Cognition Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⣷⡄⣤⣶⣶⣶⣶⣶⣶⡶⠆⣠⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⠻⠟⣡⣾⣿⡇⣸⣿⠟⣛⣉⣙⣛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣿⠟⣡⡍⠁⠾⢿⣿⣿⣿⣿⣿⡿⠟⢛⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣬⣉⣛⣉⣥⣾⣿⢰⣿⣷⣶⣬⣙⠻⡿⢋⣴⣿⣿⣿⣿⣶⣌⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⢋⣥⣼⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣟⠢⢋⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣏⡤⣙⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢛⣩⣥⣭⣙⠻⣿⣿⣿⡿⠟⣛⣿⣷⡘⢿⣿⣿⣿⣿⡿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠏⣴⣿⣿⣿⣿⡿⣷⡌⢫⣤⣶⣿⣿⣿⣿⣿⣶⣬⣭⣭⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⢰⣿⣟⠻⡿⢋⣴⣿⣷⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⣠⣾⣿⣿⣿⣆⢻⣿⣧⣴⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣆⢻⣿⣿⡇⢸⡇
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⣌⣛⠛⢛⣋⣥⣾⣿⣿⣿⡿⢛⣉⣭⣍⡘⠿⣿⣿⣿⣿⣿⡆⣿⣿⡇⣼⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⡿⣷⡘⣿⣿⣿⣿⣷⣾⣿⣷⣿
⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⡟⢿⡿⢋⣴⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣦⣴⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⡛⠻⠛⣋⣥⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣤⠄⣠⣄⡀⣀⣤⡄⣠⡀⡄⣠⢠⣤⡄⣀⢠⠀⣠⢠⣤⠀⢀⣤⣀⢠⣤⡀⣀⣤⡀⢀⣤⡄⣤⣤⢀⣤⠀⣠⡄
⠀⢿⡀⠸⡇⢨⠗⣯⣰⡆⣿⢳⡇⣾⠀⣿⠀⣿⠸⣧⠇⢸⣒⠀⢸⡷⠛⢸⠾⡇⣯⣀⡿⢿⣀⡀⣿⠒⠘⢳⡆⠛⣦
⣶⣾⠿⠷⠿⠿⣶⣾⣿⣷⣿⣾⣿⣿⣶⣿⣶⣿⣶⣿⣶⣿⣿⣷⣾⣷⣶⣿⣶⣿⣾⣿⣷⣶⣿⣷⣿⣿⣾⣿⣶⣿⣷
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣂⠡⠄⣐⣒⣒⣐⣀⣀⡠⠤⠀⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣰⣤⢼⠞⠛⠉⠙⠏⠉⠙⠁⠈⠉⠙⠓⠶⢤⣁⠀⢑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⢂⣵⠞⠁⠀⠀⠀⠀⠀⠀⠀⣖⠂⣀⠀⡆⣀⡀⡀⠀⠈⠙⢦⡁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡜⠉⠀⠀⠀⠀⣄⢠⡠⣴⡃⠀⡈⡷⠳⠀⠟⠧⠰⢯⢧⡖⢀⣀⢳⡌⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠄⠀⣆⢺⡙⠮⠛⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠈⠀⢿⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠃⠀⠀⠀⡰⣟⢮⠓⠁⠀⢀⡀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣃⠱
⠀⠀⠀⠀⠀⠀⠘⠀⢁⡟⠀⠀⢀⣸⢽⡼⠄⠀⠀⠔⢉⠐⠂⠐⠒⠰⠄⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠉⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠂⠑⠞⠀⠁⣀⣠⣼⣄⠆⢀⡠⠄⣠⠄⢀⠠⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⣐⣂⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠛⠲⢂⡴⠛⠋⠛⢙⣿⡃⡾⠁⣰⠧⡄⠈⣧⠀⣿⠛⠲⠶⠒⣛⠩⠔⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠰⡀⠼⣆⠀⠀⢀⡾⠁⠀⢀⢀⡀⠙⣿⠁⠀⠀⠀⣽⠀⢙⠶⠇⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢁⠁⠙⣆⣠⣾⡧⠀⡁⠈⠈⡁⢽⡟⠀⢠⣄⣼⠃⠀⠀⠄⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠻⣿⣇⠀⠀⠜⠀⠃⣸⡇⠀⢠⡏⡤⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠂⠠⢍⠛⣧⣴⣤⣶⣾⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠉⠉⠙⢿⣿⡂⠐⡿⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠐⣾⢻⡇⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄⠉⢫⡇⠈⣇⡑⠷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⠨⢈⡟⢀⡷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠆⢒⣋⣸⡇⢸⣃⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠂⡠⠞⣿⠀⣿⡹⠆⢖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⢛⢰⠇⢰⡇⣆⡂⢠⢠⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠠⠤⠄⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠏⠩⢍⡿⠀⣼⠇⢉⠛⠅⠂⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠓⠒⠓⠛⠒⠚⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⣀⠀⢻⡖⠛⠛⠉⠉⠉⠉⠉⠙⣻⡷⠖⠛⠛⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢤⡀⢀⡴⠋⠀⠀⡇⠀⠀⣀⣠⣤⣤⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠻⠋⠀⠀⠀⣰⢧⣤⡞⠉⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠳⣤⣀⣀⣀⣤⠞⠃⢰⠏⠙⠓⠶⣤⣀⠀⠀⢠⡴⠛⠉⠉⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠉⠁⠀⠀⣀⣼⠄⠀⠀⠀⠀⠉⠛⢰⠏⠀⠀⠀⠀⠀⠀⠀⢈⣷⣀⡀⠀⠀⠀⠀⠀⢿⣄⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡝⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣴⠞⠉⠉⠉⠙⠳⣄⠀⢀⣠⡤⠖⠛⠀⠀⠙⢦⣄⡀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⡟⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢠⡄⠘⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⢸⡇⠀⠀⠀⠸⡇⠀⠰⣆⢀⡴⠋⠀⠀⣽⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⢠⡦⠦⣴⠦⠴⠞⠋
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢻⡄⠀⠈⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⡇⠀⡇
⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣤⡴⠞⠁⠀⠀⠀⠀⢀⡤⠔⠒⠲⠿⣄⡀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡇⢰⡇
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⠁⠀⠈⠁⣼⠃
⠈⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⡀⠀⣠⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠻⠾⠋⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣼⣿⣿⣿⣿⣆⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⣩⣴⣶⣶⣶⣮⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢃⣾⣿⣿⠏⣤⠻⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢸⣿⣿⡇⣈⣉⣁⢹⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⢸⣿⡿⢡⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣿⣿⣿⣸⢟⣵⣿⡿⠻⣿⣿⣿⠟⠀⢀⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⡋⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠹⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡘⢳⣶⡛⢡⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡘⣿⢡⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣷⡀⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⡇⢻⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠛⠛⠛⠃⠘⠛⠛⠋⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠙⠛⠛⠛⠿⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠉⢀⣀⣤⠶⠴⠖⠛⠛⠳⠖⠶⠶⠤⣤⣀⡀⠉⠻⢿⣿⣦⡀
⠀⠀⢀⣴⣿⣿⣿⠋⠀⡠⢾⠛⠀⠀⠑⠀⢀⠀⠲⠃⠀⡀⠀⢠⠀⢣⡍⠳⢤⡀⠙⢿⣿⣆
⠀⢀⣾⣿⣿⠛⠁⣀⣎⢀⡠⠀⠀⠐⠆⠀⠰⠧⣀⠀⠠⣄⣀⠘⠀⠈⠑⣄⠀⠹⣆⠀⠻⣿⣧
⢀⣾⣿⡿⠁⢠⠞⠁⠀⠈⡇⠀⠀⠀⠀⠛⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠙⠧⠀⠀⢌⣧⠀⠹⣿⣧
⣸⣿⡿⠀⢰⠋⠛⢄⡆⠀⡇⠀⠀⠂⣀⣀⣄⠀⠀⠸⠀⠀⠋⠓⠂⠀⢄⠀⠀⠃⠀⠋⡇⠀⣿⣿⡆
⣿⣿⠃⢀⡟⠆⢠⣄⠀⡰⠛⣄⠀⠈⠀⠀⠈⠙⠢⣤⡀⠀⠀⣆⢀⡀⠀⠑⣦⠄⠀⢸⡇⠀⣿⣿⡇
⣿⣿⠀⢸⡤⠆⠈⠈⠀⠀⠀⢘⡂⠀⠀⠀⠤⣤⠄⠈⢻⠉⠉⠛⢯⡀⠀⠚⠁⣠⠀⡼⠁⢰⣿⣿⡇
⢻⣿⠀⢸⡀⠀⠀⠀⢐⠂⠈⢁⠉⠀⠀⠀⠐⠇⠀⠀⣠⠀⠐⡄⠀⠹⣶⣬⡭⠖⠚⠁⣠⣿⣿⣿⡇
⠘⣿⣆⠈⢳⡀⠤⠉⠉⣄⣀⣈⣁⡄⠠⠤⠦⡤⠦⣀⠈⠀⠀⠟⠀⣸⠃⢀⣀⣤⣶⣾⣿⣿⣿⣿⠁
⠀⢹⣿⣆⠀⠙⠒⢶⡚⠉⣉⠙⡿⣟⠒⠒⣾⢤⣀⣉⠀⠀⠚⣀⡤⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢻⣿⣿⣶⣄⠈⠻⢮⣴⣄⣂⣉⣛⡿⣷⠥⠚⠉⠉⠉⠉⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠈⣿⣿⣿⣿⣷⣦⣄⡀⠀⣠⠏⢀⠞⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇⠀⠛⠶⠋⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠉⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠰⢦⣄⣤⡾⠀⣀⣤⠶⠚⠳⠀⠀⢚⣉⢉⣀⣀⣠⣴⣖⠻⠀⠰⡆⠀⠀⠀⠀⣐⡂⣴⠂⢰⣻⠆⠂⠺⠿⠀⣀⣠⣤⠤⠤⢤⣤⣿⡿⠓⢺⡏
⠀⠀⢀⡴⣿⣤⣿⣿⣭⣀⣀⣠⠴⠛⠉⠉⠉⠉⠉⢉⣀⡈⠛⠛⢋⡀⠓⣠⠖⢒⣶⡤⢿⡧⣤⣀⣀⣀⣠⡤⠖⠛⡩⡀⠀⢴⡷⠀⣠⣽⣷⣌⡸⣟⠂
⠀⠀⠾⠀⡏⠀⠉⠻⢷⣍⣉⣁⣤⡀⢶⡶⠿⠿⠒⠚⠿⠟⠚⠛⠻⠿⠂⠘⠰⠋⠠⠶⠋⠀⠀⠈⠉⠉⠀⠀⠀⠀⠓⠃⠀⢀⡤⠞⢉⣀⡉⠉⠿⠛⣇
⠀⠀⢠⡀⠙⢰⠷⣄⠀⢘⣿⡙⠩⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣞⠋⠉⠙⣳⣆⠀⢹
⠀⠀⠈⢿⠲⣞⡓⠊⡥⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣄⢿⣿⣶⣿⡏⣿⠀⣼
⠀⠀⠀⠀⢻⡁⠉⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢹⡎⠈⠿⠿⠿⠀⠁⠾⠙⡇
⠀⢀⣶⣦⣀⣁⣠⣶⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⠁⠀⠀⠀⠀⠀⠰⠋⢷
⠀⠀⡙⠉⢠⣿⣬⠯⡥⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣦⣼⣤⣶⣦⠄⢠⡏
⠀⠀⠸⠄⣀⠉⢡⢀⡀⠀⣸⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢩⣿⣭⣿⣧⣼⠁⢨
⠀⠀⣻⣶⣬⣧⣿⢎⣷⣤⣿⠀⠀⠀⠀⠀⣸⡏⠉⢸⡏⣿⣿⡏⠁⢘⣧⣾⣿⣿⢘⡿⠀⣿⠺⣷⠀⣿⢻⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⢿⠿⠿⠿⠿⠯⠒⠃⣌⠀⠀⠀⠀⠀⢹⠁⠀⢻⣧⣿⠻⣁⣀⣿⣿⣿⡏⢟⢰⣷⠀⣿⠃⣻⣦⠋⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢀⣤⡤⠴⠶⣶⣤⠄⢻⠁⠀⠀⠀⠀⠻⣧⠀⢬⠸⠏⠸⠿⠟⠿⠏⢿⡇⠿⠸⠿⠀⠾⠂⢸⣿⠃⠈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣿⣶⢠⣶⡄⠀⠀⣷⢶⡄⠀⠀⠀⠀⢀⣠⣤⣄⠀⣠⣤⡄⣀⠀⣠⣄⠀⢠⡤⠀⣤⡄⢠⣤⠄⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⢀⣙⣇⣠⣤⡹⠄⢷⠀⠀⠀⠀⣿⣏⠀⠀⣼⠿⠀⠀⢼⣾⣿⣇⠀⣠⣾⣦⣿⣇⣿⡇⠀⢰⣿⡀⠀⠀⠀⠀⢠⡏⣀⣠⣞⣠⣶⢠⣄⣠⣼
⠀⠸⣂⣸⡟⡛⣿⠋⠹⣷⡄⢸⠀⠀⠀⢀⣀⠉⣶⣦⢿⣇⠀⠀⣾⣿⣟⡃⠀⣿⣿⣿⣿⠹⣟⡀⠀⢸⣿⠁⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⣿⠀⠳⣖⡿
⠀⠀⠹⠾⣷⣶⣿⣷⣾⡏⢡⠟⠂⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠁⠈⠉⠉⠉⠀⠉⠁⠉⠉⠀⠈⠉⠉⠈⠉⠉⠀⠀⠀⠀⢸⡆⠀⢻⣿⣿⡇⠀⢀⡜⢧
⠀⠀⢠⡄⠉⠛⠛⠛⠃⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣧⡘⠿⠿⠃⠀⢸⡿⣿
⠀⠀⠀⠧⠀⢀⣶⣷⣦⣀⠀⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣾⠟⠶⠦⣤⡀⠸⠀⠘
⠀⠀⢠⣶⣾⠉⠉⠛⠁⠈⠻⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢿⠄⠀⠀⠄⣷⠌⢸
⠀⠀⠀⠙⠁⣠⡄⠀⣀⠙⢦⣀⣼⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⣧⠀⠀⠀⡏⠀⠉⣶⢸
⠀⠀⡴⢲⠀⢸⡀⢰⣿⠀⢸⠟⣯⠀⠈⠳⣶⡄⠀⠀⠀⠀⠀⠀⣤⠤⢤⡤⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⣶⣷⣀⣠⣿⢸⣷⠂
⠀⠀⠀⠸⠇⠈⢀⡏⢿⢀⣏⣀⢹⢀⣷⠀⠀⠘⣨⡆⠀⠀⠀⣠⠴⢦⡀⠀⣘⡷⢬⣝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⠛⢻⡟⠀⠘⠹⢸
⠀⠀⣀⣀⣀⣀⡞⠀⢸⠘⣥⣨⣤⡞⣿⣶⡶⠶⢉⡇⣤⠆⠀⡇⢀⡀⡇⢀⣀⡴⡄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡷⠐⠸⠷⠈⢀⣀⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠟⠃⠉⠀⡟⠀⠀⠀⢠⡟⠁⠀⠀⢻⢸⣷⠇⠸⣽⣟⣿⣦⣄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣖⣶⣯⣶⣿⠀⢀
⠀⠀⠤⣤⣤⣠⣴⡶⠈⣿⠀⡴⣦⠀⠃⢀⠤⠶⢤⡀⠀⠀⠀⠘⣿⣿⠰⣧⠜⣾⢿⡻⣿⡧⢈⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⣿⣾⣿⠀⢸⡁
⠀⠀⢤⣿⠶⠛⠻⠿⠀⠏⠀⢙⡃⠀⠀⣧⠀⠀⢀⡷⠾⠿⠿⠇⠻⠟⡾⢿⡀⠺⣟⣷⠉⠁⢾⣇⣈⣿⣤⣀⡀⠀⠀⠀⠀⢸⣧⣤⣼⡿⠿⠛⠀⠀⠁
⠀⠀⠀⠀⠶⠖⠻⠆⠀⠀⠀⢠⣶⡄⠀⠈⠛⠚⠉⠀⠀⠴⠶⠤⠖⠀⠁⠀⠈⠙⠲⠤⠤⠖⠚⠋⠹⠇⠀⠀⠉⠛⠛⠻⠤⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠿⠿⢿⣶⣄⠀⢀⣴⣾⠿⠿⢷⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠉⠉⠀⠀⠀⠀⠈⣿⠀⢸⡏⠀⠀⠀⠀⢙⣿⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠁⣠⣶⢶⣦⡀⠀⠀⠀⣿⠀⢸⣧⠀⠀⢠⣤⣾⠟⠀⠀⠉⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠛⠛⢿⣧⣀⣿⠇⠀⠀⠀⣿⠀⢸⣿⣦⣀⣈⠉⠁⠀⠀⠀⢀⣠⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⢹⣿⠉⠀⠀⠀⢐⣿⠀⢸⡇⠉⠛⠛⠀⣠⡄⠀⠀⣾⡏⠁⠈⢻⣷
⠀⠀⠀⠀⠀⠀⢀⣾⡧⠀⠀⠀⢀⣀⣀⣸⣿⠀⠘⠛⠛⠻⣿⠀⢸⡇⠀⠀⠀⣰⡿⠁⠀⠀⠘⠟⠀⠀⠠⣿⣇
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢸⡟⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠙⠿⢶⣶⣦⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⢀⣾⠟⢿⣦⠀⠀⣿⠀⢸⡇⠀⣀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⣠⣤⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣸⣇⠀⠘⢿⣦⣼⡟⠀⠀⣿⠀⢸⡇⠀⢿⣦⣤⣀⠀⠀⢾⡇⠀⢀⣾⠟⠁⢹⣿
⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣾⡟⠻⣷⡀⠀⣿⣧⣀⣀⣠⣿⠀⢸⡇⠀⠀⠉⠉⠻⣷⠀⠀⠀⠀⠘⣿⡀⢀⣾⡟
⠀⠀⠀⠀⠀⢰⣿⠋⠉⠋⠙⢻⣧⣴⡿⠁⠀⣿⡏⠉⠉⠙⣿⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⣰⡷⠀⠉⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠛⠃⠀⠀⠀⣿⠀⢸⣗⣀⣤⣤⠀⠀⢿⣦⣴⡿⠃⢀⣀⣤⣀⣼⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⣀⣀⣀⠀⠀⠀⣿⠀⢸⣿⠟⠉⠀⠀⠀⠀⢻⡏⠀⠀⠛⠋⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⢸⣿⣶⣾⡟⠉⢻⣧⠀⠀⣿⠀⢸⡏⠀⠀⠀⣠⣤⣴⣿⡃⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡆⠀⠀⠀⠀⠘⠿⣶⡿⠃⠀⠀⣿⠀⢸⡇⠀⠀⣾⠟⠉⠈⠙⣿⡄⠀⠀⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣤⡀⠀⠈⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⡀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⣿⡀⣀⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣤⣤⣤⣴⡿⠀⠸⣿⣦⣤⣤⣤⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠁⣴⡀⠀⠀⠠⡀⠀⠁⠂⢘⣛⣻⣷⠀⣾⣿⠃⠀⠀⣼⣿⠇⠀⢠⣿⣿⡈⠋⠀⢾⣟⢳⣿⠀⢸⣿⣿⣿⡄⠀⢰⣴⣵⡥⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠨⠀⠂⠀⠂⠘⠛⠈⠉⠀⠀⠛⠀⠀⠀⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠋⠋⠀⠈⠻⠿⠛⠀⠀⠘⠿⠟⠋⠀⠠⠧⠁
⠀⠀⠀⣠⡤⣄⢀⣤⢤⡀⢠⡤⣄⠀⣤⡀⡄⢠⡤⣤⣤⢄⡄⣤⠀⣠⢠⡤⢤⠀⠀⢀⣤⠀⢠⡤⣤⡀⣤⢠⡄⠀⢠⡤⣤⣤⠄⡄⢠⡤⠤⢀⡤⣤
⠀⠀⠀⣿⠀⣈⢸⡇⢨⡇⣿⡠⣾⠀⡟⢷⡇⢸⡇⢸⡇⢸⡇⠸⣧⡏⢸⡗⠒⠀⠀⣾⣽⣇⢸⡗⢺⡅⣿⢸⡇⠀⢸⡇⢸⡇⠀⡇⢸⡗⠒⢘⠳⣶⡄
⠀⠀⠀⠈⠙⠉⠀⠉⠋⠁⠈⠙⠉⠀⠁⠈⠁⠈⠁⠈⠁⠈⠁⠀⠉⠀⠈⠉⠉⠀⠀⠁⠀⠉⠈⠉⠉⠁⠉⠈⠉⠉⠈⠁⠈⠁⠀⠁⠈⠉⠉⢈⡙⠉
⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⣄⠀⠀⠀⠀⠄⠄⠀⠀⢀⣗⣰⠂⠀⣠⣄⡀⠀⠀⢀⣶⣈⠃⠀⣀⣀⣀⡀⠀⠀⠀⣴⣆⢠⣦⠀⠀⠀⠀⡄⠿⠿
⠀⠀⠀⠀⠘⠋⣻⣿⡆⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢠⣿⣿⣧⠀⠀⠘⣿⠀⠀⠈⠻⣿⡟⠃⠀⠀⣴⡌⣡⣎⢩⣦⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠈⠉⠈⠉⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠬⣟⣛⣛⣛⣛⢛⣙⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠠⠤⣀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢰⣠⡎⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠁⠒⠄⠀⠀⠀⠀⠐⠬⣀
⠀⠀⠀⠀⢀⢸⠃⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠀⠀⠑⢢⠀⠀⠀⠀⠀⠠⠵⠄
⠀⠀⠀⠀⠽⠁⠀⠀⠐⠉⠀⠀⠠⢆⣰⣶⡟⠀⠛⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠠⠂⡀⠀⡀⢡⠄⠀⠀⠐⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣶⠀⠀⠰⡀⠀⠀⠀⠀⠀⢈⣷⣀⣀⡤⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⡈⠙⠢
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠣⠀⠴⠂⠔⠊⠀⠈⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢉⡠⠀⠱⠀⡰⠶⠒⠶
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⢁⣀⠆⢠⠐⠀⠄⠀⠀⠈⡀
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⢧⠀⠀⠤⠆⢣⠰⡀⠀⠀⢃⡠
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠴⠣⠀⠀⠂⠀⠀⠀⠈⡏⠋⠀⠹⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠂⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠐⠀⠀⠄⠐⠁⠀⡴⠂⢀⣄⢠⠄⢠⡀⢰⢨⠐⡞⠉⡀⢠⠟⡣⢰⢦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⣯⣸⢿⣸⠾⠙⠚⠘⠀⠛⠀⠛⢈⠋⠁⠈⠀⠉⠀⠀⠀⠀⣀⡄⠀⠀⢀⣤⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠠⡀⠀⠀⢀⠐⢏⠡⠀⠘⠒⠚⠉⠐⠾⠋
⠀⠀⡔⠲⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠉⠁⡇⠀⢀⣀⠤⠖⠀⠀⠠⠶⣄⠀⡀
⠀⠀⢟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠤⠄⠀⠶⢤⠇⠠⢿⡶⠔⠒⠁⠠⠃⢀⠀⢁⠟⡀
⠀⠀⠈⠑⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢴⣶⡍⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡀⢀⠏⠀⡀
⠀⠀⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡥⠶⠚⡋⠩⠁⠀⠀⠀⠀⢰⢇⡀⢒⣠⠄⠀⠀⠀⠉⠾⠀⠁
⠀⠀⢆⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠉⠂⠅⣨⠙
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢉⠛⠀⣁⡄⢀⡠⠀⠓⠀⠀⠀⠀⠙⠂⠀⠀⢄⡈⣁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠙⡓⠳⠜⠀⠀⠀⡀⠀⠒⠠⡀⠀⢄⠀⠀⠀⠀⠨⠀⠈⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⠀⠀⡀⠀⠳⠦⡄⢢⡀⢀⡀⠑⠄⠀⠀⠀⠀⠉⠈⠀⠀⠀⣀⠀⠰⠶⠞⠿⠛⠛⠛⠓⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⢀⠐⢄⠑⡄⠈⠉⠉⠀⠀⠀⠀⣀⡀⠀⡤⠋⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠠⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠻⢿⡿⠛⠀⠀⠂⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠉⠀⠰⢦⡰⠃⡀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⢁⠀⠴⡂⢰⠃⡀⠀⠔⠀⠄⠀⠀⠰⠀⠀⠠⡄⡂⠀⠠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠈⠁⠀⠙⠢⡑⠊⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠈⢳⣿⠀⠀⠀⢠⠄⠀⣀⠄⡀⠀⠷⠟⠀⠀⢂⡀⠀⠘⠑⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠂⠀⣴⡆⠀⠀⠈⠀⠀⠀⠘⠈⢿⡄⠀⠄⢹⣤⢚⣁⡀⢁⠀⠢⢰⡀⠀⢢⣰⣀⠀⠈⠉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠋⠉⠰⢉⠳⠀⠱⠦⢄⣀⣈⣧⠠⢁⡾⠁⠀⠙⠁⠜⢁⠀⠀⢀⣤⡔⠊⠁⣀⡀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣄⠀⠀⠀⠄⠀⠀⠸⣆⣾⠀⠀⠀⠰⠀⢠⠞⢻⣴⡏⠀⠀⠀⢸⠭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⢢⠰⣄⣀⠇⠀⠀⠀⠀⡀⠀⠹⣇⠀⡀⠀⠙⠁⢨⠨⠸⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠷⠤⠴⠶⢶⡶⠶⠿⣶⣤⣻⣦⠁⠐⠀⠀⠩⢛⠓⢾⡇⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠔⠀⢀⠞⠁⢄⠀⠀⠀⡬⠙⢿⣷⡀⠀⠀⣐⡙⠁⢈⡇⢠⠀⠵⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⡀⠰⣠⠇⠈⢉⡠⠃⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠕⠀⠄⢀⠂⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⢋⢁⠁⢠⠿⢶⠶⠛⠁⠑⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠐⠃⢀⡀⠒⡀⠤⢢⣀⣀⣠⣄⣀⡛⢸⣿⣆⣡⡾⠃⢀⠄⠠⣎⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⢐⠚⠉⠉⠉⠀⠈⠙⠻⢿⣿⡿⠋⠀⠐⠊⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠏⠑⢀⠰⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣄⡰⠾⠂⠈⢠⠰⡴⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠁⠀⣤⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⣤⠀⣥⣴⠀⢷⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠙⣀⠀⠪⠠⡁⠄⢽⠗⣶⠊⠑⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠉⠄⢬⠀⠀⠀⢤⡋⢘⡤⣿⡀⣀⣿⡜⠃⠠⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠞⠀⠰⠂⢠⡄⠆⣈⠀⠝⣿⡤⠘⣌⠓⢼⡀⠀⣶⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠉⠀⢖⠀⠀⠀⢬⡀⢼⣷⣰⠧⢿⢸⡆⢻⣀⠉⣸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠜⡳⢀⡀⠁⣀⠉⠁⠉⠻⣾⣿⠫⣄⠛⠔⢣⠸⢾⢤⠀⢸⡇⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢰⠻⠆⢋⡦⢡⠈⠉⠠⠜⠀⠀⢹⣿⡅⠘⣦⡄⣽⠀⠚⠀⠓⠈⠁⠋⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⢴⢛⠆⠀⠈⠁⠼⠪⠄⠒⢠⠸⣶⣷⠎⠹⠇⣾⠀⢁⡼⠉⢧⠹⡄⡁⢹⣞
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⡃⢠⣠⡈⠙⢰⡌⠘⠂⢃⡀⠙⢀⢸⣿⡿⠀⣠⢆⠀⣲⡞⣀⣀⠸⡆⡿⡗⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⡈⠛⡁⠰⠈⣤⡁⢀⡒⠀⠀⣀⢸⣿⣷⣿⣷⢻⡀⣹⠀⢸⣇⣽⣓⠲⠧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢦⠚⡰⠚⠁⠒⡈⡀⠩⠠⠀⠙⢿⣿⣿⣯⠀⢷⣛⣿⠠⣸⠇⠳⢤⣖⣲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠩⠀⠙⠃⠄⠗⠀⠀⠸⣿⣿⡄⠀⠐⠜⠻⠴⠛⣾⣉⡉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠖⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⡶⣿⣿⣿⣧⣿⣿⣿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠆⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢩⠀⢠⠀⠀⣹⠣⠛⠀⠀⠃⠀⠀⡀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⢈⠀⠈⠀⠀⠀⠀⠀⠀⠁⡀⠐⠀⠀⠀⠀⠀⡬⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠁⠀⠃⠐⡰⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠁⠈⠐⠀⠀⠠⠄⠀⠀⠀⢠⡠⠼⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⢀⠘⠆⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⠐⠀⠀⠀⠀⠐⢀⠀⠐⡊⠩⠉
⠀⠀⠀⠀⢀⡈⡁⢀⡠⠀⠀⠀⠈⢠⠀⠠⠄⠈⠐⠀⠀⠒⢄⠀⠀⠐⠐⠀⠈⠠⠀⠀⢩⢤⠁⠀⢔
⠀⠀⠀⠀⠀⠀⠙⡋⠀⠀⠀⠀⠀⠀⠠⠷⠀⠀⠘⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⡈⠊⢀
⠀⠀⠀⠀⠀⡀⠀⠧⠀⠂⡀⠀⠀⠀⠃⠀⠀⢠⠀⠐⠐⠐⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠜⠀⠀⠄⠀⠀⠀⠀⢂⢀⠀⠃⠀⢀⠀⠂⠂⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠬
⠀⠀⠀⠀⠀⠐⠀⠐⢉⠈⠁⠈⠀⠀⠀⠀⠀⠈⢄⠐⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠁⠀⠘⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢠⠀⠀⠀⠀⠤⠠⠇
⠀⠀⠀⠀⠀⠀⠒⠂⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠡⢄⠘⢁⡂⠄⠰⢌⠀⠐⠀⠀⡰⣆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠰⠈⡶⠉⣥⠰⠀⠀⠀⠁⠀⠀⠉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⢀⠀⠀⠀⠀⠀⠲⠨⠅⠶⠀⠇⠀⠶⠂⠉⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠂⠲⠀⠀⠀⠗⠄⠀⠀⠑⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⠀⣷⡐⢾⠀⠠⠂


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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠒⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠗⠁⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⡜⡆
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠦⢸⣇
⠀⠀⠀⠀⠀⢼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠂⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢢⢈⣽⡿
⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠇⠀⡤⡆⠁⠄⠠⠀⠀⠠⠐⠀⠀⠀⠐⠀⠀⠀⠐⡂⢸⠈⡈⣯⡇
⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣚⣉⣍⣭⡅⠀⠀⠀⠡⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⢸⡏⡿⣾⡂
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠨⠀⠀⠀⠀⢥⡤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⣆⢷⢘⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡃⣨⢸⡏
⠀⠀⠀⠀⢸⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⡅⣧⡇⠁
⠀⠀⠀⠀⢸⠘⢣⣖⣒⣒⣢⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢄⣄⣔⡈⡇⣷⠃
⠀⠀⠀⠀⠀⢇⣉⠉⠉⠉⠉⠙⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠿⠿⠛⣛⡃⡛⠉⠘⠋⠘⢑⣱⣿
⠀⠀⠀⠀⠀⠀⢀⣈⠙⠩⠮⠁⠀⠀⠀⠀⠀⠀⠒⣊⡹⠿⡿⣶⣶⣾⢟⡿⢝⡱⠶⠀⠒⠪⠍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠐⠛⠒⠉⠲⠴⠶⠦⠤⠤⠤⠶⠖⠒⠲⠾⠷⢴⣿⣹⣽⡾⠿⠛⠒⠛⠓⠒⠂⠂⠄⠰⠰⠦⠶⠾⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣶⣲⣆⣴⣲⣆⣴⣒⣦⣀⣶⣦⣠⡶⣦⢠⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⡇⠀⠸⡇⠀⠀⣧⠀⠀⣿⠀⠀⠘⣏⠉⠹⣏⠉⢻⡉⠉⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⡤⢀⣀⡀⢀⣀⠀⠈⠀⠈⡁⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢠⠏⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⢀⡎⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠒⠛⠒⠒⠒⠒⠲⠆⠀⠀⢰⣿⣿⠁
⠀⠀⠘⡏⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠲⠶⠶⠶⠶⠤⠤⠤⣤⣤⣾⢷⡇
⠀⠀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠟⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠻⠿⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠾⠿⠛⠛⠛⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣶⠀⢰⠏⠁⠉⣧⠀⢰⡖⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢹⡇⢸⡄⠀⠀⣼⠀⣾⠇⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠈⣿⠛⠙⠒⠚⠉⢻⡿⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣇⠀⠀⠀⠀⣾⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣈⣿⣀⣀⣀⣰⣟⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⣛⣛⣛⣛⣛⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⣿⡟⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⠶⠿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠿⠶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⣿⡇⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣄⣀⡈⠻⣷⣿⣷⡿⠋⣀⣀⣤⣤⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

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

⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⢀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⣀⠀⢀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⢀⠀⣼⡏⢉⡁⢈⡉⠉⣿⠀⣿⠀⣿⡇⣿⡏⢉⡁⢈⡉⢉⣿⠀⣿⠀⣿⠁⣿⡏⢉⡁⢈⡉⢩⣿
⠀⠀⢀⢀⣈⣅⣸⡇⢸⣯⣀⣉⣀⣿⠀⣿⣇⣉⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣀⣈⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⢠⠄⣉⣏⢉⠁⠈⣉⣉⣉⢉⡉⠀⢉⣉⣉⢉⣉⠁⠈⣉⢉⣉⣉⡉⠀⢉⣉⣉⡉⣉⡁⠈⣉⣉⡉⡉
⠀⠀⢠⠤⢛⡟⢻⡇⢸⣿⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⢠⠀⣼⡇⠘⠃⠘⠃⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢐⣿⠀⣿⠀⣿⠀⣻⡇⠘⠃⠘⠃⢰⣿
⠀⠀⢠⡀⠚⡿⠛⠃⠘⠻⠛⠛⠀⠛⠀⠛⠃⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠻⠛⠛⠃⠘⠛⠛⠟
⠀⠀⠠⡀⣼⡿⠿⠆⠐⠿⠿⣷⠀⣿⠀⣾⠆⣾⡿⠿⠆⠰⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢠⡄⢼⡇⢰⡆⢰⡦⠀⠿⠀⣿⠀⣿⡇⠻⠇⢰⡆⢰⣦⠐⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢀⡤⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠶⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⢠⣤⣦⡤⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣦⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶
⠀⠀⠠⡄⣤⡤⢸⡇⢸⡯⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣠⡄⢸⡇⢸⡇⢠⣤
⠀⠀⠀⠄⣼⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⡇⣿⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⠀⣿⣧⣤⡄⢠⣤⣤⣿
⠀⠀⠠⠤⣤⣤⣠⡄⢀⣤⣤⣤⠀⣤⠀⣨⡀⣨⣤⣤⡄⢀⣤⣤⣄⠀⣤⠀⣠⡀⣠⣤⣤⡄⢀⣤⣤⣄
⠀⠀⠀⠀⣼⡯⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠇⣿⡏⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠀⣽⡏⢉⡁⢈⡉⢩⣿
⠀⠀⠀⢀⣈⣥⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣇⣈⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣄⣉⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⠀⣈⣉⣍⢉⠁⠈⣉⣉⣉⣉⣉⠀⢉⣉⣉⣉⣉⠁⠈⣉⢉⡉⣉⡉⠀⢉⣉⣉⣉⢉⠁⠈⣉⣉⣉⡉
⠀⠀⠀⠈⢛⡿⢻⡇⢸⡟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⠛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⠀⠄⣼⡇⠘⠃⠘⠋⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢈⣿⠀⣿⠀⣿⠀⣿⡇⠘⠃⠘⠃⢸⣿
⠀⠀⢀⡄⠸⡿⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠁⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠛⠛⠛⠃⠘⠛⠛⠛
⠀⠀⢀⡄⣼⡿⠿⠇⠐⠿⠿⣿⠀⣿⠀⣾⡆⣾⡿⠿⠇⠐⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢀⡄⠼⡇⢰⡆⢰⡖⠀⠿⠀⣿⠀⣿⠇⠻⠇⢰⡆⢰⣶⠀⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢠⡴⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⠠⣤⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶
⠀⠀⢠⡄⣠⡄⢸⡇⢸⡿⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣤⡄⢸⡇⢸⡇⢀⣤⠀⡄
⠀⠀⠀⠀⠹⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠇⠿⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠄⠿⠧⠤⠄⠠⠤⠴⠿⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣮⣉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⢣⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⡄⠀⠀⢀⣀⠤⠒⠒⠉⠉⢱⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣠⡔⠒⠒⠒⠒⠒⠒⢺⣿⣷⢶⠊⠉⠀⠀⠀⠀⠀⠀⢸⡽⣤⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠓⢄⠀⠀⠙⢶⣿⣇⣀⣀⣀⣤⣾⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⠀⠀⣹⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜⠓⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡷⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡎⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣸⡿⣿⡇
⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⠶⠯⠓⠒⠒⠚⠒⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⡖⠒⢲⠶⣶⣾⣿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠾⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠿⠿⢿⣿⣷⣦⡀⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⣉
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣉⡛⠻⠿⣿⠿⠟⢛⣉⣡⣤⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⢠⣍⣉⣙⠒⠶⠒⢛⣉⣉⣭⣤⣤⣤⣬⣍⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⣰⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣶⣶⣿⣿⡿⠛⣡⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⡉⢩⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⢨⣤⣤⣴⣶⣶⣶⣦⣤⣤⣬⣉⠀⣉⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣍⣁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠶⠿⠿⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠱⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠦⣤⣴⣶⣶⡶⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠶⣶⣖⠛⠂
⠀⠀⠀⠀⠀⠀⡠⣟⡂⢤⣀⠀⠀⠈⠙⠓⠦⢤⣀⠀⢹⣯⣄⣀⣀⣀⣠⡤⠤⠶⣒⣚⣩⣭⣽⡿⠷⠶⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢐⠀⠀⠉⠂⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠚⠿⢿⣿⣿⠷⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⣘⣇⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⣌⠑⠢⢄⡀⠀⠀⠀⢀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢶⣒⣛⣯⣽⠅
⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣈⠑⠢⢀⣸⣿⣖⣀⣀⣀⣤⣤⣶⣖⣻⣽⣿⡶⠾⠛⠛⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡄⢈⣙⣻⣭⡽⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠂⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⣿⣧⠄⠀⠀⠀⢀⣀⣀⡤⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡷⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⢀⠀⠀⠛⠂⠠⣄⠀⠀⣀⢀⠠⣀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣉⡁⠀⢠⡁⠈⠘⠹⣟⣦⠗⠁⡀⠘⡄⠀⠐⠁⠀⡠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠃⠈⠗⢂⠸⠀⠀⢀⠉⠁⠰⠀⠀⢱⠀⢁⣀⠤⠐⠂⠁⠘⢄⠀⠸⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠄⢀⡀⠀⢸⠆⠄⠂⠀⠀⠀⡀⣀⠀⠀⠀⠄⠱⠃⠀⠀⠀⠀⠀⠠⡊⠈⣗⣫⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢉⠀⢀⢈⡀⡀⠓⠜⠀⠨⣝⢦⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠊⠀⢠⠈⢄⠈⠐⠃⠀⠀⢀⠤⣈⠈⠑⠄⠀⢦⢸⣀⡎⠀⠀⠀⠀⠀⡜⠁⢀⡴⠀⠔⠈⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⠀⠀⠈⠒⠄⢀⠀⡀⠀⠐⠁⠀⠀⠑⡄⢠⠀⠈⠃⠔⠁⣀⠀⠀⠀⣀⠆⠠⡄⠀⢨⠀⣤⠔⠖⠲⠢⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠴⡄⠈⠀⠀⠓⠘⠀⠠⣁⠄⠀⠐⠤⠄⠀⠤⡈⠛⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠘⢄⠰⣀⠀⣓⠌⠰
⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣠⠤⠄⠀⠀⠄⠄⠀⣶⠀⠄⠶⠀⠀⠈⡃⠀⠀⠉⢳⢄⠀⠀⡀⢓⠈⠃⠀⠀⠆⠀⠀⠂⢰⢰⡅⠀⠁⢮⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠘⢟⠁⠀⠀⠀⠀⠂⠀⠀⠀⢠⣿⠀⠀⠀⠀⠎⠀⣄⠀⠀⡋⠀⠀⠉⢀⠐⠒⠒⠀⢄⠀⠀⢀⢘⠒⡃⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⠀⠀⡠⠀⠀⠀⢢⠀⢰⠄⠀⠀⢀⠠⠀⠁⣀⢀⣀⣁⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢫⠉⢱⠁⠀⠀⢁
⠀⠀⠀⠀⠀⠀⣬⢤⠀⠊⠁⠁⢀⠄⠂⡐⠀⠀⠀⠀⠀⠀⠀⠹⠠⠄⠀⡄⠀⠀⠆⠐⠀⡀⠠⢀⠀⠀⢀⣀⡀⢃⠀⠀⡆⡟⠀⠀⢥⠀⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡀⠚⠀⠄⢀⡆⠐⠀⡀⠡⠀⠘⠃⣀⡁⠘⠑⠀⠀⡇⠀⠙⠈⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠁⠀⢠⠓
⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠒⣁⠀⠉⠂⠃⠀⠀⠁⠀⠁⠀⠀⢁⢀⠐⠂⠲⠀⠂⠁⠔⡢⠀⠀⡠⠀⠀⡤⠀⠲⣁⡈⢠⠀⠀⠀⠰⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠛⠀⡔⠁⠉⠒⠉⠈⢡⠄⠀⠀⠀⠈⠁⠀⠑⠁⠤⠀⠀⠀⠀⠀⣄⡐⣄⡠⣀⣄⣤⣠⡈⠁⣈⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠈⠁⠈⠀⠀⠉⠁⠀⡀⢘⠀⠀⢾⡦⠀⠀⢀⣀⣀⠀⠀⠀⠀⢄⣁⣀⡀⡻⠋⠙⣛⠁⢺⣭⣝⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣙⡀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⠶⣀⠈⣁⣨⠁⠦⠀⠷⢦⠀⢀⠣⠄⡀⣄⢴⡦⠀⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡠⠆⠉⣤⢋⢈⣄⠰⠆⠀⠁⠀⡀⢄⠀⠘⠪⠭⠄⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⣤⠀⠈⠀⠀⠀⠢⣀⡀⠀⠀⠀⢡⡘⠐⠃⢀⡀⢉⡦⠀⢄⣑⡀⠡⠄⠀⠀⢂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠠⢄⠀⠄⠀⠀⠀⠀⣀⡉⠘⠃⠈⠡⡔⣡⠀⠀⠫⠐⠀⠄⠄⠀⠍⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠂⢜⡠⠽⠂⠪⢂⠀⠙⠉⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠪⢕⡈⠀⠀⢸⢱⡋⣲⣏⠁⢎⠑⠂⠚⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠈⠛⣶⣈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⣏⠀⠀⠀⠀⠀⠀⠄⢰⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢰⣤⡀⠀⠀⠀⡀⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣈⡈⢛⣥⢀⣟⠙⠗⢦⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⢙⣏⠀⠀⢈⣤⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠉⠈⢉⠑⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⠻⣿⣷⣦⠴⠀⠀⠠⡄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠉⠉⠀⠀⠘⣿⣧⣷⣿⣦⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣴⡄⠀⠀⠀⠀⣠⣼⣿⣿⡏⣬⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣛⢛⣻⡄⠀⢀⣰⣿⣿⢻⣿⡏⢈⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣽⣿⣿⣦⣾⣿⠋⢀⣼⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⢽⠃⠀⠀⣰⣿⣿⣿⣦⣄
⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠛⠉⢀⣾⣟⣿⣿⣷⣿⣿⡀
⠀⠀⠀⠀⠀⢂⠀⢸⡀⠀⠀⣠⠀⠀⠀⠀⠀⡴⢿⡿⠟⠛⠉⠁⠀⠀⠀⠠⠟⠛⢋⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡄⠀⣧⠀⠀⢻⠀⠀⠀⢀⣴⣿⣿⣧⣤⣄⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠼⠄⠀⣼⠀⠀⢠⣾⣻⣿⣿⣿⠿⠋⠀⠀⣠⠾⠛⠉⠁⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⣰⣷⡐⠛⠊⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠙⠀⢹⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠆⠀⠀⠀⠀⠀⠢⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠛⠛⠓⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠾⡄⠾⡆⠰⢷⠰⠦⡄⠸⣆⠰⢶⠰⢰⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠞⠁⠶⠃⠰⠞⠰⠖⠁⠶⠃⠰⠊⠰⠞⢸⡇⠀⣼⠃⡝
⠀⠀⠀⢀⣤⣤⡶⢒⢶⠖⠲⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠁⣼⣤⣄⣀
⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡰⠓⠚⣧⢻⡄
⠀⠀⠀⢸⣿⣻⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡰⠁⠀⠀⢸⢸⠇
⠀⠀⠀⢸⡟⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠰⠁⠀⠀⠀⣸⢾
⠀⠀⠀⠘⣿⣾⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣰⠁⠀⠀⠀⠀⣟⣿
⠀⠀⠀⠀⣿⣽⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣿⠀⠀⠀⠀⢀⡏⡏
⠀⠀⠀⠀⣿⢳⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣾⠀⠀⠀⠀⢸⢧⡇
⠀⠀⠀⠀⢻⣧⣇⠀⠀⠀⠀⠈⣿⣧⡀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⢸⡷⣿⠀⠀⠀⠀⠀⣿⣦⣝⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⢸⡿⢹⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿
⠀⠀⠀⠀⢸⣇⣼⣧⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣟⡇
⡤⠤⠒⠒⠐⠛⠛⠚⠓⠒⠾⠾⠷⠿⠿⠯⠶⠬⠾⠥⠼⠥⠼⠿⠿⠿⠤⠴⣷⣽⣿⡵⠫⡴⠇⠤⠤⠤⢄
⠙⠒⠂⠴⠤⠤⠤⠤⠤⠤⠤⠦⠴⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣠⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠣⠀⠀⠻⠋⢀⠀⢀⠀⠀⠀⠀⠀⣤⣾⣷⣤⡀⠀⠀⠀⣌⠀⠀⠀⠸⠟⠃⢰⠀⠀⠀⡀
⠀⢀⠠⣑⠦⠥⠯⢔⡨⠀⠐⠀⠀⠀⢠⣴⣶⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠁⠀⠀⢠⣿⣿⣿⣶⣧⣆
⠀⠀⠑⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠟⠍⠩⠯⠩⠻⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⢾⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣗⣛⡯
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠀⠀⠀⠀⠘⠋⠁⠈⠁⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⣿⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⠦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣯⣟⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠿⠿⠿⢀⣃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⠧⢼⠤⢤⢤⠧⢼⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣷⣿⣛⣿
⠀⠀⠀⢺⣿⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⢞⡒⣖⢞⡲⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠿
⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠄⠄⠀⠀⠠⠀⠀⢀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠢
⠀⡀⢃⠔⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠁⣄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⢀⠀⠀⠰⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡜⢩⡠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠐⢂⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⢲⡆⠀⠀⠤⢀⣈⢀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⢄⢲⠐⠂⢱⠈⠊⠆
⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠠⠨⠀⡀⠈⡄⠀⡌⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠲⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠳⠳⠁⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠁⠊⠐⠀⠓⠓⠁⠁⠁⠀⠀⠁⠁
⠀⠀⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⠁⣹⣏⣻⣻⣛⣙⣿⣹⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠀⠀⠀⠀⢧⡤⠶⠶⠶⠶⢲⠖⠛⠛⠛⠛⠛⠛⠁⠀⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⣰⠛⣆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠦⡴⠁⠀⠈⢦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠈⠙⠛⠛⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠁

⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠄⢀⣀⠀⠄⠀⠢⠤⠄⠤⠄⠐⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠈⠉⠈⠉⠲⢄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⡀⠀⡁⠂⠀⠑⡄
⢀⡂⠀⠀⠀⠘⠟⠀⠏⠀⠀⠀⢱
⢠⠑⡄⠢⠄⣀⠀⠀⠀⡀⠀⠀⠘
⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⡌⠑⠢⢄⣀⢀⠀⡄⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠢⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⢄
⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠁
⣶⣶⡶⡶⠶⢶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⣀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠁⠠⠁⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠴⠀⠺⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡡⠀⠄⠀⣔⣭⣄⡁⡀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣳⡀⢀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢠⠉⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠠⠼⠀⠀⠓⠲⡖⣦⠀⠊⣴⠆⢀⠀⡔⡸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⠂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⣀⢉⠗⢦⠀⠁⠀⡠⠋⠀⠆⠀⢀⡀⠀⡀⠀⠄⠀⠀⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠐⠁⠀⠀⠀⠀⠧⠆⠀⡀⠀⠀⢈⠙⠒⠊⣐⣢⡞⣒⢂⣀⢈⡤⢦⡍⠀⠈⠁⠀⢰⠖⢺⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⢀⠀⠀⣤⠀⢀⠰⣾⡷⠄⢐⠉⠙⣇⣭⣒⣩⡀⠉⠁⠁⠀⠀⡔⠂⠘⠢⠜⠳⣆⢱⡦⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⠑⢄⠀⠀⠛⠀⠃⢀⡀⠀⠠⢻⡍⠀⠀⣿⠂⠀⠀⠀⠀⠐⡇⠀⠢⢰⣦⣄⡀⢠⠀⠈⠀⠀⡂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢈⡉⠐⠀⣠⢁⠄⠤⠀⠀⠙⢆⢀⣏⡴⠶⢗⠀⠄⠀⠀⠀⠆⠈⠛⠋⠁⢀⡖⡚⠂⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣀⠈⢉⡀⣲⢤⣾⡀⢘⣿⡋⠀⢀⣨⠟⠂⢀⡀⠀⠈⠊⡰⠀⡄⠜⠀⠀⠠⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠥⠤⣀⠴⠛⠃⠣⠀⠀⢹⡟⠁⠉⠛⣠⣥⣄⠀⢀⣉⣀⡀⠀⠀⠀⠏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠀⢀⣠⠖⠚⣧⡀⠀⠀⠀⠖⢮⡇⢠⠤⠤⡻⠿⠟⢀⡀⣸⠋⠙⣀⣠⡞⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠈⢰⠊⠉⠓⢤⣾⡇⠀⢀⡇⢈⠂⠀⡧⠀⠀⠣⢸⢹⡞⠉⠁⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠿⣿⡇⠀⠀⠀⠀⠘⠛⠛⠛⣶⣾⣇⠹⠶⣶⣷⠶⠶⣤⠾⠋⠙⠦⠆⠁⠐⠁⠈⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⣠⠆⠄⠠⠈⢿⣿⣦⣼⠇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣒⣦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⢼⣷⣦⣌⡙⠂⠄⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈⠭⠍⠁⠀⠀⠀⠐⠀⠀⠀⠠⢄⠀⣀⣀⢀⡠⠼⠛⠛⠿⣧⣤⠀⠄⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠐⠂⠀⠈⢉⣀⡠⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠀⠉⢉⣉⣁⣀⣀⣀⣀⣐⣒⣀⣒⣀⣤⣤⣤⣿⣗⣋⣐⣿⡿⠦⠁⠀⠀⠀⠉⠩⠭⠭⠭⠭⠥⠄⠙⠛⠛⠓⣒⠨⠇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠁⣤⠀⠀⡀⠔⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠀⡄⠄⠐⠑⠀⡃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⡘⠀⢈⡄⠀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠄⠊⠀⠀⠠⠈⡀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠂⠐⠒⠠⠂⠀⠈
⠀⠀⠀⠳⠀⠀⠀⠀⢀⠎⠈⠀⠀⠉⠘⡄⠀⠠⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠎⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⣀⠄⠄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠤⣠⠊⠁⠀⠉⠦⠠⠤⡀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⢠⠊⠁⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⠀⡀⠀⠰⠉⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠢⡀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⢲⢀⡔⢦⢠⡖⡄⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⢄⣀⡀⡀⡀⢀⠝
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠈⠣⠞⠘⠋⠐⠆⠀⠀⠀⠘⠀⠀⢀⠀⠄⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣀⢀⡀⡸⠀⠀⠀⠀⠀⠀⢠⠛⠀⠠⡆⠰⠗⠰⢒⠀⣂⠀⣖⠀⠠⠀⠆⠀⡜⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠤⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⡀⠀⠀⠀⠀⠀⠔⠀⠀⠘⠀⠁
⠀⠀⠀⠀⢠⣀⡀⠀⠈⠀⠀⠀⠀⠀⠈⣄⠀⠑⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠒⠘⠁⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠈⠒⠀⠒⢄⣀⣀⠔⠓⠂⠂⠀⠀⠀⠀⡰⠤⠤⡀⠀⠀⠈⠂⠀⠉⠀⠉⠀⠩
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠀⢀⠂⠈⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⡀⠀⠀⢠⠋⠀⠀⠀⠀⠉⠠⠀⠀⠀⠐⠂⠁⠀⠠⠄⠄⡀⠀⠀⠀⣰⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠱⡀⠀⠀⠀⠀⢀⡪⠀⢀⠄⠐⠐⠀⠂⠠⡀⠀⠀⠀⠉
⠀⠀⠂⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠉⠓⠒⠢⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠑⠂⠅⠉⠀⠀⠀⣴⠀⠀⠀⠀⠈⠢⠀⠀⣀⠄⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡀⠈⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⣒⠄⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡤⠀⠊⠋⠰⣶⣤⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠁⠀⠀⠀⠀⠈⠘⠷⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠇⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⣼⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠹⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠺⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠎⠿⢷⣤⡄⠀⢠⣤⣄⣶⡀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣄⠀⣍⡵⠀⣴⡍⠈⠃⠹⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢻⠋⠀⠀⠀⢀⣤⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠶⠉⠛⠒⠒⠒⠛⢫⡄⢀⡼⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⢀⡰⠀⠈⠉⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣯⣷⣤⣆⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡿⠃⢹⣿⣿⣿⠟⠛⠁⣀⣀⢀⡆⢠⡾⠿⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢹⣿⣤⣷⣾⣷⣶⠄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣴⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢉⣹⣧⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡗⠚⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣠⣬⣍⡂⠀⠀⣶⣿⣿⠿⠃⢿⣿⣧⣀⣼⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣼⣯⡙⣿⣶⣾⣿⡟⠁⢀⣴⣿⣿⣿⣿⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⠁⠀⠀⠀⢰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣉⣯⡀⠄⠀⣠⣿⣿⣿⣿⡿⣿⣷⡄
⠀⠀⢀⠀⠀⠈⡄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⣼⣿⡙⣿⣿⣿⣷⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⣠⠿⠼⠟⠛⣻⣯⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢹⡀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣾⣶⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠐⠄⠀⠀⣄⠀⠀⠓⠀⠘⣧⠀⠀⠀⡇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣾⡿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠤⠿⠆⠀⠀⡇⠀⠀⠀⠴⡻⣛⡿⣿⣿⣿⠾⠛⠀⠂⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⢀⣾⡳⠤⠛⠧⠈⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣿⣻⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡌⠻⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣾⠶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠓⠀⠀⠀⠀⠀⢀⠀⠸⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡜⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⡄⠀⠀⠀⠀⠀⠀⠈⠣⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠾⠧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠶⠿⠟⠾⠟⠲⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠙⠛⣻⠖⠲⠶⢶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⡛⡛⠟⠛⠛⣿⣦⣄⢰⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡋⣹⣧⣯⣿⡇⠀⠀⠁⠠⠂⠀⢀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⢀⡗⠘⣿⣿⣿⢿⡷⢶⡄
⠀⠀⠀⠀⠀⠀⢠⠾⠻⡀⠀⠀⢉⣿⡟⣙⣋⡀⠂⠀⠀⠀⠀⠀⠡⢂⡔⡠⢀⢀⠄⠠⢀⠔⠀⢸⠃⢸⡿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠐⣧⣤⣤⣷⠀⢠⣯⣸⣗⣿⣿⡷⠀⡴⢠⢀⠀⠀⠁⠉⠘⠀⠉⠈⠀⠀⠈⣠⠊⢸⠆⢸⡇⣿⡏⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡏⠀⠀⠉⣹⣟⠛⣻⡁⠞⠃⠓⠃⠀⠀⠐⠀⠀⠠⠀⢀⠀⣀⢀⠉⡡⢹⠀⣸⡇⣿⡃⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡃⣿⣇⠀⠀⣾⣹⢫⢼⣿⣻⠂⢰⡖⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⢷⣄⠟⡀⢹⣷⣿⣇⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣴⣿⣿⠀⠀⠻⢿⣿⢟⣋⣍⡀⣼⡃⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠈⣧⡴⠀⣾⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢸⣿⢋⣼⣿⠀⠀⣴⢻⡟⢹⣿⣯⡿⢸⡇⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣿⣾⣿
⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⠀⠀⠻⢾⣿⠿⢋⣟⡀⢺⡄⠀⠚⠲⠒⠒⠚⠛⠛⠛⠛⠋⠀⢰⣿⣀⣂⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣥⣿⣿⠀⠀⣴⡟⣿⡣⣟⡟⡗⢸⡇⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠀⠈⣿⡧⠭⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡗⣿⣿⠀⠀⠈⠻⣿⡷⢋⡩⠆⠸⡇⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⣟⠃⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣞⣿⣿⠀⠀⣲⡿⣿⠄⣿⣻⣿⠆⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠿⠉⡤⠀⢿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⣿⣯⣻⢾⠀⠀⠻⠷⣷⣶⡿⢛⣋⠀⠑⠹⠿⠶⢶⠒⡒⠒⠛⠟⠛⠋⠉⠋⡡⠀⣮⠀⢹⡿⢸⣧
⠀⠀⠀⠀⠀⠀⣿⣿⢿⢿⠀⠀⠀⣴⣷⡇⣰⣿⣷⣆⠀⠀⠀⠀⠀⡀⢀⠀⠄⠀⠄⠀⠈⠀⠔⠀⢾⠀⢸⣷⢺⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⠀⠀⠸⣷⣿⣧⡿⢿⠿⠟⠀⠂⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⢠⠄⠀⢺⠀⢸⣿⣸⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠀⠀⠀⢀⣿⣟⠀⣠⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣔⣋⣴⢀⢸⠀⢸⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠀⠀⠀⣿⣿⣿⣴⡷⡿⠃⠀⡠⠀⣀⣀⢀⡔⢀⠴⠤⣴⠾⣫⡿⠽⠭⠼⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⢿⠿⠀⠀⠀⠀⢿⣿⣏⣬⣤⣶⣶⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢛⡛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠙⢧⠐⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠄⠠⢄⠀⢀⠤⠀⠠⢄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠊⠉⢨⠀⠀⠀⠀⢱⠃⠀⠀⠀⢀⠍⠉⠲⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢼⢆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠈⠢⢄⡀⠀⠀⠀⡀⢸⠀⡄⠀⠀⠀⢠⠠⠔⠈⠑⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⢰⠀⠀⡄⠀⠀⠁⠀⠀⠀⠃⢸⠀⠁⠀⠀⠀⠊⠀⠀⡀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠸⡀⠀⣈⢀⡀⠀⠴⠂⠀⠀⢸⠀⠀⠀⠐⠤⠀⢰⢀⡁⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠑⡌⠁⠀⠁⠀⠀⠀⠎⠠⢼⡦⠄⠢⠀⠀⠀⠉⠀⠈⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⢀⡀⠀⣠⠋⠳⡀⠀⢀⠀⠀⣠⠄⡰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⠕⠊⠁⠀⠀⠈⠒⠤⠤⠤⠔⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠤⣒⣉⠡⠤⠄⠒⠒⠢⢤⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⡴⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡖⠒⠒⠒⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋

⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⡆
⢀⠟⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠈⣞⡵⡊⠉⠑⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⢘⢾⡀⠻⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣞⣀⣷⠀⢷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠈⠉⠉⢳⠙⢿⠑⢕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢀⡕⠈⣧⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢸⠄⠀⢸⡆⢹⣄⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠑⠚⢁⣽⡟⢻⡍⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡞⠁⢳⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠻⣆⡈⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠈⠉⠙⡧⢚⢽⡒⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠐⣇⠈⢷⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠘⣆⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⡦⢚⣡⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⡇⠈⢧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢸⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠁⠉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢻⡈⢎⡦⢤⣒⣒⠒⠈⠉⠉⠉⠋⠉⣉⣉⣀⣤⣤⠤⠶⠶⠚⠛⠛⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⢤⣤⣤⣀⣒⡐⠤⠬⠭⠒⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀⣀⣀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀
⠀⠈⢙⣶⣾⣥⡴⠛⠉⣉⡤⠬⠤⠤⠤⠒⠺⣯⡥⠸⡅⢀⣀⣀⣪⣜⠁⠛⠃⠁⢙⡡⠴⠒⢒⣞⠛⢺⣏⠈⣹⡄
⠀⠠⠏⡟⠙⠻⣯⡓⠛⣁⠀⣤⢴⡶⠤⠿⠶⠀⠶⠶⠀⢃⠖⢩⡝⠋⠉⠛⠛⠋⠁⠀⠢⠂⠉⣩⠔⢋⡛⠓⠾⢆
⠀⠀⣄⠁⣔⢦⡀⢹⠏⠩⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⣼⠱⣞⠁⢉⣳⡄⢸
⠀⠀⠘⣽⠯⠉⢂⣉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣯⡣⢻⣿⣿⠃⢇⡼⡄
⠀⢠⣄⡘⠀⣠⡈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⠄⠇⠀⠉⠉⠀⠰⢦⠁
⠀⢈⠋⢡⣯⠽⠥⠾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⡃⣄⣤⣦⣶⣤⠀⡞
⠀⢀⣃⢠⢀⣦⡄⠀⡇⢸⣿⣿⣿⠟⣿⡿⠿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣿⢿⢻⠿⢻⣿⡆⠀⠸⣭⣿⣿⠷⠞⠀⠂
⠀⢰⣿⣿⣿⣷⠿⠋⠃⢸⣿⣿⠃⣿⣿⠃⠀⢘⢼⣾⡇⠃⣿⣼⡏⣿⢸⠈⡿⢘⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⣠⠤⠤⠤⣤⣼⠇⠀⣿⣿⣷⣼⣿⣦⣴⣿⣮⣼⣧⣦⣿⣼⣷⣿⣼⣦⣷⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠸⠷⠰⣶⠀⠐⣟⢧⠀⢸⣿⣿⡟⣛⣻⢛⡛⣿⢿⡿⢻⣿⡟⢿⢻⢟⣻⡟⣽⣿⣿⠇⠀⢄⠀⣀⡄⡀⠀⠀⢀
⠀⠠⢠⣴⢾⡶⢶⣌⢸⠀⢸⣿⣿⣧⢿⣿⢸⣿⣿⢀⣿⣨⣿⡇⡄⢸⢸⣿⡏⣽⣿⣿⠆⠀⡏⣴⣾⣷⣿⠘⣖⢼
⠀⠘⠯⣿⣼⣷⣾⡟⡼⠆⠸⣿⣿⣟⣥⣿⣶⣭⣿⣾⣿⣬⣿⣷⣷⣾⣦⣼⣷⣽⣿⣿⡄⠀⣇⠉⢿⣿⡟⠀⢨⢧
⠀⠀⡆⠈⠛⣋⠁⠀⢶⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣷⣘⠛⠃⠀⡿⢽
⠀⠀⣙⣠⠼⠿⠟⠢⣼⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⠉⠙⠓⣆⠁⡄
⠀⠀⠛⠃⣤⠀⡐⠢⣉⣰⣂⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⠀⠈⣿⠈⠀⢰⠀⠛⡄⡇
⠀⠠⠲⠀⣿⢠⣿⠀⡟⢿⠀⠙⠶⠄⠈⠉⠉⠉⠭⠥⠤⠄⡄⢀⡄⠘⢿⣿⣿⣿⣿⡇⠀⢀⣘⢠⣷⣾⡤⣴⡇⡷⠂
⠀⠀⠈⠁⢀⡎⢈⠸⠲⠘⣸⣇⣀⣨⢾⠀⡀⢠⠚⠑⡄⠠⢟⠒⠾⡄⠀⠻⣿⣿⣿⡇⠀⣠⡀⢸⡉⠋⠀⠀⠛⠇
⠀⠀⠒⠒⠊⠀⢸⣸⠗⠛⢹⠏⠉⠀⢸⠋⠁⠸⡔⣶⠃⣾⣿⣧⣀⠙⣄⠀⠈⢻⣿⣧⠀⠉⠀⣨⣥⣴⣾⣶
⠀⠠⣤⣤⣠⣶⠘⡏⢰⣦⠘⢀⠤⠤⢄⠀⣀⡀⢻⣿⢰⡌⢿⣿⣿⡧⢸⣷⣾⠀⠙⠻⠀⠀⠀⡇⠉⣿⣿⣿⠈⣇
⠀⠐⠛⣋⣉⣙⠀⠃⠈⣅⠀⠘⢄⣀⠜⠛⢛⡛⢘⠃⠏⠣⣜⡿⠃⢀⣻⢿⡿⠶⠤⣄⣀⣄⠀⠓⡒⠛⠛⠉⠀⠈
⠀⠀⠀⠉⠁⠉⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠈⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⣹⣷⠰⣿⣿⣿⣿⣿⣿⠟⣠⣶⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡙⣡⣾⣿⡿⠸⠿⢋⣥⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⠿⠿⠿⢛⣱⡗⣠⣬⣙⠻⢿⣿⣿⠟⣋⣥⣤⣤⣍⡛⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣾⣿⠿⢃⣿⣿⣿⣿⣶⣾⣇⣾⣿⣿⣿⣿⣿⣿⢌⠿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⡟⣙⢻⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⣛⣁⣛⠿⣿⣿⣿⣿⡿⠟⣿⣆⠻⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣡⣾⣿⣿⣿⣷⣎⠛⣩⣵⣶⣿⣿⣿⣷⣬⣙⣛⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⡟⢿⡿⢋⣾⣿⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⡟⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣆⠻⣿⣦⣴⣿⣿⠟⣸⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣧⠹⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣬⣙⣛⣋⣥⣾⣿⣿⣿⡿⣋⣭⣭⣥⣙⠿⣿⣿⣿⣿⣇⢻⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣧⠹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⠻⠟⣱⣿⣿⠀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣷⣾⣿⡿⢋⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⣠⠶⣄⣠⠴⠄⣤⣠⡄⣤⢶⡶⢠⢠⡄⣤⢠⠶⠀⢠⠦⡄⡴⣤⢠⠴⣄⢠⡴⠆⡴⠆⣤⠆⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡄⠻⠤⠟⠻⠬⠇⠏⠻⠇⠗⠸⠇⠸⠀⠷⠇⠸⠭⠀⠸⠋⠁⠟⠳⠹⠤⠾⠹⢦⠄⠯⠅⠨⠿⠠⠽⠂


⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡄⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢉⣉⠙⣿⣿⣿⣷⡄⢠⣾⣿⣿⡿⠉⣄⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⣁⣤⡉⠛⠛⢉⣀⡈⠻⠛⢉⣠⣄⣉⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡙⠛⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢁⣴⣿⣿⣿⣷⡀⠻⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠃⢸⣿⣿⣿⣿⣿⡇⢰⠆⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣻⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠙⠛⠛⠿⢶⣦⣶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣼⡿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⢰⣿⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣾⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⢰⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⣸⣿⣾⠁⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣀⠀⠀⢸⡇⠀⠀⢀⣀⣤⣤⣤⣴⣦⣤⣤⣤⣤⣀⣀⠀⠤⠀⢹⡏⣿
⠀⠀⠀⣿⣿⡏⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣼⣇⣴⣾⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣷⣦⣼⣿⢻⡇
⠀⠀⢠⣿⣿⣷⣿⠿⢟⣛⣉⣉⣭⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣯⣭⣤⣤⣦⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠙⣧⣴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠶⣶⣦⡴⠶⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⡀⠀⠀⢀⡼⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠻⣦⣴⠟⠁⠀⠀⢸⡇⠀⣠⡴⠞⠛⠛⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠈⠁⠀⠀⠀⣠⡾⠻⣾⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠻⢦⣤⣤⣤⡴⠾⠋⠀⢸⡇⠀⠉⠛⠳⢦⣄⣀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇⠀⠀⠀⠀⠀⠈⠉⠠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣀⡀⠀⠀⠀⠀⠀⠀⠸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣤⠶⠾⠻⠶⢦⣄⡀⠀⠀⠀⠀⣀⣤⡴⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢳⣄⣤⠶⠟⠋⠁⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⡴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣠⡶⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠏
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢿⡀⠀⠙⢧⣀⣴⠏⠀⠀⠀⣼⠿⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⢷⡄⠀⠀⠀⠀⢾⠶⢶⡶⠶⠛⠛⠉
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠛⠁⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⢸⡇
⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢾⣧⣀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣾⠁⢸⡇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠟⠀⣼⠃
⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣴⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠺⣦⢀⣴⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⠛⠁⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⢀⣀⣀⣤⣴⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠉⠩⠙⢋⡀⡈⠉⠉⠩⠰⡒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢄⣠⠀⣠⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢌⠀⠣⡀⠉⢙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡤⠊⢀⠞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠱⡀⠈⡗⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃⢁⠎⠀⠀⠀⣠⣴⠒⡶⣋⠶⢆⠀⠀⠀⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠖⠂⢸⠀⠀⠀⠘⡄⠓⠀⠓⠉⢁⠎⠀⢀⠀⠂⢈⠀⢀⠕⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣔⠁⡀⢸⠀⠀⠀⡀⢸⡀⠀⠀⠀⡎⠀⠀⠀⠀⠁⡸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⡄⠈⢆⠀⠀⠠⢀⡇⠀⠀⢸⠀⠀⠀⢠⠐⢠⠃⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠎⣀⠀⠈⢦⡀⠀⠀⠛⠀⠀⡃⡀⡄⡆⠀⡰⠃⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠤⠤⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⢱⣠⡄⡉⠢⢄⣸⡀⢑⣇⣠⠤⠔⠋⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠑⠀⢇⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠰⣎⡀⢘⡽⢟⡆⠈⠑⢦⡜⠁⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠠⠔⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣉⣉⣤⡴⢞⡧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡘
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢠⣀⣴⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡌⠀⡜
⠀⠀⠀⠀⠀⠰⡁⠀⢀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠀⠀⠀⠀⠈⠻⣙⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠘⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠰⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡔⠉⠉⠲⡀⠀⠀⠀⢻⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢥⣌⣊⠔⠰⠀⠆⡔⡠⢠⢀⠎⡞⠠⠃⢋⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⠊⠀⠀⠀⠀⢸⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⠤⠤⠤⠤⠤⠔⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣱⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢣⡸⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⢈⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡴⠆⣨⢔⠵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣷⠆⢐⠵⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠕⡪⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣩⣂⠅⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣩⣿⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⣻⢹⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢕⡏⢸⠀⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡠⢀⠴⣂⢔⡢⢊⠤⣂⠔⡪⠔⡡⢐⡠⢂⠴⣂⠔⣢⢔⡠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠥⢻⡰⣛⢕⡾⢳⣄⣿⢔⡠⢂⠄⡢⡤⠔⠔⠠⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⠟⠁
⣿⣿⠟⠁
⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠶⠀⠠⠄⠠⣤⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠴⠀⠠⡄⠀⣤⠀⢠⡄⠀⣈⠀⠀⠁⠀⠘⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠶⠀⠰⠆⠀⣤⠀⣠⡄⢀⣀⠀⣈⠁⠀⠛⠀⠘⠃⠀⠖⠀⠰⠄⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠠⡄⠀⣤⠀⢠⣄⣀⣈⣁⣈⡋⠀⠛⠃⠘⠟⠀⠶⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠰⠆⠀⣤⠀⣠⡄⢀⣤⠖⠉⠀⠀⠀⠀⠀⠉⠛⢿⡀⢾⡦⢠⣦⠀⣤⡄⢀⣀⠀⢈⠁⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⣤⠀⢀⡄⠀⣈⠀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⠖⠁⠀⠀⠀⠀⠘⣏⠀⠛⠁⠘⠇⠀⠶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠤⠀⢠⡀⢀⣀⠀⢈⡁⠀⣻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⠀⢶⠆⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡩⠊⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠀⠀⠄⣠⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠈⠁⠀⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠖⠀⠠
⠀⠀⠀⠀⠀⠛⠀⠰⠆⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠄⢠⡄⠀⣀
⠀⠀⠀⠀⠰⠦⠀⣤⠄⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠀⢀⡀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⣀⠀⣈⡉⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⢀⡁⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠁⠀⠙⠁⠘⠟⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢉⣉⠀⢈⠁⠀⠛⠀⠘⠂
⠀⠀⠀⠀⠉⠀⠘⠃⣴⠟⠋⠙⠣⢰⣶⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠛⠀⠛⠃⠐⠗⠀⠰
⠀⠀⠀⠀⠀⠀⠺⢸⣏⠀⠀⠀⠀⣤⣤⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠇⠐⠷⠀⢰⠄⠀⡀
⠀⠀⠀⠀⠀⠀⣦⠀⢻⣦⣤⣤⠖⣉⡁⢈⡛⠻⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⠆⢰⡆⠀⣤
⠀⠀⠀⠀⠀⣾⠁⢀⣄⠀⢀⡁⠀⠙⠁⠘⠛⠐⠿⠛⢿⡿⢿⣿⡿⣿⣿⢩⣭⡁⣙⡻⣿⣶⣤⣤⣤⣤⣴⣶⣿⡟⢠⣦⠀⣠⡀⢀⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠘⠃⠘⠛⠀⠾⠇⢰⣶⠀⣴⡄⣠⣭⢀⣭⡁⢈⣛⠀⠛⠃⠸⠿⠛⢿⠟⢻⣿⠁⣤⠀⢠⡄⠀⣈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠰⠆⠀⣶⠀⣠⡄⢀⣭⠀⣈⡉⠈⡛⠁⠻⠟⠘⠿⠃⢾⡷⢠⣶⠀⣤⡄⢀⣌⠀⢈⠁⠀⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣄⠀⣈⠁⢈⡋⠀⠛⠃⠘⠿⠀⠿⠇⢰⡶⠀⣴⡄⢠⣄⠀⣉⠁⠈⠉⠀⠛⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘⠋⠐⠿⠂⠲⠖⠠⣶⠀⣤⡄⢀⣌⠀⢈⡁⠀⠉⠀⠘⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠰⡶⠀⣤⡄⢠⣄⠀⣈⠀⠀⠉⠀⠙⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡀
⠀⠀⢀⠱⣀⣠⣄⣤⠔⣂⠥⠤⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⡈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣤⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣓⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠒⠓⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣟⣛⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⣀⣤⣶⣄⡀⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⢽⣽⢽⠤⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠞⠻⠻⠛⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣛⣿⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣩⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣶⣶⣦⣶⡾⠿⡇
⢀⡀⠀⣀⣾⣿⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣡⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣼⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠄⢀⡸⠂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⠀⢢⡘⠋⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠘⠣⠀⣴⠴⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡃⠐⡀⣠⠟⠀⠀⠀⢀⠤⠊⠀⠑⠄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⢄⠀⢘⡬⠋⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⢆⠠⡶⠋⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢏⠁⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠠⠀⢀⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡔⠁⠀⢡⣠⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⣠⠞⣇
⠀⠀⡠⠐⠺⡁⢱⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠞⠁⣠⡿
⢀⡾⠂⠄⢀⡼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⡠⠂⠀⠀⠀⠀⠀⡠⠚⢁⣴⡾⠋
⠈⠳⣄⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⠋⢀⣤⠿⠋
⢾⡢⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠙⠦⡀⢌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠀⠀⠈⠳⣕⢌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡠⣠⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣕⢌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⡄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡈⠳⢄⡀⠀⠀⡠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠶⠋⠀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡤⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡿⣻⠋⣉⢻⣿⣿⡿⠂⠉⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣤⣿⣁⣻⣌⣉⣾⣂⡟⠀⠀⢀⣀⠀⠀⠀⢧⠀⠀⠀⠀⠦⢤
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡭⢹⣋⣽⠃⠀⠉⠻⡟⢋⣉⣿⡆⡠⢤⡈⠼⠃⠀⣠⣾⠀⣀⣤⡶⠄⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠂⠀⠀⢀⡁⣠⣄⣀⡀⠳⣉⣉⣿⣿⣜⣻⢷⣿⣷⣴⣿⢿⣿⣿⣏⠀⠀⠀⢢⠀⠱⠟⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⣀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡉⢻⣷⡧⠀⡹⠀⠈⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠈⢻⡟⢋⣹⣿⡧⣙⣿⣿⣿⣿⣶⣧⣾⣿⠴⠔⠁⠠⠤⢀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⠽⠗⣂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⣿⣿⣿⣯⡽⠟⠁⠁⠀⠧⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠊⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣤⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣉⠙⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⢿⣿⣶⣤⣈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣁⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣷⣶⣶⣦⣤⣀⣉⠙⠻⢷⣦⡈⠳⡄⠀⠀⠀⠀⠀⢀⡞⢋⣤⡾⠟⠋⣁⣈⠻⢿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠛⠛⠉⠉⠛⠛⠛⠛⠛⠿⠿⢶⣦⣌⡙⠦⠈⠀⠀⠀⠀⠀⠉⡴⢛⣡⣴⡶⠿⠿⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣠⣤⣤⣶⣶⣄⠻⣿⣿⣿⣿⣿⣇⣀⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⣤⣴⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⠟⠘⢿⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⠻⠟⢛⣡⣾⡇⠀⠙⠛⠛⠋⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃⠈⢿⣿⡿⠃⠀⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣄
⠀⠀⠀⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠨⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢀⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢨⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⣿⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢸⣿
⠀⢐⣿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⡏⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣿⣿
⠀⢐⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⡇⠀⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡏
⠀⢐⣾⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠀⠛⠚⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡇
⠀⢐⣶⣿⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡥⠀⢠⣿⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣻⠇
⠀⠀⠀⠘⢷⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⡀
⠀⠀⠀⠀⣰⣿⠟⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠉⠉⠉⠉⠻⢿⣦
⠀⢀⣴⣾⠿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣴⣶⣶⣶⣾⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣶⣶⣦⣄
⠀⣾⡟⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⡀⠉⠻⣿⣄⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠀⠉⠻⣿⣄
⠘⣿⡇⠀⣾⡟⠉⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⢘⣿⡆⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠹⣿⣬⣿⡇⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⣷⣆⠀⠈⠉⠀⢀⣠⣄⡀⠀⣀⣠⣤⣿⡇
⠀⠀⠈⢻⣿⣿⡶⠀⠘⢿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠛⠀⠀⠀⣼⡿⠛⠛⣿⣿⠟⠛⠛⠛⢿⣦⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠙⣿⣆⡀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠈⠁⠀⠠⣿⠇⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⢸⣿⣧⣤⣤⣤⡀⠀⠀⠀⠀⠻⠿⢿⣷⡄⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⣸⣿⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⢠⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠋
⠀⠀⣸⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁⠀⠀⢠⣾⡟⠉⠉⠉⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠘⢻⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠙⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⢸⣿⡟⠛⢛⣿⡏⠀⠀⠀⠀⠈⢿⣧⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⢸⣿⣇⣤⣾⠟⠁⠀⠀⠀⣠⣤⠈⠻⣷⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣾⡟
⠀⠀⠀⠘⣿⣟⠛⠁⠀⠀⢀⣠⣴⡿⠃⠀⠀⣿⣿⠀⠀⢀⣴⣿⣶⣶⡿⠟⠀⠀⠘⠿⣷⣶⣾⣷⣄⠀⠀⢀⣾⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠙⢿⣷⣶⣶⡿⠿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠰⣿⡏⠀⠀⠀⣀⣤⣴⣶⣤⣄⡀⠀⠀⠈⣿⡷⠀⠈⢿⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣿⣿⣶⣾⣿⣿⣿⣿⣶⣤⣿⡇⠀⣰⣿⠟⠋⠉⠉⠉⠛⢿⣷⡀⠀⣿⣇⠀⠀⠈⠙⠛⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡋⠁⠀⠀⠀⠀⠀⢠⣾⡿⠟⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⣷⣦⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⣠⣾⠟⠉⣈⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⣰⣿⣯⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡀⠸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣰⣿⠟⠋
⠀⠀⠀⢰⣿⣿⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⣷⣦⣤⣤⣤⣶⡿⠏⠀⠀⣿⡇
⠀⠀⠀⣾⡟⣩⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣀⣀⡀⠉⠉⠋⠉⠁⣀⣀⣀⣠⣿⠟
⠀⠀⠀⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣛⣛⣿⣷⣀⣀⣰⣿⠟⠛⠻⠟⠁
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡴⠚⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⡀⠀⠀⠁⠂⠄⠀⠒⣀⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡿⠄⠀⠀⠀⢀⡿⠁⠢⡀⠀⠀⠀⠀⢳⠀⠠⡤⠐⠂⣀⡀⠆⡀⠀⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢼⣧⠀⠀⠀⠈⢣⡀⠙⢦⡄⠈⢡⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠆⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡁⠀⢀⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⢥⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡦⠀⠀⠀⣀⣤⠶⠋⠀⠀⠀⢴⠉⠙⠓⠲⣶⠤⡄⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠁⠘⡀⠀⠀⠈⣬⠦⠀⠂⠀⠀⠀⠊⠁⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⢁⣨⡟⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠃⠒⠚⠟⠓⠒⠶⠀⠀⢱⠀⠠
⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠘⡏
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡁⠂⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠇⠐⠀⠀⠀⠀⠀⡇
⠀⢀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠔⠖⠒⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠐⠄⠀⢠⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡐⣢⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠃
⠀⡄⠀⠈⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡀
⡘⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢨⠰⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠹⡄⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⡤⠖⠁⠀⠀⠀⠈⢂⣀⣀⣐⡄⠀⠀⠀⠀⠀⣯⠀⠁⠀⠀⠀⠆
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠉⠀⠀⠀⠀⠒⠁⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⡀
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠒⠛⠃
⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠈⠙⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠠⠄⠀⠒⠒⢤⠄⠀⠀⠀⠀⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠰⡁⠸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢱⠀⠀⠀⠀⠀⠨⠹⠦⠴⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣄⣠⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⣿⢶⡶⠒⠈⠉⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠖⠖⠒⠺⡿⣷⣶⡶⢶⢃⠟⡩⢉⡬⠍⠄⠼⠛⠉⠀⣩⢞⣀⣌⡠⠊⠀⡀⠀⣈⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠟⠿⣾⡿⠿⠷⡞⢾⠛⠻⠛⠻⠉⠉⠉⠉⠁⠘⠓⠀⠐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⡂⠁⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⢐⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⡠⢀⡀⢄⣀⠤⠀⠔⠒⠀⠉

⠀⠀⠀⢰⡄⠶⠆⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⡀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠟⠉⠁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡤⠤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⣄⣀⡀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠛⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣶⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠠⠿⠉⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣤⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡠⠀⣠⢀
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠐⠂⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠤⠤⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⢠⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠢⢀⠤⠺⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠰⢿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣰⣿⣿⡆⠳⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⢯⣽⡅⠢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣿⡽⢠⡾⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⡧⣤⡀⡆
⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡾⠁⠸⠇⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣜⢿⡄⠐⣠⣌⡅
⠀⠀⠀⠀⠤⠠⠶⠿⠆⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⣴⣆⣠⡀⡀⡀⡙⠿⢿⠛⠋⠀⢠⡖⠀⠋⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠠⠆⡐⡿⣶⣷⠿⠄⠈⠆
⠀⠀⠀⠀⢄⡀⠄⣤⡀⣠⡀⠀⠀⠀⠀⢠⣿⡹⠯⠘⠓⢽⠟⠁⠰⢄⠈⢉⣀⡈⠀⠀⢣⣼⡇⠀⠀⠀⠀⠀⠀⢀⣶⢀⡟⠁⠀⠀⡀⢈⣧⣄⣁⣠⡀⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⡀⢧⣄⢀⡀⠸⠟⠃⠚⠀⠀⣬⣧⣟⠀⠺⠖⣤⣤⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠐⢾⠋⡠⠁⠀⠀⠀⠙⣾⣏⢡⡄⠈⠢⣠⡆
⠀⠀⠀⠠⠭⠤⠀⠿⠃⢀⡶⠚⠻⣄⠁⠀⠐⠂⠀⡀⢰⡄⠹⠃⠒⠢⢀⣀⢘⡗⠀⢰⠇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⢈⠀⠂⡶⠉⣰⣿⡄
⠀⠀⠀⠀⣠⠀⠀⣤⡀⠘⠣⢤⡾⠛⠇⠀⣴⣖⣿⡷⠄⠀⠀⠀⠀⠀⠀⠅⠘⠃⠀⠀⠀⠀⠀⢀⠚⡅⢹⡄⢀⡄⠀⠀⠀⢀⣙⣏⠀⠈⢺⠞⠁⣰⠏⠉⡃
⠀⠀⠀⠀⠁⢀⠘⢭⡗⢀⢀⡞⡃⠀⠀⠀⠿⣿⣿⡷⡆⠀⠀⣀⠆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⣴⡄⡾⣾⣷⣘⢿⡧⠙⢿⣡⢒⡁⠀⣸⠏⠠⣶⡄
⠀⠀⠀⠐⠛⢻⣿⠿⠃⢀⡴⠞⢘⣷⡧⢀⣺⣿⡿⠀⢖⠀⢠⡇⠀⠘⢷⡀⠀⠀⠘⠀⠀⠀⡶⠄⠀⣺⣿⣧⣿⣿⣿⣿⣿⣦⡀⠸⠇⠻⠇⡸⠁⢀⣤⣍⠁
⠀⠀⠀⠠⣤⢸⣷⣴⣄⡾⠋⣸⠃⠙⣳⣿⣿⡿⠧⠐⢠⣄⢸⡇⠀⠀⠀⣷⡀⠀⣆⠀⠀⠀⠂⠀⠈⠛⢹⣿⣿⣿⡿⢻⣿⣿⣷⣤⠀⠀⣠⢷⣦⡼⠻⠠⠆
⠀⠀⠀⢀⣀⢀⣩⣿⡟⠁⢸⠇⠀⠐⠋⢦⣿⠀⠀⡇⠀⣙⣸⡇⠀⠀⣀⣈⣁⢼⣿⢁⣶⠀⠀⠳⠀⣼⣿⣿⣿⣿⣟⠛⢻⣯⣿⣿⣁⣰⡟⣷⠟⠀⠐⡦⡄
⠀⠀⠀⠈⠛⠂⢾⠿⣧⣀⣅⡀⠀⠀⣠⣤⣶⣶⣤⣴⡟⢿⣿⡷⢀⠐⠿⢟⣧⠀⣃⢸⣿⠀⢠⣤⣾⣛⢿⡿⣿⡿⢿⣤⣾⠘⡻⠚⣿⣿⣿⣿⣄⣠⣾⣤⡀
⠀⠀⠀⠠⡶⣤⣤⣶⣿⠁⢠⣷⣶⠋⢩⣽⡏⠙⠻⢿⣿⣿⣿⠁⠀⠀⠀⢠⣿⠇⠃⣸⣿⣦⡀⠀⢨⠿⠧⠸⠛⠁⠘⣿⠏⠀⢀⣾⣻⣴⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠫⣹⡿⠧⢾⣿⡆⢀⣾⣿⠀⠀⠀⠀⢧⠀⣿⠀⠀⠀⠀⠀⢰⡼⠚⣹⠉⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠀⠞⣼⡟⠿⣾⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⡈⢳⡶⠒⠚⠛⠛⠒⣲⠶⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠲⣤⠞⠁⢀⡇⣠⠴⠒⠲⠞⠁⠀⠀⠀⠀⠀⠈⠲⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣄⣀⣀⡠⠎⣹⠓⠦⣄⣀⠀⢀⡴⠒⠒⠒⢦⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠁⢀⡤⠟⠀⠀⠀⠉⠁⠋⠀⠀⠀⠀⠀⣽⡤⠀⠀⠀⠀⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠾⡗⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡴⠋⠉⠉⠓⣄⣀⡤⠖⠃⠀⠙⢦⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼⠀⣄⠀⡰⠂⠘⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠸⣄⠈⠛⠀⢀⡼⠁⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⢧⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋⠀⠀⠀⡠⠚⠉⠉⠓⢤⠀⠀⠀⠸⠆⠀⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⣠⠖⠈⡇⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠛⠁⠀⣠⠃⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⣀⣀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠠⠤⣾⣦⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠈⠉⢻⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢐⣒⣿⣗⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠉⢹⠁⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠨⣯⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣀⣸⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢠⣤⣼⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠛⢻⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⣶⣾⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠉⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣀⣿⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠨⠿⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⣀⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠓⣿⠚⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠠⣤⣿⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠙⣿⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠹⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢉⠳⡤⠔⠒⠒⠒⣤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠲⡴⠋⢀⣇⡠⠖⠒⠚⠁⠀⠀⠀⠀⠙⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠢⠤⠴⠊⣸⠉⠒⠤⠄⡴⠊⠉⠉⠲⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠊⠁⠀⠀⠀⣠⡄⠀⠀⠀⠀⠛⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡠⠒⠛⠢⣄⢀⡠⠔⠀⠣⣄⡀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢸⠁⣄⣠⠆⢘⣅⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠈⢦⣈⣁⣀⠜⠀⠀⢀⣀⣠⡂⠀⠀⠈⢆⠀⡿⢸
⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡰⠉⠀⢀⡈⢦⠀⠀⠈⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠲⠊⠀⡼⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⡶⠾⠟⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣤⣶⠶⠶⣿⠆⢰⣦⣤⣀⠀⠀⠀⠀⠙⢻⣦⡀⠀⠀⠀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠃⢀⣼⠏⠀⠀⠀⠀⣤⣄⢈⣿⠀⠀⠀⠙⠷⠿⠃⠈⠛⢷⣄⠀⠀⠀⠀⠙⢷⡄⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢀⣾⠋⠙⠛⠋⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⡉⣠⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣿⣇⠀⠀⠀⢀⣴⠾⠛⠛⠛⠿⣦⡀⠀⠙⠛⠛⣿⡀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣈⣹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⣿⠀⠀⠀⢠⣿⡟⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢰⡿⠛⠃⠀⠀⠀⠀⣿⡀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⣰⡟⠙⣿⠀⣀⠀⠀⠙⢷⣄⣀⠀⢀⣠⡾⠋⠀⠀⠘⢿⣶⠀⠀⠀⠀⠀⠸⣷⡀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣴⡟⠀⠀⠹⠟⠛⣷⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⣀⡀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠏⠀⠀⠀⠀⠀⢾⣿⡀⠀⠀⣠⣄⡀⠀⠀⠀⣾⠛⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣾⡏⢙⣿⣤⣤⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⣀⣄⣀⠀⠈⢻⣦⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⣼⡟⠉⠙⣷⠀⠀⣿⡆⣠⡾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠘⢷⣤⣾⠟⠀⢀⣿⡿⠋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠉⠀⠀⠀⠀⠀⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠚⠉⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⡤⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣧⣴⣯⣤⣄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠂⠀⠐⠒⠒⠒⠀⠠⠬⠭⠭⠭⠉⠉⠛⢉⣙⣛⠛⠛⠒⠺⠶⢶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⠀⠀⠀⠀⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠉⣐⣒⡒⠠⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠰⢠⠀⠖⢀⠄⠉⡁⠁⣒⠒⠠⠭⠅⢁⣒⠒⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠤⠐⡠⠄⡀⠐⢂⠀⠙⠁⡄⠈⣠⠒⢁⡠⢄⠒⠤⠀⢉⡉⠁⣒⠂⠤⠭⠭⢁⣒⠂⠴⠃⡇⡀
⠀⠀⠀⠀⢀⡠⠒⠉⡀⠀⢁⠀⠈⠠⠄⠀⠡⠀⠈⠀⠒⠈⠔⠂⠄⠈⢡⠐⠈⠀⠠⠆⠀⠤⠈⢀⠀⠀⠆⠀⠤⠀⣀⢥⣟⣉⠇
⠀⠀⠀⠀⢨⣉⠉⠒⠒⠂⠥⠴⢅⣀⡐⠀⠄⠁⠀⠀⠀⠈⠁⠀⠀⠆⢀⡄⠀⣁⠀⠊⠀⠔⠂⡡⠀⠈⣁⠤⠒⣉⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⣀⣉⠉⠑⠒⠂⠤⠤⢄⣀⡠⠂⠐⠆⢀⠀⠈⡀⠀⠃⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣉⡉⠐⠒⠒⠤⡤⠒⢉⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘