Secondary education Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠲⠶⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⡛⣃⡀⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣭⣭⠁

⠀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡀⠀⣀⣀⠀⢀⣀⣀⣀
⠀⣿⡿⣽⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣀⠉⢻⣄⠀⣽⣿⣿
⠀⣯⣧⣟⣻⣟⣻⡿⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⣿⡟⡿⣿⠟⠛⠛⢓⡶⢶⠿⠟⣸⣿⣿⠉⣹⣿⣿⣿⣦⣨⣿⣿⣿⣿⢀⠈⠉⡄
⠀⠛⢛⣍⣴⢾⣿⠑⠀⢀⣿⣿⣾⣷⣾⣿⣿⠻⣷⣤⣼⣿⣿⠋⢹⣿⣿⠀⠑⠀⣀
⠀⢶⣿⣿⣷⡾⠟⠠⠄⠸⣿⣿⣿⣿⣿⡍⣿⣷⡿⣿⣿⣿⠿⣿⣿⢛⣿⠜⣻⣿⣿
⠀⠛⠋⣉⣥⣤⣴⠶⠖⠂⣈⣙⣿⣿⣿⣧⣋⡄⠀⡈⠿⢋⣾⣿⡿⢿⡿⠀⠀⢠⣿
⠈⠈⠀⠉⠉⠈⠉⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⡛⠁⠳⠀⠉⠉⠁⠀⠀⠁⢤⣤⣼⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⡯⠄
⠀⠀⠀⣀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠹⠣⢉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⡀⠀⠘⠻⢡⡜⠀⠀⣀⢀⡀
⠈⠙⠃⠀⡠⠏⠀⠀⠀⠀⠀⣤⣀⡠⠀⠐⠂⠀⠘⠉⠿⠆⠀⠀⠀⠙⠓⠀⠄⠀⠀⠀⠀⠀⠀⠁⠐⠃
⠀⠀⠀⠢⣈⡤⠀⢠⠀⢠⣏⢈⣻⣛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠗
⠀⠀⠀⢀⠁⠁⠠⡄⠀⡿⠛⠿⠿⠛⠏⠀⢠⠞⠓⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⢠⣤⣈⠂
⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣴⠋⠀⠘⠛⢷⣄⠀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠋⠒⠈⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣶⣶⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣀⢉⡀⠉⠀⠀⠀⠀⢰⣿⣿⣿⠟⢛⠁⠀⠀⠀⠀⠈⣩⢉⢩⡯⡁
⠀⠀⠀⠀⠀⠀⠘⠟⠲⡀⠀⣄⠀⠀⠀⠀⠘⠛⠛⠛⠀⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠒⠂⠀⠀⠀⠀⠀⠀⠀⣙⣛⠐
⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣾⣷⡄⠀⠀⠀⠀⠀⢂⣀⡘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣽⣿⣿⠃⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣀⣀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⠻⢟⠻
⠀⠀⠀⠀⠀⠀⢸⠻⠿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠛⢻⡟⠛⠃⠀⠀⠀⠀⣼⣿⣿⡄⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⣄⣠⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡿⠛⠉⠓⢿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣤⡀⠉⠂⠀⠀⠀⠀⢸⡟⡘⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⠀⠁⠤⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠇⠈⠻⠟⠁⢀⣤⠀⠀⢠⣤⢼⣿⣿⣿⢤⡄
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢘⣦⣀⠀⣀⣼⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣤⣀⠀⠘⠋⠀⠀⢸⣿⣀⣿⣿⣗⣻⣗
⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠈⠉⠁⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁
⠀⠀⣠⣤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⢦⡄⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⢠⡄⣤
⢠⠿⣧⣿⣾⣧⠼⡇⢸⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣸⡇⠀⠀⠀⢠⠛⢧⡀⠀⠀⠀⠀⠀⢼⣿⣿⣯⠀⠀⠄
⠘⢷⣿⡿⣿⡟⢳⠇⡞⠀⠀⠀⠀⡼⠀⢧⠀⠀⠀⠀⢸⣇⣺⣇⣸⣇⣹⡇⠀⠀⠀⠘⢄⣈⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠄⢸
⠀⠀⢻⣤⣾⠿⠋⠠⠇⠀⠀⠀⠰⠃⠀⠸⠄⠀⠀⠀⠸⣯⣺⣧⣸⣇⣼⠇⠀⠀⠀⢀⣤⣿⣤⡄⠀⠀⠀⢨⣿⢭⣭⢽⡇⠸

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢠⡆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶
⠀⠀⠀⡄⠱⡄⠁⠀⠀⠀⠀⠀⠀⠳⣤⠏⠀⠀⠀⠀⣠⠇⢀⣟⡄⠀⠐⠲⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⣠⢴⢻⠦⡄⠀⠀⠀⠃⢋⠃⠀⠰⠆⠛⠒⠀⠀⠀⠀⠉⢽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⠘⠦⠜⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⣠⠞⠛⠲⡄⠀⠀⠀⠀⠀⣠⣤⣙⠻⣦⠀⠀⠀⢠⡄⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣬⣷⢄
⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⢯⠘⠿⢀⡿⠀⠀⠀⠀⣰⣿⣿⣻⣿⡦⠀⠀⠀⢸⡇⣿⢳⡀⠀⠀⠀⠀⠀⣦⣠⣾⣿⣏⠙⡆
⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⢨⣿⣾⣯⡁⠀⠀⠀⠀⣌⠻⢷⣿⡿⠃⠀⠀⠀⢸⡇⣿⠀⠻⡄⠀⠀⠀⠀⠘⢿⢈⣹⣿⠞
⠀⠀⠚⠛⠛⠛⠛⠂⠀⠀⠀⠀⠈⠙⠈⠋⠁⠀⠀⠀⠀⠘⠛⠒⠀⠀⠀⠀⠀⠀⠘⠃⠛⠒⠒⠛⠂⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⣴⣦⣠⣿⣿⣷⠀⠀⢘⠉⠀⠀⠉⠉⠉⢉⡇⠀⠀⠀⠀⡸⠀⡀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⡆⠀⢠⣴⣾⣿⣿⣿⣶⣤
⠀⣾⣿⣿⣿⣿⣿⡟⠃⠀⢈⣉⡉⠁⢀⣈⣉⣉⡇⠀⠀⠀⢀⡇⢸⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⡿⠛⠛⠻⡏
⠀⠹⣿⣿⣿⣿⣿⠇⠀⠀⠠⠀⠠⠤⠤⠤⠤⠤⡇⠀⠀⠀⠘⠣⠞⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠒⠒⠚⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠒⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⠚⠓⢦⡀⢤⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢼⢉⡩⢙⠞⢥⣱⣸⠃⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡜⢧⣰⠃⡔⣢⢈⣿⡄⡇⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⢦⢰⠟⠢⣌⣡⠏⣸⠀⡇⠀⠀⠉⠉⠀⠀⠀⠟⠀⠋⡟⠂⢠⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⣀⣠⡞⠉⢙⠟⠂⡇⠀⠀⠒⠒⠀⠀⣀⣀⣀⣀⣧⣔⣃⣦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡄⣇⠀⠀⠀⠀⠈⢉⣁⠀⠀⡇⠀⠀⠤⠤⠀⠀⠀⢀⡴⠋⡇⠀⠀⠚⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠖⣛⠢⡀⠀⢰⠚⡇⢹⡬⠇⠀⡇⠀⠀⣀⣀⠀⠀⢀⡎⠀⠀⡇⠀⠀⡱⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⠼⢃⢧⡀⠈⠉⢀⣀⣀⠤⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⠉⠣⡙⢄⠈⢉⣀⣀⡤⠄⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠙⢎⠳⡈⠀⠀⠀⠀⡇⠀⠀⠶⠶⠀⠶⠆⠰⠶⠶⠶⠶⠶⠆⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⣴⢦⠤⠳⡜⢦⠰⠒⠂⡇⠀⠀⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡋⠙⠚⠀⠀⠈⢶⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃





⡏⢱⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢜⠉
⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⢱⢸⠀⡏⢀⣞⡄⢸⢼⠂⡆⠀⡧⠀⠀⠮⡀⠈⡏⢸⢉⠂⡎⠈⡆⣧⡊⢸⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣦⣮⣼⣤⣧⣬⣤⣥⣴⣼⣥⣧⣤⣶⣦⣤⣴⣯⣤⣥⣼⣤⣧⣵⣶⣥⣧⣼⣬⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⢠⣤⣌⣉⣈⡀⠒⠂⠂⠆⢀⣤⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⠄⠀⠀⠒⢶⣭⣙⡳⢦⣤⡀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣭⣝⣓⠶⢤
⠀⠀⢀⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠺⡟⠋⠉⠁⠀⠀⠀⠀⠀⠈⢿⡟⠀⠖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒
⠀⠀⣙⣛⡀⢀⣿⠀⠀⠀⠀⣿⡟⠛⢻⠛⠻⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⣿⣿⣿⠉⢹⣿⣿⡟⠉⢻⠉⠙⡏⠉⢻⡇
⠀⢸⣿⣿⣿⠠⠿⠀⠀⢀⠀⠿⠿⠿⠿⠿⠿⠿⢀⢧⠤⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⢛⣀⣿⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠀⠀⢸⣿⠛⠛⢿⡏
⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⢀⣀⣀⣀⣸⣷⣌⣀⣾⣇⣀⣀⣀⣀⣀⣀⣀
⠀⣴⣿⣿⢷⣰⣶⣶⣶⣶⣾⡿⣿⡿⣿⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⣸⢷⣿⣿⢻⡜⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⡏
⢠⡟⣿⣿⣿⡇⢿⠿⡿⢿⣿⡇⣸⡀⣸⣿⡿⣿⢿⠿⣿⠁⠀⠀⠀⠀⠀⣿⢿⣿⣿⢿⣧⣉⣉⣹⣿⡟⢋⢙⠻⠛⢋⣉⣉⢉⡉⡉⡇⡀
⢸⣿⣿⣿⣿⣿⣼⣤⣧⣼⢫⣭⣭⣯⢭⣭⣤⣭⣬⣤⣭⣤⡄⠀⠀⠀⠀⠷⢿⣿⣿⡟⢇⣬⣤⣼⣿⡇⠀⠰⠰⠒⠐⠐⠒⠂⠒⠰⠐⠀⡂
⠈⢨⣿⣿⡇⠁⠉⠉⠉⠉⠸⢿⠧⠶⠴⠤⠶⠤⠴⠶⢾⡿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⡿⠄⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⢸⡿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⢀⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠆⢀⣠⣤⡀
⠀⠀⡘⠿⠿⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⡈⠿⠿⠃⠀⠀⠀⠀⣤⣤⣤⡄⢠⣴⣦⡄⠀⢀⣠⣼⣿⣿⣿
⢰⣿⡿⣾⣿⣿⣆⢀⣏⢉⠉⠉⠉⠉⠉⡉⠉⠉⠉⢹⡃⠀⠀⠀⠀⠀⢠⣿⡿⣼⣿⣿⣦⠀⢀⣀⣯⣍⣹⡇⠀⠀⠀⣤⣾⣿⣿⣿⡷⠇
⣿⣿⣼⣿⣿⡏⣿⢹⡏⠉⢉⣉⣉⠉⢉⣉⣉⡉⠉⢹⡇⠀⠀⠀⠀⠀⣾⣿⣴⣿⣿⡯⣿⠀⣸⣿⣿⣗⣾⣷⣶⣂⣶⣆⣀⣀⣃⣘⣁⣀⡀
⠻⢿⣿⣿⣿⡇⠿⣸⣧⣤⣬⣥⣭⣭⣬⣬⣭⣥⣤⣼⣇⣀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡇⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⢸⣧⣬⣤⣭⣤⣤⣽⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⡷⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯
⣀⣀⣹⣟⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣟⣻⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡶⠋⠀⠀⠀⠀⠀⠈⠳⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⢤⡀
⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣤⣤⣀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠐⣯⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⡏⠀⠀⢹⡆⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣽⠂
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠞⠁⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⡅⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡶⠶⠶⢶⡄⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠄⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⣴⠶⠶⠶⣦⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⣇
⠀⠀⠀⠀⠀⡇⠀⠀⣇⣀⣀⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⣀⣀⣿⠀⠀⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠈⠳⠶⠤⠤⠤⠤⠤⠦⠤⢾⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠴⠤⠤⠤⠤⠤⠤⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣿


⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⢒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿
⠀⠀⣠⠖⠒⠒⠒⠒⣲⠞⢁⡀⠙⢶⠒⠒⠒⠒⠲⣄⠀⠀⠀⣠⣤⣤⣤⣤⡤⣠⣾⣿⣷⣄⢤⣤⣤⣤⣤⡀
⠀⠘⠳⡖⠒⠂⣠⠞⠁⠀⢯⡸⠂⠀⠙⢦⡐⠒⠲⡾⠃⠀⠈⢩⣭⣽⡿⣫⣾⣿⣇⣀⣽⣿⣷⣝⢿⣭⣭⡍⠁
⠀⠀⢸⡇⠀⠀⢹⡆⢠⠤⡄⢠⡤⣄⠀⡟⠁⠀⠀⡇⠀⠀⠀⢸⣿⣿⣦⢹⣿⠿⢿⣿⡿⠿⣿⡏⣵⣿⣿⡇
⠀⠀⢸⡇⡖⢲⢸⡇⠸⣤⠇⠸⣧⡿⠀⡇⣶⠲⡆⡇⠀⠀⠀⢸⡟⠙⣿⢸⣿⣀⣸⣿⣇⣠⣿⡇⣿⠉⢻⡇
⠀⠀⢸⠁⠳⠞⢸⡇⠀⠀⠀⡀⠀⠀⠀⡇⠻⠶⠃⣧⠀⠀⠀⢸⣧⣤⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣤⣼⡇
⠀⠀⢸⠀⠀⠀⢸⡇⠀⢰⡏⠉⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⢱⣶⡎⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠈⠒⠒⠒⠺⣇⣀⣸⣇⣀⣿⣀⣀⡗⠒⠒⠒⠋⠀⠀⠀⠈⠉⠉⠉⢸⣿⣿⠸⣿⡇⣿⣿⡇⠉⠉⠉⠁






⡟⠉⣿⣛⢻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣛⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣄⣈⣂⣂⠀⠀⠠⠀⣠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠲⣭⣓⡢⢤⣀
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣹⣿⣿⡿⠀⠀⠀⠀⢠⣤⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣷⣾⣭⣟⣲⠆
⠀⠀⠀⠀⢴⡦⠀⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⢹⠉⠀⠀⠀⠀⠀⠀⣬⣿⣄⠉⣿⣿⠿⢿⣿⣿⡿⣿⠿⣿⢿⣿⡏⠁
⠀⠀⠀⢠⣾⣷⡄⣿⡀⢀⠀⢸⣧⣤⣧⣤⣿⢸⣄⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⣿⣿⠀⢸⣿⣿⣤⣾⣤⣾⣤⣾⡇
⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣤⣤⣤⣤
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣇⠀⠀⠀⢸⣏⠀⢸⣿
⠀⠀⠀⢠⣿⣿⣄⣤⣤⣤⣤⣾⣿⣿⣷⣤⣤⣤⣤⣄⠀⠀⠀⠀⣸⣿⣿⣿⡜⠛⠛⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⡇
⠀⠀⠀⣿⣿⣿⣿⠻⠻⡟⣿⣇⣸⣀⣿⡟⠟⢿⢻⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⢭⣭⣿⡟⡉⠙⠛⠋⠉⠉⠉⠉⠃
⠀⠀⠀⠿⣿⣿⠿⠾⠿⠷⠏⣿⣟⡛⡛⠒⢲⢒⣲⣶⡆⠀⠀⠀⠉⣿⣿⡏⠑⠒⠒⠛⠃⠀⡍⠁⠀⠈⠉⠈⠉⢡
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⠩⠉⠉⠉⠉⠉⠩⠋⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⢰⣿⡶⠂⠀⡖⠾⠿⠿⠻⠿⠛⠿⠟⣶⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⠋⠉⢠⣶⣦⡀
⠀⠀⠀⣴⣶⣽⣶⣄⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⠀⣴⣦⣿⣶⡄⠀⠀⣺⣻⣿⠀⠙⠛⣡⣴⣾⣿⣻⠋⠉
⠀⠀⢸⣿⣽⣿⣿⢿⣺⡗⠛⣛⣛⠚⣚⣓⡓⠓⣿⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⢀⣿⣿⣷⣿⣶⣆⣤⣍⣉⣉⣙⣀⣀
⠀⠀⠈⠻⣿⣿⣿⠺⣼⡧⠤⠤⢤⣤⣤⣤⢤⣤⣿⣀⠀⠀⠀⠘⠿⣿⣿⣿⢻⣿⣿⣿⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠠⠿⠧⠀⠀⠼⠶⠶⠶⠶⠶⠿⠿⠄⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⡿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠴⠚⠁⠀⠀⠈⠛⠦⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣠⠞⢉⣀⣀⡙⠳⣄⣀⣀⣀⣀⣀⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⢶⡄⠀⡴⢦⠀⡏⠀⠈⢹⠀⡴⢦⠀⢠⡶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⡇⠀⣇⣸⠀⠻⠦⠴⠟⠀⣇⣸⠀⢸⣇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣨⣭⡁⠀⣩⣭⠀⣤⣤⣤⣤⠀⣭⣭⠀⢨⣭⣅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⠀⡇⢸⠀⡇⠀⠀⢸⠀⡇⢸⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠁⠀⠛⠛⠀⡇⠀⠀⢸⠀⠛⠛⠀⠈⠛⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣤⣤⣤⣤⣤⣤⢴⣧⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⡼⣧⡄





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠂⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡂⠐⢖⣲⣒⣒⡀⣤⣤⡤⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡴⠈⠉⠉⠉⠉⠉⠉⠁⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⡆⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⠀⠀⠀⠀⠀⢷⣇⣀⡀⠀⠀⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⣠⠞⣳⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⡖⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⡴⠁⢠⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⡇⠀⠀⢳⢡⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠙⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡰⠁⠀⠀⠀⡎⢦⠀⠀⠀⠀⠀⢀⠼⣿⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠁⠀⠀⠀⢠⢱⢻⠳⢄⣀⣀⡤⠊⢸⡏⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡇⣇⠀⠀⠀⢸⢈⡞⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠦⢃⠏⠀⠀⠀⢸⠎⢇⠀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⠀⠀⣠⠞⡆⠈⠲⠤⠄⠀⠼⠀⢸⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⣀⣀⠀⠀⠉⠉⠀⠀⠘⠢⠤⠤⠤⠤⠤⢲⠌⠒⠉⠓⠤⠤⠒⠁⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠉⠉⠉⠒⠒⠒⠒⢢⠀⠀⠀⠀⠀⡸⠀⠀⡠⠤⠒⠊⠉⠉⠉⢉⡩⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⡀⠀⠈⡆⠀⠀⠀⢰⠁⡴⠋⢀⣀⣀⡤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢻⡒⠒⢲⢛⡞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣒⣮⣤⣯⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⣭⣶⣿⣿⣿⣿⣿⣿⣷⣯⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⠻⣿⣿⣿⣿⣿⣿⠉⠁⣻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠄⠈⢻⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠐⠛⠉⠉⠙⠻⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⣾⣿⡟⠁⠔⠦⠄⠀⣠⡤⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢴⡟⠁⠀⠈⠁⠁⠀⠏⠉⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠆⠀⠀⠐⠔⠂⠲⠖⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡈⢤⡀⠀⠀⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠛⢷⣶⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣺⡀⠀⠀⠀⠀⠈⣽⠋⣿⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠚⣡⣶⣿⣿⣷⠀⡀⠀⠀⠠⠀⣠⠃⠀⢠⣉⠓⠶⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⣁⣄⣤⣿⣿⣿⢩⠤⠖⠋⠁⠀⠀⠀⠀⠃⢠⠀⣼⣿⣷⣷⣾⣶⣦⣍
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣤⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⠀⠀⠀⣿⣿⣦⣀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣅⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠸⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⠃⠀⢀⡿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠟⠛⢛⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣟⠿⢿⣿⠏⠀⠘⠃⠈⠩⠔⢻⣿⣿⣿⣿⣿⣿⣿⡗⠒⠀⠀⢘⡄⠀⢸⣿⣷⡀
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⡟⠀⠀⠀⠀⠀⠀⡟⣩⣿⣿⣿⣿⣿⣿⣿⣏⣉⠥⠀⢸⠆⠀⠀⠻⣿⣧⡀
⠀⠀⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⣩⢼⣿⣿⣿⣿⣿⣿⣿⣧⣤⣠⣄⣞⣀⡀⠀⢀⢠⠉⠳⡄
⠀⠀⢰⣧⣿⣿⡿⢟⣿⠿⠟⢋⡉⣴⣾⡆⠀⠀⣠⣤⣴⠒⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣾⣦
⠀⢀⣿⣿⡿⠋⠖⣟⣁⣴⣾⣾⣿⣿⣿⣗⣠⣾⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡄
⠀⣾⣿⠿⣁⣈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣼⣿⣿⣿⣷⡀
⠀⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛⠻⠈⠃⢻⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡃⠹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⢿⡿⢛⣛⣭⣥⣤⣥⠤⠤⠴⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣇⣿⣿⢻⡿⠃⠀⠈⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⠿⠻⠟⠛⠋⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

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




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

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠛⠛⠓⠒⠠⢀
⠀⠀⠀⡠⠐⠈⠀⠀⠀⠀⠉⢰⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⡠⡠⠂⢀⠔⠈⠀⠀⠀⠀⢀⡆⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⣀⣠⠄
⠐⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⠌⠘⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠉⠉⡉⠐⠃⡒⠜⢁⠀⠄⠂⠁⢆⡄
⡆⠀⠀⠀⠀⠀⠀⡠⢂⡠⠋⠀⠀⠐⢄⠨⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠃
⠇⠀⠀⠀⠀⡐⠡⠀⠁⠀⠀⠀⡀⠀⠀⠑⢿⠂⠀⠀⠀⠀⠀⣀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠈⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠰⢂⠀⣇⠉⡄⠀⠄⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢸⢠
⠀⢲⣦⠀⠀⠘⠃⠀⠀⠀⠀⡄⠀⠀⠀⡰⠃⡼⠊⠀⠍⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠃
⠀⠙⠌⣦⡀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣮⠐⠉⠀⢀⠌⠀⠀⠀⠀⠀⠀⠁⠂⢠⡄⠀⢀⣀⠀⠠⠐⠈
⠀⠀⠀⡇⠏⠒⠀⠀⠀⢠⠀⠈⣔⠰⠇⠱⡄⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⠱⠌⣰⣊⠜⠢⢄
⠀⠀⠀⢇⠀⠈⠀⠀⡰⠏⠓⠖⠄⠔⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠀⠀⠀⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⠐⠐⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠐⠤⡯⢄⠀⡀⠀⠀⠀⣨⣐⠆⠀⡁
⠀⠀⠀⠀⠀⢠⢴⡀⠀⠛⠢⠔⢢⠀⠐⠀⠀⠀⠘⢢⣀⠀⠈⠁⠂⠀⠆⠚⠀⣀⣒⣀⣠⣿⣦⠷⠊⠀⠁⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠒⠐⠠⠒⠀⠀⣓⢈⣭⡤⠤⢴⣀⡀⠠⠤⡤⢭⣭⣭⡭⠭⢥⣶⣖⠒⠚⠛⠛⢻⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠙⠛⠋⠉⠉⠁⠀⠀⠉⠢⡀⠀⠀⠀⠀⢇⠀⠀⠀⢲⠤⡀⠀⠀⠑⠦⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠻⢄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣀⣀⢰⠀⠀⠀⡆⠤⢵⠀⠀⠀⠇⠀⠸⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠰⠘⠢⠀⠀⢤⠀⠀⠒⣆⢁⡌⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⡔⠄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠱⡄⠸⡀⠀⠀⡅⠸⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⡆⠀⠈⠓⠚⠀⠁⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠠⠀⠀⠀⠈⠢⠵⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢋⠙⡿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠐⠀⠂⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣦⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠀⠀⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣄⣈⣂⣂⠀⠀⠠⠀⣠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠲⣽⣓⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⣿⣿⡿⠀⠀⠀⠀⢠⣤⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣷⣾⣭⣟⣲⠆
⠀⠀⠀⠀⢴⡦⠀⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⢹⠉⠀⠀⠀⠀⠀⠀⣬⣯⣄⠉⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⢠⣾⣷⡄⣿⡀⢀⠀⣸⣧⣤⣧⣤⣿⢸⣄⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⣿⣿⠀⢸⣿⣿⣤⣾⣤⣾⣤⣾⡇
⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣤⣤⣤⣤
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣇⠀⠀⠀⢸⣏⠀⢸⣿
⠀⠀⠀⢠⣿⣿⣄⣤⣤⣤⣴⣿⣿⣿⣶⣤⣤⣤⣤⣄⠀⠀⠀⠀⣸⣿⣿⣿⡞⠛⠛⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⡏
⠀⠀⠀⣿⣿⣿⣿⢻⠻⡟⣿⣇⣸⣀⣿⡟⡟⢿⢻⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⢭⣭⣿⡟⡉⠙⠛⠋⠉⠉⠉⠉⠃
⠀⠀⠀⠿⣿⣿⠿⠾⠿⠷⠏⣾⣟⡛⣓⢒⢲⢒⣲⣶⡆⠀⠀⠀⠉⣿⣿⡏⠑⠒⠒⠛⠃⠀⡍⠁⠀⠈⠉⠈⠉⢡
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⠩⠉⠉⠉⠉⠉⠩⠋⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⢰⣿⡶⠂⠀⡖⠾⠿⠿⠿⠿⠟⠿⠟⣶⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⠋⠋⢠⣶⣦⡀
⠀⠀⠀⣴⣶⣽⣶⣄⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⠀⣴⣦⣿⣶⡄⠀⠀⣾⣻⣿⠀⠙⠛⣡⣴⣾⣿⣻⠋⠉
⠀⠀⢸⣿⣽⣿⣿⣿⣾⡗⠛⣛⣛⠚⣚⣓⡓⠓⣿⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⢀⣿⣿⣷⣿⣶⣆⣤⣍⣉⣋⣙⣀⣀
⠀⠀⠈⠻⣿⣿⣿⠻⣾⡧⠤⠤⢤⣤⣤⣤⣤⣤⣿⣀⠀⠀⠀⠘⠿⣿⣿⣿⢻⣿⣿⣿⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠠⠿⠧⠤⠠⠼⠶⠶⠶⠶⠶⠿⠿⠅⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⡿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⣠⠤⡀⠀⠀⠀⢠⠀⠰
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠀⠸⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡔⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠆⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⢧⠆⠀⠀⡤
⠀⠈⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⢀⠞⠁⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠂⠀⠀⡴⠋
⡀⠀⠀⢀⣤⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠋⠠⠜⠁⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠁
⠁⠀⠀⢁⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢸⠀⣠⠄⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⡀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠
⠀⠂⠀⢀⠀⠀⠈⠠⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢰⡀⠀⠀⠀⠀⢀⠠⡄⣰
⢀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⡄⠀⠀⠀⠴⠃⠸⠗⠁
⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢦⡙⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠛⠛⠿⡿⠀⣿⡇⠀⠀⢀
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣷⣶⣒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠁
⠀⠈⠀⢀⡀⠰⠀⠀⠀⠀⢰⠏⣰⣫⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠐⠄⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣀⠠⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡹⠀⠀⠀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠈⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⢱
⠀⠀⠀⠰⠘⠚⠛⠛⠛⠀⠀⠄⡤⠀⠀⠀⠀⠣⠎⠀⠀⠀⠀⠀⠀⢰⡆⢰⠆⠀⠀⠀⢸
⠀⠀⠀⠘⠒⣶⣶⣶⣶⣶⠖⠲⠧⠀⢀⣀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⡖⢲⣿⣟⣛⠛⠛⠢⠐⣔⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠐⠀⠈⢀⠠⠊⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣒⣒⣂⡏⠀⠀⣀⣀⣀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⡓⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠑⠚⠛⠛⠛⡏⢉⠠⠄⠚⠉⣍⠉⠃
⠀⠀⢱⣴⣶⣶⣷⣷⣶⣶⠢⠆⠤⠀⢰⣞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡍⠉⠐⠂⠀⠀⡀⠀⡀⠀⠀⠀⠰⠦⢤⣭⣭⠥⠄⠀⠐⠶⠶⠤⢾⣬⡭
⠀⠀⠀⠁⠉⠚⠃⠀⠀⠀⡄⠀⡐⡂⢀⣶⠄⠀⠀⠀⠀⠀⠒⠤⣤⣀⣀⠀⠁⠀⠀⠀⠀⠐⠈⠁⣉⠦⠠⠀⠐⠶⠶⣦⠦⠀⠀⠀⠠⣤⣭⣥⣤⣤⡤
⠀⠀⢸⠀⠤⢤⣤⣴⣤⡤⠵⠠⠅⠀⠐⠃⢀⡚⠒⠢⢤⣤⣤⣄⣀⠈⠉⠉⠀⣀⣀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠒⢒⣶⣶⡶⢒⠂⠀⠀⠉⠉⠉⠙⢻⡆
⠀⠀⠦⣤⣶⣾⣿⣿⣿⣥⡶⠉⠉⠉⠉⠉⢹⢷⡁⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠅⠀⠀⠀⠤⠤⠤⠤⠄⢆⡠⣤⣴⣶⣮⣭⣤⠄⠦⠬⢭⢭⣬⣿⠞⠇
⡄⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘
⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣄⢀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠇
⣶⣶⣶⢾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣄
⠀⠀⠀⢠⣴⣶⣶⣶⡶⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⠂⠈⣻⣠⣶⣶⣶⣶⣶⣶⣶⣶⣦⡦⠤⠤⢤⣤⣤⣤
⠀⠀⠀⣭⣭⣿⣿⣶⣿⣤⣤⣌⡙⠈⠉⠉⢹⣮⡻⣟⠀⠂⠸⠯⢿⢿⡉⠉⠉⠋⢹⡿⢍⣿⠀⠛⠁⢹⣛⢛⠂
⠀⠀⠀⢀⣠⣴⣿⣿⢁⣠⡼⠿⠆⠀⠀⠀⠸⠃⠁⠙⠀⠀⠰⠀⠈⠸⠃⠂⠀⠀⢸⣇⢸⡏⣰⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣽⣿⡏⣀⣤⣟⣻⠀⣀⣶⣦⠠⣶⣶⡆⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡙⡟⣻⣻⣿⣿⡏
⠀⠀⢀⣉⣉⣁⠈⠈⠉⠉⠉⠁⣠⣿⣶⣿⣿⡇⠀⢀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠾⠿⠛⠛⠛⠛⠛
⠀⠀⢸⣿⣿⣿⣷⣿⣶⣶⣮⣽⣭⣿⣿⣿⣿⣿⣿⣿⡿⢇⣀⣒⣛⣒⣆⣐⣓⣤⣞⣉⣘⡒⣀⣀⣀⣀⣠⣤⡞⠀⣀
⠀⠀⠀⠸⠿⠿⠏⠉⢿⣿⣿⣿⣻⣿⣿⣿⣿⡋⠅⠐⢠⣾⣿⣿⣿⣿⡿⠻⠟⣿⡿⠛⢛⡛⠛⢛⣿⡿⠿⠛⣤⣶⣿⠃
⠀⠀⢸⡷⠲⣶⢤⢤⡄⢻⣿⡿⠟⢙⡿⠿⠿⠏⣰⣶⣿⣿⣾⣛⢉⡿⣻⣤⣿⣿⣿⡟⠛⠻⠛⠙⢿⡟⣀⣶⡾⠛⠁
⠀⠀⠀⠁⠀⠻⣆⢸⣿⣿⡿⠁⠀⠙⣷⣀⠀⠀⠙⢿⣿⣿⣏⣽⠟⠁⠈⠻⣿⡿⢩⣿⣿⡅⠀⣰⣾⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠿⠁⠀⠀⠀⠉⠅⠀⠀⠀⢸⣿⠛⠛⠛⠀⠀⠀⠀⠛⠀⠙⠻⣿⠇⠀⠈⠛⠛

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣦⣶⣶⣤
⣿⣿⣴⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢰
⠀⠀⣀⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⢹⣿⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣨⣿⣿⣀⣀⠀⠀⠀⠀⣼⣿⣶⣤⠀⠀⠀⠈⠁⠀⠀⠀⣼⣿⣿⡄
⠀⠀⢸⣿⣗⢲⠶⠒⠺⠗⠒⠂⠠⣶⣾⣿⣿⣿⠿⠿⠟⠛⠉⠀⣿⣿⣿⣿⡗⠒⠛⠛⠒⠠⣶⣶⣿⡿⠛⠛⠛⠒⠂
⠀⠀⠀⢹⣿⣿⣭⣿⡟⣻⢧⠀⠀⠀⣿⣿⣿⣿⣾⢒⠶⠀⠀⠀⢻⣿⣿⣿⣬⣍⣉⠟⡀⠀⢡⣿⣿⣿⣿⣿⡛⠢
⠀⠀⠀⠘⠛⠛⣿⣿⡏⣿⣿⣆⠀⠐⠛⢻⠛⢻⣿⠁⠀⠀⠀⠀⠀⠉⣿⡟⢻⣿⡇⠀⣠⣤⡘⠙⠛⠋⠉⣿⣷⡀
⠀⠀⠀⠀⠀⠠⠿⣿⣅⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡂⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⡇⢠⣿⣿⡀⠀⠀⠀⠀⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⢀⣸⣿⣷⣶⣶⣶⣤⣄⣀⠀⠀⠒⠾⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄
⠀⠀⠀⠉⡙⢻⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡅⡹
⠀⠀⠀⠀⠧⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⡿⢹⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠉⠁⣇
⠀⠀⠀⠀⠀⠀⢉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠛⠃⠀⢉⣉⡉⠉⠀⠛⠀⠀⠀⢀⣀⣤⣴⣶⣤⣤⣀⡀
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣶⡤⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⠛⠉⢀⣠⡀
⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⡉⠻⠿⣿⠿⣛⣁⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠛⠀⡄⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣼⡀⠀⠀⠀⠸⠆⠀⢤⣶⣿⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠉⣉⠙⠛⠛⠀⠀⣤⣤⣤⣶⣶⣾⣿⣶⣄⡀
⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠠⠴⣶⣶⣿⣿⣿⣿⣿⣷⣶⠦⠄⠉⢹⡿⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠋⠈⠅⠀⠀⠀⠠⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠓⠶⠤⣭⣭⠽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇⠛⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣴⣾⣿⣿⠋⢀⠈⢻⣿⣿⣶⣤⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⠀⢸⠀⢸⣿⠀⢺⠀⢸⡇⠀⡇⠀⣿⠰⣦⡒⡢⣦⣒⣔⣢⡆⢸⡇⠀⡇⠀⣿⠀⢸⠀⢸⣿⠀⣿⠀⢸⡇
⠀⠀⠀⣿⣄⣸⣀⣸⣿⣀⣼⣀⣸⣇⣀⣇⣀⣿⣶⣶⣷⡶⠖⠷⣶⣶⣾⣾⣇⣠⣇⣀⣿⣀⣸⣀⣸⣿⣀⣿⣀⣼⡇
⠀⠀⠀⣿⠟⢿⠟⢻⣿⠟⢿⠛⢿⡿⠻⡿⠻⣿⣿⣭⣤⣤⣤⣤⣤⣬⣽⣿⡟⠿⡿⠿⣿⠟⢿⠟⢻⣿⠟⣿⠛⢿⡇
⠀⠀⠀⣿⠀⢸⠀⢸⣿⠀⢹⠀⢸⡇⠀⡇⠀⣿⣿⣿⡇⠀⢸⠀⠀⣿⣿⣿⡇⠀⡇⠀⣿⠀⢸⠀⢸⣿⠀⣿⠀⢸⡇
⠀⠀⠀⣿⣦⣼⣦⣼⣿⣤⣾⣤⣾⣷⣴⣷⣤⣿⣿⣿⡇⠀⢸⠀⠀⣿⣿⣿⣧⣴⣧⣴⣿⣦⣼⣦⣼⣿⣤⣿⣤⣾⡇
⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⡇⠀⢘⡀⠀⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠄⠀⠀⠀⢸⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢉⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⢱⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣇⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣭⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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


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

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


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣘⣡⣤⣤⡀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣬⣤⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠿⡿⠟⠛⢻⣿⣿⠃
⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⡻⣿⣿⣿⣿⣷⡶⠂⠀⠀⠀⠀⡇⢸⣿⠿⡀⢸⣿⣿⠀⠀⡇
⠀⢲⠿⠛⠟⣯⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⢿⠛⠚⠙⠃⠀⠀⠈⠉⠉⡇⢸⣻⣀⣁⣸⣿⣿⠰⣾⣧⡀
⠀⢸⢸⣿⡇⣿⣱⠞⠒⠀⠀⠀⣈⣉⣀⣀⣀⣠⣤⣤⣤⣤⢤⣶⣶⣶⣷⡶⣶⣶⢶⣶⣿⣻⣟⢿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⣄⣀
⠠⣼⣾⢿⣷⣿⣶⣶⣾⡿⡝⡿⠿⢯⡩⡻⢟⡛⢯⣷⡝⠛⠛⢻⣞⣹⣿⡛⣛⣾⣋⣉⣿⣉⡩⢯⡿⠥⠼⡧⠟⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠨⡷⡽⠿⠉⠽⠯⠭⠟⠋⣥⣶⣦⡐⢠⣽⣯⣥⠉⣾⣿⣿⣷⠀⢸⣛⠉⠉⢠⣼⣤⠀⠀⠀⡇⢸⠀⠀⠃⢸⡟⣿⡿⣿⢿⣿⣿⣿
⠀⠐⡠⡀⢇⢀⡀⣾⡷⣀⠀⣿⡿⡻⠠⢹⣿⠯⠟⠆⣽⣿⠃⠛⡔⢨⣿⠒⠒⢸⣿⡏⠀⠀⠁⡌⢱⠉⠉⠈⢱⣿⣿⣿⣿⣾⣿⣯⣿
⠀⠰⠀⠀⡆⠀⠀⡟⠀⢸⠀⡿⠀⣇⡇⣿⢹⢐⠀⠂⡏⠸⠀⠀⠀⢸⣯⠀⠀⢸⢸⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢐⠀⠤⠀⠀⠄⠡⠴⠘⢄⠡⠧⠿⠣⠉⠾⠼⠆⣠⡃⠾⠸⠂⣠⣈⡿⠆⠂⠘⠚⠂⠒⠒⠒⣑⣎⠒⠒⡴⣮⣿⣿⣿⣿⣿⣿⣿⣿
⢚⢹⢍⣤⢌⠤⡤⢤⡤⣬⣊⠀⣀⡀⣱⣃⢀⣀⣀⣛⣁⣀⣀⣄⣓⣛⡻⠤⠤⠤⠤⢠⡤⢄⡤⢙⡉⣄⣀⢛⣩⠿⢿⣿⣿⣿⣿⠿⠿
⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠂⠀⠃⠀⠀⠘⠛⠂⠐⠀⠐⠀⠐⠀⠀⠀⠐⠙⠁⠀⠘⠂⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠿⢿⣟⣛⣿⣭⣭⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢙⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣶⣭⣟⡿⢿⡿⢿⣻⣭⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣿⠀⠀⠀⠉⠉⠙⠛⠛⠋⡉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⢸⡀⠀⣴⣿⣿⣷⠀⠀⣿⣿⣿⣷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠳⣄⠙⠟⠛⠁⣰⣆⠈⠛⠛⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠿⠿⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡉⢲⣦⣄⡀⠀⠀⠀⣏⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣴⡖⢉⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣀⠈⠛⠓⠒⠛⠉⣀⣤⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣄⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣿⣿⣿⣯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠉⠉⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣟⣻⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⣠⣤⣴⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠉⠙⢻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠛⢋⣷⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠺⣿⣿⡿⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠙⢿⣿⡇⠀⣠⣤⣤⣄⡀⠀⠀⢀⣀⣤⣤⣄⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⡇⠀⢉⣤⣬⡍⠁⠀⠀⠈⢉⣥⣤⡌⠁⠈⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⡟⢳⡜⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢋⡾⠛⡆
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢳⡘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡸⠁
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⢲⠿⠿⡒⠂⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠀⠈⠉⠉⠁⠀⠀⢀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⣦⣄⡀⠀⠀⢀⣠⣴⠟⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠈⠻⢿⣿⣿⡿⠟⠁⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣯⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣱⣿⣿⣶⣦⣤⣀⡀
⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣠⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣇⢸⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣤⣥⣄⠐⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢈⠁⠀⢶⡶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⢫⡁⠀⢠⡌⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡟⠀⡰⠁⠀⠀⠀⢹
⠀⠀⠀⢠⡆⠀⢸⣿⠇⠀⠀⠀⢧⠀⢸⣧⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠸⡇⠀⢸⡟⠀⠀⠀⠀⢸⠀⠸⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣼⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⢸⡇⠀⢸⠁⠀⠀⣦⣀⣇⣀⣀⣀⣀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⡇⠀⢹⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣙⣁⣀⣀⣣⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠘⠃⠀⠘⠛⠂⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠐⠛⠀⠀⠛

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

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

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠉⠛⠻⢿⣿⠿⣿⣛⣯⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢠⣷⡏⠙⢛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⣾⣿⣷⣶⣭⣟⡻⠿⣿⣿⠿⣛⣫⣭⣶⣿⣿⡄⠀⠀⠀⠀⢀⡴⠊⠉⠲⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣔⠋⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⣠⣾⣿⣿⣷⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⢀⡴⠋⠉⠙⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⢀⡠⠛⢿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠙⢆⢀⡴⠋⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⡟⠻⠛⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠗⠀⠀⠘⠿⠿⢿⣿⣿⠿⠿⠿⠿⠡⠀⢊⡇
⠠⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡢⠕⠒⠲⠀⠉⠉⠉⢉⣼⣿⡯⣠⠎⢁⣼⣥⢈⠀⠀⠀⠘⠀⠠⠼
⣀⠀⢠⣾⣶⣶⠀⠀⠀⠀⠀⠂⠀⣟⠤⠄⠠⠐⠄⠢⠄⠀⠈⠉⠙⣧⡃⠀⢤⠠⠞⠙
⢹⠃⢸⣿⣽⣿⡆⠀⠁⠀⠀⠀⠀⢀⢀⢁⠀⠉⠁⠀⠀⠀⠈⡀⠀⠃⠀⠀⠀⡀
⠄⠠⠉⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤
⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠀⠉⠀⠀⠀⠀⠉⠐⠂⠄⣀
⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⣀
⠀⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⣀
⠀⠯⣓⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠾⢅⣀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠉⠒⠭⣐⠤⣠⡤⠶⠚⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠝⣦
⠀⠀⠀⠀⠀⠀⠀⠙⡿⠭⣒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢐⣀⠩⠄⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⠉⠒⠬⣐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣒⡈⠥⡖⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠼⢇⣸⠀⠀⠀⠀⠀⠉⠒⢍⡒⠄⡀⠀⣀⠠⠄⢒⡈⠥⠐⠊⠉⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢸⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⢯⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡔⠁⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠠⠤⠔⠒⢄⠀⠀⠀⠀⠀⠀⡠⠐⢬⡇
⠀⠀⠀⠀⠀⡧⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠘⢞⠍⠁⠂⡀⠀⠀⢨⠀⠀⠀⠀⢀⠎⠀⠀⠠⢧⠀⠀⡀⠔⠊⠒⢄
⠀⠀⠀⢀⣔⠕⢀⢀⡜⠀⠈⣒⣤⡀⠀⠀⠀⠀⠀⠑⠨⢄⡓⣦⡰⠞⠒⢤⠤⢒⣁⡠⠤⠄⠀⠀⡏⠁⠀⢀⡀⠤⠐⠋⢱
⠀⠀⠀⠈⠉⠈⠉⠁⠀⡐⠩⠋⠉⠋⠉⠐⠒⠂⠀⠄⠠⠤⠤⠤⠗⠤⡤⢎⢙⣓⡢⣤⠀⢒⣡⠜⠂⠈⠉⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡆⠀⠘⡇⠁⠐⠒⠒⠂⠀⠁⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠳⠐⠒⠂⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⢻⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢸⠀⢀⢇⣀⠤⠴⠒⢂⠞⠀⠄⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣀⡤⠧⠐⠊⠁⠀⠀⠀⠀⡎⠀⠀⣡⠀⠀⠱⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠔⠢⣌⠢⡀⠀⠠⢖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠒⠒⠉⠁

⠀⣶⠶⢶⡶⠶⢶⣶⢶
⠀⣿⣏⢩⡅⢸⠛⢝⣿⢠⠄⠀⠀⠀⠀⡀⣤⠀⠀⢠⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⠀⠀⠀⢠⡄
⠐⡦⣟⡃⠎⠘⣿⠋⠻⠸⣽⠻⠟⠻⠀⠇⢯⡷⡇⢸⣧⠜⢋⠢⠶⠩⠾⠰⠅⣿⠐⠷⠀⡇⢾⠸⠿⠦⠞⢸⠸⢷⠀⡇⢾⠸⡮⡽⣗
⢰⠓⠀⡱⣆⠨⠉⡶⢽⠀⡄⡀⢀⣀⠀⣠⡀⡀⠀⠬⠁⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠈⠋⠀⠈⠛⠉⠋⠀⠋⠀⠁⠀⠋⠉⠃⠈⠋⠃⠘⠃⠘⠃⠋⠈⠘⠀⠋⠃⠘⠘⠉⠋⠁⠛⠃⠘⠉⠀⠚⠁⠘⠐⠃⠃⠃⠛⠍⠃⠛

⠀⠉⠛⣶⢌⣿⠟⡶⢬⣍⠙⢿⣯⠋⡏⠀⣻⠉⣤⡹⣿⣏⣩⣽⣿⣍⣉⣹⣿⡿⠿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⡈⠋⠉⣉⣻⣶⣿⢟⢳⣋⢂⣦⣯⡀⠙⠉⣁⠂⠉⠛⢻⣅⡉⠉⠀⠉⠁⠒⠇⠐⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠱⠀⠀⠀⠄
⠀⠢⠋⠉⠀⠀⠀⠁⠹⠿⣷⣮⣝⣿⠙⠛⢿⡀⡄⠀⡄⠀⠀⠀⠸⠀⠀⠉⠉⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠀⠀⠀⠀⠀⠐⠟⠿⣷⣽⡟⠋⠡⣤⠀⢻⣧⠁⢀⡃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠀⣐⣶⣿⣿⣿⣿⣿⡿⢿⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⠻⢬⠎⠃⣙⣻⢌⣛⠀⠀⠙⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠛⢋⣭⣍⡉⠉⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠘⢀⠸⠁⠀⠐⠀⠀⢀⣭⡿⣘⣯⠸⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⡍⢻⡄⠀⠀⠀⠀⡃
⠰⠄⢀⠀⠀⠠⡤⠄⠐⣦⠢⣤⠀⠀⣹⣧⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠁
⠀⠠⠁⢀⠀⣀⣀⠀⢀⡁⠀⠀⠀⠀⠻⢿⡆⠀⠀⠀⣠
⢀⠠⠀⠁⡜⠋⠉⠗⢀⠁⠀⠀⣰⣿⣟⣀⠀⠐⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠄⠀⠷⠀⠀⡸⠳⠂⠀⢠⡟⢻⣿⣿⣷⠂⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡟⢧⣄⢀
⢠⠀⠀⠀⠀⠠⠾⠃⠌⠀⠀⠈⠀⠸⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠛⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠦
⢘⢲⡀⠀⠸⣦⣄⣾⠁⡆⠘⣦⡀⠀⠻⣿⣿⣿⣿⠓⢄⠀⠀⠀⠶⠞⣁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣾⡿⣳⣦⢀
⢘⡨⠁⣀⣀⣈⣛⣛⢀⡇⠀⣿⣷⣄⠠⠙⠯⠒⠉⠃⢰⠀⠀⠈⠐⠆⣹⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠸⣇⠻⣷⠾⠃⠀⠀⡀⠀⢀
⠸⠧⡆⢈⠿⠻⠃⣿⣌⠀⠀⣿⣿⣿⡿⢦⣴⣤⣴⡄⠾⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⣿⣶⣶⣷⣽⣿⣿⣻⣿⡃
⢘⠻⠇⡈⢒⡀⠒⢿⡇⠀⢸⣧⣹⡁⠀⠀⠙⠉⠈⠁⠀⢠⡦⢀⠀⢀⠸⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠉⠙⠛⠉⠉⠁
⢘⣻⣛⡙⠻⣧⣗⠺⠇⠀⢸⣿⣿⣧⠀⠀⠘⠀⠀⠀⣆⠀⣇⠟⠀⣴⠃⠀⠀⠀⠈⢟⠃
⠘⠁⢿⣏⣵⣿⣾⣾⠃⠀⣸⣿⣿⣽⠀⠀⠀⠂⠀⠀⣿⡀⠘⠀⠘⠏⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠠⠞⣿⣿⣿⠀⠠⡿⢻⣿⣿⡆⠀⠀⠒⠀⠀⣿⣿⠷⠒⢴⡄⠀⠀⠀⠀⢰⣆
⠀⠀⠸⢃⣬⡿⣿⡏⠀⢸⡇⣾⣏⣽⡟⠀⠀⠀⠀⠀⣿⣿⣇⣐⣈⠉⠉⠀⠀⠀⠈⠙⠓⠓⠒⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠐⠲
⠀⠀⠀⢴⠁⠡⡭⠀⣠⣺⠇⣿⡏⠉⡇⠀⠀⠀⣶⣿⣿⣇⢿⠋⠻⣟⡀⢀⣀
⡀⠀⠀⢸⣴⣶⠇⢰⡟⣿⠀⢸⣧⠀⠃⣄⠀⠀⢫⣤⣿⡟⠀⠀⠀⠈⠁
⠄⠀⠦⠿⠿⠋⣠⠟⠀⢽⠀⠈⠻⠚⢺⣿⣷⣾⢛⡿⠉⢇⠀⠀⣠⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠄⠄⠀⠀⠀⡠⠄⠘⠉⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣶⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⣀⣀⣀⣀⣀⣀
⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⣀⣤⣄⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠚⠷

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⢾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⢰⣾⡄⠀⠀⠀⠀⣶⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⢀⡏⠁⠀⠀⠀⢀⣀⡀⣴⠀⠀⠠⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⡀⠀⠀⠀⠀⠐⣯
⠀⠐⠀⠀⢀⣸⣿⣿⣤⡄⠀⢸⣿⠀⢀⣀⠀⢸⣿⣿⣁⣀⠀⠖⣷⠂⢹⡀⠀⠀⠀⠀⢸⣀⡀⢶⡀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⣾⣇⠀⠂⠀⠀⣤⡄⠀⠀⣹⣇
⠀⠀⠠⣾⣿⣿⣿⣿⣤⡀⠀⢸⣿⣧⣿⣿⠇⠀⣿⣿⣿⣿⣷⡄⣸⣤⣾⣷⣾⣿⣦⠀⣼⣿⡿⠀⣹⣄⢻⣶⣾⣷⡄⠄⠀⠀⣴⣿⣿⣇⠀⠁⠸⣿⣟⣠⣾⣿⡇
⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⠄⠘⣿⣿⣿⣿⣿⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⢳⡿⢾⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⢼⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢿⣿⣿⣿⣿⣿⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⢀⡼⡅⠀⠠⠈⢄⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⢀⣤⡅⠀⢐⡀⠀⠀⠀⠀⠀⠀⠉⢭⠉
⠀⡏⠉⠕⠢⠀⠁⡈⢂⠀⠀⠈⠎⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⣺⠄⠀⠉⠀⠀⠻⠍⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡾⠂
⠀⣧⣄⠈⢢⠀⡀⠁⠀⠂⠀⠀⠀⢱⠂⠀⠀⠀⠀⠀⡴⡄⠀⠀⠈⠁⠀⠘⠻⠁⠀⢰⡭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡾⠛⠖
⠀⠅⠉⠀⠀⠡⠈⠄⠐⠎⠁⠀⠣⠜⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠱⣾⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠏⠀⠀⣀⡀
⠈⢿⠛⠳⣤⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠤⣀
⠀⠀⠰⡀⠂⠈⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠀⠀⡆⠀⢢⠀⠱⣄
⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡤⠤⠴⠊⠀⡃⠀⡁⠁⠀⡁⠀⡄⢸⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠇⠠⠂⠈⠙⠷⣶⡀⠀⠀⠀⠃⠀⠔⢸⡇⠀⠀⠀⠀⠤
⠀⠀⠀⡦⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣶⣿⣿⡆⢰⡃⠘⠀⣠⡟
⠀⠀⠀⡸⠂⠀⡀⠺⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣤⣤⡆⠀⠀⠀⠙⢌⠂⣶⣶⣿⣿⡿⠃⠀⢑⣲⠿⠋
⠀⠀⠀⠀⠊⠁⠀⠰⠈⣧⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠭⠭⠭⠤⠴⠊⠉
⠀⠀⠀⠆⠀⠠⠤⠀⢀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣧⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⢃⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠠⠔⠂⠈⡀⠄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣷⠀⠀⠴⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠦⠀⠀⠣⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣇⠁⠀⠀⠀⠠⠀⠀⠐⡀⠀⠁⠀⠀⠈⠀⠀⠀⣀⢤⠁⠀⠀⠀⠀⠀⠀⡠⠪⠀⠀⠀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠈⣁⣠⠤⠗⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡠⠤⠞⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⢁⠒⠀⠀⠐⠈⢀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣒⣀⣉⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⠀⠈⠀⡴⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠓⠄⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡄⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠀⠉⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢙⡃⠈⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠁⠊⠉⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠰⠖⠲⠾⠀⠀⠀⢠⠀⠆
⠀⠀⠀⠀⢈⠁⠀⢀⣀⣠⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠰⠖⡆⠀⢢⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠚⠀⠄⠀⠀⠀⠀⠶⠚⠓⠀⠀⠀⠀⠰⠺⠆⢠⠀⠀⠀⠸⡈⠓⠂⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠁⠀⠀⠈
⠀⠀⠀⢶⣴⠀⠘⠀⠀⠀⠾⠁⠐⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠤⠄⠈⠁⠂⠀⠀⠀⠄⢀⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠏⠀⠀⠠⠰⠀⢀⣀⣠⡀⡀
⠀⣄⢀⣸⡷⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠒⠀⢈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠔⠀⠀⠼⠏
⠀⠀⠁⠀⠀⠀⠀⠳⠖⠺⢃⡀⣀⡀⠀⠀⠀⠀⣠⡄⢰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠒⢲⣶⠃⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡿⢦
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠆⠁⠈⠁⠀⠀⠀⠈⠀⠀⠀⠈⠉⡀⠀⠀⠓⠀⠀⠀⠐⠉⠀⣾⠈⠀⢠⡀⠀⠒⠀⠀⠀⠀⠃⠀⠀⠸⠇⠀⠀⠠⠀⠰⠆⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠑⠆
⠀⠀⠀⢀⣠⡤⠄⠠⠐⢦⠀⠀⠀⠀⠀⠀⠀⣤⠲⡄⣀⠀⠀⠀⠠⠖⠀⠀⠀⢠⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⡏⠀⠀⠀⠤⣤⣀⡀⠀⠀⠀⠀⡀⠀⣀⠜⠏⠸⢠⡄⠈⠀⠀⠀⠀⠀⣀⣀⡄⢲⠤⠔⢨⠃⠽⠂⠀⠀⠀⠀⠰⡙⠶⡂⠾⠤
⠀⠀⠃⣨⡿⠦⣆⠀⠀⠀⠉⣁⠘⠀⠀⠀⠱⡆⠙⠀⠠⠊⢠⠗⠠⡦⢈⡀⠀⣤⣠⢤⡄⠀⠀⠀⠠⡷⠀⠀⠆⠀⠀⢠⢀⢠⣄⡁⠀⠀⠀⠀⠂⠀⢀
⣉⡀⠀⠉⠀⠀⠼⠀⠸⡇⠀⡍⠀⠀⠀⠀⠀⠀⠐⢀⠀⢉⣹⠀⠀⠸⠟⠀⠀⠀⠀⠀⠁⠂⣀⡀⠀⠃⠸⠃⠀⠀⠀⠘⠇⢀⡝⠇⠀⠀⢀⣸⢟⠙⣿⣄
⢀⠈⠀⢨⠀⠰⣀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⣀⡀⢠⣀⣤⡝⠀⠀⠀⠠⠷⠀⠀⣀⠀⠰⣤⡤⢼⠆⠀⠀⠀⠀⠀⠀⠐⠀⢾⣹⡆⠀⣀⢀⣾⢻⠀⠀⢹⣆
⠈⠁⠐⠛⠀⡀⠙⠐⠃⠒⠀⠾⠀⠀⠀⠀⠠⠏⠙⠀⠓⠒⠂⠀⠰⠷⠀⠓⠀⠘⠋⠒⠎⠠⠀⠴⠂⠀⠀⠀⠀⠀⠀⢠⡦⡿⢙⣄⣉⠍⠘⠏⠘⣀⠀⠈⡏⠁
⣀⣀⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣇⣀⣁⣀⣀⣀⣀⣀⣀⣘⣋⣀⣀⣀⣀⣰⣄⣀⣀⣐⣁⣀⣀
⣿⡿⣿⡿⣿⠿⢿⠿⠻⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⡿⠟⣻⡿⢿⣿⣿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣤⣀⣾⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⢸⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣼⡇
⠀⠀⣿⣿⣷⠀⠀⠀⣾⣿⣿⠀⠀⠀⢰⣿⣿⡇⠀⢸⡀⢸⣿⣿⡆⠀⠀⢰⣿⣿⣗⠀⠀⠀⣾⣿⣿⡇
⢸⣿⣿⣿⣿⡍⣧⣿⣿⣿⣿⣏⢹⣾⣻⣿⣿⣿⡙⣏⣿⣿⣿⣿⣯⢹⣿⣻⣿⣿⣿⡝⣯⣟⣿⣿⣿⣯⢻
⣾⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿
⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿
⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⢸⠀⠴⠒⠒⠒⠒⠒⠒⠲⢤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢺⠁⡤⠖⠒⠛⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⢾⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠋⠉⠛⠓⠒⠲⠤⣄⠈⡇
⠀⠀⠀⠀⠀⠀⢿⣂⣀⣤⠤⢶⣶⣶⣶⢶⡆⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⡁⠈⠀⠀⢀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢸⠅⠀⠈
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠛⠀⠴⠀⠀⠰⢾⠇⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠁⠀⡂⠀⠘⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠶⣶⣤⠤⠄⣀⡀⢀⡿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠀⠀⠐⠆⠘⠻⣿⡿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣬⣭⣬⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⢿⡿⠁
⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠐⡂
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣧
⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢣⠛⠿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⠛⠁⠁⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣿⣿⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠛⠻⠿⠟⠻⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢸⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠘⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠀⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠃⠛⠛⠛⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠐⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠠⣉⠒⠬⢍⡛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠒⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠠⢍⡓⠦⢌⣑⠂⠈⠙⠛⠶⣤⡀⠀⢀⣀⡤⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣐⠨⢍⡒⠮⢍⡒⠌⠉⠒⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠙⠳⢮⣍⠒⠤⢁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠛⠛⠛⠷⢦⣤⣀⡀⠀⠀⠀⣠⡾⠁⠀⣀⣠⣤⡴⠶⠞⠛⠛⠋⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⢶⣴⣦⣦⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡸⠱⣞⠻⣦⠤⠤⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⡠⠴⢶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠁⠀⠹⣦⡿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢶⡿⢀⣠⣤⣤⣤⣽⣿⣿⣿⣿⣿⣛⣋⣉⣉⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣀⣴⠶⠛⠋⠉⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠭⣭⣭⣍⣉⣉⣉⣛⣛⣛⣛⠛⠓⠶⠶⠶⠶⠶⠶⠶⣶⠿⠇
⢀⡾⠋⠀⢀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠚⠓⠒⠒⠒⣒⡒⠒⠶⠶⠾⠏⠭⠭⠁⠀⠀⠀⢰⠛⠲⢤⣀
⣾⠁⠀⠀⠀⠈⠙⠒⠄⠀⢸⡇⠀⠀⠀⠀⠠⠤⠄⢀⣀⣀⣀⣉⣛⣓⣒⣒⣒⠒⠢⠤⠤⠭⢭⣭⣍⣉⣁⣀⠀⠀⠀⣿⠀⠀⠀⠈⠛⢶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣂⠤⠤⠤⠤⣤⣤⣤⣤⡀⢀⠈⡉⢽⠿⠍⠭⠉⠛⠓⠒⠂⣤⣄⣠⣒⣂⣀⠀⢿⣀⣀⣤⣤⣴⣶⡾⠿⠃
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠶⠶⠶⠿⢤⣤⢦⣥⣥⣬⣥⣼⣿⣿⣷⣶⣦⣄⣀⣒⣲⣶⣶⣀⣈⣹⣏⠉⠀⠀⣾
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠁⠀⠀⢰⡏
⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠁⠀⠀⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠀⠀⢸⡇
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣤⣤⠤⠶⠶⢒⣒⣀⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣈⡻⣝⡒⠀⠀⠀⠀⠐⠒⠚⢛⣋⣉⣭⣭⡥⠶⠶⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠈⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⡌⠛⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠄
⠀⠀⢀⠃⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡸⠀⠀⠀⡎⠘⠛⠉⠐⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⡀⠀⠑⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⡇⠀⠀⠊⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠈⠉⠁⡄⠀⠀⠀⠀⠱⢤⠤⠴⡟⡀⠀⠀⠀⠀⡜⠁⠈⡆⠀⠀⠀⢠⠃
⡜⠀⠀⠀⠒⠠⠤⠀⠀⡤⠠⠌⠒⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠓⣄⠀⡏⠈⠙⡆⠀⠀⠘⠢⠾⠁⠀⠀⠀⠀⠢
⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠦⠤⠃⠀⠙⠒⠊⠀⠀⠀⠀⠀⡜⠉⠑⡄
⢰⠋⠈⣷⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠴⠃
⠀⠑⠒⢉⡞⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠉⠙⣦⢸⠀⠀⠀⢠⠂⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠈⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠔⠉⠈⠳⠤⠖⠁⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠘⠛⠃⠓⠀⡤⠀⠀⠀⠀⠠⣶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⣤
⡤⠞⠁⢆⠀⠀⢠⡔⠒⣄⢀⣀⡴⠛⠲⡄⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠈⢂⠀⠈⠓⠒⠉⠀⠘⣄⠀⠀⣽⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡀
⠀⠢⠀⠀⠀⠱⣀⡀⡀⠀⠀⠀⠈⠉⠉⠀⡞⠁⠀⠱⣄⡀⣞⠁⢘⠄⠀⠀⢢⠈⠀⢸
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠙⠦⣄⠴⠁⠀⠀⠁⠁⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⠈⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀
⠀⠀⡇⠀⡀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠇
⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠐⠒⠚⡄
⠀⠀⠀⠈⠂⠤⢀⣀⡀⠄⠊⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠀⠑⢤⣤⠤⠞⣀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⣧⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⢀⠔⠛⢄⠀⡎⠉⠉⡆
⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠞⠀⠘⠒⠚⠁
⠀⠀⠀⠀⠘⡀⢀⡅⠀⠀⠀⠀⠉⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠨⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⢢⣔⢁⠀⠔⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠦⠤⢤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢐⠄⠈⠩⠩⠉⠉⠉⠀⠐⠀⠀⠈⠀⠰⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠱⢚⠃
⠀⠀⠀⠂⠀⠀⠀⠀⠉⢸⣳⠀⢿⠁⣼⠉⢧⢰⡇⠀⢰⠏⢳⡄⡞⠉⡃⠈⢧⣰⠃⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠂⣼⡀⢿⣤⡞⠘⣧⣄⠸⣄⡼⠃⢧⣀⡟⠁⢨⠇⠀⠀⠈⠐⠀⠐⠂
⠀⠀⠀⠀⢠⠀⠠⠐⠀⡀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠄⠀⠀⠈⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢠⠀⢀⠀⡀⠀⠀⠠⡀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠈⠀⠀⠐⠄⠂⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠀⠀⠐⠋⠀⠑⠂⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠈⠉⠉⠉⢻⠉⠉⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠉⠓⠂⠒⠀⠤⠀⣀⣰⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠴⠶⠶⠆⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠒⠀⠀⠠⠀⠀⠐⢶⠐⠒⠒⠒⠒⠂⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠤⠂⠑⠂⠀⠀⠄⠠⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠐⠠⠤⠤⠄⠀⠰⠖⠂⠒⠂⢀⣀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠠⠤⠄⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠊⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠐⠒⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠬⠛⠉⠁⠀⣀⠤⠶⠊
⠀⠀⠀⠀⠀⣀⠀⢠⣄⡀⠀⠀⠀⠠⠄⠉⠀⢈⠙⠛⠲⠶⠶⠤⠤⣄⣀⡀⢀⣄⡄⠐⠂⠁⠀⢀⠀⠀⠀⠀⠀⣀⣤⡀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠙⠫⣝⣓⡒⠢⠤⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡄⠈⠀⢀⠀⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠨⠁⠐⠈⠭⢙⡒⠢⠄⢀⠀⠉⠉⠉⠛⠒⠒⠲⠶⠭⢄⣀⠄⡡⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⣼⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐⠀⠄⠀⠀⠀⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⢠⡀⠈⡉⠛⠒⠂⢤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊
⠀⠀⠀⠀⠀⠈⠑⢳⣠⢕⠂⠀⠀⠭⣓⠠⡂⠄⣀⠠⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠠⠂⠁⢀⠀⠁⠪⢀⠀⠀⠀⢈⠀⠀⠠⡌⠐⠂⠈⠀⠀⡈⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⡐⠚⠉⠁⠀⠀⢀⠀⠀⠄⠀⠈⠙⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠡⢀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠀⣀⡤⠖⠉⠀⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠑⠡⡀⠄⡈⢷⣶⡮⣤⣀⠐⠱⠦⣀⣸⣃⡀⠠⠐⠒⠉⠁⠀⠀⢀⣠⠔⠛⠁⠀⠀⢀⣠⠴⠋⠈⠀⠀⠀⠀⠀⠀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠈⠙⠓⠮⢍⡓⠲⠤⣭⣑⠊⠴⢂⠄⠀⠀⠴⠚⠉⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠔⣠⡑⠨⢙⠲⡠⢀⡀⠀⣄⢀⠄⠊⠀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⡑⠤⡉⠙⢁⠀⠁⠉⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⠄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡠⠖⠉⡠⠀⠈⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠃⢸⠂⠀⠀⣠⠾⠓⢣⡄⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠾⣀⣿⠀⠀⣼⠁⢀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠬⠀⠀⠀⠂⠁⠀⠀⠀⠀⠰⠓⠊⠉⠀⠀⢹⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠲⡏
⠀⠀⢀⠀⠀⠈⠃⠇⠀⠀⠁⠀⡀⠀⠀⡆⠀⠀⠀⠈⠀⠁⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠱⡀⠱⢠⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠘⠒⠂⠁⠐⠀⠀⠀⠀⠀⡀⢠⣾⣿⠿⣯⡆⡀⠀⠀⡰⠂⠀⢀⠄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⢆⠛⣵⣻⡉⠅⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠖⠋⠀⠀⢀⠀⠀⠀⠀⢀⢀⡀⠁
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠑⢄⣉⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⠁⠀⠘⠀⡀⠀⢀⠀⠀⠃⠀⠠⠓⠒
⠀⠀⢞⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠠⠀⠀⠀⢈⡔⠃⢀⡀⠀⠀⠀⠀⠄⡐⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠄⠀⢀⡄⠉⠀⠀⠀⣠
⠀⠀⠀⠀⠄⠀⡀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠆⠁⠀⡀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢘⠀⠀⢁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠚⠦⣤⣐
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠴⢦⣄⡀⠀⠈⠹⡖⠀⠀⠉⠂
⠀⠆⠚⠅⠀⢸⡥⠄⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⢧⡀⠘⠉⠛⢒⠇
⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⢰⠂⣀⡀⠀⢀⠂⠀⠀⠘⠋⠀⠰⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⠀⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⠄⠴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠒⠁⠄⠀⠄⠀⠱⠀⠉⡀⠀⠀⠐⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠸⠀⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠢⠄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠂⠀⠀⢁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠔⠋⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⡴⠲⡤⠀⠀⠀⠀⢴⠒⡦⠀⠈⠀⠀⠀⠀⠠
⠀⢀⠀⢑⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⣀⠀⠀⢀⢟
⠀⠐⠠⠐⠀⠀⠀⠐⢢⠀⠀⠀⠙⠢⡡⠊⠁⠀⣀⣤⡖⠁⠀⠀⠐⠆⠀⠀⠀⠀⠆⠈⠉⠁⠀⠀⠀⢈⢀⠀⠐⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⡤⠴⠚⠛⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠂⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⢂⣸⠔⠊⠀⠀⠒⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠌⢆⠀⠡⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣶⠚⠁⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡀⠀⠀⠀⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⢂⢠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡠⠀⡈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⢀⠠⠄⠠⠀⡁
⠀⠀⠀⠠⠊⠀⠀⢅⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⡂⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢣⡀⠀⠀⠀⠀⠈⡀⠀⠂⠉⠀⠀⠘⡎⠤⠄⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠌⡡⠁⠀⠀⠊⠈⠐⠒⠀⠀⠀⠀⠀⠂⠄⡀⠀⠱⣄⣠⠊⠀⣎⠀⠀⡀⠠⠐⠂⠈⠀⠀⢀⠀⠈⠐⠠⡀⠀⠐⠈⠀⠀⢠
⠐⠀⠀⠀⠀⡐⠁⠀⠀⢀⠔⠀⠀⠠⠀⠀⠀⠀⠀⡘⠒⠢⡀⠀⠀⢀⠜⠓⠈⠀⢀⠀⠐⡶⣄⠀⠀⠉⠐⠠⣀⠈⠢⡀⠀⠀⠀⠈
⡄⠀⠀⠀⡜⠀⠀⠀⡴⠃⠀⣰⠃⠀⠀⠀⠀⡠⡚⠁⠀⡀⠘⡄⠀⠆⠀⠀⠀⠀⢹⠀⠀⠠⠠⠑⡄⠀⠀⠀⠈⢓⠄⠘⡄
⡇⠀⠀⠰⢀⠀⢀⠊⠁⠀⠄⠄⠀⠀⠀⢀⠔⢠⠀⠀⠀⠁⠀⠐⡄⠀⠀⠀⠀⡈⠘⠀⠀⠜⠀⠀⠈⠉⠀⠀⠐⠒⠈⣄⡁
⠀⠀⠀⠃⠀⠠⠃⠰⠄⠪⠎⣀⠀⠤⠒⠁⠀⠘⢄⠀⠀⡆⠀⠀⡇⠀⡄⠀⢀⠅⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠸⡇
⠀⠀⠀⢻⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠑⡄⠀⡇⠀⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⠤⢂
⠀⠀⠀⠀⢳⠀⠀⡱⠆⠀⠀⢤⠀⠠⠤⠀⠀⠀⠀⣔⠀⠀⢸⡘⠀⢢⠟⠀⠀⠀⢶⠀⠀⠀⠘⠥⠆⠀⠀⠀⠀⣠⠊⠈⠆⢹⢄⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⢀⡀⠴⠎⠁⠀⣁⠠⠤⢚
⠀⠀⠀⠀⢪⡏⡖⠒⠀⠈⠒⠶⣤⠄⠀⣤⠤⠔⣒⠈⠁⠀⠀⠰⢒⠆⠄⠀⢰⢘⠓⠒⠒⠒⢒⡖⠊⢿⡇⠀⠋⠑⠀⠀⠠⠠⠄⠀⢠⠴⠉
⠀⠀⠀⠀⠀⠓⠓⠒⢀⠜⠁⠐⢩⡀⠀⢰⠤⡀⠀⠸⢱⢆⡀⡐⠞⠚⠛⠀⠈⡍⢄⠰⠤⡂⠁⡄⢈⡊⢓⠔⠀⠈⠒⣤⠤⠐⠂⠈
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢯⣀⡙⠀⠸⠀⠀⠑⠣⠌⠉⠀⠈⢘⣦⠀⡇⢔⢇⠀⠀⠀⠀⠉⠇⠀⠀⠈⣂⢀⡠⣎⠙⢦⢰
⠀⠀⠀⠀⠀⢀⠻⠒⠠⣀⠀⡄⠀⠀⠀⠀⠳⡀⠀⠀⠀⡶⠢⢠⠄⠟⠀⡅⠀⠨⠀⠀⠀⠀⠠⡖⠋⢈⣉⣁⠠⠔⠉⠉⠉⠁
⠀⠀⠀⠐⡘⣇⣀⠤⠒⠃⠬⢁⣰⠒⠂⡯⠝⠀⠀⠀⠀⠑⠀⠈⠈⠰⡤⠃⠀⠀⠀⠀⠀⠀⠀⣜⡲⠂
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣽
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣀⣠⣴⡄⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠚⠉⠓⠀⡁⢠⣤⣤⣀⡀⣀⣀⣀⣠⣄⣤⣴⣾⡏⠉⠈⠓⠈⠁⠸⢷⣄
⠀⠈⢉⣉⡉⢹⣿⢿⢾⡟⣾⣷⣿⣿⣿⠛⣿⡟⢸⣿⢿⡿⠇⠹⠸⡇⠸⣿⣿⢻⢹⢹⡇⢘⣿⡟⠁⠀⠛⡟⠃⠀⠀⠘⡟⣢
⠀⠀⠘⠻⠁⠈⠀⠉⠸⠇⠸⠿⠿⠿⠟⠀⡿⠃⢸⣺⡾⠇⡇⢀⠸⠇⠀⠙⠙⠘⠘⠘⠃⠀⠛⠋⠀⠀⠀⠃⠀⠀⠀⠀⠿⠋
⠀⠀⢘⠒⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⣀⢀⣤⡀⢨⣤⣤⣤⠅⠈⢠⠄⠀⣤⣤⣤⢤⣤⡀⢠⣤⠀⠀⠀⠀⠁
⠀⠀⠸⠶⠂⠀⠀⡀⢸⡇⠨⠙⢻⢈⢿⢸⣿⠇⠀⠈⣿⠀⠁⠀⢸⠀⠀⣏⠠⠚⠐⢿⠄⢸⣿⠄⠀⠀⠒⠀⠂⠀⠀⠀⡁⡀
⠀⠀⠀⠀⢀⠠⠘⠠⠄⠀⠀⠀⠀⠀⠈⠀⢩⣆⣠⡀⣿⠀⢄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡒⡏⣀⡀⠀⠀⠀⣀⡈⢉⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠐⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣋⠉⠙⠻⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠁⠀⠀⠀⠁⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⢋⠁⠀⠀⠀⡐⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠳⣴⣤⠎⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡘⢃⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠚⠒⠒⣵⡆⡇⢸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣉⣁⣁⣠⠀⣿⡇⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⢛⣛⠛⠉⠀⡿⠇⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⡇⠀⢰⠋⠉⢳⠀⢰⣅⡀⣼⢧⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠦⡤⠞⠀⠀⣿⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣠⣈⠈⠀⠄⡠⢻⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⢸⡇⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⢸⡇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠇⠸⠿⠿⠶⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠭⣓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢕⡢⠭⣐⠈⠙⠓⢦⣄⠀⠀⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⢴⣪⠥⣂⠅⠀⠈⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠠⠬⠿⠶⣥⣀⡀⠀⠀⠀⠀⣼⠀⠀⢀⣀⣠⡤⠶⠶⠿⠯⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠿⠗⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⣶⢶⣶⠶⠶⠶⠶⠖⠒⢒⣒⣒⣒⣶⣶⠶⠶⠓⠛⠛⠛⠛⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠁⢳⣸⡟⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⠐⠊⠙⢦
⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃⣀⣠⣤⣿⣷⣶⣶⣶⣖⣒⡚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⢀⡴⠖⠛⠉⠁⠀⣿⠟⠉⠀⠀⠀⠀⠐⠋⠭⠭⠭⠭⠭⠍⢉⣛⣛⢛⣛⡒⠒⠒⠒⠲⢶⡾⠓
⣰⠋⠀⠀⠤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⠉⠡⠭⠭⠭⠤⠄⠀⠀⠀⣼⠉⠓⠦⣄⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⣀⣀⣈⣉⣉⡀⠐⠒⠒⠒⠠⠭⠍⠉⠉⣀⣀⣀⠀⠀⣇⠀⣀⣀⣈⣹⣷
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⣀⣭⣽⣯⣤⣈⣿⠋⠉⢹⡇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⣠⠴⠖⠒⠛⠉⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡘⢷⡠⠀⠀⠀⠀⠤⠤⠶⢒⣒⣛⣫⡭⠭⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠋⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⣠⣤⣶⠶⠶⠶⠶⠶⢶⣶⣤⣤⣀⣀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢀⣀⣠⣤⣤⣶⡶⠶⠶⠶⠶⢶⣦⣄⡀
⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⡟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠉⠛⠻⢶⣦⣄⡀⣀⣤⣴⠾⠟⠋⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤⣿⠿⠿⣯⣄⡀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠙⠻⢷⣤⣀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣀⣴⠿⠋⠀⣿⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠛⠿⣦⡀⠀⠀⠀⠀⠀⢿⡇⠈⠛⢷⣤⡀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣦⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⡿⠻⣷⣄
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠙⢷⣦⡀⠀⣿⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⣼⡇⠀⣠⣴⠟⠉⠀⠀⠈⠻⣷⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⢻⣶⡶⣦⣄
⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠉⠛⢷⣦⣀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣾⣏
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠈⠙⠻⣶⣤⣴⡾⠛⠉⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣀⣤⣴⠾⠟⠋⠙⠻⢶⣦⣄⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠹⣧⡀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠀⣀⣀⣿⣷⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠈⠻⠷⣶⣤⣤⣿⠁⠀⠈⣿⠾⠿⠛⠛⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠛⠛⠿⠶⢶⣶⣤⣤⣤⣤⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⡾⠋⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣤⣴⠿⠋