For my friend FIRE Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠢⡀⠀⠀⠀⢠⡄⠐⠃⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⠀⠀⠀⢸⠙⢄⠀⠠⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠊⠀⠀⠀⠀⢇⠁⠙⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠺⠀⢀⠀⠀⠀⠈⠣⡈⠀⠉⠲⡀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠐⢸⠆⡞⡄⠀⠀⠀⠀⠙⢎⡂⠀⠙⡄⠀⠀⡠⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢄⠨⡀⣽⠀⠑⠜⣢⢳⡢⡀⠀⠀⠯⠀⠴⢹⠀⠀⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠐⢠⡈⠸⡄⠀⠀⠈⢀⠇⢘⡆⢀⡏⢰⠀⠨⠀⠀⣴⠁
⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⢹⠀⠰⠘⢀⠈⠦⡀⢀⡌⠁⠀⣧⠞⠠⢈⠀⡹⠀⢸⠏⡇
⠀⠀⠀⠀⠀⢃⠉⠲⣄⠀⠀⢣⡀⠡⠀⠑⢄⠹⡎⠠⠻⠀⠁⠔⠁⡐⢠⠃⠀⡼⢀⢽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣁⠀⠨⡄⠀⠀⠑⡆⠱⡀⠀⢂⢠⠁⠀⡧⠊⠀⡐⢀⢏⣠⠞⠁⠀⢸⠂⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠈⠻⣕⠦⣄⠀⠀⠘⡀⠀⠀⠘⡇⠀⠈⠀⠀⠰⠀⠟⢉⡀⠔⠆⠀⡽⠀⠀⢊⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡈⠳⡀⢀⡇⠘⠀⠀⠁⠀⠀⠀⠀⠀⡠⠂⠁⠠⠊⢀⡞⠀⠀⠘⡡⢔⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠄⢣⡼⢀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⣰⠏⣠⡖⢠⠞⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠇⠈⡈⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⢼⢹⢀⠗⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡟⢰⠘⡾⡘⠁⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⢢⡁⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠕⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⠠⠀⠄⡀⠀⠀⠐⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠄⠀⠐⠠⠂⠀⠂⠄⠀⠐⠠⠀⠄⠢⠒⠅⠂⠄⠠⠊⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⣷⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠠⡄⢸⠃⠀⠘⢟⡿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠁⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡜⠀⠀⠀⠀⣿⣿⡂
⠀⢠⣶⡶⠄⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⣼⡟⣨⠀⡀⣀⠀⣿⡂⠀⠀⣰⣾⣦⣀⡰⣦⣿⡻⣿⣿⣿⣦⣄⠀⠀⠀⠀⣿⣿⣿⣡⡄⠀⠀⣰
⠀⣿⠇⣿⣰⡿⣯⡆⢹⡇⠀⠰⣿⡟⠛⢋⣄⡋⢹⢶⣿⣿⣿⣿⣿⣇⣤⠄⢠⣿⣿⣿⣿⣧⠀⣠⣿⡀
⠀⣿⣿⣿⣿⡾⠋⠶⠀⢿⡆⠘⣿⣥⡿⢬⣀⣀⣸⣃⡴⠻⣿⣿⣿⠟⠃⢰⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇
⡔⢿⣿⣹⡟⠀⠀⠀⢲⣼⣿⣄⠘⢿⣷⡀⡿⠛⢻⣿⣥⠀⢠⣾⣷⡶⡾⠟⠉⠀⠛⣿⣷⣟⡟⠛⢿⣟⠀⠀⢀
⣿⣼⣿⣿⠇⠀⠀⠀⢨⠾⣨⠛⠶⠄⢉⣬⣽⣷⢚⡉⢫⡄⠈⠛⣿⣷⠄⠀⠀⠀⠀⠘⣿⣿⡟⠁⠈⠛⠀⡠⠋
⣿⣿⣿⣿⣦⠀⠀⣥⣾⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣾⣧⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡗
⣿⣿⣿⣿⣿⢸⣯⣇⣸⣟⣿⡄⠀⠀⠀⠘⢿⣧⣿⣿⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⡇
⢿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣿⣧⣄⠀⠀⠀⠀⠉⠀⠀⠀⠹⣿⠈⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⣸⢻⡇
⠘⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠻⣿⠷⣀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⣄⣾⡇⡿
⠀⠹⣿⣿⣿⣿⣻⣿⣿⣿⣿⣶⣾⣦⣌⡁⠀⠄⠀⣠⣠⡄⣾⠻⢿⣷⣦⣤⣤⣶⠿⠁⠻⠿⣿⢿⣿⣴⠃
⠀⠀⠘⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡀⠠⠾⢿⣂⣤⣌⣿⣋⣍⣀⠀⠀⠈⠠⣤⣿⠟⠁
⠀⠀⠀⠀⠙⢿⠟⠙⠟⠻⡿⢿⣿⣿⣿⣿⣿⣗⠈⣿⣠⣴⡟⠋⠉⠁⠀⠈⠉⠛⢷⡄⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠈⠛⡿⣿⣿⣿⣿⠀⣿⣿⠁⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠓
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⢀⡀⠀⠁⠙⣿⣿⣿⣿⡟⣿⣦⣿⡀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡦⡁⠀⡰⢢⡘⣀⠈⣿⣸⣿⣇⣹⡿⠋⠉⠁⠀⠠⢀⣾⡿⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠐⢷⣮⠳⣼⣳⣴⢂⣠⠈⡄⠈⢹⣿⡏⢿⠀⠀⠀⣠⣠⠆⢸⣿⡧⠀⠀⠀⠀⠀⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣿⣬⣿⣽⣿⣧⡀⠠⣿⣷⣿⣶⣠⣾⣿⠃⠀⢻⣽⠃⠀⠀⠀⠀⠀⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡆⣿⣿⡿⠷⡟⠋⠁⠀⠀⢿⡿⣂⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⢿⡿⣿⣿⣿⡟⠁⢻⣰⠀⠀⠀⠀⠘⣿⣿⣤⣠⣆⣤⣿⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢶⠀⢰⣿⣿⣦⠹⣯⢻⡤⠻⡏⣇⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠛⠀⣀⠻⣿⣿⡆⢿⡐⠇⠀⢱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣄⠨⣑⣿⡝⢿⡬⣧⢡⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⡁⠛⢈⠳⣿⡆⢷⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠈⠓⣻⣿⡈⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⢄⠀⠀⠹⣿⣇⢸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⢿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠀⠄⠀⢀⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠁⠀⢸⡇⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠂⠀⠀⢀⠀⠸⡁⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠘⠢⡀⠀⠀⠀⡀⠀⠄⠁⠠⡀⠀⠀⠀⠠⠀⢆⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⡓⠈⠀⠀⠀⠀⠄⠈⠁⢀⠈⢂⢀⢀⠐⠀⠌⠂⠀⠀⠀⠠⡄⠀⠀⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⠀⠀⡠⠀⠀⢆⠠⠀⠀⠂⠈⠣⠁⠀⠀⠀⠀⠆⢀⡈⠂⠀⠀⠀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠀⠬⠂⠀⢢⢂⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢄⠂⠀⠐⠀⠀⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡔⠀⠀⣧⠀⡀⢀⠀⠀⠀⠱⡀⠐⠀⠂⡑⢄⠀⢠⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⡀⠁⠀⠈⢢⠀⠠⢠⢠⢧⣀⠀⡴⠀⠀⠀⢹⠀⢀⠂⠐⡺⡄⠀⠠⠀⠀⣌⠄⠀⠑⠀⠘⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠠⠂⠱⡇⠈⡍⠀⠑⢮⠇⠀⠀⠀⢸⠀⠘⠀⢺⠃⢹⠀⢠⠀⢄⠋⢄⠀⠀⠇⠑⢀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢂⠀⠂⠀⡅⠀⠀⠈⡇⠀⠀⠀⢸⡀⠀⢠⠃⠀⢸⠅⠀⢚⠋⠀⢸⡆⠀⠁⡀⢰⠀⡄⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠠⢼⠑⢤⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠍⠀⠀⢸⠔⠁⠂⠀⠀⡸⠀⢀⠔⣀⢀⣀⠃⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⠸⡀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠘⡂⠁⠀⠀⡼⢡⠅⡁⢬⡄⠂⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠄⢀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⢸⠅⢘⣨⠜⢡⡏⠘⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠈⠀⠥⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡤⠋⠀⢠⠎⢠⡆⡁⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⢀⠂⠈⠀⡀⠀⢘⠢⣠⠁⢄⠀⣀⠀⠀⠠⠊⡥⠘⠀⡀⠀⠁⠀⠀⢉⡠⢪⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣁⣀⣐⣒⢶⣦⠥⠀⠀⠀⠀⠀⠒⠀⢐⡂⡈⠋⠂⠠⠀⢄⠀⠘⠉⠁⠀⠀⠀⠒⢋⣁⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣭⢹⣧⣤⠄⠀⣠⢲⠞⠫⡉⠀⢀⣀⣠⢶⠶⠦⣤⡆⡾⠠⠠⡒⣦⠁⠒⢲⠦⣌⠁⠚⠛⠛⢻⠯⢭⡠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠾⠶⠾⠶⠖⠛⢣⣚⠀⣶⣷⣿⣋⡻⣧⣸⡕⣤⡼⠁⣟⠃⡇⠀⢹⣧⡃⠘⡓⣿⣨⣈⡀⡶⡾⢺⣭⣥⠴⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠛⠷⠚⠀⠀⠹⢦⡶⠶⠯⠍⠙⠷⠥⠯⠭⠈⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠞⢋⣩⠉⠉⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣡⣶⡞⠁⠀⠀⠀⠰⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣿⢋⣠⣦⣴⣶⣤⡄⠘⠛⠛⠻⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣭⣹⣿⣿⣿⣿⣯⣀⣤⣄⣤⣄⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠋⢳⣦⣿⣿⣿⣿⣈⣿⣿⣿⣯⡭⣭⣿⡃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠹⣿⣿⣿⡿⠛⣿⣿⣿⣟⣭⣽⣼⣿⢦⡀⠀⣠⣶⣿⡿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠸⡿⣷⣼⠋⣸⣿⣿⣧⣼⣿⣿⣿⣿⣿⡀⣿⣿⣿⣟⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠇⠀⠀⢸⣃⣿⡿⢴⣿⣿⣿⣿⣷⣿⣿⣿⡿⣿⢿⣿⣯⠙⠋⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠀⢀⣀⡀⢨⣭⣷⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⣾⣫⣭⣭⡡⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢫⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠻⠷⣿⣌⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣏⣼⠉⣿⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠸⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠉⢙⣿⠛⣄⢤⠄
⠀⠀⠀⠀⢀⠀⣾⠂⣨⣿⣄⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣽⡆⠀⣠⣼⣿⣤⡿
⠀⠀⠀⠀⢸⠄⣱⢠⣾⣿⣿⣿⣿⣯⣿⣛⣻⣭⣽⣿⣿⣿⣿⣿⣯⣿⣿⠏⠀⢀⣿⣷⣿⣿⠿
⠀⠀⠀⠀⠈⠀⠛⠁⠈⣭⣿⣿⣿⣟⡛⣿⣿⣿⣿⣿⢻⡿⣿⣛⣿⣿⣇⠀⢀⣾⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿⠿⢿⣯⡿⡷⣾⣿⣾⣿⣿⣷⣶⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠃⢙⣿⣿⣿⣿⡾⢻⣿⣿⢿⣷⡊⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠘⠁⠀⠙⣾⣿⣟⡿⣱⣾⣾⣿⣸⣿⣽⣗⣾⡿⢽⣿⡏⣧⡙⠻⡷⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⣽⡟⢱⣿⣿⣿⣿⢿⠉⠁⢻⣿⣿⣷⣾⣿⣿⣿⣖⣧⣸⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣴⣿⣿⠃⣾⣷⣾⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡟⡟⠉⠙⠉⠙⠋⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇⠀⢸⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣾⡇⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠛⠋⠉⠉⠙⠛⠁⠀⠻⠿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢰⣿⣿⣿⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⣼⣿⣿⣿⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡇⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣧⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⢰⣿⣿⠋⣿⣿⡏⠀⢻⣿⣿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣿⣿⣿⣾⣿⡏⠀⣿⣿⡇⠀⠈⣿⣿⠈⣿⣿⣿⣿⣿⡇⠀⣧
⠀⠀⠀⠀⠀⠀⣿⣆⠀⢠⣿⣿⣿⣿⣿⡇⠀⢸⣿⣧⠀⠀⢿⣿⠀⣿⣿⣿⡏⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣦⣸⣿⡏⢻⣿⣿⣇⠀⠈⣿⣿⡄⠀⢸⡟⠀⢸⣿⡿⠀⣿⣧⣾⣿⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⢸⣿⣿⣿⡄⠀⠸⣿⡇⠀⣼⠃⠀⣸⣿⠁⢰⣿⣿⣿⣿⣠⣾⣿⠇
⠀⠀⠀⠀⠸⣦⡀⢹⣿⢿⣿⡇⠈⣿⡇⢻⣿⡄⠀⢹⣿⠀⠁⠀⢀⣿⠃⢠⣿⣿⠋⣿⣿⡿⢡⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⢻⣷⠀⠹⣿⡄⠙⣿⠀⢸⡿⠀⠀⠀⣼⠃⠀⣾⣿⠃⢰⣿⣿⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢿⠀⠀⢹⣿⠀⢹⠀⠸⠃⢀⣀⡈⠁⠀⠀⣿⠇⠀⢸⣿⡏⠀⠈⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡀⠸⠃⠀⠸⣿⠀⢀⣴⣶⣿⣿⣿⣿⣶⡀⠀⢿⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠙⣿⣧⠀⠀⠀⠘⢃⣴⣿⣿⣿⠻⣯⡙⢿⣿⣿⣦⡀⠀⠀⠈⣿⠇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠹⣿⠀⠀⠀⢠⣿⣿⣿⣟⢦⡑⣎⢻⣦⣽⣿⢿⣷⣄⠀⠀⡿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⣿⠀⠀⠀⢸⣿⣿⣷⣝⢦⡹⣎⣻⡿⣾⣿⢷⡙⢿⣧⡀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠛⠀⢀⣴⢿⡿⠻⣿⣬⣷⣿⣿⣿⣿⣼⣿⢷⡝⢶⣿⣿⣆⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣄⣴⣟⣵⣿⣶⣿⣿⣟⡭⢟⣩⣿⣿⣿⣿⣮⣻⣦⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⢟⣋⣭⡵⠾⠛⠁⣘⣭⠵⢛⡹⢿⣿⣿⣛⣛⣛⡻⠿⢿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⡿⢋⣥⡾⢋⣡⠄⣀⠠⠶⡋⢩⣶⣛⣥⡾⢿⣿⣿⠉⢭⣉⣛⠷⠦⣬⣝⣛⣯⣝⣛⣷⣶⣤⡀
⠀⣀⣴⣾⣋⣵⣿⡵⠞⠋⠵⢚⣩⠶⠛⣩⡶⣾⣷⣛⣽⣛⣥⣶⣿⣿⡿⠛⡳⠶⣭⣍⣛⣀⣀⢉⣉⠙⢻⣿⠟⠛⢿
⣼⡿⠿⣿⡿⢋⣠⠴⠃⣖⣛⡉⢠⣾⣿⡿⢟⣫⡽⢟⣹⣿⣿⣿⠟⠉⠶⣤⣙⡛⠾⢿⣿⣿⣯⣁⣉⣙⣿⡏⣴⡦⣼
⢿⡔⣧⡌⢿⣟⣡⣤⣾⣟⣩⣶⢿⣿⣷⣶⣯⣵⣾⣿⡟⣿⢭⣻⣷⣤⣀⡀⢩⣙⣃⣰⣼⣟⢹⣿⣙⣻⣿⣇⡽⢣⡿
⠘⣷⡙⣿⣸⣿⣿⣶⡾⢟⣫⣴⣿⣿⣿⣿⣿⣿⠷⣼⣿⣿⠶⣬⣉⠳⢶⣭⣁⣈⣉⣻⠟⡉⣿⣯⣭⣿⣧⣉⣵⡿⠁
⠀⠈⠻⣮⣙⣡⣿⣿⣿⣿⣿⠟⠛⠉⠀⠈⠙⠛⠿⢶⣭⣽⣿⢿⣯⣛⣿⣭⣙⡻⣿⡟⣀⣿⡿⠛⠛⠛⠿⠟⠋
⠀⠀⠀⠈⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣾⣿⣷⣬⣍⡛⣿⣇⣩⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣬⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡄⡏⠀⠀⠀⠀⠀⠳⢻⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⡜⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡔⢻⡀⡇⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀⢇⣠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠁⠀⠀⠀⢸⠁⢱⠀⡀⠀⠀⠀⠘⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⣀⣯⠀⠀⠀⠀⠀⡜⠑⠃⠀⠈⠛⠹⡀⠀⠀⠀⠀⢱⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡶⠇⠀⠀⠀⠀⠀⠀⠀⠸⠊⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠈⠒⠤⣄⣀⡠⠴⠊⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⣟⠒⠤⢄⣀⣀⡤⠤⢄⣀⣀⡠⠤⠒⣾⡵⢮⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣯⠟⣿⡀⠀⢀⣰⠁⡆⡞⢝⡟⣄⣀⡀⠀⡿⣝⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠯⠼⠒⠋⠉⠘⢄⠧⣉⡽⢃⠇⠀⠈⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢄⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡌⠀⠀⠀⢠⠇⠀⡠⣠⠀⡠⢖⡨⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⢰⢰⣾⠀⢸⣄⠀⠀⠁⣎⣶⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣇⢢⠸⡜⡜⣿⡄⢆⠉⠳⣄⢦⠙⠯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⢀⠠⠌⡢⣧⡱⣜⡘⣿⣎⣇⡀⢹⡆⣇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠀⠀⠀⢎⣦⢿⣜⢿⣿⣷⣘⣿⢸⣿⣿⢃⡟⠄⠀⣸⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠙⡎⢻⣿⣿⣿⡇⣾⣿⡵⢋⣤⣢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠇⠀⠀⠀⡀⢠⣇⢀⠻⣿⣿⣿⣿⣿⣴⣯⣍⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡛⢳⠶⡗⡤⢿⣦⣷⣿⣿⠿⠛⠛⠛⠉⠩⠑⠂⠫⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⠇⣸⠀⠈⠁⢂⡼⠛⠁⠁⠀⠄⠀⠀⠠⠆⠀⡀⢀⡞⡐⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠿⠧⠀⣶⠯⣤⣄⠀⠀⠀⢠⣞⣅⢠⠖⠶⠖⣾⣮⣧⣑⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⣰⡖⠄⠀⣷⠀⡘⣮⡇⣀⠼⠛⠛⣯⣼⠀⠂⢠⡿⣿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣊⣛⣠⣴⣿⠶⢤⣾⣟⣿⣆⠐⣀⣼⠟⠙⠏⠉⡁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠚⠉⠶⠀⠀⠈⠋⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⢀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀
⠀⠀⠀⣄⣾⣿⠏⠻⣿⣃⡀⠀⠀⠀⣤⣸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇⣄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣣⡀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣦
⠀⠀⠀⣿⡿⠏⢀⡀⢽⣿⡇⠀⠀⠀⣿⣿⡟⠉⢿⣿⣧⣶⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣸⣿⡿⠀⢹⣿⡇⠀⠀⠀⢠⣦⣽⡿⠃⠈⣿⣿⡆
⠀⠀⠀⢿⡇⢠⣿⣧⠀⣿⠇⠀⠀⠀⢿⣿⠁⠀⠀⠿⣿⠏⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠸⣿⠛⢀⣦⠈⢹⡏⠀⠀⠀⠘⣿⡟⠀⠀⠀⠋⣿⠇
⠀⠀⠀⠀⠙⠦⠍⠥⠚⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠂⠻⠿⠃⠋⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⠀⢠⡄⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄
⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⠀⠀⣄⣸⣿⣿⣿⣷⣿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⡀⣾⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⢠⣻⣿⣿⡆⢀
⠀⠀⠀⠀⣾⣿⣿⣿⠃⣦⠀⠀⠀⠀⢿⣿⣿⡟⣿⣿⣿⣡⠀⠀⠀⠀⣾⣿⣿⠏⢿⣿⣷⠀⠀⠀⠀⣿⣷⣿⣿⠈⢿⣿⣿⡆⠀⠀⢠⣀⣿⣷⣿⣿⣧⣾⡀
⠀⠀⠀⢦⣽⣿⣿⣿⣿⣿⠆⠀⠀⠘⣾⣿⡏⠀⡀⠙⣿⡿⠀⠀⠀⠐⣿⣿⠋⠀⢸⣿⣿⠀⠀⠀⠀⢿⣿⠻⠁⣀⠸⢿⣿⠇⠀⠀⢸⡿⣿⣿⣿⡿⣳⣿⡇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠿⡄⣼⣿⠄⠿⠁⠀⠀⠀⠀⠻⡧⠀⠀⠛⢸⠏⠀⠀⠀⠀⠈⠻⠀⣴⣿⡄⠸⠋⠀⠀⠀⠀⠻⢿⣟⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀
⠀⠀⠀⠀⡄⣰⣿⣿⣷⡄⠀⠀⠀⠀⢠⡀⠸⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢀
⠀⠀⠀⣼⣿⣿⡿⢿⣿⣿⡄⠀⠀⠀⣸⣧⣀⣿⣿⡏⢀⡀⠀⠀⠀⢀⡀⣿⣿⠙⢿⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⡀⣿⣿⣿⣷⣿⡇
⠀⠀⠀⣿⣿⠛⠁⠀⢻⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⡇⠀⠀⠀⢸⣷⣿⠋⢀⡀⢻⣿⡇⠀⠀⠀⢀⣸⣿⡟⠻⣷⣴⣷⠀⠀⠀⢸⣿⣿⣿⠟⢿⣿⣿⡇
⠀⠀⠀⠹⣿⡀⠀⠀⢸⡿⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢻⣷⢠⣿⣧⣸⡟⠀⠀⠀⠀⠈⢿⡏⠓⠀⠟⣿⠏⠀⠀⠀⠘⣿⣿⣿⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠂⢰⡆⡦⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡿⣮⢻⡄⢯⡅⠸⠇⠷⠀⠀⠀⠀⢼⠁⠀⡰⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⢮⡙⠦⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠡⠏⡠⢺⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⡉⢷⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⠃⣠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠴⠆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⢤⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠁⠈⠳⣤⡀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠢⡀⠹⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⢂⡷⠀⣸⢀⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣷⠀⡦⢸⡟⡔⣵⣊⠄⢀⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡰⠋⡠⢣⡿⠙⣇⠻⡇⢀⣼⡇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⠴⣋⣴⡏⠀⠀⠘⣦⣈⡎⢀⠈⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⡤⠊⠁⡞⡿⣿⣧⠀⠀⠀⣼⣿⣧⠀⠱⡄⠀⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣟⡀⠀⠀⢧⢁⢸⠁⢀⣴⠞⣿⡟⢻⡇⠀⢸⢀⣴⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣹⣧⠀⠀⠈⠻⣝⢰⡏⠀⠀⣯⣴⡿⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠢⣼⣿⡇⠀⠀⣿⣿⡷⣿⡇⠹⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⣭⣿⡷⣦⣿⣿⡇⠀⠀⣿⣿⠀⠘⣇⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡾⢛⣽⠀⠈⣿⣿⡇⠀⠀⣿⣷⠀⠀⢿⡰⡄⠙⠿⣷⣂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠵⠞⠁⠀⢞⣿⢀⣤⣾⣿⡇⢀⢠⣿⣿⣧⡀⢸⣿⣷⡀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠗⠛⣈⠙⣿⡿⡧⡄⠻⠿⣿⡟⠋⠚⠃⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⠃⠷⠞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠂⠁⠀⠈⠀⠀⠀⠉⠘⠀⠀⠋⠃⠀⠀⠂⠁⠀⠙⠘⠃⠁⠀⠁⠁⠙

⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠁⠈⠉⠻⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣠⡾⠷⣶⡀⠀⣼⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠏⠀⣠⡿⠁⢰⣏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃⢸⣿⡶⠾⠋⠁⠀⠈⠻⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⠀⠀⠀⠀⠀⣰⣦⣹⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣿⣄⣀⣰⣟⣣⣼⠛⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⣉⣉⣉⣉⣉⣋⣉⣴⠇⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡉⣿⠀⣉⣤⡜⠛⢋⣤⡜⢷⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣙⣉⡛⠷⠦⠜⠛⢿⣋⣡⠾⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⣿⠁⠀⣠⣤⡀⠘⢉⣁⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⠛⠻⣤⠄⠙⠛⠻⣤⠙⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⣿⣉⣁⠴⠶⠶⢦⣀⠈⠻⣤⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣍⡛⠛⢛⣭⡭⠿⠒⠲⢮⡝⢧⡄⠈⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⠿⠀⢠⣴⡛⢳⣦⠹⣾⡵
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣉⡻⠷⢄⣀⣈⡿⢋⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣄⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⠻⣿⡟⠋⢩⣭⠕⠛⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⣠⣴⠶⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⠿⠋⠉⠀⠀⠘⢿⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣤⣤⡾⠛⠉⠀⢀⣠⣴⣾⣿⣷⡄⠹⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣠⣶⣾⣿⠿⠟⠛⢱⣿⣿⠆⠹⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⢁⣽⣿⣿⠀⢿⣿⣯⣭⠀⠀⠀⠀⢀⣉⣤⣤⣄⠻⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣤⣄⡀⢴⣦⣤⣦⢀⣾⣿⣿⣿⡁⠈⢻⣛⣫⣤⣶⣾⣿⣿⣿⠿⢻⣿⣄⠻⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣋⣴⣿⣿⣿⣿⣿⣷⡀⢸⣿⡿⠟⠛⠉⣿⣧⠀⠀⠀⠻⠟⠰⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⠻⠛⠻⠛⠛⠛⠻⣿⣿⣿⠻⣷⠀⠛⠃⠀⣀⣀⣈⣁⣤⣤⣤⣴⣶⣄⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢹⡇⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⢸⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠃⠸⡇⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⠈⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠟⣿⣿⣿⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢠⡟⢀⣬⣍⣉⣉⠉⠉⠉⠉⠉⠉⢹⣿⢀⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⣁⣤⣄⠻⣿⣿⢟⣼⣿⣿⣿⣿⡅⢸⡇⢸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⢸⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⢠⡾⠀⢙⠛⠛⠛⠛⠻⣿⣯⣉⣉⣿⣿⠟⣸⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⠁⠀⠿⣿⣶⣤⣄⠀⠀⠙⠻⠿⠿⠋⢠⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣿⣿⣿⠃⢀⣤⣤⣿⣿⣛⣿⣿⣿⣶⣤⣀⡀⠀⣼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⣋⣥⣶⣶⣌⠛⢿⣿⣿⣿⣿⠀⢀⣭⣛⠛⠻⠿⠿⠿⠿⠿⣿⣿⡿⣤⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⣿⣷⣤⣙⣿⣿⣿⣠⣾⡿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⣠⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⠟⠿⢿⣀⡀⠈⠙⠛⠿⢿⣿⣿⡆⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠻⠶⣤⣄⠀⠀⠈⢉⣼⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢀⣴⠟⠙⠻⢶⣴⡿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠟⠉⠉⠁⠀⠀⠀⠙⣷⡀⠀⢀⣴⣿⣁⠰⠶⠶⠦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⣿⠛⣭⡍⠛⠛⠒⢶⣭⠷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⣧⠀⠐⠿⠟⠤⠟⢠⣾⡃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢛⣉⣉⣭⡱⠶⢄⣀⣀⡔⢣⡼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣁⣿⡁⠛⠻⠍⠩⠛⠋⠁⣠⠞⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⣛⣉⣛⣉⣤⠶⠄⠀⠰⠞⣡⠾⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡁⣿⠉⠉⠙⠶⠖⠛⠳⠚⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⢛⣀⣤⣤⢀⣰⣟⡛⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⣿⠉⠉⠲⣦⣏⣉⣛⣃⣼⠆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡙⠛⠛⠓⠳⣆⣀⡙⠻⡅⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣿⠛⠋⠸⣟⣙⣛⣀⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻⡿⣷⡟⠀⠀⠀⠈⠉⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠸⣿⢦⣤⣄⠀⠀⠀⣰⡟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢻⣄⠀⠛⣷⠀⢸⣏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⠻⢶⣤⣿⠃⠀⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣤⣀⣀⠀⣀⣠⡾⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠛⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠒⠒⠒⠚⠛⠒⠒⠛⠓⠒⠚⠒⠒⠓⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⠀⠀⣠⠀⠀⠀⢀⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⣠⣿⣿⣿⣿⡿⠃⠀⢠⣾⣇⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡄⠀⠀⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢿⣿⣿⣿⠃⠀⠀⢹⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⡀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠙⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⠇⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢰⣿⣿⣿⣷⣤⠀⠀⢻⣿⣿⣿⣿⣷⢸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⢠⣿⣿⡇⠈⣿⣿⣿⡿⠁⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⡇⠀⢹⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⡇⣸⣿⣿⣿⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⣿⣿⣿⣿⣄⣠⣾⣿⣧⣿⣿⣴⣿⣿⣿⡿⠀⣼⣿⣿⣿⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣼⣿⣿⣿⣿⢇⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⡇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⡙⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠛⠿⢿⣿⠿⠟⠛⠛⠛⠿⠿⠟⠛⠉⠉⠉⠛⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠿⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣤⣤⣤⣤⣤⣤
⠛⠛⠛⠛⠻⠶⣦⣄⣀⣀⣀⣤⣾⠿⠋⠁⠀⠙⠻⠷⠾⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣽⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠀⠀⣀⣴⡾⠿⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠶⠶⠶⠶⠾⠿⠟⠋⠁⣀⣤⣶⣿⣛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⠶⠶⠶⣦⣤⣤⣄⣀⡀
⠀⠠⠤⠶⠶⠶⠶⠶⠟⠋⠁⠀⠀⠈⠉⠛⠳⠶⣤⣤⣤⣤⣤⣤⣤⠶⠶⠾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⠶⢶⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⢀⣠⣴⠶⠶⠶⢶⣤⣄⣀⣀⣀⣀⣠⣤⡶⠞⠛⠉⠉⠉⠛⠛⣛⣲⣶⣶⡶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠓⠲⠤⢤⣀⣀⣀
⠚⠋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠉⠉⠁⢀⣠⣴⠶⠟⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣄⡉⠉⠛⠛⠛⠿⠿
⠀⠀⣠⣤⣶⠶⣶⣦⣄⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⢀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣉⣲⣤⣤⣤⡀
⠾⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠶⠿⠛⠋⠉⠉⠀⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣤⣶⣶⣦⡀⠀⣀⠴⠊⠉⠀⠀⢀⣀⣀⣌⣉⠁
⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⣀⡀⣀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⡴⠶⠶⠛⠋⠅⠀⠖⠈⠉⠙⠿⢶⣶⡶⠶⠞⢛⠉⣉⢈⡉⠛⠶⣤⣤⣤⣤⣤
⠀⢤⣤⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⢛⠛⠛⠛⠉⠉⣉⠁⠄⠐⢂⣘⡡⠊⠐⠊⣇⠘⠃⠀⡀⠰⠀⢠⠄⣈⠀⠀⠄⠀⢀⠀⠀⢠⠀⢠
⠀⢀⡀⢀⡀⠀⠠⠠⠆⠠⠂⠄⠂⠄⢩⠀⠄⢋⡍⢂⡈⢁⢈⡰⠀⠒⡤⢀⠲⠶⠆⣤⣀⣀⣈⣉⡵⠆⠀⠆⠐⠖⢀⡩⠔⠒⡈⡔⠂⠸⡁⣈⢁⢀⢈
⠀⠆⠉⠒⢀⣁⣤⡬⠐⠂⢀⣴⣦⣶⡀⣴⠾⠟⢐⣠⣴⠆⢀⣁⡉⠀⢃⠨⠜⢠⡆⠙⡫⣴⠋⠀⣠⣿⠁⠉⠈⠉⠀⠀⠀⠀⠰⡄⠍⠠⠠⠈⢄⠂⣠
⠐⢂⣴⣿⠿⣉⣁⣴⣶⢠⣿⢛⣿⣿⣷⢄⣾⠿⠿⠋⢁⣴⣿⣿⣿⣷⠈⡁⠀⢸⣇⠀⠀⣿⢀⠜⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠠⠕⠐⢀⠄⠆⣀
⠀⢼⠿⢃⣴⣿⣿⣿⠏⠘⠁⠿⢛⡍⢁⠙⢁⡄⠀⢃⠎⠉⠙⢿⣿⠟⠀⢀⠄⠘⣿⡄⣰⣿⠈⠰⣿⡇⢣⠀⠀⠀⢀⠠⢒⢩⡰⠌⠃⡐⠔⠂⡃⠒
⢀⡀⣠⡾⢿⡛⠉⣀⡀⣰⣶⡶⣠⣤⠉⣤⠀⠰⠆⡈⠀⠀⢀⠀⠙⠟⠀⠢⣄⣿⡈⢿⡽⠇⠀⠂⠈⠑⠶⠄⠐⠈⠆⠰⠘⢡⡡⠄⡁⢈⠀⡑⠉⡀⢒
⠈⠁⢀⣤⡌⠁⢾⡿⢇⣈⢉⣀⡉⢡⣶⢡⡦⢀⡀⡇⠀⠀⠀⣽⣦⡔⠀⠶⣿⢿⡶⢺⡏⠀⠀⠀⣰⣿⣿⣿⣿⣶⡛⡀⢓⠀⢤⠰⠄⢀⠁⡁⠦⠉⣈
⠸⣿⠎⠛⢣⣶⣶⡀⢙⣯⣬⢹⣶⠴⡷⢀⣤⣬⣥⠰⣦⣤⣤⣿⢿⣿⣿⣻⠋⠈⣁⣭⣶⣶⣶⣿⣿⣿⣿⣿⣿⠟⢠⡉⣨⡁⢄⠀⠆⢈⡠⡍⠰⠆⣉
⠀⣰⣶⣦⠘⠻⢛⣀⠘⠛⢡⣶⡿⠿⠶⢛⣥⡈⣵⣦⠙⠛⠛⣡⣤⠉⢉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⡛⠰⠀⣬⠁⣤⡰⣦⢲⢀⣄⠛⣠⡉
⠀⠙⠛⠡⣿⡷⠘⠟⣁⢀⣴⣤⠰⣿⡇⠻⣿⠇⠈⢡⣾⡆⠀⠙⠋⣾⣿⣿⣿⣿⠟⢩⡄⠰⠏⣛⠛⣟⠉⠶⠉⠷⠛⡀⠶⢠⠆⠠⠄⣬⠡⡍⠖⣠⠘
⢀⣴⣶⡆⢠⣤⠀⣼⡏⠘⠿⠟⠀⣴⣶⡄⢠⣶⣦⡈⢙⡁⢰⣿⠆⠙⠛⠛⣩⠘⠇⣨⡅⡶⢈⡛⢀⣉⠸⠆⣛⡐⢊⡋⣶⠛⠐⠟⠐⠆⢺⡘⠇⢰⣦
⠻⠟⣫⡄⠘⠛⠀⠙⣿⣦⠀⣤⣤⠙⢋⣤⡀⠙⠋⢰⣿⡿⠀⣁⡀⣿⢇⣤⠉⠰⠆⣩⢁⣀⠘⢃⡈⢋⣤⠀⢹⣥⠙⠃⢠⣶⡄⣶⡆⠿⠇⠁⠟⠀⣤
⠀⣾⡿⢣⣶⣿⣷⠀⣼⣿⠀⢿⠿⠁⠻⡿⠇⣴⣶⡄⣉⡀⢸⣿⠇⢰⠄⠁⢴⠆⣤⡉⠈⠋⣀⡙⢋⡈⢋⡀⢛⣏⢠⣶⡮⠉⢸⠎⢁⣀⠘⠿⠀⣀⠙
⠀⠀⠀⠈⠛⠛⠁⠀⢨⣷⣤⠀⢾⣿⡷⠀⠀⠙⠋⠰⣿⠟⠀⠀⠶⠀⠐⠿⠀⠀⠀⠀⠛⠀⠉⠁⠛⠁⠻⠋⠘⠛⠀⠉⠁⠿⠂⠀⠘⠿⠃⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡄⠀⠀⠀⢀⠜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣇⣠⠤⠴⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⣄⠀⢠⡟⠀⠀⠀⠀⢀⡜⠀⠀⡀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠢⣼⡇⠀⠀⠀⢀⡎⠀⠀⣸⠇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⢀⠔⡙⠀⢰⠀⣀⠴⠂⠉⠉⠉⠉⠉⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡀⠀⠀⠘⢦⠏⢀⠇⠀⢸⠞⠁⠀⠀⠀⠀⢀⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠑⠲⢤⡀⠀⢣⠀⠀⢀⠀⢇⠀⠀⠀⢸⠀⠀⡇⢠⠏⠀⢠⢠⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠱⡄⠀⡇⠀⢸⡄⢸⠀⠀⠀⢸⠀⠀⠑⣼⠠⡀⢠⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⢀⠇⠀⢠⡟⢦⡀⠀⠀⠈⡄⠀⠀⠈⢇⠱⠘⡆⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡏⠀⠀⢀⡇⠀⠈⠳⡄⢠⠁⠀⡀⠀⠈⡄⡆⣇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⡁⢀⡄⠀⠀⡆⠀⣾⠀⢀⡄⠀⢸⠀⠀⢾⠇⡀⠀⣇⣠⠟⠏⢘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⢸⡇⡄⠀⣃⡞⢹⠀⡎⠀⠀⡎⠀⠀⢶⢀⡆⢠⠙⢇⠔⠉⡠⠊⠵⠒⠒⠂⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣆⠘⣾⡻⡀⠀⣿⢀⠘⡄⠃⢀⠈⡇⠐⢦⡸⡼⣧⣼⠀⡞⠀⢸⠀⠠⡀⠀⣠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢦⡀⠙⢷⡀⢿⠸⡄⢱⠸⠼⡆⢧⠀⠘⣷⡙⢹⣿⣮⡇⠀⢸⠀⢷⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡏⠒⠦⣝⠺⣷⠀⠘⠂⣸⠃⠀⣀⡼⢡⣇⠿⢟⣿⠀⠀⠀⡆⣸⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠹⣆⡘⣌⢳⡈⠦⠀⣴⣿⢠⠞⠁⠀⡞⠹⡄⠙⠁⠀⣤⡀⡃⢡⣤⣾⣄⣶⡤⢒⠶⣠⣄
⠀⠀⣀⣤⡄⢤⣖⣶⡶⢒⢺⡛⣋⣉⡙⢦⡈⠙⠪⠳⡇⠀⢸⠘⣿⢸⠀⠀⠀⢻⢃⠆⣴⣾⣼⠟⣠⣿⣿⣿⣿⣛⣯⠭⠼⢞⣽⣯⡃
⠀⢼⠁⢻⣿⣷⠲⠷⠿⠰⠚⠙⠁⠀⡩⠟⠉⢑⡒⣾⠁⠀⠀⠣⠹⠉⠖⠀⠀⣫⢵⣬⣯⣭⣴⢾⣯⣭⠥⣈⣭⡋⠀⠀⠀⠀⠙⣿⠇
⠀⠀⠙⠋⠦⠶⠶⠶⣶⢶⣶⡶⠖⠋⣀⣤⣶⡿⠟⠇⠀⣴⠋⠙⠦⣘⠉⠉⠳⠭⠽⠯⠭⢿⣟⣛⣓⠧⢺⡁⢸⣷⡶⣶⢾⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣵⣄⣤⣾⣿⣟⣥⠴⣮⣿⠷⣦⣿⡓⣤⡈⠑⠢⢄⣀⣀⠀⠀⠀⠀⢀⣈⡉⠉⢣⣘⠙⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⠒⣿⣿⡿⠟⠹⠶⢾⣿⠿⣥⠿⣿⣿⣝⣿⣦⡀⠀⠉⠻⢿⣿⣿⣿⡛⠂⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠻⣽⣿⣿⣿⣶⣴⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⢫⠟⠞⢳⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣏⢟⣦⡀⠙⠁⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⠶⣤⣄⡈⠻⢞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢵⣚⡽⢶⣄⠉⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⢹⡭⠖⢚⣯⠽⢧⡀⠀⠀⠀⣰⣯⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⡀⠀⣴⢟⣿⠀⠀⠀⣯⠅⣦⡄⠚⣩⣧⠀⠀⠀⣏⠐⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣻⠃⣸⣋⣩⡭⢷⡀⢠⣟⠀⡇⠹⡄⠴⢸⡆⠀⣶⣜⢦⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡟⠀⡿⢅⠐⣫⠭⣧⣾⠅⢠⠇⠀⣇⣨⢽⡇⠀⣿⣚⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣻⡀⣿⣸⡷⣄⠋⠿⣆⢀⡞⢴⠂⡏⣒⣿⠀⣰⣛⠅⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣟⡱⡌⠳⢦⢁⡞⠐⡯⢸⠃⢽⡇⣴⣫⠥⠐⠒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡘⢿⡷⢦⡀⠀⢸⣇⣹⡄⠈⢻⡇⢨⠆⣾⢐⣻⣿⡥⠖⠀⣘⣩⠏⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⡄⢹⡭⢿⡄⣸⢗⠂⡇⢃⡄⠀⠔⠁⣿⠨⢜⠿⠚⢁⣨⡽⠋⢰⡯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢸⣗⣩⠿⣛⡉⣸⠃⣫⠆⠀⠋⠀⠸⡄⠉⣰⡟⣠⡼⢁⣄⠘⢧⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢱⡔⢯⠵⣰⠃⣘⡵⠂⠀⠀⢠⣆⠹⣴⠃⣷⠴⡇⢸⣏⡷⡄⢳⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⣴⣛⢃⡞⢹⡈⡽⠃⠠⢟⠉⠀⠀⠀⠈⣾⡆⠈⠀⢹⣚⣿⣸⠷⣚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠥⢠⡞⠀⢸⡟⠁⠀⣚⡅⠀⠀⠀⠀⠀⠀⠳⣣⣄⠈⣷⢺⠿⣊⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣛⢡⠏⢀⠄⠘⠃⠀⠀⠘⠃⠀⠀⠀⢚⠀⠀⠀⠻⠋⠀⢘⢁⣿⣵⠟⣱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠅⣾⠀⣴⡃⠀⠀⠀⠀⢀⠄⠀⠀⠀⢈⠖⠀⠀⠀⠀⠀⣼⢴⡿⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⣀⣈⣷⡂⢹⡀⠞⠔⡀⠀⠀⠀⡴⡂⠀⠀⠀⠱⣢⠄⠀⠀⠀⢸⡯⣽⠃
⠀⠀⠀⠀⠀⢠⡞⡄⣀⠀⠀⠈⠉⠙⢶⡄⣧⠀⠈⠀⠀⠀⣰⡯⠁⠀⢀⠀⠘⡵⢋⠀⠀⢀⡘⣧⢾⡙⠛⠒⠛⠉⠉⠙⠛⢦
⠀⠀⠀⠀⢀⣸⣷⡇⡇⣿⣼⣤⠶⠖⠚⠓⠋⠙⣆⠀⠀⢚⣡⣴⠖⠚⠉⠳⠤⡔⣫⢴⡏⠉⠉⠉⠙⠳⢦⣴⣠⡄⣆⣄⢠⢸⡆
⠒⢻⠯⠭⠽⣾⣿⣷⣷⡯⣛⣉⢍⡳⡄⢀⢀⢀⠉⠙⣲⠟⣩⠭⠭⢛⠲⣴⣄⢻⣴⠋⡄⡤⡄⢀⠀⢠⣾⣷⣍⣧⣏⡟⣸⢸⣧⣄⣀⣀⡀
⠀⠀⠀⠉⠉⠶⠒⣚⡗⣇⢳⢬⡇⢱⣿⢸⢹⢸⣷⠀⡇⡴⣻⢮⣽⣊⢳⠘⣿⣾⠿⠾⣷⣷⣇⣿⠃⣿⣿⠙⣿⣿⣷⣷⡿⠯⠭⠭⠭⠭⠥⠤⠶
⠀⠀⠀⠀⠀⠀⡤⠭⢷⣼⣌⣿⠴⣫⢿⣞⣼⣾⣾⣞⣧⡻⣌⣛⣳⢊⡞⢠⣿⣿⠛⠉⠩⡽⠾⣿⡄⡿⡿⢻⣿⣗⣂⡐⠒⠛⠛⠛⠛⠛⠛⠙⠋
⠀⠀⠀⠀⠀⠐⣒⣋⡭⠽⠶⠛⠚⠿⣟⠱⠶⠖⠀⠰⠶⢛⣲⣬⣭⣭⣶⠛⠒⠂⠀⠀⠀⠀⠋⠩⠽⠿⠶⣾⣿⣿⣓⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠽⠽⠏⠉⠁⠀⠀⠀⠀⠀⠀⠭⠵⠶⠒⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠐⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠕⠀⠀⠂⣸⡀⠀⠀⢀⠠⡀⠀⠀⢘⠀⣁⠀⠀⠀⠀⠂⠀⠀⠈⣿⠉⠀⠀⠘⠶⠀⠀⠀⠈⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⣄⠀⠀⠀⣴⠄⠀⠀⣀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠤⣄
⠀⠀⠀⠁⠀⠀⠀⠑⠒⠊⠀⠀⠀⠁⠀⠀⠀⠨⠂⠁⠀⠀⠀⠒⠶⠀⠀⠀⠀⠹⠀⠀⠀⠈⠙⠀⠀⠀⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠴⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠈⢟⠁⠀⠀⠀⠙⠂⠀⠀⠀⠈⠇⠀⠀⠈⠀⠁⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⣀
⠀⠀⠦⠤⠀⠀⠀⠘⠐⠃⠀⠀⠀⠾⠀⠀⠀⠈⠉⠁

⠀⠀⠐⠒⠀⠀⠀⠰⠀⠀⠀⠀⠈⡖⠀⠀⠀⠠⠀⠀⠀⠀⠀⢖⠁⠀⠀⠀⠄⠀⠀⠀⠀⠒⠂⠀⠀⠀⡊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠂⠀⠀⠀⠐⠶⠖⠀⠀⠛⠡⠎⠁

⠀⠀⠀⡀⠀⠀⠀⣠⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠄
⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣅⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠛⣷⣄⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠄⠀⠈⠻⣷⣄⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠈⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠙⢷⣄⠀⠀⠀⠀⠀⣠⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠙⢷⣄⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣀⣀⣤⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣶⠾⠟⠛⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣛⢿⣍⠉⣛⢶⢾⡇⢸⡷⠶⣛⠉⣩⡿⣛⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣾⣋⠉⣭⠭⠭⢍⡛⠷⠈⣿⢺⡇⢸⠗⣿⠁⠾⢛⡩⠭⠭⣭⠉⣙⣷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⢷⠈⠆⠀⠀⡞⠀⠀⠿⣤⡙⢋⣤⠿⠀⠀⢳⠀⠀⠰⠁⡾⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠘⣧⣼⠀⠄⠀⠸⠗⣚⣭⣭⣭⣿⣿⣭⣭⣭⣓⠺⠇⠀⠠⠀⣧⣼⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠅⣀⡀⠀⠀⠀⣀⣠⣼⠏⠀⠀⡼⠃⠘⢧⠀⠀⠸⣧⣄⣀⠀⠀⠀⢀⣀⠠⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣟⡴⠞⢹⢀⣤⢖⣫⣽⡶⠒⠀⠀⠈⢧⠀⠀⡾⠁⠀⠀⠒⢶⣯⣝⡲⣤⡀⡏⠓⢦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡵⢿⡸⢟⡴⠋⢰⠏⠀⠀⠀⠀⠀⠈⡇⢸⠃⠀⠀⠀⠀⠀⠹⡆⠙⢦⠻⢇⡿⢮⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢧⡼⠁⠀⢸⠀⠘⣟⠻⣷⡄⠑⠇⠸⠊⢠⣾⠟⣻⠃⠀⡇⠀⠈⢧⡼⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣇⢀⣀⣀⠁⠀⢀⠘⢧⠀⠛⠥⢍⢻⡆⠀⠀⢰⡟⡡⠬⠛⠀⡼⠃⡀⠀⠈⣀⣀⡀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣉⣥⡄⡼⢠⣤⠼⠆⢸⡇⠀⠀⠀⡿⠁⠀⠀⠈⢿⠁⠀⠀⢸⡇⠰⠥⣤⡄⢧⢠⣬⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠋⡀⡇⢷⠘⣆⠀⠀⠈⢧⡀⢇⢠⣧⣤⡤⢤⣤⣼⡄⡸⢀⡼⠁⠀⠀⣰⠃⡾⢸⢀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢸⢧⣻⡘⡄⠈⠲⠤⠄⠀⠑⣾⣈⠦⠘⢧⡼⠃⠴⣁⣷⠊⠀⠠⠤⠖⠁⢠⢃⣟⡼⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠧⠘⠖⠛⡆⠀⠀⠀⣯⣓⣤⣀⡸⢅⣀⣤⣚⣽⠀⠀⠀⢰⠛⠲⠃⠾⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠴⠊⢳⠀⠘⢤⡀⠀⠈⣿⣛⠉⠀⠀⠉⣙⣿⠁⠀⠀⣤⠃⠀⡞⡑⠢⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣅⡼⢻⢸⠀⠀⠈⠁⢀⡀⠻⣧⡄⢀⡀⢠⣼⡟⢀⡀⠈⠁⠀⠀⡇⡟⢧⣨⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⣿⢀⣤⢶⡆⣸⣁⡀⠹⠿⣿⣿⠿⠏⢀⣈⣇⢰⡶⣤⡀⣿⢻⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣇⣾⠁⢸⣇⣬⡆⣏⠀⠀⠈⠁⠀⠀⣸⢰⣧⣸⡇⠈⣷⣸⢏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠿⠀⠀⣿⠁⣧⠸⣠⠀⠀⠀⠀⣄⠇⣼⠈⣿⠁⠀⠟⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠈⠀⠹⣆⠉⢀⡠⢄⡀⠉⣰⠏⠀⠁⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠀⠀⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠂⠀⠀⠀⠀⠀⠀⠆⢸⢢⡀⠀⠉⠳⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⣀⡁⠱⠄⠀⠀⣀⡀⠀⠀⢠⡯⠔⢹⠀⠀⢠⢃⡀⠠⠀⠀⣠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣟⠓⢄⠀⠀⠈⡏⠳⣠⠏⢀⡄⢸⠀⠰⠂⡸⠙⡆⠀⠀⠃⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⡅⠀⠀⢹⠀⠈⣆⠀⠀⡿⠔⡛⠰⠁⡅⢸⠀⠀⡼⠋⠀⢹⠀⢠⠁⠱⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠘⡟⠲⢤⣀⣻⠂⠂⠘⢆⠀⡏⡀⢀⠀⠀⠁⠘⡄⡼⠁⠀⠀⡔⢠⣿⠐⠁⡀⣀⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠤⣠⠾⠛⠣⠀⠦⡈⠳⠇⠈⠂⠀⠀⠈⢄⠹⠃⠆⠀⢰⡴⣣⠃⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠐⡂⢤⠀⠑⠠⡀⠂⠀⠀⠀⠈⠄⠠⠀⡀⠙⢹⢃⡴⠋⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⡀⠸⡄⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠂⠀⢠⠆⢸⠊⠄⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢙⣄⡇⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠘⠘⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠌⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣣⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢰⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡵⠁⠀⣀⡀⣀⢀
⠀⠀⠀⠀⠀⠸⡟⣄⠈⠉⣁⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠮⣁⣤⢤⣤⣀⣀⣠⡆
⠀⠀⠀⠀⠀⢰⠛⠛⠋⠉⠉⠁⠀⡠⢴⠪⣦⡀⠀⢀⣰⡂⠄⢴⣄⣦⡀⠀⠀⠒⠒⡲⢦⣤⣀⡀⠀⠀⢀⡎⠎⣧
⠀⢀⣀⣀⡀⣉⢀⣤⡀⠠⠂⠒⣼⠀⠺⠐⠄⢑⠀⢹⡏⠄⠠⣂⠩⠳⣿⠠⣄⡀⠀⢠⡁⠀⣿⣧⣤⣦⣼⣴⢛⡟⠠⠀⠤⠊⠒
⠀⠀⠀⠀⠈⠉⣄⣻⣥⡤⠾⠛⠿⣦⠀⡀⠠⣸⢠⣾⣥⠆⠀⠜⠀⣀⣯⣇⠈⡙⠳⡶⢂⢼⣿⣃⠀⠀⡏⣕⣿
⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠘⠳⠷⠟⠛⠁⠀⠱⣾⣈⠀⣾⡦⠀⠉⠳⢧⣄⣑⠎⢹⠏⡉⠉⠛⠛⠛⠁⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁



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

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢃⣴⣿⣷⣾⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⡀⠛⢠⡤⠿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣦⣣⣤⠒⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣋⣤⠖⠀⢀⠀⠀⠲⢶⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠠⠶⠾⢿⣭⣭⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠓
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⣿⣦⡝⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⣯⣍⠉⠉⠀⢠
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠌⠛⠃⠠⣰⡆
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⡇⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠁⣆⠀⠈⠀⢰⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡟
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡇⠒⣿⠇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣿⣿⠀⠘
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠃⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⡇⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⠃⠀⠀⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⡏⠀⠀⠀⣼⣿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣼⡿⠁⠄⠀⢠⣾⣿
⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠁⠀⠀⢀⣾⣿⡇⠀⠀⠆
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣼⢟⣿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠜⢁⣾⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣶⡾⠋⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⡿⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⠁⠀⡀⠀⠀⠛⠁
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠒⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⢰⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠐⠲⡄
⠞⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣦⡹⣿⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠂⡀⠀⠀⠀⢿⣿⣷⢹⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠐⢦⣄⠀⠀⠀⠸⠃⣼⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠋⠇⠀⠀⠀⢸⣿⣿⣦⣤⣴⣿⣿⡏⠀⠸⣷⣄⠀⠈⣿⡷⠈⢷⡄
⠀⠀⠀⠀⠀⢀⣆⠀⠀⢸⣿⡇⠀⠀⠀⠘⣦⣸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣿⣿⣇⠀⠸⡇⠀⣼⡗
⠀⠀⠀⠀⠀⢸⣿⣶⣴⣾⣿⠁⠸⣦⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡄⣼⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣾⣿⣿⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣰⣧⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢰⣆⢀⣤⣤⣤⣤⡄⠐
⠀⠀⠀⠀⠀⢀⣸⣿⣵⡿⢀⣿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠟⠁⣿⠗⣸⣿⣿⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡄⠘⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣤⣀⠀⠀⣿⣿⣦⣴⣦⣤⣤⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡷⣶⣿⡖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣦⣹⣿⣿⣿⣿⠟⢛⡻⣿⣁
⠀⠀⠀⠀⢀⣴⣿⣿⢧⣿⠋⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠛⣿⡏⠁⣴⡾⠃⢉⡻⠀⠠⠊⠉⠙⡆
⠀⠀⢀⠰⣿⣿⣿⣿⢸⣿⡄⠀⢻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⢷⣤⣴⣶⣿⣿⡿⢛⣿⣿⣾⣿⣇⠀⠀⠀⠰⠟⠁⠀⢀⡤⠒⠊⠁
⠀⠠⠀⠀⠀⢻⡙⠻⢸⡿⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢹⡿⡵⠛⠻⣤⡉⠉⠹⣷⣦⣄⡀⠀⠠⠆⠠
⠀⠀⠀⠀⠀⣀⡀⠀⠉⠀⠀⠘⠻⠾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢀⡀⠀⠀⣠⠆⠀⠉⠙⢿⣿⣶⡤
⠀⠀⠀⠀⣀⡘⢧⠀⠀⠀⠀⠸⡆⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⣼⢁⣀⣀⡀⠀⠀⠹⠟⠏
⠀⠀⠰⠿⠿⠿⠿⠷⠄⠀⠀⠀⠹⣤⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣾⣿⣦⠙⠛⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣴⣶⣦⣄⠀⠀⢠⣴⣿⣿⣿⡇
⠀⠀⠀⣤⣤⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⣿⣶⣿⣿⣅
⠀⠀⢴⣿⣿⣿⡿⣿⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣰⣿⣿⣟⠀⢘⡉⢹⣿⣿⡇
⠀⠀⠀⢿⣿⣿⣿⣻⣦⠀⣶⣺⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠈⢡⣿⡿⠻⠀⢸⡅⠀⣿⡿⠁
⠀⠀⠀⢨⣿⣿⠏⣿⠋⠀⠹⣧⢹⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⢀⣰⡿⠃⠀⠰⢶⣶⣿⣿⡿⠃
⠀⠀⠀⠸⣿⡇⠀⠋⠰⠂⠀⣤⠾⢿⣿⣿⡿⠋⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⠀⠀⠀⠀⠀⠈⠙⣿⣦⡀
⠀⠀⠀⠀⠈⠛⣆⠀⢠⠈⠄⠀⢀⠈⠻⠿⠷⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣶⣤⡀⣠⡾⠛⠉⠻⡄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠑⠠⠀⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠘⢷⡦⠀⡀
⠀⠀⠀⠀⠀⣼⣟⠀⢀⣦⡀⠀⠐⣄⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠙⠛⠋⠀⠉⠻⣆⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠠⣿⡅⠀⠀⠉⠀⠁⢠⣿⣇⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣠⣤⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠃⠀⢀⡀⠟⠒⠛⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠄⠀⠀⠠⣾⣿⣿⣶⠖⠋⡉⠛⣇⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⠃⣸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⡈⠻⠧⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⣿⣿⡇
⠀⠀⠀⠀⡀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿
⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⢀⣿⣿⠇
⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⢠⠲⡄⠀⠐⠀⢀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢃⠀⣄⠀⠀⠌⠀⠀⠆⠀⠀⠉⠀⠀⠀⠀⡄⠀⠰⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠈⠆⠊⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠄⠑⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠀⠀⠀⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠀⡰⠄⠐⠁⠀⠁⠀⠀⢀⠀⠀⠁⠀⠀⢀⣀⠀⢀⠀⠼⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⠀⠀⠄⠀⠗⠀⢀⣤⣤⡶⠒⠐⠈⠀⠀⠱⢆⠙⠧⠴⠄⠀⠀⠁⠚⠲⣥⠀⡀⠀⠀⠐⠀⠈⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠰⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⡚⠀⠀⠉⠃⠘⠀⣰⠒⠀⠀⠀⠀⠀⠑⢆⠀⠀⠂⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⢠⣍⠁⠘⣠⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠏⢀⣀⡀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⠒⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠞⠹⠁⠀⠀⠀⠀⠀⠀⠹⡤⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⣢⠄⠄⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡟⠢⠄⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠷⠀⣠⠈⠀⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣏⢁⣄⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠄⠀⠢⢺⠇⠀⢀⡥⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠡⠀⠀⠰⡄⢀⠀⠁⠀⠀⠍⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⣥⡌⠀⡀⡿⠃⠀⠀⠀⠀⠀⠁⠌⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠋⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡴⣤⡂⡹⠋⢁⣘⣊⡥⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠐⠈⡄⠀⠀⠀⠀⡘⠀⠂⠀⡀⠀⡄⠁⠀⠉⠰⡀⠒⠤⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⡒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡉⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠂⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⣠⢆⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣄⠀⠀⠀⠀⠀⠺⣄⡀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡀⠔⠈


⡿⢿⡿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠸⣄⡄⠂⠀⠀⠀⠀⠀⠀⡾⡇⢠⠇⠀⠀⠀⠀⠀⠀⠈⣉⣰⠀⠀⠀⠀⠀⠀⣴⡴⡀
⠀⠀⢠⡘⡰⢪⡕⠚⢀⠀⠀⠀⠀⣄⡇⢷⡾⣴⠀⠀⠀⠀⠀⣄⣠⣏⡍⠀⠀⠀⠀⠀⠐⠛⠛⣥
⠀⠀⠀⣷⢃⣾⠳⣉⠛⠀⠀⠀⠀⠻⠃⣰⣄⠹⠀⠀⠀⠀⠀⠈⢢⡻⣧⠔⠀⠀⠀⢰⣟⡆⣳⠛⠟
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠈⠋⠠⠈⠂

⠀⠀⠀⠀⠀⢘⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⢾⣿⢊⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⡀⠀⠀⠀⠀⠀⠠⢂⣿⡄⡄⠀⠀⠀⠀⢀⢠⣆⡿⡸
⠀⠀⠀⠀⢾⡘⢶⠀⠀⠀⠀⠀⠘⠹⡶⢆⠛⠓⡃⠀⠀⠀⠀⣦⣺⠙⣷⣷⡀⠀⠀⠀⠀⠹⡉⢘⠃⡆
⠀⠀⠀⢲⣸⡷⣾⡇⠀⠀⠀⠀⠀⣟⠃⣼⢆⡙⡇⠀⠀⠀⠀⢻⡏⢣⡅⠻⠁⠀⠀⠀⢠⣷⠿⡏⠲⡄
⠀⠀⠀⠀⠀⠾⠙⠀⠀⠀⠀⠀⠀⠉⠘⠋⠈⠋⠀⠀⠀⠀⠀⠀⠁⠋⠉⠀⠀⠀⠀⠀⠀⠘⠰⠙⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠈⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠾⠳⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⣠⡤⠶⠖⠛⠉⢁⣀⡤⠤⣄⠀⠘⢧⡀⠘⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡥⠴⠚⠋⠉⢀⡠⠤⠔⠚⠉⠉⠀⠀⠀⠈⢆⠀⠀⠻⣄⠈⠒⠡⠙⠡⢐⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠁⠀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⡜⡄
⠀⠀⠀⠣⡀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⣠⠏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⢀⠎⠀⠀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡤⠚⠁⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠘⠀⢀⡞⠀⣠⣤⡀⠀⡇⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⣨⣧⠔⠋⠁⣀⡤⠚⠁⠀⠀⠀⣠⠾⡆⠀⠀⠀⢇⢇
⠀⠀⠀⠀⠀⠀⡀⢀⠊⢀⠄⢀⡞⠀⠀⠀⠈⢻⣤⡷⠤⣄⣀⣟⠶⠶⠶⠶⢒⣫⠕⣃⡤⠖⠋⠁⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠈⠢⣕⡂⠤
⠀⠀⠀⠀⠀⠔⠠⠁⠰⠁⢀⡞⡴⠾⣯⠞⠉⠙⣿⢧⣀⣀⣀⣈⣩⠭⠭⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠱⠀⠀⠀⡞⠰⡁⢿⡏⠀⣾⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠉⠳⣤⣬⣿⣡⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠑⠒⠒⠲⠦⠤⠤⠤⢤⠶⢶⣶⣖⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⡠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣠⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣸⠃
⠣⣈⠢⢄⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠶⠚⠋⠓⠚⠋⣡⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣠⠴⠊⠁
⠀⠀⠑⠢⠬⣙⡒⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣬⣿⣿⠥⠤⠶⠒⠚⠋⣉⣁⡤⠴⠖⠋⠉
⣀⣀⣀⣀⣀⣀⣈⣉⣙⣒⣲⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣖⣒⣚⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣛⢏⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣦⣿⣿⣧⣴⣶⣶⣾⣥⣾⣮⣿⣷⡄⠘⠓⣦⠀⠛⣒⣶
⠀⠀⢀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣧⡀⢸⡻⣿⣿⡇⠀⢀⢀⠐⣾⢈⢦⣠⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣴⣶⢟⡤⣿⡋⢐⡆
⠀⠀⣼⣿⣿⣷⣷⡀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣇⠀⠁⠘⠋⠀⢠⡌⣎⢠⢦⡟⣿⣯⣾⣦⡰⠀⠉⠉⠉⣉⢉⢉⣉⣉⣉⡭⣁⡀⢀⡻⣿⣬⣽⠟⢻⣟⣾⡋
⠀⣼⣿⠟⠁⣿⣿⣧⠀⠀⠘⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⢀⢰⣾⣧⣿⣿⣾⢻⣿⢿⣼⣟⣷⣷⣤⡀⣜⣿⣏⠻⠬⣟⠚⠋⠛⠃⠀⠻⣷⠾⣷⣴⣿⡿⠟⠃
⠀⢻⣯⠀⠀⢨⣿⡿⠀⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠺⠿⠿⠛⠿⠻⠿⠻⠟⠛⠛⠛⠟⠛⠿⠛⢸⣿⣿⣿⠿⠇⠛⠆⠀⠀⠀⠀⠀⠀⠟⠛⠃
⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠀⡄⠠⠀⠀⠀⠀⢀⠀⡀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡆⣱⣸⠸⣄⡏⢴⠀⣆⡄⠀⠀⠐⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠿⠟⠛⢁⠀⠅⠆⠀⠀⠘⢧⣀⡴⠀⠀⠀⠀⢀
⠀⠀⠀⠸⡄⠀⠀⣸⢸⣭⣏⣷⣿⣾⣧⣿⣼⣿⣥⣧⠀⠀⠀⠀⠀⠀⠀⣰⣆⣄⣀⡀⠀⠀⠀⠀⠀⠰⠄⠄⠸⢸⣿⠏⠃⣨⢿⡿⣿⠴⡿⠀⠀⢀⡈⣲⡄
⠀⢀⠄⣼⠇⠀⠀⠘⠿⠿⠿⠟⠟⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⢀⣰⣺⣷⣿⣿⣿⣿⡀⠀⠀⠀⢀⡵⣴⣠⡆⠀⠀⠀⠀⠙⣿⣿⣶⡞⠁⠀⣠⠬⡝⢿⡄
⠀⣾⠘⣯⢰⡇⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⢠⡠⣄⠠⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⢰⢿⡐⣤⢿⣄⡤⠀⠀⠶⠶⠿⠿⠿⠷⠴⠆⠙⠓⣿⣿⠏
⠀⠸⣷⢸⣿⣻⡆⠀⢀⡴⣫⣿⣿⢧⠀⠀⢠⣌⢿⣾⣡⡘⢶⡄⠐⠉⢻⣿⣿⣿⣿⣿⡟⠠⣏⣿⣼⣏⣼⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿
⠀⣼⣏⣸⣿⣿⠁⠀⣼⡿⠁⠿⠿⣾⡇⠀⢢⣹⣿⣸⣟⣹⣤⠇⡄⠀⣼⣿⡿⢿⣿⠗⠀⠀⢹⡿⣿⣿⣿⣿⣾⣷⠀⠀⠀⠀⠘⣄⠄⠀⠀⠀⠘⠿⣷⣤⡀
⠀⠻⣿⣿⣿⡿⠀⠀⠙⢷⡀⠀⣰⠟⠁⠀⠸⣿⡿⢿⢿⠿⢿⣾⠟⠀⠈⢻⠃⢹⡿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⠀⢠⣴⣾⣿⣿⣶⣦⣤⡄⢀⣤⣾⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⠿⢿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠉⠁⣀⣀⣀⠀⠀⣀⣀⣀⠈⠉⠛⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⣀⣤⣶⣾⡿⠿⢿⣿⠀⠀⣿⡿⠿⢿⣷⣶⣤⡀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⣠⣾⡿⠋⢀⣤⣾⡿⠛⠉⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠉⠛⢿⣷⣤⡀⠙⢿⣷⣄
⠀⠀⢀⣼⣿⠋⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠙⣿⣧⡀
⠀⢀⣾⡿⠁⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠈⢿⣷⡀
⠀⣼⣿⠁⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠈⣿⣧
⢰⣿⠇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠸⣿
⣾⣿⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣿
⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹
⣿⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼
⢿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣠⣾⠀⠀⣷⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⠸⣿⡆⠀⢿⣷⡀⠀⠀⠀⢀⣴⣿⠟⠁⣠⣾⣿⣿⠀⠀⣿⣿⣷⣄⠈⠻⣿⣦⡀⠀⠀⠀⢀⣾⡿⠀⢰⣿
⠀⢻⣿⡀⠈⢿⣧⡀⢀⣴⣿⠟⠁⣠⣾⡿⠋⢸⣿⠀⠀⣿⡇⠙⢿⣷⣄⠈⠻⣿⣦⡀⢀⣼⡿⠁⢠⣿⡟
⠀⠀⢻⣿⡄⠈⢿⣷⣿⠟⠁⣠⣾⡿⠋⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠙⢿⣷⣄⠈⠻⣿⣾⡿⠁⢠⣿⡟
⠀⠀⠀⠻⣿⣆⠀⠙⠁⣠⣾⡿⠋⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣷⣄⠈⠋⠀⣰⣿⠟
⠀⠀⠀⠀⠘⢿⣷⣄⠀⠙⠿⣷⣦⣄⡀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⢀⣠⣴⣾⠿⠋⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠈⠙⠻⠿⣿⣶⣾⣿⠀⠀⣿⣷⣶⣿⠿⠟⠋⠁⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣄⣀⡀⠉⠉⠀⠀⠉⠉⢀⣀⣠⣤⣶⣿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠻⡆⣆⠈⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⠛⢡⣿⣿⠏⠘⣷⡈⢧⡑⣼⣟⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣡⢧⢀⡻⣿⣦⣻⣾⣿⡀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣵⣿⣿⢟⣥⢞⣳⣿⣼⣿⣿⣿⢿⣿⣿⣷⣵⣄⡻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⡄⠀⢉⠛⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⣻⢿⣿⣿⠟⠋⠙⠻⣿⡿⠋⢡⠀⣼⣷⣤⣼⣧⣴⡈⡛⢿⣻⣙⠻⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣥⣾⣿⣶⣇⢠⡄⣠⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠗⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣧⢤⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⢸⡿⣇⠀⠀⠀⠀⠀⠀⡔⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣼⣿⣿⠁⠀⠀⠀⠀⢀⢷⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣶⣴⡄⠀⠀⠀⠀⠈⡏
⠀⠀⠀⣾⠿⠿⣿⣰⠀⠀⠀⢠⡾⠿⠿⠷⠆⠀⠀⠀⠀⠀⠈⠛⠙⠋⠉⠉⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠲⠦⠠⠖⠂
⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠏⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢾⣦⡏⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣴⣿⣿⣼⡯⠀⠀⠀⠀⠀⣾⠱⢀⡤⠀⠀⠀⠀⠀⠀⣇⣿⡰⣤
⠀⠀⠀⢀⣶⣶⣶⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⣾⢿⡿⣿⣿⠟⠁⠀⠀⠀⣴⣿⣷⣿⣿⠘⠃⠀⠀⢀⣰⣾⣿⣼⣷⣿⣄⣀
⠀⠀⠀⠘⠛⠟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠘⣈⠙⠀⠁⠀⠀⠀⠀⠘⠛⠛⠿⠟⠋⠀⠀⠀⠀⠚⠛⠛⠟⠛⠛⠛⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠶⠦⠤⠤⠤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠋⠉⠁⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣈⣈⡉⠉⠙⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠴⢶⢺⠛⢹⠙⠟⢉⣡⠈⠐⠛⠉⢉⣸⣇⡉⠉⠛⠂⠑⣌⡉⠻⠫⡏⠛⡓⡶⠦⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣦⡜⠂⠿⢀⣤⠿⢇⠄⠀⢐⣾⣿⠏⠻⣿⣷⡂⠀⠠⡸⠻⣤⡀⠿⠘⢣⡶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢾⡰⠀⣿⣿⡤⢃⣤⢶⣿⣯⠉⠀⠀⢉⣽⣿⡶⣤⡘⢤⣿⣿⠀⢆⡷⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢻⠀⠘⡗⠀⢰⠀⢀⣾⢁⣾⡧⠈⠿⡆⠴⠟⠁⢿⣷⠈⣷⡀⠀⡆⠀⡾⠣⠐⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⠀⠀⠸⣇⢸⠿⠀⢙⣷⠀⣠⡆⢰⣤⡀⣾⡋⠀⠟⡇⣸⠇⠀⠁⣤⣾⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠻⣾⡀⠀⠸⢿⣿⣁⡀⢀⣈⣿⡿⠇⠀⢀⣷⠏⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⡈⢣⠐⡄⡼⣻⣿⡷⢾⣿⣟⢧⢠⠂⠞⢁⣠⣤⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣾⠛⢍⣋⡃⢘⣙⡹⠛⣷⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣉⠉⢉⣉⡉⠉⢉⣉⣉⣉⣉⣉⡉⣩⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡷⠂⠴⠖⢻⢿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡤⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⡀⠀⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡁⠀⠀⠀⠀⢈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠾⠟⡛⢰⣏⠀⠀⣹⡆⢙⡿⢿⢶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣠⣿⣀⠈⠀⠈⠉⠉⠉⠓⠚⠉⠉⠉⠁⠈⠁⣠⣿⣄⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⣟⣧⣿⣛⣿⠶⣶⣶⢤⣤⣤⣤⣤⡤⣶⣶⢾⣿⣟⣿⣿⣻⠷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢭⣋⡱⠿⠏⢽⣿⠗⣱⣿⡷⠺⣿⣎⠺⣿⡯⠹⣿⣎⣙⡭⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢓⣿⠚⠂⠀⠀⠀⠀⠀⠀⡠⡆⣉⣉⠀⡦⡀
⠀⠀⠀⠠⣿⠋⠻⢟⡀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣠⣾⣿⠃⠀⠀⠀⢰⣤⣼⣿⣧⣀⠀⠀⠀⠀⠀⢸⠤⠿⣦⣤⡾⠧⢴
⠀⠀⢸⣿⠂⠾⢆⠹⣷⠀⠀⠀⠀⠀⣿⣆⠀⣿⣿⣿⡏⠀⠀⠀⠀⠘⠉⠙⠛⠋⢹⣇⠀⠀⠀⠀⠰⡀⢣⣸⣿⣠⠃⡰
⠀⠀⠀⠻⠿⠟⠺⠿⠋⠀⠀⠀⠀⠀⠙⠷⠄⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠈⠰⠮⠩⠶⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⣠⣶⣶⣿⣿⣶⣶⡄⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠺⣯⡄⢰⡆
⠀⠀⠙⠿⠿⠻⢿⡿⠋⠁⠀⠀⠀⠀⠻⢿⣏⠀⣴⣶⠀⠀⠀⠀⠀⠀⣿⠫⣄⠈⢻⡇⠀⠀⠀⠀⠀⠀⣀⣐⣿⣷⠟⣱
⠀⠀⠰⠀⠆⠰⠠⠆⠆⠀⠀⠀⠀⠀⠀⠻⢯⣿⠉⠉⠀⠀⠀⠀⠀⢸⣿⣧⣍⣥⣿⣷⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⡿⠋
⠀⠀⠀⠉⠈⠁⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠉⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⡇⠀⠀⠀⠀⣴⠒⠒⠒⠒⠚⣿⡷⠀⠀⠀⠀⠀⣤⣶⣾⣿⠷⠆
⠀⠀⠀⢀⣠⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⡟⠛⡛⠛⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡆
⠀⠀⠀⠀⢿⣯⣿⠇⠀⠀⠀⠀⠀⠀⠀⡇⣾⣷⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣶⣶⣆⣤⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣄⢰⠐⠣⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣟⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⢠⡀⠟⣠⡾⠀⠀⠀⠀⠀⠀⠃⠀⠈⡏⠀⠈⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⣼⣁⣄⣴⠃
⠀⠀⠀⠀⢙⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⢹⣿⣿⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⣿⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠙⢦⡀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣉⣦⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢹⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢷⣄⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣉⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⡴⠞⠃⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⣠⡶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣄⣀⣾⣉⣀⣼⣧⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⣿⣤⡾⠛⠛⠛⠛⢻⣶⡟⠛⠛⠛⠛⠛⠛⢷⣤⡀⢰⡟⠛⠛⠛⠛⠉⠻⠀⣸⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠈⠇⠀⠀⠀⠀⠸⡇⣽⠀⠀⠀⠀⠀⠀⡄⠘⣧⣾⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠇⠀⣇⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⣠⡆⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⡇⠺⠶⠶⠶⢶⠀⠀⠀⠀⢀⣀⣄⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠻⢤⡀⣿⣻⠁⠀⠀⠀⢞⡧⠀⠀⠀⠰⣄⣀⣀⡀⢸⠀⠀⠀⠀⢻⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣼⠃⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⣷⢸⠀⠀⠀⠀⢸⠌⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⣼⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠇⠀⠃⣸⠀⠀⠀⠀⢸⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠏⠉⠹⣆⠀⠀⢀⣸⣧⡀⣿⣤⣤⣤⣤⣼⠂⠀⠀⢹⡇⢀⣤
⠀⠀⠀⠀⠀⢀⣼⠋⢳⣶⡶⠒⠈⢁⣴⣿⠒⠒⠒⠒⠖⠊⣠⠷⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠒⠋⢀⣤⣄⠉⠉⠉⠉⠉⢉⣿⣿⡀⠀⠀⢠⡏⠀⠀⠀⠀⢻⠟⠙
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠛⠛⠋⠉⠁⠘⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠙⣿⠛⠛⠛⠛⢻⠉⢻⠛⠛⠛⠋⠁
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡰⠋⠀⠘
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣠⠔⠋⠀⠀⠀⠀⢺
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⠶⣄⣀⠀⠀⠀⠈⠓⣤⣤⣤⣀⠀⣀⣠⠶⠶⠶⢶⣄⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⣠⠶⠶⠦⣴⡏⠀⠀⣀⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣛⣛⣋⣀⣀⣈⣉⣛⣉⣁⣀⣀⣀⣀⣉⣻⣶⣶⣾⣛⣉⣁⣀⣈⣉⣻⣶⣴⣛⣁⣀⣀⣀⣈⣙⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⢠⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⡜⠀⠐⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠉⣐⡉⠈⠀⠄⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠂⠀⠀⠀⠀⠀⠂⠀⡉⠃⠀⠐⢀⠀⠴⠁⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣯⣤⣬⣉⣁⣂⡐⠀⠀⠤⠿⠽⠛⠂⠴⢂⣄⣀⣀⣐⣷⣃⠟⠀⢀⣀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠉⠙⠋⠙⠉⠉⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⡏⢹⡅⢰⡊⠁⣀⣠⣄⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⣒⣚⣻⣿⣽⡂⠨⡂⣼⡠⠊⠁⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣚⣋⡭⠭⣖⠀⠆⣼⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠈⠥⠀⢒⣒⣛⡭⠅⢘⠁⡇⠀⠀⠀⠀⢀⠜⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠍⠩⠭⠵⣖⣒⠀⠐⠀⡇⢀⠠⠤⠒⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⣨⡭⠭⠽⣒⠈⠥⢠⡗⢀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⣒⣺⠭⠭⠗⢀⠅⡼⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢂⣴⣾⣿⡟⠡⣡⠞
⠀⠀⠀⢀⣀⡠⠤⠤⣶⠒⠒⠒⢒⣈⣹⣦⣅⡀⠀⠀⠐⠄⠶⠀⢶⣾⣿⡟⣇⣡⣾⣿⣿⣿⣖⠒⠶⠶⠦⢤⣄⣀⡀
⠀⠀⣞⢉⢀⡀⠀⠠⡂⠀⠀⠀⠀⢄⠀⠉⠫⢉⣀⣀⠀⠀⠘⡛⢫⣭⣭⣍⣭⠽⠋⣀⠄⠈⡉⠉⠉⢓⡦⠀⣀⠙⡯⣷
⠀⠀⠈⠓⠿⣟⣒⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣅⣉⣀⣀⣴⣶⣶⣦⣤⡤⠬⠴⠖⣚⣫⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠦⠤⣤⣤⣤⣤⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣤⣤⣤⠤⠴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣧⡄⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠛⣿⣦⣭⣦⡀⠀⠀⠀⠀⠀⣀⣤⠞⢻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢻⣦⡇⠀⠀⠈⠋⠀⠀⠀⠈⢻⡍⠁⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣛⣻⣧⣀⣀⣀⠀⠀⠀⠀⣠⣿⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣎⠹⣩⣛⢿⣿⡿⠇⣄⣀⢠⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢁⣀⣼⣿⣿⣿⣿⣷⣄⣻⣯⣙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣶⣶⣿⡈⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣤⠀⣠⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⣼⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⣭⣽⣿⣿⢿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⢘⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣧⡠⠿⣬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣧⣴⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠉⠉⠛⠛⠛⢣⣼⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠈⠉⠉⢩⣽⣿⣿⣿⣟⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣧⢦⣄⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣹⣿⣧⡀
⠀⠀⠀⠘⢷⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢹⡇⢩⣿⣿⡇⠀⠀⠀⠤⠁⢸⣿⣿⣻⡿⠁⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠺⣿⣇⠀⠀⠀⢀⡀⠀⢸⠀⢾⣿⠘⡇⠀⠀⢀⠀⠀⣿⣿⡿⣿⡇⠀⣤⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢿⡗⠀⣀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠘⠁⠀⠉⠁⠙⣇⣀⢸⡀⣺⣿⡆⢧⠀⠀⠀⠀⢰⠁⣿⣿⣿⡇⠀⠉⠛⠇⠙⢿⣿⡀⠀⠀⡠⠀⠀⢰⣤⡀⣸⡟⠋
⠀⣠⡄⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀⠀⢹⠛⠻⢁⣿⣿⡇⠸⡄⠀⠀⠘⠎⠀⢻⣿⣿⠃⠀⢠⡴⡄⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⣿⣿⣧⣸⣿⣧⠄⠀⢀⣴⣶⠾⠃⣀⠀⠈⠀⢘⣿⣿⣿⠤⣿⠀⠀⠀⠀⠀⣸⣿⣿⣔⢿⣷⢴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣷⣦
⣿⣿⡟⠛⣯⣿⡀⠀⠛⠻⠿⠆⠀⠃⣤⣤⡤⠤⣿⢿⣿⣤⣘⣧⠀⠀⠀⠀⢻⣿⣿⡄⢘⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡿⠋⠉
⠉⠻⢥⣤⣿⣿⠿⠶⠀⣀⠀⠀⠀⣠⠚⠋⠀⢦⡙⡟⢉⣏⡉⣿⡆⠀⠸⠀⣿⣏⢹⣿⣿⣿⣿⡇⠈⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠺⠾
⠃⠀⠸⠃⠘⠁⠀⠀⠀⠀⠀⠲⠿⠋⠀⠀⠀⢠⣿⠛⠙⢻⣧⢸⣿⡦⠀⠀⠀⢿⣾⣿⣿⣿⡿⠇⠀⢀⠂⠀⠀⠀⠈⠓
⣤⣄⠀⣀⣴⣶⣦⣄⠀⢀⣠⡀⠀⠀⠀⠀⠀⠐⢒⡉⠹⢟⣻⣿⣿⠀⠒⠈⡧⢸⣿⣿⣿⡿⠁⠐
⣿⣿⣿⣿⣿⣿⣟⣉⣸⣿⣿⠁⠀⠀⠀⣀⣠⣿⣿⣿⣤⣼⣿⣿⣿⣰⣦⣤⣴⣾⣿⣿⣿⣷⣾⣿⣷⣦⣤⣄⣀
⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣷⣦⣄⣠⣴
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣏⣉⡑⢻⠈⣿⡛⣭⣽⡍⣙⣿⣿⣿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠟⠛⠛⠙⠿⢿⣿⣿⣿⣯⣤⣤⣤⣠⣤⣿⣧⣠⣤⣤⣍⣬⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠿⠟⠛⠛⠛⠓⠂⠀⠀⠀⠀⣀⣰⣿⣿⣟⣿⣿⣛⣙⣟⣫⡌⢹⣿⣏⣉⣁⣀⣼⣿⣿⣧⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣯⡍⠛⠟⠛⠛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⠛⠏⠿⠿⠿⣿⣿⣶⣾⣭
⣀⣀⣈⣉⣛⣛⣻⣿⣟⣙⣁⣨⣍⠧⢤⣄⣠⣄⣀⣀⣀⣐⣠⣼⣿⣿⣿⣿⣿⣯⣿⣯⣄⣨⠙⣿⣿⣿⣿⣿⣿⣿⣧⠤⠀⠠⣾⣿⣿⣿⣿⣿
⠉⠉⠉⠉⣡⣄⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⡿⠿⠿⠋⠉⠁⠀⠉⢻⣻⣿⣿⣧⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣈⣁⣉⣉⣛⣛⣛⣛⣋⣉⣽⣭⣁⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣶⣦⣽⣶⣄⠀⣤⣾⣯⣴⡶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⡿⠿⠿⠿⠿⡆⠙⣿⠋⢸⠿⠿⠻⠿⢿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣻⣿⣿⡿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⣟⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⣒⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠰⣿⣿⣥⣶⣾⣿⣿⡷⠄⠀⠀⠀⠀⢻⣄⠀⣰⡏⠀⠀⠀⠀⠠⢾⣿⣿⣷⣶⣬⣿⣿⠆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣿⢟⡋⠉⢉⣿⣦⡄⠀⠀⠀⠀⠀⠈⣿⣴⣿⠀⠀⠀⠀⠀⠀⢠⣴⣿⠉⠉⣙⡻⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢀⣾⣿⠟⠀⣄⣀⠀⢸⡄⠀⢸⣿⡇⠀⢰⡇⠀⣀⣠⠈⠻⣿⣧⡀⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⡿⠋⢠⣿⣿⣿⡇⢠⣬⢽⣿⣿⠀⠀⠈⣿⠀⠀⠈⣿⣿⡯⣥⡄⣸⣿⣿⣷⡀⠙⢿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠘⣿⣿⣿⣏⠀⠙⠶⣥⣿⠀⠀⠀⠛⠀⠀⠀⣿⡭⠞⠃⠀⣿⣿⣿⣿⠁⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣠⣴⠀⠀⠘⣿⣿⣿⡄⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣠⣿⣿⡿⠃⠀⠀⣦⣄⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣄⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣰⣿⣿⡟⠁⠀⠀⠀⢹⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⢀⣴⡆⠀⠀⠈⢿⣿⣿⡟⠀⠀⡀⣀⡀⠀⢀⣀⢀⠀⠀⢻⣿⣿⡟⠀⠀⠀⢰⣦⡈⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⠀⢄⡀⠈⣿⣿⡁⠀⠀⠙⠻⣿⣿⣿⠟⠋⠀⠀⢸⣿⣿⠁⢀⡤⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠈⢳⣄⢹⣿⡇⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⢸⣿⡇⣰⡟⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣿⣿⣿⣿⡇⢀⡆⠀⢿⣾⣿⣿⣄⡀⠀⠀⣀⣤⣀⠀⠀⢀⣠⣿⣿⣿⡟⠀⢰⠀⢻⣿⣿⣿⣿⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢁⣼⡇⠀⠘⣿⣿⣿⣿⡿⠋⠉⠁⠀⠈⠉⠙⣿⣿⣿⣿⣿⠁⠀⢸⣧⡘⣿⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⡇⠀⠀⢹⡿⣿⣿⣷⣤⠀⠀⠀⠀⢀⣤⣿⣿⣿⢿⡇⠀⠀⢸⣿⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢁⣼⣇⠈⠁⠙⣿⣿⣿⣷⣧⣤⣶⣾⣿⣿⡿⠃⠈⠁⣸⣧⡘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠻⣿⣿⣿⣿⣿⡄⢲⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⡆⣰⣿⣿⣿⣿⣿⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢻⣿⡟⢹⣿⣿⣌⣿⣶⣤⡙⢿⣿⣿⣿⠟⣃⣤⣾⣿⣱⣿⣿⠏⢻⣿⡟⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⢻⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠏⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⡈⠙⠿⠋⢀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠉⠻⣿⣦⠀⣴⣿⠟⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⠁

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢀⣼⣿⠇⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⣿⣿⡏⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠀⣿⣿⣿⠀⠀⢸⣿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⢸⣿⣿⣿⡀⠀⠈⣿⠀⡿⠃⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣦⠀⢸⣿⣿⣿⣷⡀⠀⠸⡄⠁⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠻⣆⠹⣇⠘⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠙⠀⢹⣿⣿⣿⣿⣿⡀⠀⣰⡆⠀⠉⠀⠀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠲⣤⡀⠀⣿⣿⣿⣿⣿⡇⠀⣿⣧⠀⠀⠀⢰⠟⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⢻⣿⡀⢸⣿⣿⣿⣿⣧⣴⣿⣿⡇⠀⠀⠈⢼⠋⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⡿⣿⣿⣿⣿⣿⡇⢀⣾⠃⠀⣠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣦⡀⠀⣸⣿⣿⣾⣿⣿⠇⠈⠻⣿⣿⣿⣀⣼⣿⡄⡰⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⢻⡄⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠀⠞⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠁⢿⣿⣿⢻⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⣿⣿⡄⠙⠿⣧⠀⠀⠀⢸⣿⣿⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⣸⠟⠁⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⢀⣼⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⢹⣿⣿⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣸⣿⣿⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣿⠉⠉⢹⣿⣿⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⢛⣿⣿⠛⠛⢻⣿⣿⠀⣿⣤⣤⣼⣿⣿⣤⣤⣼⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣨⣿⣿⣀⣀⣸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠟⠛⢻⣿⡿⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢀⣸⣿⣿⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠙⣿⣿⠉⠉⢹⣿⣿⠀⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣰⣿⣿⣤⣤⣼⣿⣿⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣾⣿⣿⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢨⣿⣿⠀⠀⢸⣿⣿⠀⣿⣿⡟⠛⠛⣿⣿⡟⠛⠛⣿⣿⡇⠸⢿⣿⣿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣴⣿⣿⣤⣤⣼⣿⣿⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⣿⣿⡟⠛⠛⣿⣿⡟⠛⠛⣿⣿⡇⠈⢹⣿⣿⠉⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⠶⠿⠿⠶⠖⠚⠛⠛⠀⠛⠛⠃⠀⠀⠙⠛⠁⠀⠀⠛⠛⠃⠠⠼⠿⢿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣁⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣉⡁
⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣴⠏⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢿⣆⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠈⢿⣦⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⢻⣇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠋⠀⠀⠤⠤⠫⠤⠀⠀⠀⢉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢂⣉⣩⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣌⣉⣁⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⡷⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢸⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⠐⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠡⠤⠤⣔⣂⠁
⠀⠀⠀⠀⡤⠒⠊⠉⣁⣤⣴⣶⣶⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⣶⣶⣦⣤⣈⠉⠑⠒⢤
⠀⠀⠀⠀⠑⠢⠤⣀⣉⠛⠻⠿⠿⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣉⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢖⡐⠒⠀⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠄⠀⠒⢂⡲⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⣼⡟⠁⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⢀⣼⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⣼⣿⠁⣸⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⣠⣾⠇⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⢀⠃⢰⣿⣿⢀⡇⢸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⢀⣼⣿⡏⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⣿⠀⢸⣿⡟⠸⡇⠋
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⣰⣿⣿⡏⠀⣠⣾⣿⣿⣿⣿⠋⢠⠂⠀⢰⡿⠀⢸⡿⢁⡆⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⣸⢿⣿⡟⢀⣾⣿⣿⣿⣿⠟⠁⡰⢃⠆⢠⣿⠇⠀⡞⢁⡾⢱⠆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠁⢸⢻⡿⢰⡏⣸⣿⣰⣿⡿⢋⣾⡿⠋⠄⡼⣡⠟⣴⡿⠃⠀⠀⣰⣟⣷⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⠏⠀⠀⣼⠈⡇⣿⠀⣿⣿⣿⡟⢀⣾⣿⢃⡏⣼⣷⣯⣾⡿⠁⢀⣴⣾⣿⠟⠁⠀⠀⣀
⠀⠀⠀⣶⠀⣼⢿⡀⠀⠀⢸⣿⠀⢠⠀⡟⠀⠃⠟⢸⣿⣿⠛⠀⣾⣿⠇⣾⢡⣿⣿⣿⡟⠀⣰⣿⣿⠟⠁⠀⣠⣴⣿⠇
⠀⠀⠀⣿⠀⠸⣟⣧⠀⠀⢸⡇⠀⣾⣴⡇⠀⠀⠀⢸⣿⠇⠀⢸⣿⡿⢸⣿⢸⣿⣿⣿⡇⣼⣿⣿⠃⢀⣴⣾⣿⡿⠃⢀⠆⢀⠄
⠀⠀⠀⣿⡄⠠⢹⣿⡆⠀⢸⠃⢠⣿⣿⠃⠀⠀⠀⢸⡏⠀⠀⣿⡿⡇⣿⣿⢸⣿⣿⣿⣷⣿⡿⠁⢀⠼⡫⠓⢉⣠⡶⣋⠔⠋
⠀⠀⠀⣿⣧⠀⣧⣻⣷⠀⢸⠀⣼⣿⡿⠀⠀⠀⠀⢸⠃⠀⢠⡿⢰⢱⣿⣿⠸⣿⣿⢹⣿⣿⡇⢀⡾⠋⣠⣶⣿⡿⠛⠁
⠀⠀⠀⢻⣿⣧⢸⣿⣿⠀⡀⢠⣿⣿⠃⠠⠀⠀⠀⣎⠀⠀⢸⢇⡟⣼⣿⣿⢸⣿⣿⠸⣿⣿⡇⢸⠇⣰⣿⣿⡩
⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⢃⣾⣿⠃⠀⢰⠂⠀⠀⠋⠀⢸⣿⡼⢡⣿⣿⢇⣾⣿⡟⣸⣿⣿⡇⠀⢰⣿⣿⠋
⠀⠀⠀⠀⠀⢻⣻⣿⡟⠀⣾⡿⠃⡀⠀⣸⠃⠀⠀⠀⢠⣿⣿⢃⠟⣸⢏⣾⣿⣟⣵⣿⣿⡟⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⢿⣿⠃⣼⡿⠁⣼⠃⣰⣿⠀⠀⠀⠀⣾⣿⢃⢂⢠⣫⡾⣻⣿⣿⡿⠟⠋⡠⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡟⣸⡟⠀⣼⠃⣰⣿⡏⠀⠀⢀⣼⡿⠁⢂⣾⣰⣟⣽⣿⠿⠋⠀⢀⣠⠄⠀⢀⣠⣴⠾⠟⢉⣴⠋
⠀⠀⠀⠀⠀⠀⢀⢱⡟⠀⣸⢃⣼⣿⡿⠁⠀⢀⣾⠟⠁⠀⣼⡇⣏⣿⡟⠁⠀⣠⡾⠋⠀⣠⡶⠟⠉⠀⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⣿⣿⡟⠁⠀⢀⣾⠋⠀⠀⣼⣿⣇⢻⡟⠀⢠⣾⡿⠀⢀⠾⠋⠀⢀⣴⣾⢿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⣾⣿⣿⠋⠀⠀⠀⡾⠁⠀⠀⣼⣿⡟⢻⢸⡇⣰⣿⣿⠃⠀⠁⠀⠀⣰⣿⣿⣳⣿⡟⠀⡜⣯
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⣿⠃⠀⠀⠀⠀⠁⠀⢀⡾⢋⣿⡧⣀⣼⣠⣿⣿⠏⠀⠀⠀⢀⣼⣿⡿⣷⣿⠟⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣰⠏⣴⣿⣿⡇⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⡰⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⣿⣼⣿⠿⠋⠁⠀⠀⠀⣰⣿⣻⣿⡿⠋⠀⢀⡔⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡄⠀⠀⢀⣴⢟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⣠⣾⣿⡷⠛⠁⠀⣠⣴⣿⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⣼⠁⠀⢀⠞⣱⡿⠋⠁⠀⠀⣀⣤⣔⣀⡴⠞⠛⠉⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣳⠏⠀⠀⢊⡞⠁⠀⢀⣤⡾⠿⠋⠁⠉⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠏⠀⠀⠀⢋⠀⢀⡴⠟⢁⡀⠀⠀⠔⢊⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⢀⡴⢃⠤⢊⣠⠞⠁⠀⠀⣠⣾⡿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⣼⡿⠋⠀⣠⣴⣾⠟⠀⢀⣴⡿⠁⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⣼⠋⢀⣴⣿⡿⠟⠁⣠⡴⠿⠋⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠚⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠊⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⡄
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣯⣿⣆⠀⣀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⣿⣧⣤⣤⣒⡢
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⢁⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣧⣤⣤⡀
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠘⣭⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠶⠆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⠀⠰⣩⡿⡿⣿⣿⣿⣿⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠰
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡟⠉⠀⠉⢿⣿⣶⣤⡿⠉⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⢺⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⢀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠟⠛⢿⣶⣶⣶⣶⠤⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⡾⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠒⠂
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⠟⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠂⠀⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡶⡀⠑⠢⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⢠⣣⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠻⠀⣦⣠⣿⠿⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢤⡀⠉⠁⠉⠀⠀⠀⠀⠀⠣⠀⠀⠔⠁⢀⣀⢍⠄⢠⣿⠇⠈⠁⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡒⡐⢁⡄⠀⠀⠀⠀⠀⠀⠗⠶⡄⠀⠀⢀⠞⠛⣠⡮⢷⢴⠿⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠈⡀⠀⠀⠀⠀⠂⢀⠀⢸⢈⠉⢘⡂⠀⢈⠗⣾⡄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⢰⣮⡅⠀⠈⠀⢎⠡⡄⠀⣁⠀⠀⠀⠈⠁⠀⠘⢀
⠀⡄⠀⠀⠀⠀⠀⠈⡂⠀⢀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠘⠻⠸⠛⠁⡀⠀⠠⠏⠀⠀⠀⠶⠶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡲⠁⠀⠀⠒⢈⡀⠀⠀⣶⠖⠎⠑⠀⢾⣻⡔⠋⠁⠀⡠⠐⠲⢶⠀⠀⠀⠀⣠⠆⠙⠂⠀⠐⡊⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⠀⠰⠝⠄⠰⣄⣀⠀⢐⠘⠻⠶⠋⢀⡂⢄⠀⢀⡶⠋⠁⠠⠀⠀⠁⠀⡄⠀⠀⠀⠁⡀
⠀⡀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠈⠁⡾⠁⠀⡈⠙⠠⠀⠀⠋⢠⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⢩⠀⡀⠀⠀⠂⠀⡄⠘
⠀⠁⠀⠀⡌⠀⠀⠀⣠⠄⠀⠀⠀⠄⠈⡀⠃⣠⠂⠀⠀⠀⠆⠀⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠀⠀⡀⢁⠔⠀⠀⣘⡗⠀⠀⣀⠀⢠⡄⠰⡔⠗⡔⢆⠴⣈⠞⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡐⣰⠂⠀⠀⠀⢌⢸⣀⠀⣀⠿⠿⠖⠊⢤⢑⣂⠀⠺⠶⡈⠀⡇⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠡⠀⠀⠀⢐⡝
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡘⠛⠉⠉⠥⠊⠀⢀⠀⢰⠙⠙⣣⡀⠙⠛⢤⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢠⠁⣀⣤⡤⠐⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡂⠠⠚⠁⠀⠀⠘⡧⠆⠀⠰⠋⠁⡖⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠸⠿⣿⡶⠶⠊⠀⠈⠀⠀⠀⠀⣠⣿⠀⠓⠂⠀⠀⢀⠀⠀⠓⠂⢀⠀⠈⢀⣦⠟⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠰⠄⢐⠁⠀⠄⠀⠀⢀⣠⡦⠶⠉⢠⡀⢀⠀⠑⠤⣀⣠⠂⢀⣀⠀⠒⠐⣏⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⡀
⠀⠀⠀⠀⠉⠀⠀⣾⣭⣤⢼⡁⠀⠀⠀⠰⠀⠉⠈⠠⠀⠈⠀⠀⠈⠁⠀⡆⣸⠛⢄⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⢹⣆⠀⠀⠠⠀⠈⠄⠀⠈⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣾⠐⠃⠀⠀⠀⠀⠀⠀⡀⠙⠂⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠾⠿⠿⠶⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠉⠉⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣰⣿⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡀⠀⠀⠀⠀⠀⣸⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢷⣤⣀⣀⣠⣴⡟⠁⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠉⠉⠉⠉⠙⢿⣄⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠹⣷⣀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠶⣶⣶⡶⠾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠄⠔⣀⡠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⣀⡨⢄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠⠊⠀⢰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡀⠸⠈⡁⡇⡠⠀⣠⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃⣀⢵⣄⠀⠃⢞⡈⢉⡽⠠⠔⠁⠀⠉⠉⠉⠉⠉⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⣚⠳⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠉⠀⠒⠤⠄⡀⠀⣀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢑⠃⡠⠃⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡀⠑⢌⡠⢁⣳⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⡀⡈⣿⣛⡠⠊⠁⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠐⠲⠖⠀⠉⠀⣰⢿⡵⡂⢠⠆⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⣔⠀⠀⠀⠀⢀⡷⢛⡗⢞⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢄⡀⠅⠋⠘⠤⠊⡶⢯⡶⠛⠹⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠦⣙⣀⢠⠛⢄⠊⢃⠸⢷⠆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣿⠀⠆⡡⢚⠀⣆⠤⠢⠊⡜⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠠⠔⠂⣡⣂⣤⣤⣾⣽⡾⠷⠔⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠛⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⣧⡀⠀⢸⣷⡀⠀⣄⢸⡇⠀⠀⡀⠀⠀⣴⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢹⣷⣄⣨⣿⣷⣶⣿⣿⣷⣤⣾⣁⣠⣾⠇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠟⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⢸⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣏⠀⠀⢀⣀⣠⢤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠋⢠⣿⣿⣯⣻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣤⣠⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡻⢯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⣿⡻⠋⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠿⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠛⠉⠙⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡝⠘⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⣸⣿⠟⢻⡿⢿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⠎⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢠⡟⠁⠀⠚⠁⠀⣿⠟⡏⢀⡹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣠⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁

⠾⣧⡍⠛⠟⠂⡛⠴⠆⠄⠄⠂⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠄⠢⠄⠄⠶⠞⡃⠚⠿⠍⢥⡿
⣧⡁⣠⣺⡆⠂⣂⠝⢐⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⠘⢅⠂⠂⣞⣂⡀⣡
⠿⠁⠨⠉⡁⡨⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠀⠄⠀⡀⠀⠔⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⣉⠩⠀⠹
⣻⡈⣄⠔⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣉⠀⠀⠂⠀⠀⠗⠀⢈⡁⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠂⢄⡌⣼
⠈⡅⢔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠰⠍⠚⠁⠂⢁⠈⠐⠘⠎⠵⠀⠰⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⡍
⡢⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠠⠀⠄⠀⢤⡦⣐⠛⢠⠙⡡⠘⢓⡠⢦⠀⠀⠄⠠⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡
⢄⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠒⠀⠀⠀⡀⠀⠱⠀⢘⠀⢈⠄⣌⠄⣘⠀⠰⠣⠀⡀⠀⠀⠐⢃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠒⠀⠄⠀⢀⠔⠁⢥⡄⡛⠼⠁⠀⠐⠈⠒⠩⠜⣂⣤⠁⠑⢆⠀⠀⠄⠐⠒⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⢀⠀⡊⠀⠈⠀⠀⠀⠐⠂⠒⠀⠒⠒⠒⠀⠈⠀⠈⠁⠈⡂⠀⠀⠀⠀⠀⠢⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⠄⠤⠶⢖⠐⠚⠛⠍⠉⠹⠉⠉⠝⠛⠒⠒⠶⢦⠤⢌⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠖⢸⠁⡌⠐⠀⠐⠀⢀⡰⣄⣶⣆⡴⣀⠀⠀⠀⠐⠈⠄⢹⠐⠖⢑
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣬⠲⠂⠘⡆⠩⠅⢂⠈⠀⠸⣳⣍⣻⣛⣵⡳⠀⠈⠀⠀⠨⠁⡞⠀⠶⢊⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠶⠆⠌⢿⡄⠀⠈⠘⠃⠈⡿⢟⣿⣟⠿⡏⠀⠛⠈⠀⠀⣼⠇⠄⠶⠎⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠉⠀⠀⠀⠒⠙⢃⠑⢦⣀⠘⠆⢀⡑⢿⣿⣿⢗⣁⠀⠞⢀⣠⠖⢁⡛⠑⠂⠀⠀⠀⠡⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⡴⠆⢋⣝⣂⠙⡿⠶⠾⠭⠭⡤⡬⠭⠭⠴⠾⡟⢁⣺⣈⠃⠲⠄⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠚⡀⠈⠐⠁⠐⠃⠡⠾⠲⠄⠲⡀⠀⠀⠀⠀⠀⠀⡰⡂⠤⠸⠆⠁⠂⠀⠑⠈⢠⠘⢦⠀⢀
⢀⠀⠀⠀⠀⠈⠂⠀⠀⠰⠁⡰⢰⠀⢀⠀⡌⢉⠑⠆⢈⡕⣆⠀⢐⡟⣍⡀⠖⢉⠉⠀⠄⢠⢠⠀⠱⡄⠓⠀⠀⠊⠀⠀⠀⠀⢀
⠃⠅⠀⠀⠀⠀⢐⠉⠂⠀⡀⠀⠈⠀⢑⠰⠀⠠⠀⠂⢨⠁⣻⠁⠸⡅⢩⠄⠂⠀⠈⠆⠐⠘⠈⠀⠁⡀⠀⠂⢑⠀⠀⠀⠀⠀⠥
⠑⠄⢈⠀⠀⠀⠀⠉⠀⠀⠀⡇⠐⣆⢀⠀⠰⠰⠀⠀⠉⡰⠁⠀⠀⠱⡔⠀⠀⢰⠰⠀⢀⣀⡆⠀⡅⠀⠠⠈⠁⠀⠀⠀⢀⠀⠔
⣀⡃⠀⠠⠀⠀⠀⠀⢸⠀⣄⠀⠀⢹⠄⣀⡠⠃⢃⣴⠶⠁⠀⠀⠐⠀⠹⢲⣤⠂⠣⣀⡀⢼⠁⠀⢀⡀⢰⠀⠀⠀⠀⠠⠀⠂⣃
⡽⣁⠉⠃⠀⠄⠀⠀⠈⠂⠁⠁⢀⠆⠛⡲⠆⣌⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢉⡀⠲⡛⠃⢖⠀⠀⠑⠊⠀⠀⠀⠄⠠⠪⢁⡹
⣶⡄⢂⣠⡄⢆⡈⠀⠤⠤⠄⢀⡅⠀⣜⠑⣝⣄⢍⠰⠂⠰⠀⠰⠀⠰⠀⠲⢈⢅⣘⡕⢙⡄⠀⢅⠀⠄⠤⠀⠀⣀⡆⣠⣐⡂⡴
⡗⡥⠉⠼⠇⠄⠄⠤⠸⠔⠀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠛⠚⠓⠒⠒⠓⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠬⠠⠄⠄⠤⠵⠌⢁⡕
⣲⣟⣡⣴⣧⣀⣥⣺⡆⣃⣃⣂⣁⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⢂⡠⣀⣁⣀⣃⡃⣺⣢⣅⣨⣷⣄⣙⣷
⣟⠉⣹⣻⣛⣛⣛⣻⣟⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠱⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⠋⠀⠙⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⢸⠁⠟⠐⠗⢸⠄⠀⠀⠀⠀⢰⣿⣤⣿⣠⣿⡆⠀⠀⠀⠀⢰⠃⠀⣀⠀⠈⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢆⡢⠤⣂⡼⠀⠀⠀⠀⠀⠘⢿⡿⠿⣿⡿⠁⠀⠀⠀⠀⠈⢧⡸⠤⢗⡴⠃⠀⠀⠀⠀⠀⢿⣿⡽⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡜⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦
⠀⠀⠀⠀⠀⢠⠞⣄⠀⠈⢧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢳⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢸⠘⠟⠀⠀⢸⠃⠀⠀⠀⠀⢸⡟⠀⢹⣿⣿⠇⠀⠀⠀⠀⠸⡼⡄⠀⠀⢀⡇⠀⠀⠀⠀⠸⡇⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠓⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠙⠶⠿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠮⠥⠴⠊⠀⠀⠀⠀⠀⠀⠙⠶⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⣠⣫⠀⠳⡀⠀⠀⠀⠀⠀⠀⣠⠟⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⠎⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄
⠀⠀⠀⠀⠀⢰⠑⠋⠀⠀⢹⡀⠀⠀⠀⠀⢰⣿⣴⣿⣿⣿⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠘⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠘⢇⠠⠤⠄⡰⠃⠀⠀⠀⠀⠈⢿⡿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦
⠀⠀⠀⠀⠀⢀⢞⡼⠁⠈⢦⠀⠀⠀⠀⠀⢀⡼⣡⣿⣿⣦⠀⠀⠀⠀⠀⠀⡼⠁⢀⢈⢣⡀⠀⠀⠀⠀⠀⣴⣿⡿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⢸⠆⠀⠀⠀⠀⢸⡵⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⡹⡀⠙⠘⢀⡇⠀⠀⠀⠀⠰⡏⣿⣧⣷⣾⡇
⠀⠀⠀⠀⠀⠈⠳⢭⣻⠤⠋⠀⠀⠀⠀⠀⠀⠱⢬⣭⠽⠋⠀⠀⠀⠀⠀⠀⠳⢭⣤⡤⠚⠀⠀⠀⠀⠀⠀⠙⠮⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢄⣀⠀⠀⠐⢴⡟⠆⠀⠰⣾⠧⠀⢰⢦⠀⡀⢂⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠤⠄⠞⠻⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠚⠉⠐⠒⠲⠤⣀⠀⠀⢀⠄⣠⠄
⠀⠀⢸⡅⢸⡁⠀⠀⠀⢸⠀⠒⢒⠢⡀⠀⠀⠀⡠⠺⠷⠶⠀⠀⠀⠀⠀⠀⠐⠛⠺⡗⢦⡀⠐⠁⢀⡀
⢾⠀⢺⠀⢮⢠⠆⠀⠀⠈⠦⠼⠷⠤⡐⠀⢀⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠑⡄⠀⠹⠛⠂
⢻⠀⣻⠄⡿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣷⠶⠸⠀⣰⠀⠀⠀⠸⠿⣆⣸⠀⠀⠀⠸⠀⠠⣤⣄
⣹⠀⢻⠀⠗⣧⠀⠀⠀⠀⠂⢤⣀⠤⠀⢸⠀⠀⠀⠀⡀⠀⢎⡀⢤⠄⣤⡀⠀⠀⠉⠒⠀⠀⠀⡇⠀⠛⠁
⢹⠂⡠⠲⡄⢁⣀⠀⠀⠑⢄⣤⡕⠀⠀⢸⠀⠀⡠⠚⠀⢀⡀⠀⠀⠀⣀⣙⠦⠤⢤⠀⠀⠀⠀⡇⠀⢠⢄
⠈⠁⢈⠟⠓⠚⢤⡠⢦⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠈⠣⠤⣀⣀⣀⣀⣀⠤⠜⠃⠄⠀⠀⠀⢠⠃⠀⠲⠄
⠀⢰⡃⠀⢰⡀⠀⢣⠀⠀⠒⠤⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠩⢂
⠀⠈⣧⠀⠈⣧⠀⠈⡆⠀⠐⢄⣤⡶⡀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠽⠇
⠀⠀⠸⡆⠀⠺⡏⠀⠸⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⡀⠀⠔⠊⠁⠀⠀⠀⣱⠒⠒⠄
⠀⠀⠀⢻⡀⠀⢻⡆⠀⢣⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠑⠂
⠀⠀⠀⠈⢷⠀⠀⢷⠂⠈⢆⠀⠀⡠⠉⠀⡣⠀⡐⡀⡀⠀⢀⠀⠠⠂⠀⠀⢠⡀⠰⣄⣀⣛⡃⢹⠀⠈
⠀⠀⠀⠀⠘⣇⠀⠘⣧⠀⠘⡄⠂⠀⠀⠉⠀⠀⡧⠁⢣⠀⡷⠷⣧⠀⠸⡿⠂⠃⠀⠀⠀⠀⠙⠈⣄
⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⢱⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠄⡀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠐⠄⠀⠑
⠀⠀⠀⠀⠀⠀⢷⠀⠸⠿⢇⣀⠇
⣀⣀⣀⣀⣀⣀⣘⣧⣴⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣿⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣰⣷⡄⠀⠀⢀⢰⣧⠀⠀⠀⠀⣾⣧⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⣾⣄
⠀⠀⢾⠿⡹⣿⠀⠀⢰⣿⣿⣿⠀⠀⢾⠟⠙⣿⠀⠀⢸⣾⣯⡤⠀⠀⢰⣿⣿⡆⠀⠀⣽⡟⠻⣷⠀⠀⣿⠏⢻⣶
⠀⠀⠈⠘⠃⠉⠀⠀⠈⠛⠛⠁⠀⠀⠈⠃⠐⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠐⠃⠉⠀⠀⠈⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣄
⠀⠀⣰⣾⣿⣧⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣷⡀⠀⠀⢀⣿⢷⣄⠀⠀⢠⣾⢿⡆⠀⠀⣠⣿⣿⡀⠀⠀⢠⣶⢿⡄
⠀⠀⠻⢿⡽⠟⠀⠀⠰⣿⣿⠟⠀⠀⠸⡏⠘⠟⠀⠀⠸⣧⣆⠟⠀⠀⢸⡇⠸⡇⠀⠀⢻⣿⣿⡟⠀⠀⢸⣣⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣷⡄⠀⠀⠀⢰⣦⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢰⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣾⣆
⠀⠀⢴⠟⡈⣷⠀⠀⢨⡟⢹⣶⠀⠀⢰⡿⠹⣧⠀⠀⢀⣿⢿⣀⠀⠀⣰⣿⣿⡇⠀⠀⣤⣿⢿⣧⠀⠀⣤⡟⢹⣧
⠀⠀⠘⠚⠃⠋⠀⠀⠘⠃⠀⠋⠀⠀⠘⠃⠀⠃⠀⠀⠈⠃⠈⠃⠀⠀⠘⠻⠿⠋⠀⠀⠙⠋⠈⠋⠀⠀⠙⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡇⠀⠈⠀⢠⡀⢸⡆⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⣡⠀⠀⠀⢠⡇⠼⠃⢠⠀⠀⣷⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠾⠀⠀⣸⠀⠸⡁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢀⡀⠀⠀⢀⠀⠻⠆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⢸⡇⠀⠀⢸⡀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣼⠁⠀⠀⠈⡿⠀⠀⢸⡆⠀⠀⠀⠀⢰⡀⡆⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠙⠐⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡆⠘⢁⠀⢸⣇⡷⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⣻⠇⡆⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⢸⠘⠁⠀⠸⠀⠛⡟⠃⠀⡀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢠⡋⢠⡟⠀⠀⠘⠀⠀⣶⡄⠠⡄⣾⠈⠀⠀⠘⠃⠀⠰⠀⡄⠀⠀⠀⢸⡇⠘⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⣷⠀⡁⠀⠀⠀⠀⠀⠹⠃⠀⠀⠻⠀⠀⠀⡄⠀⠀⢠⠀⠀⠆⠀⠀⠐⠁⣾⠀⣆⠘⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠀⠀⠛⢣⡟⢸⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢠⡀⡇⠸⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠠⡇⢀⢀⡅⠈⠠⠀⠄⢰⠀⠀⠀⢀⠀⠠⠀⠃⢻⠆⢸⡇⠈⢠⡆⢺⠄⢀⠰⢃⡀⠀⢠⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⢠⠌⢇⡄⠀⡀⠀⢀⣈⣥⣾⣯⢠⠀⢸⡄⠁⠐⠀⡼⠀⠀⠈⠀⠘⠛⢀⡄⢀⢸⣆⠘⠃⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡈⢠⡀⠁⠘⠿⠄⠌⠛⢿⡿⢘⣿⣦⡘⠀⣠⠀⠘⡧⣀⠀⠀⠀⡀⢂⣬⠷⣿⠋⣻⣷⠶⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠑⠀⠀⡂⠀⢤⡀⠀⣢⠈⠉⠸⣿⠀⠁⠀⠰⢀⣿⠃⠀⠄⠀⣼⡏⠀⠛⠘⠛⢁⠠⡄⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣆⠈⠛⢷⠀⡄⠉⣠⣿⡞⣯⠉⣤⡙⡄⠉⡀⠀⠀⠃⠉⠙⠁⠀⠀⠀⠙⠅⢀⡤⠀⠀⣿⠀⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣦⠟⠒⠚⠛⠷⣧⡙⠛⢿⠇⠉⠀⠙⠗⢻⣦⠉⣤⣁⠀⣀⠐⢠⣿⣆⠀⢴⣶⠄⢻⠇⢰⣿⣦⡄⠾⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢰⣿⡷⠆⠉⠀⠠⢤⣄⠀⠀⣀⣼⣿⠟⠋⠉⠈⠛⢿⣿⣷⡌⠉⠀⠄⠐⠀⠷⣤⣀⡀⢠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠦⡤⢤⣤⣶⣿⡟⠁⠀⠀⠀⡀⢢⠋⠀⠀⠉⣻⠃⠀⠀⠀⠀⠀⠀⠙⣯⠀⠀⠀⠀⡆⠀⠀⠀⠹⡿⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠺⠓⠒⠛⢾⡆⠲⠦⠶⢖⣿⣄⠀⢀⡾⣿⣦⡀⠀⠀⠀⠀⣠⣶⣿⣷⢤⡴⣢⡿⣶⠠⠠⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠛⠛⠙⠿⠷⠦⠴⠿⢯⣄⣒⠂⠒⠚⢋⣋⡸⠛⠛⠛⠻⠟⠟⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁



⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠞⡠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣾⠟⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⡀⠀⠖⠚⠋⢉⡈⠁⠀⣠⠖⠃⠀⠀⠀⡀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⣷⣤⡔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡅⠀⠠⠀⠀⠀⠈⢛⣀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⢠⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⣿⣶⡆⢠⣮⡄⢀⣄⣀⣽⣷⣄⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠠⠀⠀⣿⣿⣿⣿⣿⠀⢠⣿⣿⠃⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡙⠛⠷⢿⣿⣿⣿⣷⠿⠿⣿⣿⡆
⠀⠀⠀⠀⠄⠀⠀⢹⣿⣿⣿⣿⣷⣿⣿⡿⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣦⣄⣦⣴⣿⣿⠻⠋⣁⣈⣻⣿⠠⠂⠘
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡁⠀⠘⣿⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠿⡟⠛⠛⣩⣥⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠙⠻⠟⠀⠀⣰⢏⡄⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⠀⠀⠀⠘⣩⣒⣫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠙⠻⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠃⠒⠀⠈⣹⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡄⠀⠀⠀⠘⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠘⠷⠾⠿⠻⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡞⣿⣷⡀
⠀⠀⠀⠀⠀⠋⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢣⠀⠀⠀⠁⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠐⢧⣤⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠁⠀⠷⣤⡀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣥⣾⣷⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⢐⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣶⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠂⠀⠀⣠⣾⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠋⢸⡌⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢣⣿⡟⠁⠀⠀⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠛⠿⠷⠆⠀⠀⠀⠀⠀⡢⠄⠠⣾⣿⣿⣿⡧⣿⡿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠿⣿⡇⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢱⣾⣿⣿⣶⣿⡿⡇⣴⡟⠢⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣧⠘⠼⠇⠀⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣌⡙⠇⠀⠁⠀⠀⠹⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⣶⣿⠿⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛