇⠀⣀⡤⠚⠁⠀⢠⢞⡁⢉⠜
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡄⡆⠀⠀⡇⠀⡜⠀⢠⢿⠷⠊⠁⠀⠀⡠⠖⠥⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢧⢱⢸⠀⡇⢰⠇⢠⢃⡏⠀⣀⣀⣀⡞⢑⡤⢾⣒⡹⠍⠭⣝⠉⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠸⣾⢸⠀⠀⣸⢀⡇⡼⢚⠭⠚⠢⣜⡋⠣⡝⢉⣠⠤⠖⠒⠈⠱⣄⠒⠀⡃
⠀⠀⠀⠰⠖⠶⠤⣀⠀⠈⠢⢄⡀⠀⢻⣿⢸⢀⡇⡜⣰⡗⠁⢀⡠⠚⠁⢙⣥⣬⠻⣄⣠⠴⠖⠒⠒⠉⢳⣤⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣈⡑⣾⣿⣾⡸⣸⣱⢃⢇⡴⢋⡀⠔⠊⠁⣀⣈⣢⢛⣇⠴⠖⠒⠒⢲⠎⣄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⢹⣿⣧⡿⢃⠞⠁⠉⠁⠉⠑⣖⣉⠁⠀⠈⡏⣗⠒⠦⣄⠀⡎⢠⡅⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠺⣟⠿⣤⠃⣀⡀⠠⠤⢴⣈⣀⠈⠉⠓⠲⡟⢹⡀⠀⠀⢑⢏⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⣹⠟⠺⣏⡿⡤⣄⡀⠀⠀⠀⠈⠉⠑⠢⣠⢋⢸⠛⢦⣠⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡸⠁⠀⠀⠙⠎⢳⣄⣈⠙⠓⠒⠤⠀⣀⠀⢸⠀⢸⠀⡴⣃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠑⣭⢛⢶⢤⣀⠀⠈⢳⠞⣤⡧⠞⢄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢾⣧⠈⠉⠓⠛⠚⠓⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠑⣶⣤⡀⢰⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣛⣛⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⡇⢀⢸⣿⠃⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣴⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠠⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⡀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡸⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⢸⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡏⠀⢸⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⣿⣿⣿⣿⣧⡀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣭⣤⣤⣄
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠆⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡼⠃⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣌⡙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠿⢿⣿⣿⣿⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄
⠀⠀⢠⣿⣯⣿⣿⣿⣿⣿⣦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣿⣿⣿⣿⣿⣽⣿⡄
⠀⠀⠈⢿⣷⣿⣿⠛⢿⣧⣼⣿⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡾⠟⣿⣧⣼⡿⠛⣿⣿⣾⡿⠁
⠀⠀⠀⠈⠻⣿⡿⣷⡟⠛⠻⠶⣿⠛⠓⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠚⠛⣿⠶⠟⠛⢻⣾⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠖⠂⠠⠼⢄⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⡠⠧⠄⠐⠲⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣀⣀⣀⣀⠀⠽⠦⢄⡀⠈⠻⣦⡀⢠⡄⢀⣴⠟⠁⢀⡠⠴⠯⠀⣀⣀⣀⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⠳⣾⣷⠞⣡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣽⣿⣿⣯⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢉⣉⣉⣉⡝⠒⠚⠉⣉⢽⣿⣿⡯⣉⠉⠓⠒⢫⣉⣉⣉⡉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣁⠤⡴⠋⠀⢀⡠⠊⢡⢾⣿⣿⡷⡌⠑⢄⡀⠀⠙⢦⠤⣈⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠊⢉⡠⠼⡤⠤⢲⠋⠀⡰⢃⠎⣿⣿⠱⡈⢆⠀⠙⡖⠤⢤⠧⢄⡉⠑⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣖⠉⢀⡜⠀⢠⠃⢀⠞⠀⡎⠀⣿⣿⠀⢱⡀⠳⡀⠘⡄⠀⠣⡀⠉⣲⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣎⢀⣠⠃⢀⠎⠀⣼⠀⠀⢸⡇⠀⠀⢧⠀⠱⡄⠘⣄⡀⣱⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣷⣿⣀⡼⠁⠀⠀⠈⠁⠀⠀⠈⢧⣀⣿⣾⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⣠⣄⠀⠀⠀⢠⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣯⣺⣿⢿⣟⡆⠀⠀⠈⣻⣞⠄⢾⡿⠶⣄⠀⠀⠀⠀⢀⡴⢿⡿⠀⣽⣿⠁
⠀⠀⠀⠀⠀⠀⡼⢋⡽⠟⠋⠡⣤⡇⠀⠀⠀⠸⣿⡀⠀⠑⠤⠈⠳⣀⣀⠔⠉⠠⠎⠁⠀⣻⠏⠀⠀⠀⠀⢀⣠⡤⠖⠒⣿⡗⠒⠦⣤⣀
⠀⠀⠀⠀⣠⣤⢿⣦⣯⡀⠀⡘⢳⡇⠀⠀⠀⠀⠙⠶⣤⠶⢶⠶⠶⣿⣿⡶⠶⣶⠶⣤⠶⠏⠀⠀⠀⣠⢾⣿⡇⠠⠂⣴⣿⣷⡄⠰⠄⣿⣿⢦⡀
⠀⠀⠀⣸⣿⡏⠀⣀⠈⣿⡀⠃⣼⠁⠀⠀⠀⠀⠀⣾⠇⠀⠦⠔⠈⣿⣿⢈⠲⠤⠀⠸⣷⠀⠀⠀⣶⣿⢠⣁⠀⠀⢸⡏⣸⡁⣿⠀⠀⢀⣡⣸⣻⡆
⠀⠀⠀⢻⣿⣖⣒⡚⠢⢈⣷⣴⠷⠀⠀⠀⠀⠀⠀⠹⣧⣄⡈⡰⣰⠊⠋⢆⢣⢡⣠⣼⠏⠀⠀⠀⠛⠛⠛⠿⣿⣶⣯⣤⠏⢧⣬⣷⣾⡿⠟⠛⠙⠋
⠀⠀⠀⠀⠛⠳⠲⠄⠵⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⣸⣟⠿⢛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣤⣄⣀⣿⡿⣧⡀
⠀⠀⣿⠟⠁⠹⡟⠻⣆⠀⠀⢀⣴⣾⠹⣿⠇⠀⠀⠀⢸⠛⠘⠻⣿⢤⠘⠿⣷⠀⠀⠀⠀⢀⣤⣦⡞⢻⣻⠟⠛⠒⠲⢄⣤⣠⠴⠒⠛⠛⣿⣷⠛⣴⣦⡀
⠀⠈⢯⣀⣀⣀⡉⠂⠸⣦⠔⠩⠘⠁⣾⠃⠀⠀⠀⠀⠘⣟⡷⣸⡟⢙⠀⠀⢩⠇⠀⠀⠀⠈⠛⢿⣵⡄⠈⠁⠉⣠⡴⠿⣿⡿⠷⣤⡉⠁⠁⠀⣶⣽⠟⠋
⠀⠀⣰⠟⠉⠉⠙⢻⣿⡷⠶⠶⣦⡤⠋⠀⠀⠀⠀⠀⠀⠙⣆⢸⠁⠊⣠⠞⠛⠻⢧⡄⠀⠀⠀⠀⠙⠷⣤⣤⠟⠡⠎⠀⣿⠃⠘⠮⠛⢦⣄⡶⠟⠁
⠀⠀⣟⡀⠈⢊⢅⣾⠏⠃⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⣥⡾⠋⣉⡵⠀⢨⣧⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⣀⣇⠏⢇⣂⡀⣀⣠⠟
⠀⠀⠻⣷⣤⡵⠊⠀⠀⣆⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⡐⠢⢍⣾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⡿⢿⠛⡟⢻⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣰⣿⠿⣿⡿⠻⢿⣿⣿⠋⢸⢀⡇⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠿⠿⣿⣿⣦⣄⡀⣿⡇⠀⢸⠀⠀⢸⠃⢸⣷⡟⠻⣿⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⡁⠈⢣⠀⢹⠀⢻⠻⣷⣿⡇⠀⠸⠀⠀⠘⠀⣜⣹⢷⣛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣏⠀⠙⠢⡀⢣⣈⡃⠘⠀⣹⣿⠀⠀⠀⠀⣠⠶⠚⠋⣭⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠓⠲⣤⣌⠻⠀⠙⢦⣀⢿⡿⠀⠀⢀⡴⠋⠀⠀⣼⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠑⠲⢤⣈⠙⠳⣇⠀⠀⢹⣼⡇⠀⢀⠜⠁⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡈⠛⢦⡈⠳⣄⠘⣿⡇⠀⠀⠀⣀⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢐⣬⣶⣿⣧⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠺⠿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⢀⡠⠚⣩⢻⣿⡏⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⣀⡤⠞⠉⣠⠞⠁⢸⣿⡇⠀⢀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠒⠋⢁⣤⡶⢛⠃⠀⢀⣿⣿⡇⠀⠀⢳⡀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡶⠚⠋⢉⠴⣯⣠⡔⢩⠘⣿⡇⠀⠀⠈⠳⡄⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡤⠖⠁⢠⠁⣰⠀⡸⢀⣿⣷⠀⠀⠀⠀⠙⠷⢤⣄⡟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⢀⣴⠁⢠⣏⣠⣷⣾⢻⣿⠀⠀⣴⠀⠀⡆⠈⣎⣷⢿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⠟⠉⠀⢸⣿⠀⠀⣇⠀⠀⣷⠀⣽⢿⣤⣶⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣾⣿⡿⣧⠠⡇⢸⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣶⣧⣼⣤⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣴⠉⡟⠳⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠛⡏⢱
⣿⣿⡿⣾⣿⠿⣎⡙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⣉⡾⢿⣿⡾⣿⣿
⠹⣿⣍⣿⣿⣷⣿⠙⠋⠉⠊⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠈⠉⠛⢹⣷⣿⣿⣏⣽⡿
⠀⠹⣿⣿⣿⡗⠈⠭⣶⣶⠞⠳⣾⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⡶⠛⢶⣶⡮⠭⠐⣿⣿⣿⡿⠁
⠀⠀⢹⣿⣄⣽⣷⣶⣿⣿⠿⠾⡿⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠻⡿⠾⢿⣿⣷⣶⣿⣅⣼⣿⠁
⠀⠀⠀⢻⣷⢰⣿⠟⠋⠁⠀⠀⠙⢄⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠜⠁⠀⠀⠉⠛⢿⣷⢰⣿⠃
⠀⠀⠀⠸⣿⢿⣛⡤⠄⠀⠀⠀⠀⠀⠐⠂⠀⠈⢿⣷⣄⣤⣄⣴⣿⠏⠀⠀⠒⠀⠀⠀⠀⠀⠀⠤⣜⣻⢿⡿
⠀⠀⠀⠀⢹⡶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣾⠁
⠀⠀⠀⠀⠘⢿⣿⣶⣦⣤⣶⣶⣶⣾⣷⣶⣴⡶⠶⠛⠻⣿⡟⠛⠳⠶⣶⣤⣶⣷⣶⣶⣶⣦⣤⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣩⣟⣉⠛⠄⠈⠉⣡⠛⠁⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠙⢣⡉⠉⠀⠜⢋⣙⣯⡁
⠀⠀⠀⠀⠀⠀⢰⡽⡯⠀⠀⠀⠀⢈⣍⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣈⣍⠀⠀⠀⠀⠨⡿⣵
⠀⠀⠀⠀⠀⠀⢸⣆⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣇⣾
⠀⠀⠀⠀⠀⠀⠀⢹⡧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢿⠇⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣵⣄⣀⠀⠀⢀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠑⡀⠀⠀⢀⠀⠀⢀⣀⣴⣥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⡦⣴⡬⢦⣴⠞⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⠮⣴⡤⣾⡷⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠹⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⡀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⡾⣬⡟⣦⡀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⣷⢺⣧⣿⣿⣏⣁⣀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⣀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢉⣙⣩⠿⠋⠉⠉⠙⠀⠹⠿⣿⣿⣟⣷⡀⠀⠀⠀⠀⠀⠠⡄⠰⣧⣼⠂⠀⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⣸⣆⡀⠀⠙⠿⢿⡻⣷⣦⣄⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢳⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢈⠉⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠤⢤⡀⠀⢀⡾⠀⠀⠈⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠈⠙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⣿⣿⣿⣿⣦⣀⠀⠀⢸⡶⠋⠀⠀⠀⠀⠀⠀⠀⠛⠻⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇⣠⠞⣀⣀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠄⠀⢠⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣯⠠⠤⠴⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡠⢍⠒⠤⠠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡉⠛⢷⡀⠀⠀⢙⣿⢏⣴⡀⠈⠙⠿⠿⠛⠋⠁⠀⠀⠀⠘⠿⠿⠃⠸⢶⣥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠈⠻⣦⣠⠞⠁⠀⠙⠛⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠇⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣶⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⢀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠹⠇⠀⠀⠀⠀⠀⢀⣤⢶⡟⣿⣯⣿⠇
⠀⠀⠀⠀⡀⣴⠋⢀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠄⠀⠀⢀⣴⡿⠋⠀⠀⠈⠉⠛⢶⣄⣶⡂⠀⠀⣀⣀⡐⠃⠀⣰⣶⣄⢀⡄⢠⣴⠟⣹⡽⣷⣠⡿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢹⢗⣾⣿⡿
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠙⢿⣿⡟⠛⠋⠁⠀⣠⣶⠟⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠺⣷⡄⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣧⣄⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠠⠀⠃⠀⠈⠀⠐⣇⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⡹⣿⣿⣿⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⢿⣿⣿⣽⠟⠁⠀⠀⠘⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃
⠀⠀⠀⡀⠀⠀⢄⠈⠃⠀⠀⣾⣶⠄⠀⠀⣾⡁⠂⠀⠀⠐
⠀⠈⠀⠒⠈⠀⠀⢀⠀⠀⠀⠈⠉⠂⠀⢰⠟⠁⣀⣀⡈⠈
⠀⠀⠀⠀⠀⢀⣿⠆⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠇⠀⠀⣠⣷⡤⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠁⠀⣀⣤⠀⢠⣆⠀⠀⠀⠀⢐⣾⣷⡆⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡃⠀⠨⠀⠀⠀⠀⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⢧⠀⠀⡈⠁⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠁⠀⠚⠁⠀⠀⠀⠀⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣤⣀⣀⠀⠠⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣐⣒⡂⠐⠢⣐⣢⣵⣦⣶⣀⣬⡙⣷⣤⡢⢄⡑⠢⣥⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠀⠂⠀⠈⠓⠢⡀⠙⠿⣯⣻⢿⣿⣾⣿⣿⣷⣭⡲⣴⡙⢳⣟⢿⣷⣆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠛⠛⠿⠿⣿⣯⡻⣟⢾⣝⢿⠻⣿⣧⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠢⠙⢎⢷⢩⠻⣏⣿⣧⢌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠈⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠠⡀⠀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠰⢀⣡⢰⣿⣼⣼⣆⣿⣷⣿⣿⣿⣷⣶⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣌⢇⣳⣿⢷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⣷⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⠿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋⠹⠤⠚⣿⣽⣿⣿⣾⠀⠰⣿⠀⠀⠀⠀⠈⠉⠛⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⣿⣿⣿⣿⠿⠟⠁⠀⠘⣆⠀⠀⠀⠀⠉⠙⠓⠁⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠒⠂⣤⠄⠀⠐⠘⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠂⠔⠂⡀⠀⠈⠉⠛⠓⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠍⠍⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⡀⣀⡀
⠀⠀⠀⠀⢀⠠⢤⠂⠉⠉⠉⠉⠛⠒⠒⠒⠶⠤⠤⣀⣴⡤⣀⠠⠦⣄⡤⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⢁⠀⠀⠨⠕⣄
⠀⠀⠀⡔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠅⡇⡈⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣨⡅⠈⠂
⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⡕⠪⡇⡇⢡⢳⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⡘
⠀⠀⠀⢧⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡋⠬⠐⢀⣜⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠂⢀⡀⠀⡥⡠⠃
⠀⠀⠀⠈⠣⡠⡀⠈⠋⠀⠄⠀⠀⠀⠀⠀⡠⠀⠀⣠⠾⣍⣀⣉⡘⠉⠙⠲⢤⣀⠀⢢⡀⠀⠀⠀⠀⢀⠀⠁⣢⠔⠁
⠀⠀⠀⠀⠀⠱⢀⠐⠂⠀⠄⠀⠀⠀⠀⠀⢀⡠⠞⠁⠘⢅⣀⣀⢄⠀⠀⠀⠀⠈⠳⢄⡑⠀⠀⠀⠈⠠⡀⠰⡸
⠀⠀⠀⠀⠀⠀⢣⡢⢌⠁⠀⠐⠠⠀⣨⠴⠋⠀⠀⠀⠀⣇⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢀⣴⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠠⠒⢲⠞⠁⠀⠀⠀⠀⡀⠄⢸⡉⢉⣁⡅⠀⠀⢰⢄⠀⠀⠀⠀⠀⠉⢺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢀⠞⠁⡄⠀⣍⣉⣁⣄⠀⠀⠀⠄⠁⢄⠀⠀⠀⠀⣀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠁⠊⠂⠂⠀⠀⠀⠀⠨⣀⣤⢼⠀⠀⠀⠠⡀⠀⠁⠀⠨⢐⣸⠂⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠳⡄⢚⠀⠀⠀⢀⠀⠀⠀⠐⠂⡄⢀⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠅⠂⠀⠀⠒⠚⠀⡰⠇⠨⠲⣄⠤⢀⡒⠀⠀⢀⣠⢾⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣥⣂⡀⠀⠠⣀⣠⠴⠊⠀⠈⡄⠀⠀⠉⠒⠺⠭⠍⠙⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣿⣿⣿⣭⡭⢝⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⢷⡮⡳⢔⠭⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣕⡭⣊⠕⠊⠔⠪⣦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⠀⠀⠀⢸⣿⣿⡿⣿⣝⡻⣮⢉⠓⠦⣤⡐⢿⣿⣾⣯⣉⣉⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡷⣝⢪⡲⣕⢦⡀⡳⣝⣦⣽⣿⡯⠓⣤⣄⡑⣷
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣮⡳⡱⡌⢳⡑⢈⢻⣿⣿⡟⡠⡺⠛⢽⣿⣾
⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠋⠉⠻⣗⠔⡙⣄⠙⢦⡹⣿⣿⣿⠈⢀⠀⣿⣿⡃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠘⣎⢪⣳⣽⣿⣿⣿⢸⢨⢦⢫⢸⠃
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⡳⣌⠿⣾⣿⣿⣿⣿⣿⣿⣊⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡝⣝⢮⣓⣚⣫⠵⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡻⣝⠮⣕⣒⣲⡴⠞⡽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣕⠮⣟⣒⣒⣦⠴⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡫⠖⢒⣒⣬⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠸⠀⢠⠄
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠄⠀⢠⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢠⠁⠀⠀⠀⢉⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡴⠑⠂⠁⠀⣀⠐⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠐⠀⢄⠀⠀⠀⠀⠈⡀⠀⢠⠀⠀⠀⡀⠘⡃⠀⠀⠀⠰⠄⠄⢀⡤⠪⠁
⠀⠀⠀⠀⠀⠈⠀⠀⡈⠁⠉⠀⢠⡀⠈⠁⠀⠀⠀⠀⠁⠀⠈⢷⢄⠀⠈⠀⡀⠀⠀⠀⠀⠅⡠⢊⠂⠀⠀⠸⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠘⠁⠀⠸⠀⠀⠁⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⠳⢐⠄⠀⠇⠀⠀⠀⠀⢸⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠌⠖⠀⠐⠄⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠡⠄⠓⠢⡖⢄⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠈⠙⠲⠄⡄⠀⠁⠀⠀⠀⡐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢅⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠂⠂⠀⠈⠈⠀⠀⠀⠀⠈⠀⣀⠤⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠢⠈⠸⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⠄
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠘⠐⠙⠀⠀⢲⢊⣀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠠⡀⠀⡀⠙⢆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡌⡌⠀⠀⠉⠙⠁⠈⠀⡀⠀⠀⠠⠂⠒⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠣⠄⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠒⠀⠠⢄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠠⠀⠰⠀⠠
⠀⠀⠀⠀⠀⢀⠊⠀⢀⠄⠀⠠⡀⠀⠄⠀⠂⠀⠊⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⢀⡀⡠⠢⠈⠢⠀⠀⠀⡁⠀⠀⠄⠠⠀⠂⢁⠀⡀⠀⠉⠀⠀⠀⠀⠠⡀⠀⢃⠀⠀⠀⡄⠂⠀⠀⠀⠁⠐⢀
⠀⠀⠀⠀⢰⠀⠃⠀⠋⢀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡁⠀⠈⠀⠀⠰⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⢀⠀⠀⠉⠐⠀⠔
⠀⠀⢀⠀⠀⢧⡀⠀⣠⣂⡀⠃⠀⠀⠖⠈⠀⠀⠀⠀⠁⠀⠀⢀⠤⠂⠠⠂⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠒⠐⠀⢄
⠀⠀⠀⠠⠀⠐⠬⢉⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⡆⠠⡀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⢂⠀⠈⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⠀⠈⠂⠀⠀⡠⠔⠒⠒⠒⢄⠂⠀⠀⠁⠀⠀⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⣀⣈⠀⢀⣀⡀⠀⠀⠁⠀⢡⠔⠒⠀⠤⠤⠤⠄⠀⣑⣌⠂⠠⠀⠀⠀⠊⡀
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠂⠐⡁⢄⠨⠁⠩⠀⠈⢙⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠠⡀⢱⠘⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠡⣀⣆⡂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠤⠔⠀⠀⡠⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠤⢀⣒⣀⣀⠖
⠀⠀⠀⠀⡰⠊⠄⡁⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠊⠀⠀⢀⠘⠂⠀⠁⠀⠀⠈⠊⢔⡤⠠⡀⠀⠀⠀⠈⠒⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣄
⠀⠀⢀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢠⡿⣧⡀
⠀⠀⢻⣿⡀⢿⡾⠉⠛⠳⠶⣄⡀⢀⡀⢀⡤⠶⠞⠛⠙⢿⡇⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡞⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠙⢿⣬⠃⠀⠉⠀⠀⢀⣹⣾⣷⣏⠀⠀⠀⠁⠀⠸⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠑⠀⠹⡄⠀⠀⠀⢀⣠⣴⣾⡏⢻⡗
⠀⠀⠀⠀⠈⠻⣷⣤⣤⠶⠛⠉⢨⣿⣿⡄⡙⠓⠶⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢶⣤⢤⣀⡀⢻⣠⣠⠖⠋⠁⠀⠹⢡⡾⠁
⠀⠀⠀⠀⠀⠀⣿⡀⠺⡄⠀⠁⢸⢿⢻⡇⠈⠀⣰⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡞⠁⠀⠀⠙⣿⣿⡷⢤⡀⠀⠀⣨⡟
⠀⠀⠀⠀⠀⠀⠹⣿⣄⡹⢀⡠⠋⠘⠀⠳⣄⠰⡃⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠈⡗⠀⠒⠀⣾⢻⡿⠀⠀⢸⠷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠓⣀⣠⠞⢁⡿⡇⠈⢀⡜⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠈⠁⢣⡠⠞⣠⣾⠃
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠛⠁
⠀⠀⠀⠀⠀⢀⣾⢟⣿⡿⣷⡄⠀⠀⠀⠀⠀⠀⢰⣿⠛⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⣿⡆
⠀⠀⠀⠀⢀⣾⣯⡾⠏⣠⣴⡇⠀⠀⠀⠀⠀⠀⠸⣷⢀⡿⠞⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠳⢿⡀⣾⠇
⠀⠀⠀⢀⣿⣿⡿⠙⠉⠙⢳⡇⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⠀⠈⠳⣄⠀⠀⣠⠞⠁⠄⠀⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄
⠀⠀⣀⣤⣿⢽⣄⠀⠀⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⣸⣶⣶⣇⣀⣀⣀⣀⣠⡿⠇⠀⠀⠀⠀⠀⠀⣴⣶⣾⣥⡈⢿⣆
⢀⣼⡟⢉⡠⠞⠛⢦⡀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣽⠁⠀⠂⢉⣿⣿⡉⠐⠀⠈⣯⠱⣄⠀⠀⠀⠀⠀⠀⠈⣯⣀⣙⢿⣿⣑⠛⣷⡄
⢸⡿⢀⡞⠁⢄⡀⠂⢷⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⡄⠐⠁⣸⢻⡟⣇⠈⠂⢠⡗⢨⣿⠀⠀⠀⠀⠀⠀⠀⢿⡟⠉⠉⠉⢻⣇⡼⠁
⠈⢿⣬⣀⣀⣀⣴⠶⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡓⣀⡴⠋⠘⠃⠘⢦⣀⠞⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠐⠀⠀⣼⣋⣛⠛⢷⣦
⠀⠈⠙⠉⠉⠉⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡜⠁⠀⠈⠳⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠙⢿⣾⣇⣀⠐⠁⠀⢈⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠙⣇⠀⠀⠀⠀⠀⠀⠉⠛⢯⣉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠧⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠓⣶⣦⣄⠀⠀⠀⠀⡟⠠⠀⠩⣧
⠀⠀⠀⠀⢀⣠⣴⣶⠛⢋⠉⠀⢠⣿⣿⡀⠀⠉⡛⠓⡶⣤⣄⡀⠀⠀⠀⠹⣦⡹⡏⠈⡛⠲⢄⣤⡇⠀⣀⣴⢋
⠀⠀⠀⠰⣿⣅⣙⠧⡀⠀⠀⣰⠛⣿⣿⠻⣄⠀⠀⢀⠞⣋⣨⣷⠀⠀⠀⠀⠈⢳⣇⠀⠀⢀⡠⣿⣿⡏⠀⠈⣌⢳⡀
⠀⠀⠀⠀⠈⠙⠛⣿⣷⣶⡞⠃⢠⡿⢿⡀⠙⣶⣶⣾⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠓⡚⡇⠀⡀⣿⢿⠣⡁⠘⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣬⡴⠋⠀⠀⠱⣬⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⢆⡁⠘⠈⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢚⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⣭⡗⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⢿⣏⣁⡤⣟⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣼⣷⣾⣿⣥⣴⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⣽⠧⣯⡼⠋⠉⢈⣿⣾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡿⠃⠀⣼⣡⠴⠚⠉⠙⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡟⠀⠀⣰⠏⠀⢀⣀⡤⠶⢿⣿⣻
⣠⣴⢲⡶⠶⢶⣖⣲⣶⢲⣤⡤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣰⠿⠒⠋⠉⠀⠀⠀⢀⣿⡹
⢻⣯⣯⡴⠚⢋⣻⣥⣿⡟⢣⡽⠛⠛⠿⠴⣶⣤⣀⠀⠀⠀⠀⠀⢀⡟⠀⢠⠞⠁⠀⠀⠀⣀⣤⣴⣾⣿⣏
⠀⠙⢶⢟⣟⣉⣡⡿⠛⠋⠛⠻⣤⣀⠀⠀⠀⠀⠉⠛⠲⢤⣾⣿⣞⢀⠔⠁⣀⣀⣤⣶⠿⠛⠉⢁⣹⣶⣿⣵⣦
⠀⠀⠀⠙⢿⣏⣽⣷⣤⠖⠚⠉⠉⠉⣙⠷⠒⠶⠦⠤⠤⠤⠼⣿⣿⡷⠾⠟⠛⠛⠛⠲⣄⣉⣉⣁⣀⣀⣨⣿⣾
⠀⠀⠀⠀⠀⠱⣿⣿⣇⣀⣀⠤⠖⠋⠁⠀⠀⠀⠀⣀⣤⡾⠟⢙⣿⣿⣂⠤⣀⣀⠀⠀⠀⠈⢙⣦⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠱⣿⣿⡏⠀⠀⠀⢀⣠⣤⣶⠾⢿⡟⠉⠀⢀⠆⠿⣿⣋⡑⠂⣉⠉⠙⢿⣏⡉⠀⠙⠲⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣳⣶⣶⣾⡿⠛⢉⡠⠖⡹⠀⠀⢠⠎⢸⠀⡿⣿⡗⢄⠈⠒⢦⡀⠘⠿⣆⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣾⣤⡷⠖⠉⡠⣾⠁⢀⣴⠃⠀⡏⠀⡇⠹⣧⠀⠑⢤⡀⠙⠢⣀⢈⡷⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣅⡠⠊⠀⠳⠒⠁⣼⡀⢠⠇⠀⡇⠀⢻⣧⠀⠀⠙⢦⡤⣿⣫⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⡇⠀⠀⡿⠁⢸⠀⢰⡇⠀⠘⢿⡇⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⣶⣾⣄⣀⣀⣧⣀⣼⣤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⣿⣿⣷⣿⣶⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠀⢄⠀⠄⠀⠠⠤⡄⠀⠀⠀⡀⠀⠀⠀⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠂⠀⠀⠀⠒⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣀⣐⣀⣀⣀⣀⣁⡉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢀⠤⠀⠀⢄⠀⢠⠔⠢⡀⡔⠒⢄⠁⠀⠀⠀⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡸⠂⠀⠀⠀⠀⠁
⠀⠀⡇⡎⠀⠀⠀⠀⢡⠘⢄⠤⠃⠣⣠⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠶⠲⡄
⠀⠀⡇⠇⠀⠀⠀⠀⠸⢰⠁⠉⡄⡋⠈⠁⠀⠀⠀⠀⠀⠀⡉⡈⣁⠀⠀⣀⣀⡀⠀⢀⠀⢠⠀⠀⠀⠘⢄⡀⡀
⠀⠀⡇⠈⠢⢄⡠⠔⠁⠈⠒⠊⠀⠁⠒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠈⠉⠉⡉⡉⠁⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢩⡋⠽⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠇⡄
⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⢀⠀⠀⣇⠆⠀⢹⢠⠎⡗⠀⡔⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠘⢤⣠⣤⣎⡫⠃⠀⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠢⠈⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠀⢀⡀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⢡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣻⣟⣟⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣻⣯⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣧⣾⣼⡿⠟⠁⢈⣱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⡁⠈⠻⣿⣴⣷⣼⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⠛⠃⠀⠀⢠⣿⣉⣿⣬⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣥⣿⣉⣿⡄⠀⠀⠘⠛⣿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡆⠀⢠⣤⡾⢿⣿⠛⠛⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠛⠛⣿⡿⢷⣤⡄⠀⢰⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣴⣾⠿⠶⠖⠛⠷⣤⡀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠁⠀⢀⣠⠾⠛⠶⠶⠿⣿⣦⣾⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢾⣟⣀⣀⣤⣤⠶⠖⠚⠛⠷⢦⣄⠙⢷⡀⣠⣄⢀⡾⢋⣠⡤⠶⠛⠓⠲⠶⢤⣤⣀⣈⣿⡷⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠉⠁⠀⠀⠀⠀⢀⠀⣀⣀⣈⣙⡺⣷⣿⣿⣾⡗⣋⣁⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠤⣽⣿⣿⣯⠤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣤⠔⢻⠟⠋⠉⢠⡞⠉⣿⣿⠉⢳⣌⠉⠙⠻⡟⠢⣤⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⡟⣛⣿⠷⠖⠋⠀⢀⣴⠋⠠⠀⢨⡅⠀⠀⠙⣦⡀⠀⠙⡲⠾⣿⣛⢻⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⣀⡴⢿⡶⢶⡟⠁⡰⠀⠀⢸⠇⠀⠀⢆⠈⢻⡶⢶⡿⢦⣀⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣤⠞⠋⣠⠟⢠⡟⠀⡠⢁⠃⠀⢸⡇⠀⠘⡈⢆⠀⢻⡄⠻⣄⠙⠳⣤⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣄⣼⠋⢀⡿⠀⣰⢁⠎⠀⠀⠨⠅⠀⠀⠱⡈⣆⠀⢿⡀⠙⣧⣠⣿⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡿⣽⢲⡾⣇⣠⢃⡾⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣄⣨⢷⡖⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣿⣳⢯⣟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⡽⣞⣿⠝⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣳⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⣿⠋⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣷⠾⣷⡾⢿⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣋⣰⣋⣰⣏⣤⣿⣾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⡿⠛⠉⢹⣯⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢡⢃⡜⠀⢀⡴⠋⠉⣻⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡧⠞⢻⡠⠞⠁⠀⣠⠞⠉⢻⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⡿⠀⠀⡿⠀⣀⡴⠋⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢸⣿⠃⠀⢰⠷⠚⠁⠀⢀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⢿⣷⣿⣿⣝⢻⣿⡷⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⡞⠀⠀⢀⣴⣿⣿⣿⣤⡴⢶⣿⣿⣶⢦⣄
⠀⠀⠀⠀⠈⠻⣽⡦⣈⣿⣏⣉⣻⣿⡿⢿⣿⣷⣦⣀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⠁⠀⣠⣾⣿⣿⠿⠋⠀⣀⠤⠛⠛⢻⣏⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠻⢾⡿⣥⣭⡿⠯⢍⣐⣬⠟⠿⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⢇⣴⣾⡿⣫⠟⣁⠤⠒⢉⣀⠤⠔⠒⠋⠉⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣧⣀⣀⣈⣙⣦⣄⡀⠈⠉⠛⠻⢿⣶⣤⡀⠀⢤⣤⣄⣿⣠⣿⡿⠟⣩⠞⠋⠛⠓⠒⠺⠯⣄⡠⠤⠒⠒⠚⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣍⡁⠀⠀⣀⣉⠿⠶⠦⠤⠤⣬⣉⣛⣷⣾⣿⣿⣿⣏⣡⡴⢋⣁⣀⣀⣀⣀⡤⠔⠚⠉⠓⠲⢤⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⡿⠻⣿⣯⣻⣿⣭⣉⡉⠉⠉⠀⠉⠉⠓⠲⢤⣀⠀⠀⢠⣛⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣶⣾⣿⣿⣿⣿⡿⠿⠛⣉⡴⠊⡹⢿⣿⣿⣗⠮⣉⠓⠲⠤⣄⣀⠀⠀⠀⠀⢙⣷⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣻⣿⠋⠀⠀⣰⠁⢸⠹⡛⢿⣗⠮⢕⣦⣀⠀⠈⠙⠓⠶⣤⣾⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⠞⠁⣸⡆⠀⢠⡇⠀⠸⡄⢱⡈⣿⣦⠀⠈⠉⠓⠦⣤⣤⣾⣿⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠃⢱⣤⠟⢆⠀⠀⢧⠀⢳⣹⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸⡄⢣⡈⢦⡀⠸⡄⠈⢿⡄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣵⣤⣑⣄⣡⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣻⣻⣯⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣭⣿⣽⣿⣯⣯⣽⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣴⠶⣿⣿⣿⣅⡀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠋⠀⣠⣿⢉⣧⡙⡙⢷⡀⢀⠢⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣀⡴⢡⣿⠘⣿⣧⢹⡌⢻⣄⢣⠘⢦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⣤⣤⠶⠛⣁⣴⠟⢹⡆⠙⠛⠦⣷⡀⢻⣎⢧⠀⠙⠷⣤⣌⣙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣾⡿⠛⢉⣠⣶⠿⠛⠁⠀⠀⢿⡀⠀⠀⠸⣷⠄⠹⣷⡀⠀⠀⠀⠀⠉⣉⠬⠙⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⣧⡀⢉⠻⢦⣤⡤⠖⡋⠁⠀⠀⢀⣤⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⣴⣿⣿⣿⠒⠶⣄⠀⠀⠀⣠⡴⠞⢿⣄⠀⠀⠈⠻⢦⣿⣶⡶⢖⡉⠄⠀⢖⣤⣿⣗⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣾⠁⣸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡴⢾⣿⣿⣮⢳⣦⣤⣤⣤⣴⣶⠿⣷⣶⣾⣿⣎⢿⣏⠳⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⡏⠉⠉⠀⣰⠀⠀⠀⠈⠈⠉⠉⠘⢿⣽⣿⣿⣿⣿⣿⣷⡘⢆⠹⣷⡹⣟⢿⣷⣶⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡿⢹⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡜⣿⡿⢹⡟⡞⡄⢹⣷⢹⣶⣽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⢀⣿⣿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸⢧⡿⠁⣸⠧⠀⣿⢸⣿⡇⣿⡆⢿⣆⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣷⡄⢠⣶⣶⢶⣤⡄⠀⠀⠀⠀⠀⡸⢁⣴⠟⠁⣰⠏⡀⡰⡏⢸⣿⣷⠘⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⢫⣶⣿⣿⣿⣷⡈⠛⠷⠖⠉⠀⠀⠀⢀⣤⣎⣴⠟⠁⣠⡞⠁⣰⠟⣸⡇⣼⣿⣿⠀⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠁⢠⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢀⣠⠾⢋⣠⠞⠁⢰⣿⠁⢿⣿⣿⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏⢸⣌⢻⣿⣿⣿⣷⣶⡾⣿⣿⡿⠋⣀⣴⠟⣡⠶⢋⣤⠶⠖⠛⣸⣢⠸⣿⣿⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⠻⣷⣝⣿⣿⣿⣿⠇⣿⠟⢠⣶⢟⡵⠞⠁⣠⡯⠔⣲⣯⡴⠛⠻⣦⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢳⡀⠘⣦⠑⠸⡏⢻⣿⣿⠏⣰⡏⣰⣿⢣⡞⠁⣤⠞⠁⣠⠞⣽⡟⠀⠀⢀⣼⣦⣿⣿⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠇⢸⣷⡀⠘⣧⠀⢰⣼⠟⠁⣴⣿⣀⣿⢣⡟⢀⣾⠁⠀⣴⠏⠘⣿⠀⠀⢠⠟⠁⠀⢿⡄⢧⠻⣮⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⢠⣿⣿⣷⠀⠘⣇⠈⠁⣠⣾⢻⣿⣿⣿⠸⡇⣼⡇⠀⢸⡿⠀⢸⡟⠀⠀⠰⠄⠀⠀⠈⢧⠈⢧⡈⠻⣦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢸⡇⢹⣿⠀⡀⢿⣠⡾⢿⡿⢸⣿⣿⣿⣇⢸⣿⠁⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢻⣦⡈⠛⢦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿⣠⣷⠿⠋⣠⢪⠃⣼⣿⣿⣿⣿⡌⣿⠀⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣿⢦⣄⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣷⡴⠟⠋⠀⢀⡴⢡⠎⣰⣿⣿⡟⢸⣿⣷⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢳⡍⠳⢦⡻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠟⠁⣀⣀⡤⠔⢁⣴⢋⣴⣿⣿⣿⡇⡄⣿⣿⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣄⠀⠙⢶⣄⡉⠻⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠗⠛⣉⣉⣡⣤⡶⢟⣵⣿⣿⠏⣼⣿⢰⡇⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠳⣦⣀⠉⠻⢶⣤⡿
⠀⠀⠀⠀⠀⠀⢠⠞⣁⣤⣶⣿⣿⡿⡿⠿⠿⣿⣿⣿⠋⣼⣿⠇⣼⢡⣿⠏⣿⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⣿⣷⣌⠙⠿⣿⠛⢹⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣭⣥⣴⠾⡷⢶⣶⣞⣋⣡⣾⣿⠟⠠⢣⣾⡟⢰⣿⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣻⡿⣿⠈⣷⣀⡿⢀⠆
⠀⠀⠀⠀⢠⡾⢋⠏⢰⣿⣿⡿⣡⢫⡆⢾⣿⠉⣿⣿⡿⢋⣀⣴⠿⡿⢁⣿⡇⠀⢀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⣾⠟⠀⠿⣳⡿⢋⡴⠃
⠀⠀⠀⠀⢻⣅⠈⢧⠘⣿⣿⡇⣿⢸⣧⡈⠉⢶⣮⣛⣻⣛⣋⣥⡞⢁⡾⣿⠁⠀⣸⣷⠟⢧⣀⣀⢀⣀⣠⠴⠒⠋⠉⢉⠾⠋⠁⢀⡰⣶⣯⠖⠋
⠀⠀⠀⠀⠀⠙⠳⣤⣙⠺⣿⣿⣼⣆⠻⠦⣭⣜⠛⢿⣟⣿⡿⠋⢀⠞⣽⠃⠀⣰⣿⡇⠘⣷⣬⣉⣉⣁⣠⣤⣤⣀⣤⣤⡤⠶⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⡙⠛⠿⠶⠦⠽⠷⠾⠛⠉⢀⠴⣋⡾⠁⢀⣴⡿⣡⣯⡾⠛⠛⠛⠋⠉⠀⠀⣽⣿⣿⠙⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣭⣙⠲⠤⣀⡀⠀⠀⠀⣈⣤⠞⢋⣠⡴⠻⣿⠀⣿⠁⠀⣀⠔⠒⠒⢒⣒⣚⡋⠭⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡋⠛⠻⢿⡿⣷⣶⠞⠋⠹⡇⠰⠙⠦⠬⢉⣉⠴⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠛⠛⠉⠀⠀⠀⠀⠈⠑⠒⠻⠛⠋⠁

⠀⠀⠀⠀⣴⡾⠿⣛⣶⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⢿⣛⡛⠿⡶⢄
⠀⠀⠀⠰⠶⠁⣀⡛⠋⠀⠀⠈⠛⠙⠂⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠝⠁⠀⠀⠈⠙⢣⡄⠀⠚⠆
⠀⠀⠀⢴⡶⠀⠩⣥⣀⡀⠀⠀⠀⠀⠐⠦⣉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢊⣭⠖⠂⡀⠀⠀⠀⢠⣴⡖⠀⠰⠶⠂
⠀⠀⠀⠀⠰⣶⣶⣬⡍⠀⣀⣀⡀⠀⢀⣀⣀⠀⠈⣦⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣨⣤⣀⠀⢀⣠⣤⡄⠀⠲⣾⣿⠟
⠀⠀⠀⠀⠀⠈⣡⣼⣄⣸⣿⠿⠗⠀⣀⣿⡿⠷⣄⠈⠙⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠁⡴⠿⢿⣯⣀⠘⠿⠿⠿⢠⣴⣤⡍
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠗⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠠⡀⠀⠳⣄⠀⠀⠀⠀⠀⠠⠊⠀⡀⠀⢈⠄⠀⠀⠀⢠⣶⣶⡄⠸⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⣤⣤⣤⣀⠉⢉⣀⡀⠀⠐⠒⠚⠂⠀⠀⠀⠘⢦⢠⣄⠀⠔⠀⠀⣈⠄⠚⠁⠀⠀⢀⣀⣀⠈⢩⣤⣶⣦⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢛⡁⠀⠸⣿⢿⣟⣀⠀⠐⠒⠠⠤⠀⠂⢌⣿⣿⡾⣠⢔⣋⠠⠐⠒⠀⠀⠐⢿⣿⡿⠂⠀⢍⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠞⠛⠉⡉⠁⠄⠀⠀⠀⢉⡓⠲⠔⢚⣿⣿⣿⠁⠈⠉⠉⢉⡡⠤⠤⠄⠀⣀⠈⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⠸⠿⠃⠀⡀⠀⠈⠡⠄⢀⠴⠋⠸⣿⣧⠙⠢⡀⠠⠤⠀⠀⢄⡀⠈⠻⠇⠀⠻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣀⠀⢴⣶⠀⠀⢀⣶⣷⡖⡀⠄⡸⠁⠀⠀⠘⣿⡷⡙⠄⢌⢳⡢⢀⣻⣿⣿⡍⠐⠐⠿⠆⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠾⠿⠃⢠⣤⠄⠠⠛⠛⡹⠋⢀⠔⡕⠀⠀⡜⠀⣿⡇⠱⡌⢎⢣⡱⡄⠈⠻⡉⢁⣀⡀⢶⡦⠀⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⣠⣤⠘⠋⢰⣿⡇⢀⣀⢀⠊⠜⠰⠁⠌⠀⠀⣿⡇⠀⠙⡌⢣⡹⡜⠆⢠⣕⠜⢿⡿⠀⠁⠰⣷⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⣥⠀⣤⠀⠉⠀⠻⠏⢰⠎⢰⠃⠀⠀⠀⠀⢹⡇⠀⠀⠈⡄⢳⡘⣾⠈⠛⠂⠀⠀⢴⡄⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⠐⢛⡀⠀⣶⠀⣀⠈⢠⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠀⢣⠀⢠⠀⠸⣦⢀⣤⡉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠨⢡⣎⡽⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊⠤⠷⠄⠈⠛⠁





⡿⠿⠿⠻⠿⠻⠟⠿⠿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣴⣿⣿⣹⣿⡿⣶⣶⣤⣤⣀
⠀⣿⣿⣶⣿⣷⠶⠾⣿⠀⣿⣾⢿⣖⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢿⢳⡿⣿⡖⠉⣹⣿⣿⣍⣿⣿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠙⢿⣿⣛⣿⣟⠉⠉⠉⠉⢻⣄⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠈⢻⣿⣯⣿⣿⠟⠛⠉⠉⠉⠑⣢⣄⡀⠀⠈⠙⢦⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣀⣤⡤⠖⠊⠉⠀⠀⠈⠉⠒⢦⣤⣹⣷⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⡶⠶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⣰⣿⣿⣿⣿⣿⡏⠑⠦⣀⠀⠈⠉⠛⠿⢶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠂⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠅⠀⠐⠀⡀⠀⠀⠀⡠⠊⣹⣿⡿⠏⠿⠙⣿⣄⠀⠈⠑⢢⣄⣀⠀⠀⠉⣿⣿⢿⣷⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⣀⠀⠀⠀⠀⠀⣠⠞⠀⢰⣿⣿⠡⠁⡇⠀⠹⣿⣆⠀⠀⠀⢹⠉⠙⣗⠒⠿⣏⢻⣿⠟⣙⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠔⢂⠂⢁⡔⠀⢠⣿⣿⠋⠃⢠⠀⠀⠀⣿⣿⣧⡀⠀⠈⢆⠀⠈⢦⠀⠘⣿⣿⢷⡟⢯⠙⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣖⠀⢀⠀⠀⠘⠁⠀⣿⣿⠃⠀⢀⠛⡀⠀⣘⠹⡙⣿⣿⡄⠀⠈⣆⠀⢈⣷⣤⣿⣯⣀⣷⣈⣷⣼⢷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣀⣠⡄⠀⠘⡿⠃⠄⢰⠟⠀⣧⠀⢻⠀⢱⠘⣿⣿⣄⣀⣽⣷⣾⣟⣯⣿⣭⣿⣿⣿⣿⣽⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣶⣥⣂⡤⠃⠀⡰⠀⡇⠈⠆⠀⣿⣿⣿⢿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣷⣄⣼⠃⠀⡇⠀⣷⣾⣿⣿⠏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣾⣷⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣴⢛⣩⣍⣉⣉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢋⣉⣉⣩⣙⢳⣄
⠀⠀⠀⠀⢸⣁⣠⣤⣤⣤⣤⣉⡑⠲⢍⡳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⠵⢒⣉⣡⣤⣤⣤⣤⣀⣻
⠀⠀⠀⠀⢸⣇⢀⣠⡤⠤⠤⣤⣍⣙⣶⠛⠉⠛⢦⡀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⣠⠞⠋⠙⢳⣞⣉⣥⡤⠤⠤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⢿⣄⠠⠤⠤⠤⠤⠤⣬⣿⣄⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⣼⣧⡤⠤⠤⠤⠤⠤⠀⣼⠇
⠀⠀⠀⠀⠀⠈⢿⡀⠠⠤⠶⠒⠒⠒⠶⠾⠷⣄⡀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣀⡴⠿⠶⠖⠒⠒⠶⠦⠤⢀⣸⠏
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠤⠒⠒⠒⠒⠒⠒⠒⠾⠿⠶⢤⣄⣀⠙⣄⢠⢤⢀⡞⢁⣀⣤⠶⠿⠷⠖⠒⠒⠒⠒⠒⠒⠶⢄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣖⠒⠒⠒⠒⠶⠦⠤⠤⠬⠿⠷⣾⡾⠰⣮⠾⠿⠧⠤⠤⠤⠶⠖⠒⠒⠒⢒⣦⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠛⠛⠛⠛⠓⠶⠟⠛⣛⣛⢿⣧⢠⣿⢟⣛⡛⠛⠷⠖⠛⠛⠛⠛⠯⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣠⠤⠒⠒⠚⣛⡷⠒⠊⠉⣠⢻⡇⠋⡏⢣⡈⠉⠒⠺⢟⡛⠃⠒⠢⢤⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⡠⠔⠚⠋⡽⠁⠀⠀⢀⡴⡃⢸⢇⠀⢿⠀⡷⣄⠀⠀⠀⠹⡉⠓⠒⠤⡈⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⡠⠔⢊⡷⣄⡤⢶⠋⠀⠁⡾⢨⠃⠸⡄⠁⠈⢳⠦⣄⡴⣏⠒⠤⡀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡰⠋⢀⡴⠋⠀⡜⠀⡏⠀⠀⣰⠇⠸⡔⠀⢧⡀⠀⠈⡇⠘⡆⠈⠓⣄⠈⠳⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡞⠀⠀⡜⠀⢀⠀⠀⢀⡟⠀⠀⠀⠀⠘⣇⠀⠀⣦⠀⠘⡄⠀⠘⡆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅⡀⢸⠀⠀⠀⡇⣠⡞⠁⠀⠀⠀⠀⠀⠙⣦⡀⡏⠀⠀⢸⠀⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠾⠛⠉





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

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⠒⢀⠀⠀⠒⠂⠀⠠
⠀⠀⠀⠀⠁⠀⠈⠀⢀⡀⠐⠀⡀⠀⣀⠀⠀⠈⠉⡀
⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠠⠀⠁⡀⠉⡄⠀⠀⠀⢅⠀⠁⠀⠄⠀⠀⠠⠊⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠠⠄⣀⣁⡀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠘⢀⡀⠀⢀⠀⢀⠀⠙⠉⠁⠀⠠⠛⠀⠀⠀⠀⠘⠣⠄⠀⠀⠄⠀⠄⠀⡀
⠀⠀⠈⠀⠁⠀⠀⠁⠁⠤⠤⠐⡀⠡⡔⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠠⠌⠀⠒⠀⠀⡀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠠⢆⠀⡀⠐⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢠⠎⠀⠁⢈⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠄⢗⡌⠀⠐⠀⠰⠂⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣡⠀⠊⠀⢉⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠘⡀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠐⡀⠤⢀⠀⠴⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⣀⠔⠓⠀⠐⠁
⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⣶⣶⠶⢶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠖⠉⡐⠒⢲⣿⣿⣮⣍⠉⠛⠶⣬⣣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⣫⣠⠞⠛⠉⠭⢭⣿⣿⣶⣬⣿⣿⣄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢿⡀⢰⣏⢄⣢⡴⠶⠛⠉⠉⢹⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⢏⣼⡇⢸⣛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡻⡀⠑⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⣷⢿⣿⣇⡌⣉⠁⠀⠐⠚⠒⠲⢤⠀⠹⣿⢿⡀⠹⣌⠳⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡞⣵⣿⣿⠟⠁⠙⠁⠀⠀⠀⠾⠳⠄⠀⠀⢻⣷⣝⢦⡈⢗⢌⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⡜⣟⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡧⡠⡙⠢⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⢸⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⢇⣹⣌⡛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⢀⢂⣿⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡆⡆⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢥⣙⡻⣿⣿⣦⣤⣤⡤⠠⠀⠂⠀⠀⠀⠀⢸⠀⢧⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⠟⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣜⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣡⢀⠜⠋⠻⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠁⠀⠠⣺⢿⠈⣿⣹⢿⡆⠀⡐⠁⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⡞⡱⣾⠀⣼⢋⡞⣠⢣⢟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣧⡁⢳⡀⣯⠞⡰⡱⣣⣣⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣻⣿⡇⠁⡸⠐⣱⠿⡫⢾⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣇⡇⡇⡇⡏⠞⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢻⣌⣄⠡⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣪⠿⣿⡴⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣀⣤⣤⣤⡤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠄⣤⣤⡤⢤⣀
⠀⠀⣿⡟⢈⣿⠿⠀⠉⠁⠀⠈⣭⣝⣲⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⣋⣭⠁⠀⠈⠁⠀⠘⠿⣧⢈⣿⡇
⠀⠀⢿⣯⠘⣓⠒⠀⠀⡀⠀⠶⠟⠃⠈⠉⡐⣱⡦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣏⣁⠀⠀⢙⠿⠁⠀⠀⣀⡈⠭⣛⣲⣿⠃
⠀⠀⠀⠛⣽⣳⣈⡼⠗⣀⡠⠀⠀⣹⣶⠾⣟⡛⠏⠈⣷⣄⠀⣀⡀⢀⣞⣕⡩⢛⣉⡿⢷⣾⡅⠒⠀⢤⡔⢒⣦⣽⣛⠟⠁
⠀⠀⠀⠀⠈⠹⡞⣛⣡⠖⢂⡠⠤⢹⣿⣮⣉⣉⣙⠲⣦⣝⣧⡅⣧⣿⡽⠞⠋⠡⠤⢤⣿⣯⡀⠠⢄⡓⣂⢬⣛⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠱⠾⣯⣠⠞⢃⠩⠟⠓⣂⡠⠤⠉⠩⠉⠿⣿⡻⠅⠀⠊⠉⠀⠀⣒⠚⠩⢅⣙⣦⣤⣛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣇⠎⢁⠀⠄⠀⠀⠀⠐⠁⢼⢘⢁⠀⠑⢄⠂⠀⠉⠂⢀⠀⢋⣿⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣫⣍⡀⡌⠀⢀⠠⠀⠀⠀⠎⠀⠨⠀⠄⠀⢂⠙⠦⢀⠀⢀⡧⢀⡮⣽⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣹⢋⡁⡠⠊⠸⠁⠀⢀⠂⡌⠀⠀⠐⠀⠘⡈⠀⢆⠐⠀⠘⡈⠱⡄⢈⢻⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⢯⠞⣰⢀⢃⠀⡇⡜⠰⢀⠀⠀⠀⠀⠀⠠⠀⠘⡄⢆⢠⢱⠰⡌⢲⡟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣷⢧⣎⣼⣰⣡⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠡⣳⣸⣘⣆⢷⠻⣏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠰⠏⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠛⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣆⣤⡿⢿⣿
⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢿⣿⣏⣉⢈⣙⣷⣾⣿
⢿⣿⣿⣿⣧⣀⠠⢴⣿⣿⠛⢿⣗⠦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢀⣠⣾⣯⣾⣯⠄⠚⣿⣅⣀⣈⣹⣿⣿⣿
⠸⣿⣿⣿⣯⣤⣤⣴⣾⣍⣁⡾⢿⣿⣟⡛⠦⣄⠀⠀⠀⠈⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⣠⡴⣋⣥⣾⣿⣋⣉⣽⣿⡏⠉⠉⣹⣿⣿⣿⡟
⠀⠹⣿⣿⣿⣿⣀⢀⣀⣿⣿⠶⠦⢾⠿⠿⠶⠾⢿⣦⣀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠙⠉⠉⠉⢻⣿⠷⠾⣿⣿⣿⠏
⠀⠀⠘⢿⣿⣿⡟⠋⢻⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⣿⣤⣤⣿⣿⠋
⠀⠀⠀⠀⢹⣿⣷⡶⠾⣿⣄⡀⠀⠀⣤⠷⣄⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠘⣆⠀⠀⠀⡼⠁⠀⢠⡾⠋⠀⠀⠀⠀⢀⠔⠋⠉⠉⠀⠐⠲⣿⣿⣭⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⣤⡤⣿⡟⠁⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠙⢷⣄⠀⠘⡆⠀⡸⠁⢀⣴⠟⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⣸⣉⣈⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣦⣼⣧⡀⠀⠀⣀⣀⣤⠤⠶⠓⠠⠀⠀⠀⠀⠀⠈⠳⡄⣸⣶⣇⢠⠞⠁⠀⠀⣀⠀⠀⠉⠉⠉⠛⠒⠒⠒⠀⠤⣶⣏⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠙⢿⣿⣿⠃⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢁⣠⣾⣿⣿⣆⣈⣁⣀⣀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣶⣿⠿⠿⠿⣿⣿⡿⠿⠿⢛⣛⠛⠛⣿⣿⣿⣿⣿⡛⠛⠋⣉⣉⠙⠹⢿⡟⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⢀⡴⠋⡟⣿⡟⢏⠳⢄⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⠀⢀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠶⠆⠀⠂⠀⠸⠁⠀⠀⠀⠀⣠⠔⠋⠀⠘⢀⣿⣷⠈⠄⠀⠙⠲⢄⡀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠉⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⣠⠃⠀⣀⡤⠶⠋⠁⠀⠀⡴⠀⡜⣿⣿⢀⠈⢦⠀⠀⠀⠉⠛⠲⢤⣤⠈⠷⢄⠀⠀⠀⠀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣠⠀⠀⠉⢉⠬⠋⠁⠀⠀⠀⠀⢀⠌⠁⡼⠁⣿⣿⠈⢦⠀⠱⡀⠀⠀⠐⡀⠀⠈⠑⠀⠀⠀⠈⠛⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠊⢀⣶⠃⠀⢹⡏⠀⠘⢷⡄⠈⢀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣤⣤⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣷⣾⣏⠀⠀⠀⠀⡀⠀⠀⣠⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢢⡀⠀⠀⣦⣀⣀⣀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣿⣅⣀⣴⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⣿⣿⣳⣤⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣴⣻⣿⣷
⢹⣿⢶⣿⣭⣭⣍⣻⣍⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣏⣽⣋⣭⣥⣽⣷⠾⣿
⠘⣿⡾⣯⠤⢬⠧⣮⣤⣿⣿⡟⠻⢷⣄⡀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣀⣴⠿⠟⣿⣿⣧⣬⡦⢭⠤⢤⡽⣾⡟
⠀⠈⢿⣿⣶⣼⣾⠾⠿⠿⠭⣙⣾⣿⣿⣿⣦⡀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⣾⣿⣿⣯⣞⡩⠽⠿⠿⢾⣾⣴⣾⣿⠟
⠀⠀⠈⢿⢷⣯⣤⣤⡀⠀⠀⠀⢳⡀⠀⠉⠻⣿⣦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣾⡿⠋⠁⠀⣰⠋⠀⠀⠀⣀⣤⣬⣷⢿⡏
⠀⠀⠀⢸⣿⣏⣿⠋⠉⠉⠉⠀⠈⠙⠢⣄⠀⠀⠙⢿⣦⡀⠈⠢⡀⠀⠀⡠⠊⠀⣠⣾⠟⠁⠀⢀⡠⠒⠉⠀⠁⠉⠉⠉⠹⣟⣻⣿
⠀⠀⠀⠀⣿⣿⣿⣦⣄⣀⠀⠀⠀⠤⠤⠬⡳⢄⡀⠀⠙⢿⣄⠀⣼⠀⢸⡄⢀⣴⠟⠁⠀⢀⡴⡻⠤⠤⠄⠀⠀⢀⣀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠲⢄⡀⠙⢷⡌⠀⠈⣴⠟⠁⣀⠤⠚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⡿
⠀⠀⠀⠀⠀⠸⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⠀⠀⣉⣓⣲⠀⠀⠀⢹⣒⣋⣁⠀⠀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡽⠿⠿⠿⠿⢟⣛⣛⣋⠭⠅⠚⢉⡄⠀⠀⣌⠙⠒⠬⢍⣛⣛⣛⠿⠿⠿⠿⠿⣭⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣥⠤⠴⠖⠒⢒⠏⠁⠀⠀⢀⡤⠚⣹⣿⠂⢺⡷⠉⠢⢄⠀⠀⠀⠉⢷⠒⠒⠶⠤⢤⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⣀⣤⠤⢒⠞⠁⠀⠀⣠⠖⠁⠀⡰⢡⡟⠀⠘⣧⠡⡀⠀⠑⢦⡀⠀⠀⠙⢖⠲⠤⣄⡀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡷⠋⠁⣀⡤⢏⡤⠔⠒⡾⠁⠀⢀⡜⢁⡾⢡⠀⢠⠙⣆⠘⢄⠀⠀⠙⡖⠒⠤⣄⠧⢄⡀⠉⠳⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣧⣴⠋⠁⣠⠋⠀⢠⡞⢀⠄⣠⠏⢀⡾⠁⠘⠀⡈⠀⠹⣆⠈⢣⡀⢄⠘⣦⠀⠈⢣⡀⠈⠳⣤⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣰⢃⠂⢠⠏⣠⠏⡴⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠱⡌⢳⡈⢣⠀⢢⢳⣠⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣯⣶⣾⣶⢣⠞⢁⢄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢆⢘⢦⢳⣼⣶⣮⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣻⣯⣟⣷⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣳⣟⣿⣽⣻⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣿⣿⣶⣶⣾⣭⣝⡲⣬⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣝⣛⡻⠿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⡴⡍⡊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢻⡜⣆⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡘⣿⣿⢸⣿⢸⣿⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣿⡏⣾⣿⢸⣿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣯⣿⢋⣾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣵⠟⠡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⣾⣻⠇⣼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡏⣿⣼⣯⠁⠐⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡸⣯⡘⢿⣿⣄⢠⣞⣚⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣮⡳⠌⠻⣶⣝⢻⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⣾⣿⣦⡈⣷⡝⠧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢉⣵⣿⠿⠃⠐⢵⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠾⠶⠿⠼⠶⠮⠽⡶⠀⠰⠿⠴⠾⠾⠯⠀⢰⠵⠾⠧⠗⠾⠷⠦⠾⠶⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠔⡳⠀⠀⠱⠀⠀⠀⠀⠡⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⢣⠀⠀⠀⡘⠀⣀⣀⡠⠀⠀⢀⠂⠀⠠⠈⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠤⠄⡈⠀⠀⠀⠸⠆⠀⡁⠈⠑⠂⠠⠧⣠⡛⢢⣾⠛⠛⠋⠻⠛⢻⣻⣯⡎⠀⠀⠙⣼⣟⡛⠛⠛⠛⠛⠟⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠈⠗⠂⢦⠘⠂⠀⠀⠀⠀⢀⡿⣶⢻⠁⠉⠈⠻⠶⣶⣶⠿⠟⠿⠛⠃⠀⠀⠀⠛⠻⠻⠿⣶⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⣀⠃⡰⠐⠄⠀⠀⠀⠀⢸⡷⢁⠄⠀⠀⠂⠀⠰⢈⠟⠛⠒⠶⢴⡃⠀⠀⢸⢦⠴⢶⡓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⠀⢀⠀⠀⠀⠀⠀⠄⠠⡀⠠⠀⠀⡀⠀⣸⠗⠀⠀⠁⠀⠀⠁⠀⠀⡇⠀⢀⡠⠜⣣⠀⢀⡂⠀⠀⠀⠉⠑⢦
⠀⠀⠀⠀⢀⣀⣀⣤⡀⢀⠀⣤⣌⠋⠊⡿⠣⡄⠀⠉⠀⠘⠀⠀⠉⠑⡄⠈⠹⡿⢤⠀⠀⠄⠀⠀⢀⢢⠛⠀⠀⢈⠇⠘⢿⣇⣿⠟⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⣠⣦⣭⣭⣽⣷⣞⣷⡟⢻⠟⠈⠀⠀⣄⠀⠡⠀⠇⠀⢆⠀⠐⠌⠀⠀⠈⠑⠐⢟⢠⡚⡈⠈⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣭⣍⣙⠻⠏⠀⠈⠿⠃⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⣔⢂⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢻⡿⠛⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠡⠘⡄⢀⠳⣦⠀⢀⢀⣀⣀⣠⣤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠄⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠳⡀⣥⠀⢉⠄⠀⠆⠁⢄⠉⡇⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠁⠀⠀⡀⠀⠓⠀⠀⢀⠈⠧⠘⡌⠻⡊⠄⠑⠆⠀⠸⠁⠀⠀⠈⠘⠇
⠀⠀⠀⠀⠀⠀⠲⠒⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠑⠀⠂⢀⠀⠂⢤⠀⡅⠠⠡⠀⣠⡔⠂⠀⠚⢃⠀⣾⡷⡄⠁⠀⠀⠀⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠠⠀⠀⠀⠀⠂⠓⠀⠀⠀⢀⣀⠤⠀⠂⠉⠉⠀⢀⠸⡁⡇⢰⢦⡄⠰⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢳⠀⠀⡀⠉⠐⠀⠤⡀⠀⠀⠀⠶⠆⡀⠄⠠⠀⠀⠂⠠⠀⠀⠀⠈⢇⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⡇⠠⠀⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠙⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠐⠐⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⢀⣠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠄⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠋⡛⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠄⠋⠲⠅⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⣇⠎⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠎⠰⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⣟⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⣠⢴⡾⠛⢻⣿⠶⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⢶⣿⡟⠛⢷⡦⣄
⣼⣽⡿⢿⣶⠿⢿⣦⣾⣿⣥⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣬⣿⣷⣤⡿⠿⣶⡿⢿⣯
⣯⣽⣿⣿⠿⣶⡿⠋⠉⠉⠉⠉⢻⣿⡿⠿⣿⣷⣤⡀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⣤⣾⣿⠿⢿⣿⡟⠉⠉⠉⠉⠙⢿⣶⠿⣿⣿⣯
⠹⣯⣿⣿⣾⣿⣀⡀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠉⠛⠿⣷⣄⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⡠⠚⠉⠀⣠⣾⠿⠛⠉⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣿⣷⣿⣿⣽
⠀⢻⣯⣿⣿⡿⠿⠿⠿⠿⠶⠶⠦⢤⣽⣿⣆⠀⠀⠀⠈⠛⢷⣄⠀⠀⠑⡄⠀⠀⢀⠞⠀⠀⣠⡾⠛⠁⠀⠀⠀⣰⣿⣯⡤⠴⠶⠶⠿⠿⠿⠿⢿⣿⣿⣽⡟
⠀⠀⢹⣏⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⣷⡀⠀⠘⡆⢠⠋⠀⢀⣾⠋⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣹⡟
⠀⠀⠀⢿⢻⣷⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠈⢻⣄⠀⣿⣿⠀⣠⡟⠁⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣾⡟⡿
⠀⠀⠀⢸⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢶⣄⡀⠀⠹⣶⣿⣿⣶⠏⠀⢀⣠⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⡇
⠀⠀⠀⠸⣧⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣿⣿⣿⣿⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⠇
⠀⠀⠀⠀⠻⣧⣽⣷⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⡶⠶⠶⠶⠶⠾⢿⣿⣿⣿⠷⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣾⣯⣼⠟
⠀⠀⠀⠀⠀⠈⠻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣟⣉⣉⣀⣠⣤⡤⠶⠖⠒⢛⣿⣿⣿⣿⡛⠒⠲⠶⢤⣤⣄⣀⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣧⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣋⣁⣀⣀⣀⣀⣈⣿⠋⠁⠀⠀⢀⣠⠾⢋⣽⣿⣿⣯⡙⠷⣄⡀⠀⠀⠈⠙⣿⣁⣀⣀⣀⣀⣈⣙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣻⠛⠛⠛⠉⠉⠉⢉⣵⠟⠋⠀⢀⣤⡶⠛⠁⢠⡞⠈⣿⣿⠉⢳⡄⠈⠛⢶⣄⡀⠀⠙⠻⣮⡉⠉⠉⠙⠛⠛⠛⣟⣧
⠀⠀⠀⠀⠀⠀⠀⣏⣿⢀⣠⣤⠶⠒⠉⠙⣿⠶⠶⢛⡿⠃⠀⠀⣰⠏⠀⢠⣿⣿⡄⠀⠹⣆⠀⠀⠙⢿⡻⠶⠶⣿⠋⠙⠒⠶⣤⣀⡀⣿⣹
⠀⠀⠀⠀⠀⠀⠀⢿⣻⡟⠉⠀⠀⣠⡶⠋⠀⠀⣠⡞⠀⠀⢀⡾⠃⠀⢠⠏⢸⣿⠹⡄⠀⠘⢷⡀⠀⠀⢳⣄⠀⠀⠙⢶⣄⠀⠀⠉⢻⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣄⣴⡿⠉⠀⠀⠀⣸⠟⠁⠀⣠⡟⠁⠀⣠⠋⠀⢸⣿⠀⠙⣄⠀⠈⢻⣄⠀⠈⠻⣇⠀⠀⠀⠙⢻⣦⣠⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣯⡀⠀⠀⠀⣼⠃⠀⠀⣴⡟⠀⢀⡜⠁⠀⠀⢸⣿⠀⠀⠈⢣⡀⠀⢻⣦⠀⠀⠘⣧⠀⠀⠀⢀⣽⣫⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣹⣷⣤⣾⣇⠀⢀⣼⣟⣠⡴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢦⣄⣻⣧⡀⠀⣸⣷⣤⣾⣏⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⣿⣛⣿⣟⣻⡿⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢿⣟⣻⣿⣛⣿⣧⣾⠟⠁

⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢰⡆⣤⠄⠀⠀⠀⠀⠀⠀⢠⣄⢲⠄⢰⣶⠢⠄⡀⠀⠀⠀⢀⠠⠔⣶⡆
⠀⠀⠀⠈⢳⡀⠀⠀⢀⡀⠀⠀⢀⡞⠀⠀⠀⢰⡇⣠⢆⡀⢀⡐⣤⠸⡆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢯⡀⠁⠈⣢⣀⢔⠁⠈⢀⡽
⠀⠀⠀⠀⠀⣙⠓⠀⢸⡇⠀⠚⣋⠀⠀⠀⠀⠸⠇⡇⢸⢸⡇⡷⢸⡢⠇⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠈⠃⠀⠀⠀⠸⣆⣀⠀⣈⣿⣁⠀⣀⣰⠇
⠀⠀⠀⠀⠀⣷⡀⠀⠘⠃⠀⢀⣿⠀⠀⠀⠀⠐⣯⣧⠈⢻⡟⠁⣸⣽⠆⠀⠀⠀⠀⢰⣷⡀⠀⠘⠁⠀⢠⣷⠀⠀⠀⠀⠀⢸⠃⣾⠏⠻⠹⣷⠈⡇
⠀⠀⠀⠀⠀⢈⡿⠛⠋⠛⠛⢿⠁⠀⠀⠀⠀⠀⠈⠿⣵⠀⠁⢠⡿⠃⠀⠀⠀⠀⠀⠀⠙⢻⠶⠆⠰⢾⡟⠁⠀⠀⠀⠀⠀⠈⠳⣤⡏⠀⢸⣤⠞⠁
⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⡀⠀⠁⠀⠀⠀⠘⠀⢀⡄
⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⡖⠰⠆⣄⡀⠀⠀⢀⡄⠶⠐⡆⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⢻⡀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠏⠀⠀⠹⡆⠧⠿⠸⣦⡴⠨⠧⠇⣾⠁⠀⠘⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣟⠧⠀⣤⠀⠼⣻⠇
⠀⠀⠀⠘⢻⡤⠀⠀⢸⠂⠀⠀⣴⠛⠀⠀⠀⠀⣷⢤⡀⠳⢼⠴⠃⢠⠴⡇⠀⠀⠀⢻⣄⠀⠀⠀⣿⠀⠀⠀⣰⠇⠀⠀⠀⣠⠀⠀⠀⢿⠀⠀⠀⣄
⠀⠀⠀⠀⠘⣷⣦⣀⠘⢀⣀⣶⡿⠀⠀⠀⠀⠀⠙⢦⡓⠶⠈⠰⠖⣫⠞⠁⠀⠀⠀⠀⠸⣄⠀⠀⠃⠀⠀⣰⠇⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠘⠛⠁
⠀⠀⠀⣦⠀⡤⣀⠀⠀⠀⢀⡠⡄⢠⡖⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⢦⡀⠁⠀⢠⠀⠀⠉⢠⡞⠁⠀⠀⣀⠀⠀⠀⠘⣿⡟⠀⠀⠀⢀⡀⠀⠀⠀⠉⣾⠑⠁⢠⡄⠑⢸⡇⠉⠀⠀⠀⠀⠀⢀⠸⣿⠇⡀
⠀⠀⠀⠀⢈⡓⠀⠀⢸⠃⠀⠐⣋⠀⠀⠀⠀⠙⠂⠀⠀⠀⢸⠁⠀⠀⠀⠚⠃⠀⠀⠀⠀⣜⠀⠀⢺⠇⠀⠘⡅⠀⠀⠀⠀⠀⣠⡙⠓⡛⠚⢋⣄
⠀⠀⠀⠀⠀⠳⣤⣤⠘⠠⣤⡴⠇⠀⣠⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠘⣆⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣰⠬⣉⡛⣩⣛⣉⠥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠇⠀⠀⠟⠀⠀⠀⠀⠀⠈⠛⠁⠲⠚⠓⠓⠖⠈⠛⠁
⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣀⣤⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢹⡏⢠⣿⠃⠀⠀⠀⠀⠻⣷⠈⢿⠃⠀⠸⡟⠳⠟⣢⣄⢀⣤⡚⠷⠛⡿⠀⠀⢹⡟⠛⠑⡀⠀⠠⠊⠛⢻⡇⠀⠈⢻⣯⠂⠀⠀⠀⠀⠀⠐⣿⡟⠁
⠀⠀⠀⠅⠀⣿⡀⠀⢸⠆⠀⢠⡏⠀⠈⠀⠀⠀⢷⠀⢰⣧⠈⡏⢠⣷⠀⣰⠇⠀⠀⠸⡇⠀⠀⠘⣶⠃⠀⠀⢺⠀⠀⠀⠀⢻⡄⠀⠀⣿⠀⠀⢠⡟
⠀⠀⠀⠈⢀⠘⢷⣤⡿⣇⣴⠟⢁⠈⠀⠀⠀⠀⠈⣅⠘⠃⠀⠃⠀⠛⢀⡍⠀⠀⠀⠀⣹⠒⠂⠀⡿⠀⠐⠒⡏⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠈⣿
⠀⠀⠀⠀⠈⢷⣦⣬⠀⠠⣤⣴⠟⠀⠀⠀⠀⠀⠀⢹⣶⡤⠀⠀⢠⣴⣾⠁⠀⠀⠀⠀⠹⣤⣴⠁⠁⠘⣦⣴⠇⠀⠀⠀⠀⠀⠹⣦⣼⠉⣧⣴⠏
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣶⠤⠀⠀⠀⢀⠠⣴⠆⠀⠀⠀⢀⣴⣀
⠙⠦⡀⠘⠷⢦⣤⠶⠟⠀⡠⠞⠀⠁⠀⠀⢰⡆⠀⠀⠈⠀⠀⠘⣦⠘⣦⣴⠏⣰⠏⠀⠀⠀⠀⢈⣿⡿
⠀⠀⠈⠱⣾⣿⣿⣿⡶⠉⠀⠀⠀⠀⠢⡀⠘⠀⣀⠄⠀⠀⠀⠀⠸⣷⡜⠇⣴⡗⠀⠀⠀⠀⠲⡴⢇⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⣠⡄⠀⠀⢠⣄⡀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠶⣆⠀⠀⡄⠀⢀⡶⠆⠀⠀⠸⣿⣶⡀⣠⣾⣿⠁⠀⠀⠀⠻⣿⣦⣴⣿⡟⠁
⠀⠀⠀⠸⣏⡯⠫⣏⡟⠀⠀⠀⠀⠀⠉⠉⢽⢏⠉⠁⠀⠀⠀⠀⠀⣿⠏⠷⢿⠃⠀⠀⠀⠀⠁⠀⢰⣦⡄
⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠁⠈⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠶⠛⠉
⠀⠀⠘⢶⡀⠀⠀⠀⣰⠏⠀⠀⠙⢿⣳⣦⣀⣀⣴⢟⡿⠁⢲⣟⢶⣄⠀⠀⣠⣶⣾⡷
⠀⠀⠀⢨⠃⠠⡧⠀⢫⠀⠀⠀⠀⠀⢽⡌⢹⠏⣡⡏⠀⠀⠀⠹⣄⡹⣧⣄⠏⣁⠝⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠓⠂⠀⠒⠃⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⢻⡿⠈⠁⢿⡿⠀⠀⠀⠀⠀⠻⠶
⠀⠀⣤⣦⣄⠀⠀⢀⣤⣦⡄
⠀⠀⠈⢿⣿⣷⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠙⠻⠻⡟⠟⠛⠁⠀⠀⠀⣮⡀⣀⣤
⠀⠀⠀⠀⢣⠋⠉⢣⠇⠀⠀⠀⠀⠀⠢⠄⠀⠀⠠⠞⠀⠀⠀⠀⠈⠶⠂⠀⠶⠃⠀⠀⠀⠀⠐⠿⠿⠇⠃

⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⣇⣹⣿⣿⣿⡿⢶⣄⡀
⠀⠀⠐⣿⢍⣽⣿⣿⣿⣿⣶⣝⡳⣄⠀⠀⠀⠀⠀⠀⠰⠦⣄
⠀⠀⠀⠙⠿⣿⣿⠋⣌⡈⢹⣿⣿⣮⣟⢦⡀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⢻⣸⣿⡆⠠⣽⡟⢩⣿⣿⡯⣳⣝⢦⡀
⠀⠀⠀⠀⠘⣯⣹⣿⡀⠐⠿⣾⣿⠋⣶⣿⣶⣽⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀
⠀⠀⠀⠀⠀⢿⣽⣿⣷⡡⠀⢾⣿⣿⣿⠿⣿⠍⠀⢙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢁⣹⣿⣷⣶⣼⣿⣿⣤⣤⣀⣁⠒⠦⣍⣿⡄⢀⣀⣤⠒⠈⠁
⠀⠀⠀⠀⠀⣼⣿⣯⣤⣴⡞⠁⠀⠀⠀⠀⠉⣙⣻⣿⣿⣿⣿⣾⣭⣹⣀⣀⣤⣤⣶⠶⢶⣾⠿⠿⣭⣭⣽⣯⣿⣽⣿⣲⡶⣦⣄⡀
⠀⠀⠀⢀⣰⣿⣽⣿⣥⣿⠀⠀⠀⠀⠀⠀⠉⠉⣩⣿⡿⠛⣿⣟⣿⣿⣿⣿⡤⢬⣅⠈⣸⣿⣿⣿⣻⣿⣿⡿⣿⡟⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠈⢻⢏⢸⣿⣤⡾⢧⣀⣀⣀⣀⣤⣶⠟⠛⠉⠀⣸⣿⣼⣿⢻⣿⣿⣿⢶⣤⣈⠉⠛⣿⣯⣴⣿⣟⣠⣿⠲⢤⠀⣿⣿⡿⢻⣧⣤⠇
⠀⠀⠠⢴⣿⣋⠈⢿⣿⣡⣶⠋⢁⣽⣿⠟⠁⠀⠀⠀⢠⡟⢲⣿⠃⢸⣿⡯⠛⣇⠈⠉⠳⢶⣟⣿⠿⠏⠙⠋⠈⣁⣥⣾⡿⢻⣷⠽⠛⠁
⠀⠀⠀⣸⣿⢟⠅⠛⢻⣿⣇⣠⣿⠿⠃⠀⢠⠀⠀⢀⣿⣙⡿⠃⠀⡐⠹⣷⠀⠈⠀⠀⠀⠀⢻⣿⣤⣈⣤⣶⣿⡟⣛⣯⠞⠋⠁
⠀⠀⠈⣿⣹⣷⣁⠀⢉⣿⣟⣿⣷⡶⠶⡀⡌⠀⠀⢸⠧⡿⠃⠀⠐⠀⠀⣿⣧⡀⠀⠀⠀⠀⢨⣧⠹⣿⡿⢻⣧⠿⠋
⠀⠀⠀⠉⢫⣿⣎⣠⣾⢿⡿⣿⠿⠀⠀⢸⡇⠀⠀⣸⠟⠀⠀⢀⡶⠀⠀⣿⣿⠓⣄⣀⣀⣴⣿⢹⣷⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⡛⣼⣀⣿⣀⡀⠀⢩⡇⠀⠀⠀⠀⠀⣠⣾⢠⣄⠘⣿⣿⠀⣾⡅⣹⣧⣽⣿⣯⣿⠟
⠀⠀⠀⢀⣼⠏⠈⠉⠓⠋⠉⠿⠉⠑⠤⠾⠁⠀⠀⠀⢀⡴⠏⠁⠠⢸⣷⣿⣷⣤⣿⣷⣿⠿⢻⣡⣿⣿
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⡀⠀⣿⣿⣿⣿⠛⡇⠉⠁⣴⣄⣿⡇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣻⣾⣃⣼⣿⣿⢀⣅⣰⣷⣽⡞⠉⠁
⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⣡⣿⣿⣴⣿⣻⣯⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡂⣻⣾⡿⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⢿⣷⣗⣒⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣈⣉⣋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢉⣝⣉⣀⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⣀⣀⠑⢦⡀⠙⢦⡀⢀⡀⢀⡴⠋⣀⠴⠉⢀⣀⣀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣀⣀⣀⣀⠀⠀⠑⠠⢹⣾⣷⡋⠐⠊⠁⢀⣀⣀⣀⣀⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡽⠿⠿⠟⡛⠛⠛⢺⣿⣿⡟⠛⠛⢛⠻⠟⠿⢯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⡠⠊⢀⡠⠂⠘⢻⡟⠁⠀⠀⡀⠁⠄⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠐⠖⢂⠋⠀⠀⠀⢾⡷⠀⠐⠀⠙⡀⠲⠂⢀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⡄⠀⠋⠠⠀⡠⠀⠸⠇⠀⢄⠀⠄⠙⡀⢠⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣧⡴⠁⠀⠀⠀⠀⠈⢦⣼⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣾⡟⣶⣿⣿⠻⣟⡲⢤⡀
⠀⠀⠀⠀⠀⠀⢸⣯⣟⡒⣾⡛⠫⢍⡻⣟⢶⣄
⠀⠀⠀⠀⠀⢠⣯⡼⣷⣤⣌⣉⡓⣦⣽⠛⢿⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⣼⣋⣻⣿⣅⣀⣀⡉⠳⣼⠓⠾⣶⠛⣷⡄⠀⠀⠀⡔⠀⢀⠤⠀⢀⣤⣶⣯⡛⣳⣯⠽⣿⢶⠇
⠀⠀⠀⠀⠘⣟⣹⡏⠁⠀⠀⠀⠈⠉⠻⡄⠀⠀⠙⢿⣷⠀⠀⡎⢀⠔⠁⣠⠾⣯⣜⡹⡧⠔⠒⠫⣽⡿⠋
⠀⠀⠀⠀⠸⣯⠛⠉⠉⠉⠒⠒⠒⠒⠲⠿⣆⡀⠀⠈⢿⡇⠰⢀⠂⣠⡾⠛⠉⢉⣽⣓⣉⠉⣻⣾⣷⠇
⠀⠀⠀⠀⠘⣿⣶⡒⠒⢒⣒⣤⣄⣀⠀⠀⠀⠙⢦⡀⠘⣿⣶⣦⡾⠋⠀⣀⣴⡋⠀⠀⠈⠙⣿⣹⠏
⠀⠀⠀⠀⠀⠀⠉⢙⣛⣿⡿⠿⠿⠿⣿⣶⣦⣤⣤⣝⣦⣿⣿⣋⠤⠒⠉⠀⠀⠈⠉⠒⠢⣴⣯⠟
⠀⠀⠀⠀⠀⠀⣾⣻⡿⠋⠉⠙⠒⣲⠖⠒⠛⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣖⠲⢤⣀⡼⠋
⠀⠀⠀⠀⠀⣴⣻⠛⠋⢉⣉⣭⠋⢁⣀⣠⠤⠖⣻⣿⣿⣿⠳⣄⠈⠱⡶⠭⢙⣿⡛⠛⠛⠁
⠀⠀⠀⠀⠀⣿⣷⠖⠉⢁⡠⠞⣩⠝⠋⢀⡤⣪⣾⣿⡟⣟⢧⠘⣦⣀⣙⠒⢄⣹⣻⠄
⠀⠀⠀⠀⠸⣷⣯⣤⡔⠉⢠⠞⠁⣠⠖⣩⢖⢿⣿⠟⠀⡟⡘⡆⢸⡈⠇⠑⡄⢹⣻
⠀⠀⠀⠀⠀⢼⣛⣿⣤⣴⠃⢀⡾⢁⣔⣵⠋⠘⠋⠀⠀⣧⡁⣧⠀⡇⢸⡀⢸⢿⠃
⠀⠀⠀⠀⠀⠀⠻⣧⣿⢻⣶⣋⣠⡾⠟⠁⠀⠀⠀⠀⠀⢹⡇⢿⣠⣧⣼⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣽⠧⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠉⠉⠙⣦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡆⠀⠀⠀⡠⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⢀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢠⡤⣀⠀⠀⠰⣄⠀⣠⣾⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠘⣇⠀⠀⠀⠈⠑⠂⠀⠙⣷⡿⢿⣿⡿⢀⣠⠠⠴⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠐⠀⠀⣀⠀⠘⣿⣷⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠂⠀⠉⠐⠾⣿⣿⣿⡟⠁⡀
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⠀⢀⡤⢶⣶⣾⣿⣷⣉⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠈⢳⠆⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣰⣟⢠⣟⣻⣿⣿⣇⠙⠛⣆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⢦⠀⣸⠟⠉⣾⣟⠺⢿⣿⡟⠀⠀⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠲⠆⣐⠦⠀⠀⠀⠄⠃⠀⠀⢹⣧⣴⣿⣿⠇⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⡌⢻⣿⡿⠋⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠰⢇⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠑⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡆⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠌⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠒⠀⡀⠀⠀⠀⠀⠀⢀⠈⡈⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢻⠀⠀⠀⠑⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⣠⡄⠈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠀⡇⠀⠀⠘⠧⠀⠀⠄⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⢰⠁⠀⠀⠀⡀⠳⣄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⢀⣄⠀⠀⡆⣾⠈⡌⠀⠀⡰⠃⠀⠈⠱⣄⠀⠈⠢⣆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣿⡄⠀⢸⡇⣠⠃⠀⠐⠁⠀⠀⠀⠀⠈⢷⡈⢄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠘⣷⠀⣼⠀⡏⠀⣴⡷⠃⠀⠀⠀⠀⠀⠈⣷⠼⣷⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠄⢀⠸⣧⡏⢠⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢇⠹⣆⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⢠⠄⠘⣧⡉⢀⢸⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⡇⠈⠨⣿⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⡀⡏⢢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⡿⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⠄⢹⣏⣋⡟⡿⢛⣉⣛⣏⣽⢛⣙⠋⣿⠋⠛⢉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣛⣻⣿⣟⣻⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⢀⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠄⠀⠀⠀⣾⠆⠀⠀⠀⠀⠄⠈⡀⠴⠘⡄⠁⠄⠀⠀⠀⠂⣰⠈
⠀⠀⠀⠀⠀⣠⣤⣾⢿⣧⣤⠜⠀⠀⠀⠀⠀⠠⠀⠀⠀⢘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⣴⠁⢌⡱⢐⠀⠀⢀⢈⠇⠈
⠀⠀⠀⠐⠀⢙⣿⣅⣏⣿⡟⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠈⠀⠀⢄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⢀⠒⠢⢀⠀⠀⠀⢀⠚⡆
⠀⠀⠀⠐⠀⠈⠟⠛⠛⠟⠇⠄⠂⠀⠀⢀⡨⠇⠂⢠⡈⢄⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠈⠀⠠⠈⠃⠀⠀⠀⠀⠀⠐⢧⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠐⠀⠀⠀⠀⣤⠼⠀⠀⠀⠈⢁⠀⠀⢰⣿⠛⠁⠀⠁⠀⠀⢄⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢻⡄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⣾⠁⠀⠀⠀⢀⣰⢠⡁⠠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠠⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠁⠦⠘⠘⠸⠁⠀⠀⡀⠀⠀⠀⠀⢀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⢄⠂⠐⣀⢆⠄⠀⠀⠀⣀⣸⣄⠂⠀⠠⠐⣾⡂⠀⠀⠀⠀⠁⠰⡶⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⡀⢀⠀⠀⠀⠀⣰⠇⠂⠀⢌⢄⠂⠤⡩⣚⠌⡄⠀⠈⠔⡷⠾⠉⠂⠀⠀⢸⡇⠁⠀⠀⠀⠄⠀⢰⠀⠀⠀⠀⣠⠁⠀⠀⠀⢀⡴⠂
⠀⠀⠄⢢⠼⢦⠂⠀⠀⠅⠀⠀⠠⠂⢌⠦⠐⡖⠀⢂⡱⡁⠀⠀⠀⠀⠀⠀⠀⠐⠸⣇⣀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡏⠀⢀⠀⠁
⠀⠀⠨⠜⠓⠋⠁⠀⠀⠀⠀⠀⠠⠎⠎⡜⠨⡰⢁⠃⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢏⠀⠀⠀⠀⢠⡆⠀⠈⠁⠘⠁⠀⠄⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠈⠚⠱⠱⠑⠂⠉⠀⠀⠀⠀⠀⡐⠈⡄⠀⡀⠀⡀⠈⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠙⠁
⠀⠀⢀⣀⠀⠈⠋⠈⢠⡦⠀⠀⠀⣤⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢥⢀⡖⠀⠀⡀⠁⠀⢀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠗⣀⠀⢂⠠⠀⢀
⠀⣠⡌⠋⢶⢆⡄⣁⢠⡀⠀⣠⡀⠉⢀⠀⠀⣁⠀⡄⠀⠀⠈⠢⠄⡠⢿⣿⠗⠃⢀⠀⠀⠚⠋⠀⠀⠀⠀⡀⠀⠀⢀⣄⠀⠊⠓⡌⠀⡄⠙
⠀⠀⠁⠨⢄⠈⠁⡄⠈⠳⡶⠙⢁⡀⠀⡔⢆⠉⠀⠁⠜⠠⠀⠀⢠⠀⢀⠋⠠⠀⠀⠄⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⠀⠈⡗⠋
⠀⠀⠘⠓⠀⢒⠀⢼⠀⠸⠆⠇⠘⠁⠀⠀⢀⣀⠤⠤⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡰⠖⠀⠀⠀⠀⠈⢈⠑⡼⠂⣀⢰⠆
⠀⠀⠀⣀⡀⣘⠃⠐⢦⠀⠋⠘⠓⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠴⠏⠀⠀⠄⠀⠟⢠⠀⠀⢹⠄⠒⠉⠀⠀⡀
⠀⠀⢠⡝⠃⡛⠀⠛⢀⢡⠀⠿⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣸⠛⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣇⠄⠀⠀⠀⢀⠇
⠀⠀⠈⠁⠀⣤⠁⠉⠉⠀⢳⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⠀⢸⡀⠀⠀⠄⠀⢀⠋⠉⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡸⠧⠀⠐⠀⠀⡘⠀⠀⠀⠈⠸⡟⠋⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢴⡗⠃⠀⠀⠀⠀⠀⢰⣶⡴⣾⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢸⠀⠀⠀⠀⠘⣤⢧⡤⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣾⣯⣝⣿⡷⠤⠂⠀⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠻⢅⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠈⣿⣿⣦⠀⢀⣼⣿⣿⠀⠀⠀⠀⣶⡶⠤⣀⠀⠀⣀⠤⢶⣶⠀⠀⠀⡏⡵⢆⡀⠀⠀⠀⡰⢮⢹⠇⠀⠀⣿⣿⣤⡀⠀⠀⠀⣤⣼⣿⠃
⠀⠀⠀⠀⠀⢙⡇⠹⣿⣿⠯⢜⠋⠀⠀⠀⠀⠻⢥⣠⣈⣶⣿⣁⣠⠸⠏⠀⠀⠀⠻⠿⣶⣒⣦⣤⣖⣷⠾⠟⠀⠀⠀⠸⣿⣿⣿⣦⣤⣾⣿⣿⠿
⠀⠀⠀⠀⠀⣿⠁⡖⠹⠉⠦⢸⣧⠀⠀⠀⠀⢀⣿⠷⣻⠟⠿⣛⠾⣿⡀⠀⠀⠀⠀⣶⣿⣶⢽⡏⣱⢿⣶⠄⠀⠀⠀⠀⣾⣿⣌⠿⠻⣃⣾⣿⡆
⠀⠀⠀⠀⢀⠿⡿⠀⠀⠀⠐⠿⢯⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠙⠿⠍⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠙⠿⠟
⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣾⢶⣤⡀⠀⠀⢠⣤⢶⡧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣿⠟⠤⠀⠀⠀⢀⠤⠚⣿
⠀⠀⠀⠀⠸⣿⣷⣿⣶⣴⣷⣿⡿⠃⠀⠀⠀⠈⠀⠐⠀⢰⡆⠀⠄⠀⠁⠀⠀⠀⠻⠄⣀⢈⣶⣶⣁⣀⠠⠿⠀⠀⠀⡔⠁⠙⠛⢳⣶⠘⠛⠁⠢⡀
⠀⠀⠀⠀⠀⣾⣾⡷⠻⠟⣾⣷⣷⠀⠀⠀⠀⠀⢠⠂⠀⢹⡏⠀⠠⡄⠀⠀⠀⠀⢠⣞⠁⣼⡿⣿⣧⠈⢳⡆⠀⠀⠀⠙⢳⠾⠾⢿⡿⠷⡶⡶⠚
⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⠛⠿⠋⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠽⡷⠟⠁⠂⠻⢶⠿⠀⠀⠀⠀⠀⠀⠑⠒⠃⠈⠒⠒⠁


⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢻⡉⠘⢷⣴⣴⡟⠅⣫⡇⠀⠀⠘⠧⡉⠀⠛⢾⡾⠋⠀⣉⠼⠃⠀⠀⠀⠀⠀⠐⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣶⢾⣿⣶⠸
⠀⠀⠀⠀⠀⣳⢤⣔⢿⠿⣠⢴⡏⠀⠀⠀⠀⠀⠈⢷⣤⡞⢳⣤⡿⠁⠀⠀⠀⠀⠀⠀⢀⢤⠀⠃⡤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡻⢿⣿⠇⣽⡯
⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠙⠿⠛⠁

⠀⠀⠀⠀⣴⣿⣟⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣟⣿⣷⣄
⠀⠀⠀⠸⣧⣼⡃⠀⠀⠉⠛⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠁⠀⠀⣹⣦⣿
⠀⠀⠀⠀⠻⣿⡟⣿⠿⠿⣿⣷⠿⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠷⣾⡿⠿⢿⡟⢿⣿⠃
⠀⠀⠀⠀⠀⠹⣿⠛⣶⠞⢻⣧⣀⣀⣀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢀⣀⣀⣠⣿⠛⢲⣞⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⠟⢷⠞⠉⠉⠙⠻⣿⣷⠟⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠳⣿⡿⠟⠉⠉⠉⢳⡟⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣄⣀⣠⣀⣀⠘⣇⠀⠀⠀⠈⠻⣦⡈⠉⠢⠀⠀⠀⠀⠀⢀⠊⠉⢠⡾⠋⠁⠀⠀⠀⣿⢁⣀⣠⣀⣀⣼⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⡏⠀⠀⠈⠉⠙⠻⠦⣄⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢀⣠⠾⠛⠋⠉⠀⠀⠈⣿⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣿⣶⠤⣤⣤⣄⣀⣀⠈⠙⠲⣄⡀⠀⠙⣧⡀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡴⠞⠉⢀⣀⣀⣤⣤⠤⣴⣾⣏⣉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⠁⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢮⣽⣦⣄⠘⣷⠠⣶⣦⢠⡟⠀⣠⣾⣯⡴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠙⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⣀⣀⣀⣀⣠⣀⣀⣈⣙⣿⣷⣸⣷⣿⣷⣾⣡⣾⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⣿⣿⣏⣹⣿⣀⣀⣀⣬⡽⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⢯⣄⣀⣀⣨⣿⣋⣿⣿⡿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⣀⣀⣤⡶⠛⠉⠀⠀⣠⡾⢻⣿⣿⣿⣿⡻⣦⡀⠀⠀⠉⠻⢦⣤⣀⡀⠀⢙⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠷⠛⠛⢉⣥⠞⠁⠀⠀⠀⣠⠞⠉⣰⢻⣿⣿⣿⣿⡳⡄⠙⢦⡀⠀⠀⠀⠙⢷⣍⡙⠛⠳⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⡤⠚⢁⣿⠀⠀⠀⣠⠞⠁⠀⡼⢡⠟⣿⢻⡟⣏⢷⠙⣆⠀⠙⢦⡀⠀⠀⢀⣧⠉⠢⢤⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣿⠁⠀⢀⠞⠛⡿⠖⣻⠋⠀⠀⡼⠁⣼⢰⡇⣾⡇⢻⡈⡇⠘⣆⠀⠀⠻⡗⠶⣟⠛⢦⡀⠀⠙⣯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⣷⡞⠁⢀⠞⠀⢰⠃⠀⠀⡼⠁⢠⣿⡟⠀⣿⡟⠈⢿⣿⠀⠸⡆⠀⠀⢹⡀⠈⢢⠀⠘⢷⣾⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣏⠀⣠⡏⠀⠀⣸⠃⠀⣸⡟⠀⠀⠈⠁⠀⠈⢿⡆⠀⢹⡄⠀⠀⣧⡀⠈⣷⢲⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣛⣿⣿⠶⢾⣿⣶⡶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣾⣿⡶⢶⣯⣿⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣥⣼⣷⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣾⣧⣼⠟⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⢿⠛⡿⠻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀⢀⠏⠀⠀⢃⠀⠀⡞⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢀⠊⠀⠀⠀⠱⡞⠀⠀⠀⠈⡄⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⡄⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠁⡏⠀⠀⠸⠀⠀⠀⠀⢀⡤⠖⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⣲⠟⣻⣀⣤⣄⠀⢀⠔⠉⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⣴⢛⣭⠵⠛⠉⠉⢀⣽⠔⠁⠀⢀⡞⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⠳⡀⠀⠀⢸⡀⡠⠒⢠⣾⣿⣿⣿⣯⣥⣤⣶⠶⠞⡟⠁⢀⠀⣠⢋⣤⠞⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠹⡄⠀⠀⠉⣿⡆⢸⣿⣿⣧⣿⣿⣃⢼⣿⢇⡀⠀⠰⠃⣼⡵⠋⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⣆⠠⠚⣿⢏⠈⠙⣻⠟⠻⢿⣿⢛⣥⣻⣷⠀⢁⡾⠋⠀⢀⡞⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⣀⡀⠈⠻⢦⡀⠀⢀⣀⣬⣧⡔⣛⣱⣶⠶⠚⢷⣿⣽⡿⣼⣴⠟⠀⠀⣀⡾⠞⠉⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠻⢦⣀⡿⣿⣷⣞⠻⢿⣿⣻⠷⢾⢻⣿⡾⣿⢁⢀⣤⠾⠉⠀⣠⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣛⣶⡶⠶⠶⠶⠶⣶⣿⣦⠼⣭⣿⡓⣮⣾⠛⢻⢟⡆⠹⣾⢇⣼⠟⠃⣀⣴⡾⠟⠋⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠀⠀⠀⠒⠊⠉⢁⣀⣽⣍⢿⣫⣥⣦⡟⡡⢶⡃⡩⠼⡿⢨⣾⣿⣴⠿⠿⡟⠁⢄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⣀⣠⣤⣶⣶⠿⠛⣫⣿⣿⣆⣶⣻⣶⣦⣽⢛⣿⣡⣋⣵⣿⠿⠋⠀⠀⡼⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠛⠋⠉⠉⠀⠀⣠⡴⠛⠛⠻⣿⣧⣬⣥⣿⣭⣤⡞⠋⠉⠀⣸⠋⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠴⠚⠋⠹⣤⡴⠶⡈⠀⠀⠀⠀⢀⣠⡞⠀⠀⣠⣾⣥⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠒⠋⠀⠘⣷⣾⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⣠⣤⣴⡶⠶⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠶⢶⣦⣤⣄
⠀⠛⢿⣿⣷⡶⠂⠀⢶⣶⠿⠿⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠿⣶⡤⠄⠠⢶⣾⣿⡿⠟
⠀⠀⠘⢿⣿⣧⡤⠤⣼⠃⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠘⣧⠄⢠⣼⣿⡿⠃
⠀⠀⠀⠀⢻⣿⣄⡤⢾⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⡷⢄⣠⣿⡟
⠀⠀⠀⠀⠈⢻⣿⣄⡤⠟⠓⠲⣦⣤⣄⣀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣼⠋⠀⠀⠀⣀⣀⣤⣤⠶⠚⠿⢄⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠘⣿⣯⣀⣀⠀⠴⣏⠀⠀⠉⠙⠳⢦⣀⠘⣿⣤⣤⣾⠃⣀⡤⠖⠋⠉⠀⠈⢹⠦⠀⣀⣀⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣅⡀⠀⠻⡗⠀⠀⠀⠀⠀⢈⣓⣼⣿⣿⣧⡚⠉⠀⠀⠀⠀⠀⠲⠟⠀⢀⣨⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣧⣤⠘⣷⠶⠖⠛⠉⠉⠉⢉⣹⣿⣿⣟⡋⠉⠉⠙⠓⠲⢶⣾⠃⣤⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣋⡠⠔⠀⠀⢀⡠⠊⡵⠃⢻⡟⠙⢮⠑⢄⠀⠀⠀⠢⢄⣙⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⣠⠏⢀⠞⠀⡄⢸⡇⢠⠀⢳⡀⠹⣄⠀⠀⠀⠙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠴⣄⠀⣀⣾⠁⢀⡎⠀⣸⠁⠀⠀⠀⣇⠀⢳⡀⠈⣷⡀⠀⣀⠥⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠉⡿⠉⡿⢶⡟⠀⢀⡟⠀⠀⠀⠀⢻⡀⠀⢳⣶⢿⠉⢿⠉⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣴⠀⣠⠀⣸⠁⠀⣸⡇⠀⠀⠀⠀⠸⣧⠀⠈⣇⠀⣄⠀⣦⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⡟⢲⡏⠀⢠⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡖⢻⣦⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣾⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⣷⣾⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣄⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈