Bass Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⠀⠀⠈⠓⠤⠄⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢡⠀⣆⣰⣦⣤⣷⣴⣦⣶⣤⣽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢾⢿⡽⣥⣒⡂⠤⢀⠀⠀⣀⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠻⠹⡉⠆⠳⠐⠌⠂⠂⠁⠨⡙⠠⠄⠀⠊⢡⡞⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣯⡥⢶⣶⣴⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠊⣰⣿⣿⣿⣾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⣃⣭⣶⣿⡿⡏⢉⣾⡄⢸⣇⠀⠠⠀⠀⠀⣠⠎⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⣿⠟⠫⠛⠃⠀⢸⣿⡇⣼⢉⣷⡀⠀⢀⣼⣧⠊⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⣭⣶⣶⣾⣿⣿⣿⡳⣖⢦⣀⣾⠿⠇⠈⢹⡇⡿⢶⣿⣿⣷⠇⢀⡼⠏⢩⣿⣿⣯⣍⠙⠿⣿⣿⣿⣿⣷⣌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⣾⣿⡿⢟⠋⠁⠀⠀⠀⠀⣹⣷⡌⢿⣆⢢⠀⣾⡇⣇⠈⣷⣍⣉⣴⣿⣿⣦⣤⠘⣿⣿⣿⡿⠀⠌⠋⠉⠛⣿⣿⣆⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣴⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⣸⣿⡏⣰⡿⣰⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡷⠌⠙⡌⠀⠀⠀⠀⠀⢀⠈⠿⠟⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠚⣿⠀⠀⣀⡤⢄⣀⠀⠀⠀⢰⠛⠟⣁⠔⠉⣹⢟⡿⠡⣡⡏⣰⡞⡟⠟⠉⢉⣿⡿⠏⠁⡤⠀⠂⠀⠀⠀⠉⠉⠀⠔⠀⠀⠀⡀⢧
⠀⠀⠀⠀⠀⠀⣠⠋⠛⡶⠾⢃⣀⡤⠤⠤⣄⢈⡇⠀⢀⡞⢠⡞⠁⠀⢠⠏⠞⠄⣰⠏⡰⢡⠇⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⠥⠄⢸
⠀⠀⠀⠀⣠⠾⠋⠁⠀⠄⠒⠉⠀⠤⢤⡀⢸⠋⠀⠀⠉⠀⠸⡆⣀⡴⠿⠶⢶⣟⡡⠊⣠⡟⣸⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⢔⡂⠡⠒⠋⠁⠀⠀⠀⠀⠄⠀⢸
⠀⠀⣠⠞⣡⡴⠖⢚⣫⣿⣿⣿⣿⡷⠶⠼⠋⠀⠀⠀⠀⠀⠀⠋⠁⠀⢘⡛⣁⣡⣴⡿⣋⠞⡅⠀⠀⠀⣠⣶⡄⣽⡿⠻⠝⢋⡀⠀⢀⣠⣶⣿⣷⣤⣤⣾⡇⣸
⢀⣾⣵⠟⢉⣴⣾⣿⣿⣿⣿⣿⣿⡧⢂⡤⠀⠀⠀⠀⡠⠂⠀⠀⠀⠐⢿⣿⠿⠿⣓⠊⣁⣾⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢡⡏
⡾⠡⠀⠰⣿⣿⣿⣿⣿⣿⣿⡟⠁⡴⠋⠀⠀⢀⡴⠊⠀⠀⢀⣴⣷⣠⣴⡶⠛⠋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠭⢉⠓⢿⣿⣿⣿⣟⣛⣋⣉⠀⣾⣧
⠘⢶⣄⠀⠈⠛⠛⣿⣿⣿⡟⠀⡞⠁⠀⣠⠔⠋⠀⣠⣴⣶⣿⣿⣿⣿⡿⣿⣷⣿⣿⡿⢿⡿⠛⠛⠿⠿⠿⠿⢿⢿⣿⣭⡓⠢⣄⠀⢻⣿⣿⡿⠟⠁⠀⣸⣿⢿⢷
⠀⠀⠈⠙⠶⣄⠈⠻⣿⠏⢀⠜⠀⠠⠊⢁⣀⣠⣾⡿⠟⠛⠛⢛⣩⡭⠂⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠑⣬⣉⡙⠢⠀⠁⠀⡟⠋⠀⢀⣴⡾⢻⠓⠈
⠀⠀⠀⠀⠀⠈⢱⠊⢁⠔⠁⠀⢠⣴⣾⣛⣭⠭⠖⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢠⠉⣿⣿⣿⡇⠀⠈⠀⢀⡴⠋⢸⡁⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠰⣄⠈⢁⣂⠭⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡌⣸⣿⣿⣿⠃⠀⠀⣠⠟⠀⠀⠀⠙⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣴⣿⣿⣿⣿⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣡⣼⣿⣿⣿⣿⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣔⣺⣟⣠⣾⣿⣿⣿⣿⣿⡿⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡒⠽⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠐⠀⠠⠜⢯⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠐⠋⠡⠚⢻⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⢀⣀⠀⠀⠀⠀⠿⣿⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠱⡄⠀⠀⢠⡟⠁⣼⠋⣾⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠃⣸⠃⡼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠀⡉⢠⡇⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠈⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡠⠞

⠀⠀⠀⠀⠀⣾⣿⣷⡄⣦⡠⣶⣶⣶⣤⡤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⣿⠃⠙⣿⡘⣷⡙⣿⣿⠁⢰⣿⣦⠀⠹⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇⠈⠻⠀⠑⠤⠿⢔⣴⣿⣿⣿⣿⠟⠋⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⢻⣿⣷⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡗⠀⠀⠀⣄⠀⠐⠛⠻⠿⣿⣿⣿⣇⠀⠀⠀⠀⢤⣌⠻⠿⢷⣶⣦⣄⡀⠉⠛⢿⣿⣦⣄
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠏⣸⠇⠀⣷⣤⠈⢿⣦⡀⢿⣷⣄⠙⠻⣿⣆⠀⠀⠀⠀⠙⢿⣦⡀⠈⠙⠛⢿⣷⣄⠀⠙⢿⣿⡗
⣠⣤⣶⣦⣤⣤⠤⠤⠀⠀⠏⠀⠀⣹⣿⣷⡄⢻⣷⡌⢻⣿⣷⣄⠀⠙⠆⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠈⠛⣿⣷⡀⠈⣿⣄
⠻⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠿⠟⠛⠋⠾⢿⣿⣆⠉⠻⣿⣿⠶⢤⠀⠀⠀⢀⣴⣿⣿⠀⢠⡄⠀⠀⠈⢿⣷⠀⢹⣿
⠀⠈⠑⠢⢀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠃⣠⣿⣧⡀⠀⠀⠀⢻⠄⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠦⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠁⠀⣀⣀⣉⣉⣛⠂⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣦⣤⣀⣀⣀⡀⠒⠛⠋⠉⠀⠀⠀⠀⠀⠰⠛⠉⠉⠩⣭⣭⣍⣻⣶⣦⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠈⣩⣉⣁⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⢳⣦⣄⡀⣶⣤⣿⣿⣿⣿⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠆⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⣀⣀⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣶⣸⠉⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⢸⣿⣿⡀⢸⡀⢆⢠⣴⣦⡀⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣉⠙⠦⣄⠙⠻⢷⠘⢷⣼⣆⠉⢻⣿⡆⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣅⠀⠀⠉⠀⠀⠀⠀⠉⠛⢓⣼⠟⠀⢠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠠⣽⠦⢸⣦⣄⡈⠙⠚⠛⢋⣁⣤⣾⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠙⠻⣇⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠐⢊⣉⣀⣤⣴⣾⣿⣿⣿⡿⠟⣫⠅⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡈⠉⠉⠀⣀⣩⡿⠿⠛⠁⠘⠻⠿⣶⣦⡌⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠠⠖⢒⣩⣥⠶⠊⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⣤⠉⠉⠉⠀⠀⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣩⣤⣤⣄⠀⠘⠿⠻⠿⡷⣢⣶⣿⡆⢀⣈⣉⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡉⢀⣾⣿⣿⣿⣿⣧⣤⣶⣦⠊⣴⠿⢿⣿⣵⣾⣿⣅⣶⡴⣶⣦⡉⠒⢄
⠀⠀⠀⠀⠀⠀⢀⡴⣡⣾⢁⣼⠃⠏⠿⣿⣿⣿⣿⠟⠁⢠⠀⠀⡀⠸⠛⢛⣛⣈⣉⡉⠁⠀⢠⣼⣷⣈⣀⣠⣄
⠀⠀⠀⠀⠀⢠⢞⣾⣿⣧⣾⠟⡀⠀⠀⢻⣿⡿⠁⢸⣌⣿⡆⠀⠀⠀⡰⠁⠀⠀⠄⠀⠀⣠⣿⣿⣿⣿⣧⠀⠈
⠀⠀⠀⠀⣠⣯⣾⣿⣿⣿⠃⠀⠃⠀⠀⠈⠹⠁⣤⡇⠹⣿⣿⡆⡀⠀⠃⢀⠔⠓⠠⣠⣾⣿⣿⣿⣿⡇⠈⠀⢀
⠀⢀⠔⡫⢛⣿⣿⣿⣿⡿⠀⠀⡇⠀⠡⡡⡀⠀⣿⣿⣄⠉⠻⢿⣶⡄⢠⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⡏⠀⠀⡜
⡰⠁⠀⠀⢉⣼⣿⣿⣿⠖⢀⣴⢇⠀⢄⠙⠾⠶⡻⣿⣿⣷⣦⣄⣀⠁⠀⠀⠀⣀⠀⠹⢿⣿⣿⣿⣿⢠⠃⠸⠁
⠳⢄⡀⠀⣐⣩⠵⠋⣠⣶⡿⠃⠈⢆⠀⠑⠦⢵⡈⠪⢝⣛⣛⡛⠿⠷⠂⠀⠀⠀⠉⠐⠀⡈⠙⠻⠿⡁⣠⠃
⠀⠀⢸⠿⠋⠁⣀⣀⡿⣡⠀⠀⠀⠀⠑⢄⡢⠤⠼⠤⢀⣀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⣍
⠀⠀⢸⢀⣴⣾⣿⣿⣴⣿⣷⣴⣴⣶⣿⣷⣬⣐⣤⣄⡉⠘⢒⠄⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠲⠀⠀⢀⠔⠁
⠀⣠⠞⠛⣻⣿⢿⣿⡸⠿⣿⣏⠛⠛⠿⠿⢿⣿⣿⣿⣿⢡⣄⣀⠀⢀⣀⢀⠀⠠⠐⠐⠂⠉⠁
⠼⠥⠤⡄⡀⢐⢻⣿⠿⣿⣿⣿⣟⡵⠃⠀⣾⣿⣿⣿⡟⣸⣿⣿⣿⠛⠋
⠀⠀⠀⠙⣜⣌⢧⡅⠀⠿⢛⡙⡿⣷⣀⠀⠈⠙⠿⠟⢁⣿⣿⡇⣿⣦⢤⡀⢀⣀⣤⣤⣄⣀⣀⣀⣀⣀⢀⣀⡄
⠀⠀⠀⠀⠈⠫⡳⠇⣰⠶⢶⣤⣭⣙⠛⢧⡰⣤⡀⠀⢿⣿⣿⢣⣿⣿⠀⣏⣾⣿⣿⣿⣿⣿⠧⣠⢖⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠪⢽⣷⣆⣉⣙⣛⠈⢿⣦⣝⣶⣄⠀⠀⠸⣿⠏⢰⣿⣿⣿⣿⣿⣿⡷⡰⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠻⠿⢿⡿⠷⢿⡿⣿⣿⣿⣶⣄⡀⠀⠈⠉⠈⢛⣽⣿⣿⣳⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡰⠋⠀⠈⠙⠻⢿⣿⣿⣷⣦⣤⣶⣿⣿⣿⡏⢳⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣿⣿⣿⣿⠧⠀⢣⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⡻⣮⡻⠲⢄⠈⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡈⠁⠀⠉⠢⣀⢨⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢔⠈⣢⣤⠛⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡦⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠮⢭⣒⡿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠶⠶⠬⠝⣿⣿⡋⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣥⣶⣾⣿⣿⣿⣧⡝⣦⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡾⡟⣰⣫⣾⣴⣄⠀⢀⣴⣿⣿⣿⣿⣿⠿⣛⣻⣿⡟⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢋⣾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣬⡹⣟⣿⣿⣿⣿⣿⢟⣿⠇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⣿⣿⣻⣿⣿⣿⡿⣿⣿⡿⣟⣻⣝⡗⣦⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣠⣴⣶⣟⣻⣿⣿⣿⣯⣭⣭⣤⣴⣶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡚
⠀⠀⠀⠀⠀⠀⢿⣷⣍⢄⠉⢿⣝⣿⠿⣿⣿⣿⡿⢷⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠟⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⢆⠇⠈⠻⣬⡀⠉⠠⣄⠀⠁⠙⠙⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠻⣭⣭⣁⡁⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣿⠀⡄⢹⡷⠀⠀⠈⢻⣮⡂⢸⣛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡁⠂⠀⢉⠄⢳⠿⠀⢀⡾⠻⠋
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⡾⠀⣿⣸⠇⠀⠀⠀⠀⠙⣻⢸⠿⠋⣸⣛⡿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠱⡄⠦⠇⣴⣾⣿⠷⠄
⠀⠀⠀⢀⣄⡀⠀⣠⣴⣿⡿⠗⠀⢹⡏⢀⣀⣠⣤⣀⠀⠋⠃⠀⣰⡿⠻⠷⠿⠿⠿⠿⡿⠋⠁⠀⠀⠀⠀⣀⠄⠛⠉⠀⠈
⠀⠀⠀⠈⠉⠁⠶⣤⡄⠀⠀⠀⠀⠿⠷⠿⠿⠛⠋⠉⠀⠀⠀⣴⣟⠃⣠⣄⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠚⠟⠿⢷⠿⣿⣿⣿⣿⣿⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠟⢛⣛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣤⣾⠟⠁⣰⣿⠟⠛⠻⣛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠛⣿⣿⢿⡙⠋⠉⠋⠀⣰⣿⣿⣧⣤⣄⡈⣧⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠁⣄⠲⣌⠛⠋⣠⠆⠀⠀⢠⣿⣿⣿⣿⣿⣌⠳⠌⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⠁⢀⠀⢀⣾⣿⣷⡄⠀⠚⠉⠀⠀⠀⣿⣿⣿⡟⢻⡇⢻⠂⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢋⠁⠀⠀⢀⡞⠀⢩⠟⠡⠏⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠹⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣰⡏⠀⠀⠀⣾⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣿⣿⠀⠀⠀⠀⡆⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣠⣾⡿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠇⣼⣿⣿⠀⠀⠀⠠⡇⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⡿⠁⣠⣾⣿⣿⠇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠀⣿⣿⣿⣿⠿⠿⢾⣧⣀⣹⣿⣄
⠀⠀⠀⠀⣀⣤⣴⠾⠿⠛⢛⣿⣿⠏⣠⣾⣿⠟⣿⣿⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⣼⡟⠀⠀⢀⣿⢰⣿⣿⣿⡶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣦
⢀⣠⣶⣿⣿⣿⣶⣶⡶⣿⣿⣿⠋⣴⣿⡿⠁⠀⢻⣿⡀⠀⠀⠀⠀⠙⠻⠿⣦⣤⠀⣿⣇⠀⠰⢞⣣⡾⠋⠉⠀⠀⠀⢀⣀⣤⣴⣶⡶⠶⢶⣶⣦⣾⠃
⠀⠀⣹⣿⣿⣿⣿⠶⣶⣿⣿⢃⣾⡟⠃⢀⣷⡀⠘⢿⣷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠘⠛⣫⡍⣡⠀⠀⠀⣠⣿⠟⠁
⠀⠘⣿⣿⣿⣿⣶⣾⣿⡿⢡⣿⡿⠁⣄⣼⡿⣷⡀⠀⣽⠉⠉⠉⠉⠉⠀⠀⢀⣠⠞⠀⠀⠀⢀⣤⠂⠀⣀⠀⠀⠀⢀⡼⢃⠞⠁⠀⣰⡾⠋
⠀⢀⣿⣿⣿⣿⣶⣾⡿⢡⣿⡟⠀⢀⣿⣿⣇⠈⠃⠀⣙⣳⡄⠀⣤⣤⣴⡾⠟⠁⠀⠀⣠⣴⠟⠁⣀⣼⠃⢀⣴⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⣿⣿⣿⠿⢿⣿⢡⣿⡟⠀⠀⣿⣿⣷⣿⣶⠿⠟⠛⢉⣳⣦⣄⡤⠤⢤⡴⠶⠾⠛⣋⣡⣴⠾⠛⣁⣴⠿⠁⠀⠀⡀⠀⣠⣿⠟
⠀⢰⣿⣿⣿⣿⣿⠃⣾⡟⠀⠘⣿⣿⣿⡏⠁⠀⠀⠐⢺⡿⣿⡿⢋⣿⠟⠛⠛⠉⠳⠦⣤⣤⣀⣒⣛⣋⣁⠀⣀⣴⠟⢀⣴⣿⠏
⠀⠀⣿⣿⣿⣿⡇⣼⡟⠁⠘⣶⣿⡏⠻⣿⣄⠀⠀⠀⠀⠚⠉⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠻⠿⠟⣡⣴⣿⡿⠁
⠀⠐⢿⣿⣿⡿⢰⣿⠃⡀⠰⣿⣿⠋⠀⠈⠻⢿⣦⣤⣄⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋
⠀⠀⠀⣿⣿⠇⣾⡟⢀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉
⠀⠀⢀⣿⣿⢠⣿⠇⠘⣷⡀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⣾⣿⠃
⠀⠀⢸⣿⡇⢸⡟⠀⢀⣹⣷⣼⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⢀⣾⠃⣠⣿⣿⣿⣇
⠀⠀⣾⣿⠇⣸⡇⠀⠘⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⢂⡄⣼⣟⣴⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⠀⣿⡇⠀⢷⣿⣿⡟⠧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣥⣾⣣⣾⣿⣿⣿⣿⠏⣸⣿⣿⣿⣷
⠀⣿⣿⣿⠀⣿⡇⢠⣼⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⡄⢹⡇⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠛⠋⠉⠉⢉⣿⣿⡟⠁⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡀
⣸⣿⡏⢸⡇⢸⡇⢻⣾⣿⣯⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠀⠙⠿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣴⠟⣡⣿⣿⣿⣿⡀
⣿⣿⡇⢸⣷⠘⣷⣈⣿⣿⡟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⢀⣴⠟⣡⣾⣿⣿⣿⣿⣿⡇
⣿⣿⠁⣿⣿⡆⢿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣫⣾⠿⢋⣭⣿⣿⣿⣿⡇
⣿⣿⡄⣿⡿⣿⡈⠏⢿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣿⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⣡⡶⠟⣻⣿⣿⣿⣿⡇
⢿⣿⡇⣿⡇⢹⣷⡐⢈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣾⡟⠀⠹⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠈⠀⠋⣡⣶⠿⢛⣻⣿⣿⣿⡇
⢸⣿⣿⣿⣿⢸⣿⣷⡀⠻⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⠁⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣧⡀⠀⠛⣡⠴⠾⢛⣻⣿⣿⣿⣧
⠀⢿⣿⣿⣿⣇⣿⣿⣿⣆⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣾⣷⣾⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠠⠶⢛⣛⣻⣿⣿⣿⣿
⠀⠈⢿⣿⣿⣿⣿⣿⡿⢿⣷⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣶⣿⣴⣷⣴⣄⢠⡄⢠⣀⣄⠀⠀⠀⠀⠀⢸⣿⡄⠀⢘⣉⣭⣭⣿⣿⣿⣿⣇
⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠹⡏⠻⠿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⡇⠀⢈⣉⣭⣭⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠙⠉⠀⠛⠀⠀⠀⠀⣿⠇⠀⠤⣬⣭⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣦⣤⣀⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣿⣄⠀⣀⡂⠬⣭⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠛⠛⠛⠛⠛⢿⣷⣌⠻⣷⣦⣭⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣞⠿⢏⣃⠭⠶⠖⢒⣒⣒⣒⡒⠶⠤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⡧⢞⣋⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⣷⡶⣶⣾⡟⣳⣶⣽⡷⢂⣌⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⢛⣥⣾⣿⣿⡿⡿⠟⠟⠛⠙⠉⢋⠏⠁⠈⢩⠜⠉⠻⣧⣭⣿⠃⢁⣿⣿⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡴⣫⣾⣿⠿⠛⠉⠀⠀⢀⡀⣠⠀⣴⢀⠎⠀⠀⠀⠁⠀⠀⠀⠀⠋⠀⠀⣼⣿⣿⣿⣷⠰
⠀⠀⠀⠀⠀⠀⢤⣾⢗⣼⣿⠿⠋⢁⠀⡄⢸⡤⣷⢸⡇⣿⠂⡿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠆
⠀⠀⠀⠀⠀⠀⢸⣟⣿⠟⠁⢠⣄⣿⡦⢿⣌⡧⠝⠖⣣⣬⣀⣧⡘⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⡼⣿⣿⣿⡿⠐
⠀⠀⠀⠀⠀⠀⢸⣿⠏⣀⢺⡞⣿⠘⠏⠘⠳⣄⠀⠀⠐⠚⣿⡽⣉⠀⠀⠀⠀⠂⠀⠤⢬⣄⠉⠉⣰⠻⠋⠙⠛⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠈⡏⣰⣽⡎⠃⠈⠀⠀⠀⡀⡌⠗⠒⠒⠚⠉⠁⠈⢄⣄⡀⠄⠀⠀⠀⠀⠈⠛⡒⠒⠂⠀⠬⠁⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠀⣷⠟⠃⠀⠀⡀⠄⠂⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣉⣶⣧⣤⡀⠀⡠⠚⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣤⣤⣶⠞⠋⢿⣿⣿⣭⣿⠁⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡐⠁⠀⢀⣠⠶⣾⣿⣿⣯⣁⡔⠁⣴⣾⡿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣰⣿⠿⡶⣤⣈⢙⠻⠿⢿⣿⣿⣿⣷⡀⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠷⢀⣀⠁⠉⠛⠸⢶⢴⣤⣤⣤⣼⡇⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠉⠙⠑⠒⠤⠄⣀⣀⣉⡽⠁⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⢿⣿⠷⠛⡜⠀⠈⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣴⡶⠿⠛⢛⣩⡁⣽⠂⠀⠀⠀⠀⡀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡽⣷⣿⣿⠟⠛⠊⠁⠀⠺⢿⡁⠘⠀⠀⠀⠀⠀⡀⠀⠀⠈⣝⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠞⢋⣼⣿⡿⠋⠀⠀⠀⠀⡄⠰⠄⠀⠈⠀⠄⠀⠀⠀⠀⠈⠉⣽⣿⣿⣷⣶⡅
⠀⠀⢠⠂⠀⠀⠀⢠⡆⢤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠒⠢⠄⡀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠋
⠀⠀⠸⡆⠀⣴⣾⣿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⣨⣶⣶⣶⣾⣿⡿⠋⠁
⠀⠀⢸⡇⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⣿⣿⣿⣿⠟⠋
⠀⠀⠈⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠆⠀⠹⠛⠋⠁
⠀⠀⠀⠃⠀⠈⢤⡤⡶⣶⡖⠒⠂⠀⠀⠀⠲⠶⠒⠐⠾⠂⠀⠀⠈⠈⠉
⠀⠀⠀⠈⡄⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⢠

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣶⣶⣿⣶⣶⣦⣤⣤⣤⡀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣧⣼⣤⣤⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠉⠩⠤⠤⠬⢉⣉⡩⠽⠿⠿⠿⠿⠿⠿⠿⠟⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠚⠭⠿⢷⣒⣒⡒⠒⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⡀⡒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⢢⠀⠀⢀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠃⣴⡆⠆⠀⢌⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⠫⣭⢻⣿⡏⢀⣿⣷⢰⠀⠀⡤⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡟⠃⠘⣿⣿⣧⣑⣤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠿⠟⢋⡴⠃
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡀⢀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢠⣾⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣮⡻⢿⣿⣿⣿⠟⠀⠀⠀⢀⠄⠀⡴⠃
⠀⠀⠀⣼⣿⣿⡟⣴⣿⣿⣿⣿⠿⠿⠟⠿⢿⣆⠉⠛⠛⠋⠁⠉⠲⢤⣄⣀⣀⣠⡤⠞⠁⣠⠞
⠀⠀⠀⢼⣿⠏⣼⣿⣿⡿⠋⠁⠀⠀⠀⠀⢀⡈⠳⢤⣀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⢸⡟⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⢠⣬⣿⡷⠄⠈⠙⠢⣀⡈⠁⠺⠚⢁⠖⠋
⠀⠀⠀⠀⢱⣿⣿⠃⠀⠀⠀⠀⠀⠀⢛⣫⣭⣶⣶⣦⣤⠀⠀⠀⠈⠁⠀⣠⠋
⠀⠀⢀⣤⣿⣿⠇⠀⠀⠀⢀⣠⠴⣺⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⡼⠁
⠀⣠⣾⣿⣿⡿⠀⠀⢠⠴⣫⠔⣯⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠔⠋
⢰⣿⣿⢻⣿⡇⠀⠀⠀⢘⠡⢾⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⣠⣿⣷⣄
⣿⣿⣿⢸⣿⡇⠀⠀⠀⠈⣷⣦⣤⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦
⠹⡟⢻⢸⣿⡇⠀⠀⣤⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⣰⠏⠹⣿⡿⡏⡏⠉
⠀⠀⠸⠎⣿⡇⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠁⠁⠃
⠀⠀⠀⠀⢹⣷⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⢻⡆⠀⢹⠷⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣠⣞⣮⡄
⠀⠀⠀⠀⠀⠀⠹⣄⠻⠇⠀⠀⠀⠀⠀⢰⢿⣷⣦⣄⣀⡀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠸⡀⠈⠉⠉⠙⠛⠻⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⣿⣿⣷⣶⡖⠂⠀⠠⣌⡻⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠙⠳⢬⣛⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡛⠃⠀⠀⠀⠀⠀⠀⠉⠙⠳⠿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣖⣒⣛⣿⣭⣭⢭⣭⣽⣓⣲⣦⣤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣤⣄⣀⣀⣤⠶⣚⣯⣽⣶⣿⣿⣿⣿⣿⣫⣍⢝⣛⢶⣞⢟⣱⣿⢡⣾⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣭⣛⣿⣿⣿⢟⣯⣶⣿⣿⣿⣿⣿⣹⣿⠟⠉⢡⣿⣿⢿⡽⣸⣇⣿⣿⢣⣿⢿⣸
⠀⠀⠀⠀⠀⢠⠶⣿⣶⣞⡿⣿⣿⣯⣾⣿⣿⣿⣿⡿⣫⣶⣿⠘⠇⠀⠀⠀⣛⢿⡿⢫⣿⡇⣿⣿⣸⡿⢼⣇
⠀⠀⠀⠀⡔⠛⣶⣒⣾⣿⣿⣿⣿⣿⣛⢿⣿⣿⢟⣾⡟⠻⣿⠇⠀⠀⠀⠀⠩⣿⠇⣿⣿⡇⣿⡿⡟⣧⠸⣻
⠀⠀⠀⢀⣇⣤⣤⢽⣿⣻⣿⣿⣿⢰⣿⣶⠙⠃⣾⣿⡇⢰⡟⠀⠀⠀⠀⠀⠀⠻⢸⣿⣿⡧⣿⡇⣇⡙⠀⢻⢧⠀⠀⢀⣀⣤⣴
⠀⠀⠀⠀⣏⣭⣽⣿⣿⣿⢿⣜⠟⢺⣿⣿⣿⡇⢻⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁⢰⣿⠁⢸⡿⠷⠿⢟⣿⣿⣿⡿⢟⣫
⠀⠀⠀⠀⢻⠽⣿⣿⣿⣿⢃⣿⣧⣼⣿⣿⣿⡷⢸⡼⡀⢸⣧⡀⠀⠀⠀⠀⠀⢀⣤⣿⡄⠸⣿⣇⠸⣿⣿⡿⠟⢩⣯⣿⡿⠛⠉
⠀⠀⢠⢞⣵⣿⣿⣿⢈⣋⣾⢿⣼⣿⣿⣿⣿⡇⠀⢷⢣⡈⣯⢷⣄⠀⠀⠀⠀⣿⠀⠀⢶⣾⣿⠿⠛⣉⣱⡿⠿⠛⣹⡟
⠀⣰⢣⣯⣽⣿⣿⢿⣼⣿⣯⡘⠻⠿⣿⣿⠁⠀⠀⠸⣇⠱⣜⣿⡿⢧⣤⡤⠴⠛⢠⣤⣬⣥⡤⠶⠛⣉⠀⢀⣤⠞⠋
⢰⢣⣛⣽⣿⣿⡟⠈⠻⣿⣿⡗⠀⠀⠀⠉⠀⠀⢀⣀⣹⣷⣤⣈⠉⠉⠉⠙⡆⣀⣀⣀⣴⣶⣶⣶⣿⠥⠖⠋
⡏⣼⢟⣭⣾⣿⡇⠀⠀⠙⠿⠷⠄⣠⣾⣿⣿⡿⢟⣻⠿⣻⢏⡸⢿⣶⣶⣿⣾⣵⣿⣿⣿⣿⣷⡿⠃
⡧⠗⣡⣿⣿⣏⠀⠀⠀⠀⠀⢠⣾⣿⠻⢟⣥⡾⣻⠵⣫⣵⣿⣿⡼⠃⠀⠈⠉⠉⠉⠉⣩⠟⠋
⠃⠘⢿⣿⣿⡿⠀⠀⠀⠀⠀⠘⠻⢿⣷⣄⠫⠞⣡⠞⣡⢤⡿⠁⠀⠀⢀⣀⣤⣤⣴⠊⠁
⠀⠂⠠⠼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢛⣛⣷⣿⡷⠋⢀⣠⣴⣾⣟⣿⣿⡿⣟⢧
⠀⠀⠀⠀⠸⣿⣿⣗⡀⠀⠀⠀⣠⣴⣶⣿⠿⠋⠉⠀⠀⣰⣿⣿⣿⠰⣽⣿⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣷⣹⣷⣄⠀⠀⢹⡌⠁⠀⠀⠀⢀⡤⢖⡿⣏⠋⡿⣱⢏⣿⣿⣸⡏⣿
⠀⠀⠀⠀⠀⠀⠻⣷⣭⡛⠀⢠⣤⠇⠀⠀⢀⡴⢋⣾⠋⠀⠘⢦⡰⠏⣸⣿⢧⡏⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠀⢰⡿⠀⠀⢠⢊⡴⣿⣿⣶⣒⣚⣋⡵⢦⣁⣫⣿⠖⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⢀⡽⠀⠀⠟⣿⣭⣭⣍⣉⣻⣭⣽⣷⡾⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡏⣄⣠⣤⣶⠿⣿⣿⣶⣶⣶⣶⠶⠶⠶⠛⣛⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⣿⡟⣿⡀⠈⣍⣓⠪⠭⠭⠽⠿⠿⠿⠿⠟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠉⠀⠹⣿⣿⠳⣯⣍⡛⠛⠓⠒⠒⠒⠒⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣉⣉⣀⣀⣀⣤⣤⡶⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⠑⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢁⣀⣤⣤⣶⣾⣶⡷⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⣷⡌⢻⣿⣿⣽⣶⣒⣒⡰⢤⣤⡠⣤⠀⡀
⠀⠀⠀⠀⠀⢀⣴⢖⣋⣥⣶⣟⣻⣥⣶⣶⣿⣷⣶⣤⣉⣲⣶⣄⣤⣠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡈⡝⢿⣍⢉⣻⡟⣿⣷⣬⡙⢿⣦⣉⢣⡀
⠀⠀⠀⠀⡴⣫⣾⣿⡿⠟⡿⣟⢛⣹⣭⣿⣷⣿⣿⣭⣍⣿⡯⢻⠟⣡⣾⡟⠀⠀⠀⠀⠀⠀⠀⣆⡼⡯⣿⣇⠙⢢⡈⢘⢙⢾⢻⡼⡿⢿⣾⣿⣷⣄⢧
⠀⠀⢀⣾⣱⣿⠟⡉⢶⡗⢸⡿⠀⣿⡟⣟⣿⢻⠛⠍⠉⠁⣰⣷⣾⢋⠋⠀⠀⠀⠀⠀⠀⠘⢇⠉⢩⣭⠭⢍⡳⠿⣧⣤⠌⠀⢸⣷⣶⠉⠙⢻⣷⣿⣿
⠀⠀⢸⣿⠟⢷⡸⡧⠜⡋⠀⠀⢠⣿⣿⣎⣢⠤⡓⢤⣤⠄⡈⢁⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣄⠀⠀⡀⠀⠀⠠⠖⠿⡟⠁⢀⣤⢹⠟⣿⣿⡄
⠀⠀⠸⣿⣿⠈⠉⠀⠂⠀⠀⠀⣈⡹⠏⠀⠂⢟⣙⢪⣽⡷⠥⠤⠘⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⡉⠉⠉⠀⠀⠀⠈⠉⠁⠂⠆⢼⢏⣿⡏⡄
⠀⠀⠀⢿⠀⠀⠀⠀⢀⡀⢀⣀⣀⡔⢲⣶⣠⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⣦⠦⠀⣀⡀⠀⠀⠀⠚⠚⣽⣷⠁
⠀⠀⠀⠈⢆⠀⣴⠻⢿⡀⣼⣿⣿⡁⢈⠵⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣄⠙⢿⢿⡏⢶⣄⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠻⢻⡀⠈⠛⠻⠟⢺⡜⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠴⣈⣹⣄⣹⣄⣣⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣶⣍⣣⣾⣿⣧⠀⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⡐⠢⠤⣴⡀⠘⣦⡀⠀⠀⠀⠀⠀⠠⣀⣾⣿⣷⣿⣿⣿⣿⣛⣛⣛⡷⢦⣄⡀⢄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠛⠟⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠒⠾⠿⠆⠀⠠⠀⣠⣾⣿⣿⣿⣿⠿⣿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣛⠓⢄⠀⠀⠀⠀⢠⡁⢉⣿⣦⠄⠺⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣿⣿⢿⠋⣫⠲⣷⣰⣷⢉⣶⡜⡿⣿⣿⣉⡛⣻⣿⣦⡑⢀⣀⠀⡘⠱⣿⠟⠁⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣯⢹⡸⠇⠙⠢⠁⠀⠈⢉⡍⠃⢻⠉⠙⠛⠻⠿⠃⢘⡡⣴⡾⠁⠗⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⡟⠖⠁⠀⢀⠀⡀⠠⠤⠛⠉⠡⢬⠢⠀⠈⠉⠔⠚⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣶⣴⢂⣶⣶⡄⠈⡹⣿⡆⠀⠀⠐⠀⣱⠀⠈⣻⡿⣾⡶⠙⢦⢀⣄⡲⣾⢿
⠀⠀⠀⠀⠀⢀⣤⣨⣴⣞⣹⣿⡯⢠⠏⣼⣿⣼⡇⠀⠀⣉⣦⠀⢈⡷⠂⠀⠀⠰⠋⠀⠈⠁⠀⠀⠀⠈⠳⢿⣇⠄
⠀⠀⠀⢀⠰⠿⣿⣿⡏⠉⡋⠁⠀⣯⡶⠿⣿⣻⡃⡀⠀⢹⣿⣞⠁
⠀⠀⠀⡆⠀⢄⣤⣶⣏⠂⠴⠤⠜⠁⢸⣿⣶⡶⠒⠁⠀⠀⠻⣿⣆
⠀⠀⠀⢱⣄⡘⠟⢻⡏⠳⠤⣄⣀⠈⣿⠻⠊⠀⠀⠀⠀⠀⠀⠈⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡘⠀⠆⡀
⠀⠀⠀⢸⣿⡿⠘⠉⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⢸⣿⡛⠀⠀⠀⠀⠀⢠⣏⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⣠⡞
⠀⠀⠀⠀⢻⣝⡀⠀⠀⠰⣰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠋⣙⠉⣍⡉⢩⣤⡔⢤⡍⢈⡁⢩⡀⣙⠙⡛⢻⠛⡛⣟⡲⣾⣿⡗
⠀⠀⠀⠀⠀⠙⢧⣀⣄⠅⢻⡇⢳⣽⣀⣤⣴⣶⠂⡆⠀⠀⠘⢠⠀⠠⣍⠠⠀⠉⠀⡛⠛⣠⠻⢧⢹⣇⡸⠇⡹⠃⠛⠠⠃⠘⠈⢋⣉⠋⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⣁⡀⠉⠉⢉⡟⠛⠀⡇⠀⠀⠀⠀⠀⠠⠬⠝⠁⣀⢈⡠⠞⠁⠽⣏⠀⠀⠀⠀⠀⠁⠈⠀⣠⣴⣾⣿⣿⠇⠈⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠹⡟⠛⠛⠈⠳⣄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣖⣂⡤⢤⣀⣠⣤⣤⣤⣤⡦⠤⠀⣠⣀⣀⣠⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣸⢿⣿⣿⡟⣁⣀⣄⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣶⣾⣿⣿⡛⠉⠉⢻⣿⡟⢹⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠴⣴⣿⣿⣿⣿⣿⠿⠋⡀⠠⣧⡀⠋⡏⠘⣿⣿⣿⣿⣿⡿⠃⣾⣿⣿⡙⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡟⠁⠀⠘⠚⠱⡙⢿⣦⡈⠂⠈⠛⣿⣿⣿⡇⣴⣿⣿⣿⣃⣿⣿⠗
⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⢫⢋⠊⢶⣶⣿⣿⣿⣿⠳⣾⣷⣄⠀⠀⠀⠸⠇⣰⡿⠀⠉⢛⣛⣉⠥⠚
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⢡⣎⠔⣿⣿⣿⣿⢿⣿⠗⠘⢿⡋⠁⡀⠀⠈⠊⢙⣡⣴⠟⠉
⠀⠀⠀⠀⠀⣸⣿⣿⣿⠄⢾⣿⣮⣽⠝⠓⠂⠉⣡⣴⢊⠀⣉⡿⠟⠓⠚⠛⠋⠉
⠀⠀⠀⠀⠀⣿⣿⢹⡻⣦⠈⠛⠁⣄⠈⠀⢠⣾⣿⣿⣷⣫⠏
⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠘⠘⠀⢀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⠁⠀⢀⡘⢂⠀⢀⠔⠁⠹⡻⡿⣿⣳⠃
⠀⠀⠀⠀⠀⣿⠟⠀⡀⠐⠳⠀⡠⠋⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣆⠀⣵⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣿⡆⢿⡆⠀⠀⣿⣶⣖⣶⣶⣶⣿⣿⡿⢟⡷⠆
⠀⠀⠀⠀⠀⢻⣿⡃⠹⣿⣜⣄⢊⠙⣿⣿⣿⠿⠿⢋⠔⠁
⠀⠀⠀⠀⠀⠂⠉⠀⠀⠙⢿⣮⣦⣹⣾⣿⣿⡗⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡦⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢿⡌⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠲⣴⣥⣤⣥⣴⣶⣍⢽⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡷⠾⢛⣿⠻⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠉⠉⢛⠁⠚⡟⢻⠽⢛⡟⠻⠟⠟⠛⠌⠛⠩⢯⡀⣥⣀⡰⢞⣫⠿⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣦⣄⠀⠀⠈⠀⠀⠀⠞⠁⠡⢡⡀⠀⠀⠀⢠⠈⢃⡓⠽⣆⣁⣀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣮⣿⣶⣷⣤⣀⣠⣠⣴⣿⣰⣯⣶⡶⢶⡾⣶⠾⠿⠛⠀⠉⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠿⠓⠋⣰⣿⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⢠⠟⠀⠀⢀⣿⠛⠛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡎⠀⡇⠀⠀⠀⠀⢸⣿⣀⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣫⡞⢻⡁⠀⠷⣄⢰⡇⣨⡼⠛⠚⣉⣉⣛⣒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣳⢋⡼⠌⠓⠒⠀⣀⠄⠉⡁⠀⠀⠈⠛⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣳⢃⣾⡿⠒⢪⣦⣭⠥⢖⢋⢖⡵⢂⡀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⢧⠇⢸⡏⠳⣄⣀⡴⠁⠀⠀⠈⠉⠉⣽⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⢰⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡆⢸⡿⠓⠀⠀⠀⠀⣀⣴⣤⣼⡾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⢇⣾⡏⠀⠀⠀⠀⠐⠛⠛⣻⡟⠑⢿⣿⡇⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢠⣿⠇⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠙⠀⣠⣮⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢹⣄⠀⠀⠀⠀⡄⣼⠻⣦⣀⣀⣀⣀⡞⠋⣸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡙⠀⠀⠀⢸⣷⣷⣄⣀⡉⠉⠉⠁⠙⣦⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠶⣄⡀⠘⣿⡏⠉⠛⠛⠛⠛⠳⠀⣸⠀⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠤⠤⠶⠖⠒⠛⠛⢦⡲⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆⠀⠙⠀⠶⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⢠⡄⠀⠀⠀⣿⢻⢻⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢿⠀⠀⢀⠹⣟⠻⣡⣾⣿⣿⣾⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⣿⡄⠄⠸⠀⠙⠛⢻⡿⢻⢿⣿⣿⣿⣩⣍⡙⢻⢷⣦⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠀⡆⠀⣿⣦⠀⠀⠛⠻⣿⡅⢻⣫⠛⣦⡸⣾⣩⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣒⠋⠀⠀⠁⠀⠀⠔⠀⠀⠀⠀⠙⠄⠀⢓⠀⠀⢹⠓⠋⢱⣶⠟⣷⠙⣏⡳⠾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡾⣿⠛⠋⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠈⣄⠉⣨⣿⣇⣼⣿⡜⢻⢛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⣀⠤⢀⣸⣿⠿⠃⠀⣿⢾⣿⡞⡻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠏⣰⠟⠋⣡⠀⢀⠘⠃⢈⣻⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠⠰⡀⠢⡀⠀⠀⠀⠀⠀⠀⠊⡡⠴⠈⠁⢠⣄⣀⠀⠈⠻⠉⠿⣧⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠊⠳⠈⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⢿⣦⣄⡸⡟⠴⠛⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⠀⠤⠙⠷⠀⠐⠂⠈⠙⠋⢛⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢧⡀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠄⣾⠃⠄⡀⠀⠀⠀⠨⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⢀⠛⠄⠀⡒⠀⠀⠠⠼⢃⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠈⠃⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠃⣀⠀⠀⠀⠁⢀⠀⢢⠄⡠⠈⠻⣦⡀⠛⠀⠀⠀⠀⠐⢷⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠘⣪⠄⠁⠈⡁⠂⢀⠀⣹⠀⠀⠐⠃⠀⠀⠀⣶⣀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠈⢱⡟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣦⠄⠈⠃⢾⣿⠂⢠⠀⢀⡀⠀⠀⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⣿⣧⠀⠀⠻⠉⠀⣾⠋⠉⠀⠀⢀⡄⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⢾⡟⠛⠿⢃⣸⣧⣿⡆⠀⠀⡀⠈⢰⠀⠀⠀⠋⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⡀⠀⠀⠀⠀⠄⠀⡴⠃⠀⣤⢀⡠⣤⣼⢢⡙⢿⡱⣿⡷⣤⡈⠀⠀⠀⠀⢀⡬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⡟⢫⠾⠹⡏⠀⠃⠙⠈⠁⠙⠁⠈⠻⠛⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡀⠠⠞⠛⡻⠓⠀⢀⣀⣀⣀⠀⢀⢤⡤⠀⠀⡑⠂⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠎⠁⠀⣴⠿⠛⠉⠀⠀⠀⠀⠁⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃





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

⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠒⠒⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⣀⣀⠀⠀⠐⠒⠀⠉⠉⠉⠉⠉⠉⠒⣢
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⠻⢿⣆⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡍⠒⠤⠌⠧⣀⠀⡔⣩⣭⣭⡙⢶⣤⣤⣴⠶⢦⣤⡤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⠀⣿⣿⣿⣷⡀⣿⣿⣟⣻⣿⣿⣦⡶⣦⣬⣉⠛⠿⣯⣭⣖⡂⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠟⣿⣿⣿⣇⢸⡄⠟⣷⣄⠉⡜⣿⡎⢿⣿⣿⣦⣄⠙⢿⣭⣍⣀⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢀⡾⣹⣿⣿⡄⠛⢆⣈⠉⣡⠾⢹⣴⢸⣏⢹⣿⣿⣷⣵⣮⡻⣧⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠤⣽⣾⣡⣯⣯⣿⣷⡈⣠⣿⡆⠀⠶⢸⣘⠁⡿⢃⣛⢠⣿⣿⣿⣿⣜⢷⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⠶⠶⠶⠶⠦⠝⠛⠻⢿⡶⠖⢁⠄⢠⣼⣿⣿⠂⠀⠀⢩⣿⣿⣧⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣦⣐⠢⣄⠠⡐⢤⡐⠒⠤⢄⡶⢻⣿⠟⠋⠀⠀⣤⠋⣿⠿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡿⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡑⠺⠶⠬⠶⠒⠉⠀⠀⠈⠳⠦⠖⢋⡍⢠⡁⣸⡟⣿⣿⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣛⢡⡿⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣶⠶⡀⢠⣄⣀⠀⠀⠀⠀⠀⠁⢘⢣⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⡄⠘⢮⡹⣿⣽⠒⣦⣄⠀⠀⠉⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣦⣄⡀⢻⢁⣽⣶⣤⡙⢦⣽⡄⡸⣿⣷⡄⠀⠣⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠞⠛⣻⣿⣾⣿⣿⣿⣿⣽⣷⣬⣿⣾⢿⠿⣿⡀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢠⣍⠙⣿⣿⡹⣿⣿⣿⠛⠏⠿⠎⢑⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⣴⡄⣦⠀⠀⢠⣼⣷⣾⣿⡟⠸⠿⣿⣮⣀⣀⡠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⡆⣀⣸⣿⣿⣿⣿⡷⢀⣴⡷⣿⠟⠁⠠⡍⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢳⣦⣴⣿⣿⢃⣼⣿⢿⣿⡟⣿⠁⠹⣫⣿⣯⣶⣶⣾⣿⣿⣿⡟⠧⣄
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠏⣸⣿⡟⣼⡿⣱⡧⠔⢊⡵⣻⣽⣿⣿⣿⣍⡻⣿⣤⡀⠈⠃
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⢀⣶⣿⣿⢿⣿⡿⠁⣴⡿⢋⣼⠟⡱⠃⠀⠀⣩⡾⣻⣿⣿⠟⠋⠈⠁⠉⡽⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣰⣿⢟⣭⣶⣿⣿⣀⡀⠁⣰⡿⠋⠀⠀⠀⢀⡾⢋⣾⠟⠋⠀⠀⠀⠀⠀⠀⠁⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠵⣿⡿⠟⠉⠙⠋⠀⠀⠋⠀⠀⠀⠀⠀⠉⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⣶⡟⣛⡛⣫⡿⡟⠭⣭⣭⣿⣭⣽⣓⡲⢴⣶⣿⡿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⢋⢷⡌⢿⣦⣍⣙⣷⣣⣴⣿⠶⣝⢿⣯⣽⣿⣷⣮⣝⠾⢛⡽⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⢾⣿⡌⢳⠑⡈⠻⠙⠏⠙⠁⠍⠀⡙⠛⠻⢹⣟⣿⣿⣮⡲⣛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣿⣤⣿⣦⡘⢀⣆⣼⠀⠀⠀⠈⠀⢀⡷⡤⠊⣸⢿⣿⣿⣿⣿⣮⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡟⢩⠤⠤⣄⣈⠉⠙⠋⢀⣸⡿⠶⠶⠶⢊⣭⣠⠴⢋⣌⡈⢩⣼⣿⣿⣿⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣦⡘⠒⠒⠀⠋⢙⣻⣶⣦⣤⣤⣴⣮⣅⣐⡐⢁⣼⡿⣷⣶⣽⣽⡿⢿⣷⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣥⡶⠶⢽⣿⣿⡻⣿⡛⠷⠞⠉⠀⠈⠻⣿⣿⣿⡿⢟⣫⣥⣿⡞⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠈⠁⠈⠁⣤⣤⣀⣀⠀⠀⠈⠉⠉⠀⠻⠿⢟⣽⡇⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⡀⠀⠈⠻⣿⣿⡻⣟⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣷⠦⣄⡙⣾⣿⣾⡇⠀⠀⠀⠀⠀⢩⣷⣶⡇⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⠟⠁⠀⠀⠉⠛⣿⣿⣷⠀⠀⠀⠀⢀⣛⣛⣿⢣⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠛⣻⠘⣿⣷⣆⢸⣿⣿⢋⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣶⣶⣶⡞⠋⠀⣿⢩⣿⣶⣶⣿⠟⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣛⣿⣿⡁⠀⢀⣽⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣖⣻⣿⣿⣦⣘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣽⡿⣿⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⢡⡾⣿⠏⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢦⡀⠈⠿⠿⠿⢛⡛⢻⣶⣶⣶⣤⣄⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠈⣑⠂⠨⠁⠠⡿⠟⡛⠛⠿⢿⠿⣿⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠋⠁⡦⡅⢋⠛⠀⠓⡽⡧⣫⣏⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⡤⠴⠄⣀⡠⠂⠀⠀⠀⡆⠀⠀⠀⠙⠘⠷⠺⢤⣄⢀⢋⠀⠛⠹⢮⣟⣿⣿⠉
⠀⠀⠀⠀⠀⠹⢍⡉⠁⢤⡠⠀⠀⠀⠐⠀⠐⠄⠀⠀⢀⣶⠀⠈⠯⣾⠋⠀⠀⠀⢸⢮⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠛⠀⠀⠀⠃⡀⠀⠀⢀⣞⡻⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠰⢄⣀⡀⢛⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠉⠛⢋⡉⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠃⠴⢽⣿⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠈⠓⠢⠀⠀⠀⠀⠀⠠⣽⠗⣰⢾⣿⣿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⡀⠀⠰⡄⢠⣷⡀⠀⣠⡶⡏⣥⣬⣷⣿⠯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣵⣦⣤⣿⣿⣿⣿⣶⠟⠁⣴⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣆⣉⠉⠙⠛⠉⠁⠀⠀⣠⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣨⣿⡿⠋⠀⠀⠉⠑⠒⠒⠒⠲⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢌⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⢀⠀⠀⡄⠀⢸
⠀⠀⠀⠀⡄⠀⠓⠀⢸⠠
⠀⠀⠀⠀⡇⠀⠈⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣷⣦⣀⡀
⠀⠀⠀⠀⣧⠈⠀⢸⣿⠀⠀⢀⡤⠀⣂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⢠⡇⠀⠀⢨⡏⠀⢠⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⢹⠀⠀⠀⠈⡇⠀⠘⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠘⡀⠀⠀⠀⡇⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⠛⠿⠀⠙⠀⠁⠀⣤⠈⠉⢁⣤⡤⢞⠩⢉⠿⢾⣿
⠀⠀⠀⠀⡇⠀⠀⢠⡏⠁⠀⠀⠸⣿⣿⣿⣿⠇⣀⡀⠀⠀⠀⠀⠠⡾⠻⠖⠦⠿⠉⠀⠀⢰⠎⢠⣟⡾
⠀⣀⣀⣀⡀⢀⣀⣸⣃⣇⣀⣄⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠠⣿⣷⣞⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣿⣿⣿⣿⡇⢰⣿⣿⠲⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣍⡀⢁⡿⣿⣶⡶⢒⣶⣶⣖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠒⣂⠠⣶⠄⢒⣲⣛⣴⣾⣿⣿
⠿⠛⠋⣭⣥⣤⡍⠹⠊⠙⠿⠻⡞⠁⠹⠻⡿⠛⠻⠛⠙⠷⠿⠟⠛⠙⠿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿
⣀⣤⣴⡾⣿⣿⣿⣿⣦⣀⡈⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠉⠉⠉⠀⠈⠉⠁
⣿⣿⣿⣿⡹⣿⡆⠈⠻⣿⢿⣆⠠⠀⠀⠀⠐⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⣰⣿⣿⡿⠋
⣿⣿⣿⡿⣷⡘⡟⢶⣆⢘⣿⢿⣷⣦⣄⡐⠤⣌⣷⠀⠀⠀⣴⣤⣤⣤⣤⡀⢄⠀⠀⠀⠀⢀⡀⣠⣾⣷⣿⣿⣿⠟⠀⠀⠐
⡿⠛⠉⢠⢹⣿⣾⣾⡟⢳⣿⠈⢻⣿⣿⣽⢷⣦⠹⣶⣄⠀⣺⣿⣿⣿⣿⣿⣾⣿⣶⡶⣦⡄⠻⠛⠿⠿⠿⠿⠏⡴⠆⣶⡂⠤⠤⢤⣤⣤⣀
⠀⠀⠀⠀⣳⣿⣿⣿⣷⡀⠙⠳⠀⣹⣿⣿⡽⢯⡻⣿⣿⣿⣿⣿⣿⣶⡄⠉⠑⠋⠡⣒⠻⠿⢟⠓⠂⠀⠀⠐⠉⠛⠀⠀⠀⠀⠠⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣤⡀⠲⢸⣿⡿⣶⣦⠉⠀⠙⠛⠿⠿⠿⠟⠛⠃⠀⠀⠀⠈⠑⠦⠦⢶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣽⣧⣿⣭⣤

⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠘⠮⠙⠛⠛⠛⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⢉⠵⠂
⠀⠀⠀⢁⣀⠀⠀⠀⠀⠠⣄⣙⡟⢦⣀⡤⠤⠤⠴⠶⠶⠶⢦⣄⢀⣴⠒⠒⠒⠢⠤⠤⢤⣀⡄⣡⣎⣠⠄⠀⠀⠈⠉⣀⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠁⢿⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢶⠿⠿⠀⠀⠀⠐⠲⠇⠘⢶⡄⠀⢀⡀⠀⢀⡵⠂⠠⠖⠛⠂⠀⠘⢿⠿⡶⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠙⣄⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠂⢱⣼⣦⣤⣤⣴⣦⣄⣰⡀⠀⠀⢺⡿⠀⠀⢀⣆⣠⣴⣦⣤⣤⣴⣥⣮⣤⣿⢢⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⣷⣠⠿⠟⣿⣷⣤⣀⣼⣇⣴⣤⣾⣿⠉⢻⡇⠾⠷⣖⠂⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠋⢩⡿⠿⠿⢿⡟⠻⠉⠹⣿⠟⠛⢻⡿⠿⠛⠻⢯⡙⠉⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠖⡋⠀⠈⡏⠀⠀⠀⢀⣰⣉⣀⠀⠒⠶⢦⡀⠀⠀⠁⠀⠀⣠⡤⠤⠄⠀⣀⣀⡀⠀⠀⠂⢹⠀⠀⠙⠒
⠀⠀⠀⠀⠀⢰⢁⠀⠀⠇⠀⠀⠀⠋⣤⠞⠛⢛⠋⠈⢻⣷⣄⣀⢀⣐⣊⣁⣶⡶⠒⡛⠛⠳⢬⢻⠀⠀⠸⠀⠀⡈⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠁⡀⢠⣼⠇⠀⠘⣿⠿⣿⣿⣿⠿⣿⡿⠀⠰⢧⡄⢀⠀⣧⠀⡗⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡸⡖⠁⢸⣧⡁⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⡇⠈⣆⡿⢸
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠛⠀⢻⠀⢉⡿⣹⠟⠒⠶⠶⣿⣿⣿⣶⣶⡖⣾⡟⣿⡟⠀⣿⠃⠀⠑⡜⣦⣳⢀
⠀⠀⠀⠀⠀⠀⠙⠱⡀⠀⠀⠀⠀⠀⠀⠣⠘⣧⣾⠀⣿⡗⢱⠃⠿⠻⣎⢹⡅⠸⣷⣿⢃⡜⡿⠀⠀⠁⠛⢠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠀⠳⣄⠹⣿⡀⣿⣷⡿⠷⣿⠶⢿⣾⣿⢀⣿⢿⣿⠄⠀⠀⣶⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠄⢀⠘⠳⡜⡓⢿⣏⠀⢀⠀⠀⠀⣹⡿⣾⢃⠞⡃⢠⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠐⠀⠀⢹⡄⢃⣰⠀⢛⣠⢆⡌⢠⡏⢠⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠦⠍⢁⠀⠒⠉⣚⠻⢾⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⢹⠇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣤⠀⠋

⠀⠀⡄
⠀⠀⠀⠀⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠒⠠⢔⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣇⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢁⠠⠀⠤⠤⠤⠤⡀⠀⠄⠀⠤⢈⠀⡄⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠼⣛⡽⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢤⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠀⠈⠁⠀⠀⠀⣼⠶⠿⠿⣾⣧⡀⠀⢸⣦⠀⠷⠿⣏
⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠙⢇⠼⣿⣿⡏⠁⠀⠀⠀⠉⠙⠂⠄⡀
⠀⠀⠀⠀⠀⠘⡁⠀⠀⡰⡏⠀⠀⠀⢀⣾⣋⠄⠀⠀⠀⢻⡉⠻⠙⠦⠀⠀⠀⠀⠀⠀⣀⠀⠑⢤⣀
⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⡷⠀⢸⠀⢸⢋⠴⢶⣦⣸⡄⠁⠀⢀⠀⠀⠉⠉⠢⣄⣠⣸⣯⣤⡈⣱⠋⡆
⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡈⡄⣿⡇⢳⣄⣾⠉⢻⡀⠰⡀⠑⡇⠀⠀⠀⠀⠀⠈⠉⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠘⢾⣿⠊⡦⠩⢅⣒⣟⠷⣶⣤⣹⡄⠀⠀⠀⠀⠀⠀⣠⣼⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⢀⡀⠀⠈⠈⠙⢻⠶⠛⠁⠙⣆⢰⠓⠂⠉⡛⠷⠤⠤⠔⠚⠉⣙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣡⣠⣀⣀⢹⡀⠠⠖⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣀⡴⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⢦⡀⢄⣠⡶⠒⠛⠙⢷⣚⠛⠛⠒⣋⡿⠗⠒⡲⢤⣆⢠⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢷⣄⠉⠲⢿⡦⠀⠀⠉⠛⠶⠂⠀⠀⠶⠚⠉⠀⠀⠐⣿⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡌⣻⣷⣄⠔⠀⠀⠀⠐⢦⡍⠀⠀⠀⠉⠀⢀⡀⠀⠀⠲⢄⠀⣿⣧⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⣿⣧⣤⣿⠁⢲⣦⠴⣿⢶⣄⠁⠀⠀⢰⠀⢸⢋⣴⡶⢦⡴⠀⣦⣙⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢤⣬⢿⣿⠀⠋⢹⠀⡈⠃⠓⢿⠾⢻⡆⠀⠁⠀⠈⠱⠻⢬⠷⡺⠁⠀⠈⣤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠶⢦⡞⣰⡷⠈⣲⠟⢤⡀⢠⠤⣄⣀⣠⣄⠠⠀⠙⢄⠰⣧⢫⡁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿⠀⢹⢛⣿⢻⣶⣢⠈⠚⠲⡆⢠⢰⠄⠀⢴⣾⣿⠯⢻⡉⣿⡀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⡿⠀⢘⡇⠘⢸⣿⣅⣁⡀⠀⠙⣻⠃⢀⣀⣠⣝⠛⠋⠯⢴⣿⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠂⠹⣦⣾⣁⣀⣼⡿⣿⠉⣙⣛⡛⢛⣛⣋⠙⣿⡏⠙⢻⣶⣶⠏⢩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠲⢾⠉⣹⠏⡀⣻⡝⡆⡿⠿⢿⡿⠟⣿⠸⣿⣀⢴⣉⢉⡤⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣠⡙⠉⣷⢻⡷⣄⣀⣁⣴⣿⢰⡇⢁⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⡟⣾⡇⣛⣛⣛⣃⣿⢷⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢘⠬⣛⠭⠀⠭⢝⣫⡎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⡤⠆⠀⠀⠲⠤⠹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣁⡀⣀⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠛⠹⢧⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠙⠻⣷⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠐⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣿⡿⠿⣷⣦⣄
⠀⠀⠹⣿⡌⠙⠻⢿⣶⣤⡀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠹⣿⣄⠀⠀⠈⠙⢿⣷⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢀⣀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡷⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⢀⣾⡟⠁⢀⣠⣴⣾⠟⠉⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⣾⣿⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣠⣴⣾⠿⠋⠁
⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣃⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⣿⢷⠲⢤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢻⣮⠭⠖⣚⣉⣭⣤⣤⣶⣶⣦⣤⣭⣍⣛⣛⣲⣶⣴⣒⠦⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⢀⣰⢾⡷⢚⣩⣴⣾⣿⣿⣿⣿⣿⢿⢿⡿⡿⠟⣫⡿⢻⣿⣿⣏⠾⢻⣿⠻⢡⣶⣶⡌⢂
⠀⠀⠀⠀⠀⣨⢟⣥⣾⣿⣿⠿⠟⠛⠙⠈⠁⠁⠀⠀⡴⠃⠀⠀⠔⠁⠀⠈⠛⢳⠿⠃⠀⣿⣿⣯⣿⠈
⠀⣀⡤⢒⣟⣵⣿⠿⠛⠉⠀⡀⢠⠀⣴⢠⣇⢠⡇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⠇
⢺⣿⣭⣿⡿⠛⠀⢠⡀⣶⡸⣷⢿⡇⢿⠘⡿⢸⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⢾⣿⣿⣿⡿⡰
⣸⢧⣿⠋⢡⠀⣧⢸⣿⠻⣇⠝⠚⠛⠿⢶⣶⣮⣦⣈⠀⠀⠀⢀⠐⢄⠀⢰⠀⢇⣀⡼⢸⣿⣿⣿⡁⡅
⢸⣾⠃⢠⣸⣇⢻⠀⠙⠀⠉⠢⣀⠀⠈⠉⢿⣿⡳⣤⠀⠀⠀⠀⠁⠒⠒⠰⢧⣀⣁⡼⠎⠛⠈⣉⠱⠧
⢈⡇⣸⡾⡇⠉⠀⠀⠀⠀⣀⢠⠸⠉⠒⠛⠉⠀⠀⠀⠤⢤⡄⠀⢀⠀⠀⠀⠀⠉⣙⣛⣛⣒⣒⣃⣀⣀
⠸⣷⠏⠁⠀⠀⠄⠐⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣬⣿⢷⣴⣀⢀⡠⠚⠉
⠀⡟⠀⡀⠔⠁⠀⠀⠀⢀⣀⣠⣤⣴⣶⡾⠋⢙⣿⣿⣿⣿⣶⠈
⠀⠰⡰⠁⠀⠀⣠⠶⢿⣿⣿⣿⣅⣠⢊⢠⢶⣿⡟⣿⣿⣿⣿
⠀⠀⠀⢄⣰⣿⠿⡷⣤⢀⠉⡛⠿⢿⣿⣿⣿⣿⣧⠘⣿⣿⣿
⠀⠀⠀⠀⠁⠻⢄⡀⠉⠙⠙⠓⠷⡦⣦⣤⣤⣤⣼⠀⣿⣿⣿⣧
⠀⠀⠀⠀⢢⡠⠀⠀⠉⠑⠒⠲⠤⢄⣀⠈⠉⢉⡟⢠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣄⣾⣿⣿⣿⣿⡇
⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⢿⣿⡿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⢁⡴⠋⢁⡤⠖⠋⢙⣻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣛⠽⠋⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡞⣁⢀⡞⢀⡴⠋⢀⡴⠚⢉⣡⠽⠿⢷⣶⡄⠀⠀⣰⡿⠋⡴⠃⠀⣠⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣴⣧⣾⣠⣞⣀⡴⠋⣠⠖⠋⣀⡤⢶⣾⠟⠁⠀⣼⡟⢀⡾⠁⠀⣼⠁⠐⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠿⠛⠛⠉⠉⠉⠭⠭⣍⣙⣛⠻⢷⣾⣥⣴⠋⣁⣠⣿⡇⠀⠀⣼⡟⢀⡾⠁⠀⣼⠃⠀⡴⠃⡴⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⠀⠀⠓⠲⣤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣉⠻⢿⣧⣴⡿⠁⠀⢰⡿⢀⡾⠁⢀⣾⠃⢀⡾⠁⠈⣴⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⢿⣧⡀⠀⣿⣧⡾⠁⣠⡿⠁⣠⡿⠁⢀⣾⠏
⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠙⢷⣾⣿⣋⣤⡾⢋⣤⡾⠟⣀⢀⣾⠋
⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⣼⣿⡋⣿⠀⠀⢧⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⣛⣩⡴⠞⠁⣾⠏
⠀⠀⠀⠸⣿⠹⣷⣤⡄⠀⠀⠀⠘⠻⠿⠋⠀⠀⢸⡄⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠥⠤⠤⠄⢸⡿
⠀⠀⠀⠀⠻⣷⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠬⣉⡓⠒⠒⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣾⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⢿⣷⣦⣄⠙⠲⣤⡀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣤⣤⣤⠀⠘⠁⠀⠀⣠⣾⣿⣟⢮⣙⠻⣦⡀⠀⠀⣠⣾⡟⢷⣦⡀⠀⢻⣿⣝⠻⣦⣄⠙⠦⡌⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣤⣄⣤⣴⡾⠛⠁⠈⠻⣶⣬⣳⣾⣿⣴⣿⠛⢧⣉⠳⢬⡻⣷⡄⢻⣿⢷⣄⠙⢷⣄⠈⠂⠹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⢿⣿⡿⢿⡛⠻⣟⠻⣿⠳⣤⣈⣷⣴⣿⠿⠃⠀⢿⣧⠹⣧⡀⠙⣷⡀⠀⠠⣙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⡳⣄⠙⢦⡈⠳⡙⣷⣾⠟⠋⠁⠀⠀⠀⠀⠀⢿⣆⠈⢿⡄⠈⢿⡄⠰⣌⠙⣮⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢻⡿⣷⡈⠳⣄⠙⢦⡙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⢿⡄⠀⠻⣄⢈⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠳⡜⣿⣦⡈⠳⣄⠙⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢻⣄⠀⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣜⢮⣿⣿⡲⣌⠳⡌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠙⢦⡀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠙⢿⣎⠳⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠟

⠀⠀⠀⣴⢖⣛⡓⠶⠶⠞⣛⣛⠛⠛⢛⣛⠛⠛⠛⠶⠦⣤⣤⠶⣾⣿⣦⣤⣤⣤⡄
⠀⠀⣼⢧⡟⣫⣿⣷⣷⣾⣿⣿⣿⣧⣤⣤⢩⣝⡛⢿⣶⣶⣶⣯⣽⣛⡴⣾⣿⠿⣤⣤⣤⡀
⠀⠀⣿⣼⣾⣿⣿⣿⣿⣿⣻⣟⠛⢛⣿⣿⣼⣿⣿⣷⣜⣷⣭⣭⡹⣿⠿⣷⣮⣟⠿⣿⣿⣥⣀⡀
⠀⠀⣿⣿⡟⣿⣿⣿⣿⣿⡇⢹⣷⠰⣬⣭⣍⠙⢿⣿⡟⣿⣿⣿⣿⣧⣳⡈⠙⠟⢿⣮⡓⢿⣿⣧⡀
⠀⠀⢹⣿⣧⣿⣿⣿⣿⡿⣇⣄⣿⣧⠀⠙⢿⢷⣾⣿⣿⣿⠿⣿⣿⣿⣿⣷⣶⡀⠈⢹⢿⣦⡹⣿⣧⡀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⠞⢙⡇⢿⢿⣇⠀⠈⠀⠙⠿⢹⣿⠀⢻⣿⣿⣻⣿⣟⣷⣿⡆⡈⠘⢻⣮⢿⡀
⠀⢠⣿⡿⣿⣿⣿⣟⣡⣴⠛⠉⠁⢛⣛⣿⣦⣤⣀⣐⢭⡿⠀⠈⢈⡵⢣⡏⠟⢹⣷⢿⣿⣰⡆⢹⣿⣷
⣴⢛⣿⣷⣿⣿⣛⣭⣭⣴⣖⣒⣋⣭⣭⠤⣿⡅⠲⠶⢎⣀⣀⡴⢋⣴⣿⣶⣶⣴⣗⣾⣿⠝⣛⠶⣽⣿⣇
⠙⠿⠿⠿⢿⣯⣉⠍⠉⠁⠤⠤⣤⣤⣤⣴⠿⡿⡶⣄⣀⣠⣿⣵⣿⠁⠉⣹⣿⣿⣿⣿⣿⣿⣗⣒⠈⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠛⢷⣿⣿⣶⣶⣾⣛⡤⣞⢿⣦⣤⣤⣶⣿⠟⠁⠀⢹⣿⣿⣿⣿⣿⡿⢿⣟⠊⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠛⢿⣿⣿⣿⣿⣿⡉⠑⠊⠀⠀⠀⠀⠛⢿⣟⣿⣿⣟⣧⢶⣶⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣝⡪⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⣟⠁⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣷⣯⣷⣶⣤⣤⣄⣠⣔⣶⣶⣤⣙⠻⣿⣽⣧⢘⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⡾⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣿⠛⣿⣿⡿⠿⠻⢿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⠷⣉⠈⢀⠟⠋⠑⢲⣶⣤⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢶⡶⣶⣶⣶⣶⠞⢩⡏⢋⣴⣛⠀⠁⠀⠐⠁⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠨⣷⣿⣿⣿⣿⢡⣾⡟⢿⠛⠉⠈⠀⠀⠀⠀⠀⣀⡴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⣯⣽⣿⠇⠾⠁⠀⠀⢀⣠⣤⣶⣶⣶⣶⣿⣿⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡟⣿⣿⠀⢀⣠⣤⣶⣿⡿⡿⣻⣽⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⡟⢹⣿⠋⣻⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢹⡿⣿⣟⣿⡿⠀⠀⢹⡆⠘⠛⠛⠋⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠿⣿⣾⡿⠁⠀⠀⠈⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣃⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣫⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⠋

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⠉⠓⠒⠂⠀⠉⠉⣁⣀⡄⣴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠢⠴⠤⣠⣃⡼⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣙⣄⡤⠦⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢄⠀⠙⡙⠀⠀⠈⠢⠈⢄⠠⠄⡠⠀⠖⠁⠀⠠⠋⠋⠀⣠⣴⠖⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⢀⣣⣴⣠⣄⣠⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣄⠀⢠⡀⠘⠛⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠛⢛⣦⠀⠀⠹⡏⠀⠀⣴⡛⠛⠀⠀⢿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠍⠁⢺⣿⠓⠦⠴⠳⡖⠙⣿⡷⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠙⢧⠈⠀⠀⠀⣽⠋⠷⠤⡴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡷⢶⣴⡐⠈⠀⠀⠀⢸⠀⠃⠀⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠋⡟⢁⢈⡀⠀⠀⢘⡄⠩⣄⡐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡶⡻⠁⡬⠥⢤⣀⣘⡡⠀⡾⢵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⢄⡀⠀⠀⠒⣉⢵⡱⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⢑⠚⡛⡋⠁⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠤⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠟⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⣤⣶⣤⡝⣶⠾⢻⣿⣾⠛⠛⠛⣶⡖⢻⠟⣳⣣⣄⣀⡀⠀⢠⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢫⣽⣤⣤⣶⡿⠁⠉⢹⣿⣿⡄⠀⢸⡻⡄⠀⠤⢸⠃⠀⣰⣱⣿⡟⠉⠻⣷⡟⣌
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣮⢻⡄⢹⡿⠆⠀⣙⣿⣿⣿⣦⣼⣿⣿⣦⣤⣿⣤⣼⡿⣿⣿⠀⠀⠀⣼⣳⠟⠛⢽⡆
⠀⠀⠀⠀⠀⢀⡞⠃⠐⠻⢿⣿⣤⣷⣿⣶⡿⠿⣻⣿⣿⣟⣿⣹⡟⣩⠻⣿⣿⠿⣿⣶⣯⣤⣿⣿⡏⠀⣠⣼⡿⢶⣦
⠀⠀⠀⠀⢀⣾⣧⣤⣄⣰⣾⣿⣿⣿⣹⢛⠆⢾⠿⠛⣷⣿⡳⣯⣽⠾⡇⠙⠝⠺⣍⣼⣯⣿⣿⣿⣷⢣⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢰⣯⠈⠃⠈⠉⣿⣿⣟⣿⣷⡎⢙⠀⡻⢔⣒⣥⢀⣤⣠⠖⢠⠞⡙⢫⣷⣽⠻⠿⣿⣿⡿⣿⣶⠶⠶⠖⠀⢿⣻⡄
⠀⠀⠀⢸⣍⣠⣤⣴⣿⣿⣿⣍⣛⣋⠈⣹⣂⡝⠳⠬⡟⠀⠁⠪⠂⠀⠔⠤⢾⠣⣯⡛⣽⣛⣮⣍⣿⣿⣤⣤⣤⠤⠼⠿⣧⣴⣦
⠀⠀⠀⣈⣿⣿⣍⣤⣤⣤⣽⣿⣽⣾⡆⡰⢋⡴⡇⢠⣦⣽⢛⢺⡛⢲⠄⠐⣋⣬⠋⢿⠾⢟⣿⣿⡿⢻⣭⣿⣭⣿⣿⣿⡍⠉⠁
⠀⠀⠀⠟⠻⣿⠿⠛⢻⣿⣿⣿⣿⣿⣷⣧⣹⣷⣶⣾⢋⢷⠸⡾⠛⡞⣷⣻⣿⣍⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣛⡛⠛⠢⣄⠀⢢⡀
⠀⠀⠀⠀⠀⠹⣅⣴⠾⠁⠒⠤⣾⠛⠻⡿⢝⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿⠿⢿⣿⠿⢿⠏⠙⣿⣿⣻⣿⣯⡝⢿⢶⣄⡈⢳⣄⠑⡆
⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⣴⡎⠀⠀⠀⠀⢹⣿⠃⠋⠈⠛⢹⡏⠉⠉⠛⠀⠀⠀⠀⠓⣈⡀⢀⣼⢻⣿⡿⠉⡧⢸⠇⣿⢻⡄⠹⣆⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣴⣿⣷⡎⠀⢀⠄⣿⣿⠀⠀⠀⠀⢀⡴⠀⠀⠰⣦⡀⣀⣀⣤⢹⣷⣿⣷⠿⢉⡿⠛⠇⡟⢀⣿⢨⡇⠀⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠉⠈⠛⠷⠶⠾⢾⡿⢿⣧⣴⣦⣶⣾⣧⣾⣶⣶⣿⠻⠿⢿⣿⣿⣡⠄⢀⡄⣰⡿⢣⡾⠁⢸⠇⢀⡇⠀⣿⠁⣏
⠀⠀⠀⠀⠀⠀⠀⡸⠁⣆⠀⠀⠀⡐⠀⠀⢉⡓⠀⠈⠙⠻⢹⠈⠉⠛⠋⠀⠉⠀⠀⣘⠿⠉⠉⠞⣹⡿⠃⣰⡟⢁⡴⠃⣸⠏⠀⣴⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠸⣿⠀⠀⠘⢁⡐⠀⠀⠀⠈⠉⡥⠄⠸⠌⠷⠈⣃⠀⢁⣠⡄⡀⢰⡀⡿⢢⣿⣡⣾⣡⣴⠏⢰⣿⠇⢀⡼⠛⢀⡎
⠀⠀⠀⠀⠀⠀⠀⣿⡷⠟⠁⠐⠦⠘⠃⢸⡦⣤⣸⣦⠀⠀⣄⣰⣠⣾⠉⠀⠙⢿⠿⠁⠺⠁⠀⠿⡿⠛⠻⣿⡇⠀⡼⠃⣴⠋⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⣄⣤⣘⣈⣀⡀⠐⣁⣀⠀⠾⠟⠁⠀⠁⠄⣠⣤⣤⡀⠤⠀⠀⠄⠒⣤⣼⠃⢿⠇⢠⡇⢸⡇⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⠻⢀⣷⠀⢀⣀⣠⣬⣴⣶⣶⣶⣾⣿⣿⣿⠂⠘⣦⠀⣿⠈⢷⣈⡳⣶
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⡿⠀⠈⠳⠞⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠛⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠆⠀⠹⢿⣿⣿⣿⣿⣿⡿⣿⡃⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠻⠿⠂⠀⠙⠋⠉⣁⠀⠀⠠⠈⠀⡆⣠⠀⠀⠀⠉⠉⡉⠀⢀⣙⠛⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⡷⣾⣷⣿⡇⠀⠰⡾⢯⣾⣿⣿⣿⣶⡲⠄⠻⠃⣤⣴⣿⡟⣿⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣿⡇⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⡆⠶⠈⠻⣿⣿⠁⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠛⣏⠀⠟⠾⡷⣭⠉⣿⢉⣯⣽⣿⣿⡧⡤⠴⡾⢻⡿⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢻⣾⣤⣠⡇⢹⠀⡿⠀⡷⠈⡇⢸⡇⡗⣃⣠⠾⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣿⡌⠉⢻⡾⠒⡧⠠⡿⢰⡗⠘⡇⢻⠈⣤⡟⠘⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣬⡙⠻⢮⣇⢀⣇⠀⣇⢀⣧⣀⡧⠸⠿⢋⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠂⡍⠈⢋⠿⠉⠨⠉⠉⠃⢀⠀⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⣉⢀⠀⠀⣉⣀⠁⢉⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⡉⠉⠉⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠂⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⡀⣀⣀⣀⣀⣈⣁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣄⣀⣈⣠⣤⠤⠀⠌⠥⠴⣺⠁⠀⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠀⠀⠐⠂⠀⠣⠁⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠰⠤⣀⠀⠀⢀⡴⠁⠀⠀⠀⣀⠤⠊⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠖⠉⣀⡤⠒⠈⢁⡠⢀⠔⠀⠀⠉⠉⡁⠀⠀⠁⢉⠁⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡴⠃⠀⡠⠊⠁⠀⢀⠔⠉⣠⠃⠀⠀⠀⠄⢠⠀⠸⠀⠀⠸⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡑⠀⣠⠊⢀⠂⠀⡰⠃⠀⡴⠁⠀⡜⠀⠀⠀⡎⢠⠃⡇⠀⢠⡙⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡔⠁⠀⠀⢀⠜⠁⡠⠊⠀⢰⢰⠁⠀⠀⡜⠀⡜⠀⣻⠀⢸⢡⠸⡄⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⠀⢀⡠⠚⠁⠀⠀⡠⠨⠀⠸⢸⠀⠀⡜⠀⡜⠀⢈⠉⡆⢸⢸⠀⢣⠀⠸⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⠁⡰⠁⢀⡰⠋⡰⠁⠀⡇⡆⠀⢈⠀⡜⠀⠀⢸⠀⡇⠸⢸⠀⠸⡀⢠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠜⣁⠔⠉⢀⠜⠁⢀⠜⢰⠁⠄⣡⠞⠀⡀⢀⣾⠀⡇⡇⢸⠀⠀⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⠔⢃⠠⠔⠁⣠⠃⡠⠊⠁⠀⡜⠀⡜⢸⢠⢡⠁⡌⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡇⢰⠁⠀⠀⠌⠀⡼⠀⢸⠀⠆⢠⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠐⠀⠘⠀⠀⡆⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⡀⠀⠀⠀⠀⠰⠁⢀⠊



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡔⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣤⠀⠀⢀⣀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⢁⣀⣠⣤⣤⣤⣤⣀⣁⡄⠒⠈⠛⠂⠀⠄⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠉⠉⠉⠀⠀⠈⠉⠉⠛⠷⢦⣄⠀⠶⠀⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠸⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠈⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⢀⡴⠒⢤⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠓⠀⠀⠀⠀⠀⠀⣜⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡇⢩⠀⠀⠀⠗⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢥⡀⠀⠀⠀⡀⠀⠀⠈⠀⢀⠀⢠⠄⠒⠈⠁⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠳⡀⠠⠤⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⡤⠀⠦⠀⠤⠀⠴⠆⠀⠀⠀⠁⠀⠀⢄⡀⠀⠀⠈⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠡⠋⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠀⢠⠎⠀⣠⡀⠀⠀⠀⢀⠠⠀⠀⠖⠀⠀⠀⠀⣀⡤⢒⡀⠀⠀⠀⠐⢣⡀
⠀⠀⠀⢣⠄⠀⠉⠁⠀⠀⠀⠘⠀⠀⠈⠀⢀⣠⠔⠛⠀⠀⢈⣉⡀⠀⠀⠀⠀⠈⠓⠤⣀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄⠐⠚⠉⠉⠰⠊⠀⠀⠀⠈⠉⠙⠛⠶⣦⣀⠀⠀⠀⠉⠙⠒⠂⠉⠀⠐⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⡼
⠀⠀⠀⠀⢢⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⢀⣀⣄⣀⣀⣈⣀⣀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠞⠛⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠉⠉⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠐⠒⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⢊⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠒⠛⠉⠁⠉⠉⠈⠁⠐⠂⠒⠂⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠐⣤⢀⠀⠀⠈⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡀⢀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠘⠂⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⣶⣽⠀⠀⠉⠲⠄⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⢑⣆⣠⣬⣟⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠏⠑⠒⠦⢤⠉⢉⠤⢶⣶⠃⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠇
⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⡿⣛⣛⡿⠿⠿⠿⠿⠿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣽⣿⣶⣶⣾⣶⣶⣾⣶⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⣶⣶⠶⣶⣶⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣿⣿⢶⣿⣿⢶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠾⣿⣿⠾⠿⠛⠁⠀⠀⠀⠀⣴⢶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⡿⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡆⠀⠀⠉⠀⠀⣼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡌⠷⣶⣴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣩⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡻⠿⠿⠿⢛⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⣿⣿⡿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣿⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⠛⣛⡃⠸⠿⠿⣿⣿⡇⣿⣿⣿⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⢁⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠸⣿⣿⢀⣾⣶⣶⣤⣤⣍⣿⣿⠀⠹⡆⠀⠀⠀⢀⡀⢠⣶⣿⣿⣿⠿⠟⠋⣴⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⣤⣈⠛⠛⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⠌⠛⠋⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣤⣶⣾⣿⣷⣶⣦⣭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⠿⠆⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠇⠻⠿⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠶⠦⠄⠉⢙⣳⣤⣀⣀⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⢛⣣⡤⠶⣚⡛⠉⢩⡄⠀⣀⠀⠶⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠞⠋⠁⠤⢀⣈⠀⠠⠄⠀⠀⠉⠀⠀⠈⠀⡴⠁⠀⡴⣢⡒⡄⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢫⠀⠘⠃⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⠾⠚⢁⣠⡴⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠾⣿⣿⠤⠄⣲
⠀⠀⠀⠀⠀⠀⢀⣤⣾⢃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣛⣋⠙⠓⠒⠄⠀⠀⠙⠄⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⢰⢏⢠⡇⠈⢀⣀⡤⠀⠀⠀⠀⢀⣀⣀⣼⣯⡡⠄⠀⠀⢀⣄⣀⣀⣀⣀⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⢸⣧⢸⠇⢠⡯⢒⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠙⠺⠶⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢻⡄⠀⠛⠗⠋⣰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠇⢸⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⠏⠀⠐⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡶⠖⠊⠁⢀⠈⣦⠞⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠐⡄⠘⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢸⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡼⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠤⡤⠾⢃⠒⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠈⠋⣠⠖⠀⣠⡴⠶⢊⠀⠀⠤⢤⣤⣥⣦⡀
⠀⠀⠀⢀⡠⣄⡀⠀⠀⢀⡇⠀⢸⠁⠔⡉⠀⡴⠞⢋⣀⣤⡴⠦⠄⠤⡴⢮⣆
⠀⠀⣀⣻⠷⢿⡷⠀⡄⢸⡇⠘⣄⠀⢰⠇⠀⠇⡴⢋⡀⠀⠀⠠⠴⠀⠀⠀⣉⠡⡀
⠀⠀⠋⡝⠶⡴⢶⣏⣤⣾⡇⡀⠈⣧⠸⠀⠀⠈⣰⠋⣀⠴⠶⠶⠶⠶⠐⢋⣉⣤⠐⠄
⠀⠀⠈⠛⠛⠛⢛⣻⣿⠋⢱⡘⣄⡈⠛⣆⠀⢠⠇⠀⢁⡔⢀⣀⣹⡒⠣⡈⠋⣁⡁⠈
⠀⠀⠀⠀⠀⠀⢏⠹⠌⠣⠓⣧⣬⣽⡿⠋⠀⠸⡀⢰⠋⠰⠃⡴⠛⠛⠖⣀⠈⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⡳⢷⣶⠋⠋⠉⠀⠀⠀⠀⠀⠁⡞⠠⠁⢇⢿⠏⣠⠭⠯⠽⣅⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢹⡆⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⢀⢀⣀⡙⠛⠁⠁⢀⠙⣦⠻
⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣬⡆⠀⠀⠀⠀⠀⠈⠀⢠⢀⠀⢧⢨⡈⢀⠅⠀⠀⠍⠋⢨⣸⡆⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣷⡼⠠⠀⠈⣯⡃⠀⠠⠀⠄⠀⠀⢣⡏⣀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠳⣍⡬⠁⢹⡠⢴⠖⡟⡇⠡⠖⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠻⠿⠆⠀⣔⣀⣰⡴⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠶⠶⠀⡄⠀⠀⠀⠀⠀⣠⡞⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣀⡿⠁⠛⠉⠙⠉⢻⡉⢀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠠⣈⠛⠀⠀⠀⠀⠀⣀⣾⠟⣉⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠛⠀⠀⠀⢀⡀⠈⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢄⠀⠀⠉⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣦⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠂⠀⠰⠆⢠⣩⣅⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠄⣀⣀⠀⠁⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣿⠋⢀⣴⡄⠠⣀⡀⣀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠏⠒⠛⠉⠀⠀⠀⠀⣉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢿⡄⠀⠀⠐⣦⣿⣧⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠋⠓⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠹⣿⣿⠀⠀⠀⢀⣠⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⡀⠀⠉⠁⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠁⠙⠷⣦⣄⣀⡀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⠕⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⡋⢉⢿⣿⣦⢠⠀⢀⡀⣰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣽⣿⣷⣶⣦⠿⣿⡀⠻⠟⠋⠙⣿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣫⣾⣿⡍⣝⡷⡄⠀⠀⠀⣜⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⣯⣼⣿⣿⠟⠍⢿⣯⣷⣷⠢⣄⣤⣧⣄⣤⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⢿⣿⣿⡓⣤⣼⣟⣿⣅⣤⣿⣩⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠲⣿⢿⣿⣦⢬⣿⣍⡻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢞⣿⡿⠀⠀⠀⠉⣍⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠽⣷⣦⣤⣾⣿⣿⣷⣬⣶⣆⣺⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡟⠛⢹⡿⣏⠀⠈⠙⣿⣿⣿⣿⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⣾⣤⠶⠛⠁⠨⡟⢦⣄⣈⣻⣿⣿⣿⡇⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠦⣄⠱⣌⣠⣿⠿⢿⡟⢿⣿⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠒⠰⠿⠷⠿⠟⢤⣤⣶⣼⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⡻⠮⣅⣈⣀⣠⣤⣴⣶⡾⠛⠿⠅⠉⣛⣿⣿⡟⠀⠀⠀⠀⠸⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⠉⠁⠀⠀⠀⠀⠉⠙⢻⣷⣦⣌⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣽⣶⣶⣀⣐⣨⣤⣶⣿⣿⡿⠋⣿⣿⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣿⡿⠁⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠎
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣻⠋
⣀⣤⢤⣶⣾⡿⠿⢻⡿⢋⣭⠉⠁⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠿⣶⣦⣤⣤⣤⣴⣿⣿⡽⠃
⠻⣿⠀⠫⢥⣀⠒⠛⠳⣿⣿⡀⠀⠀⠀⠌⠉⠁⠃⠀⠁⠀⠁⠁⠉⠉⠁⠀⠄⠀⠠⠀⠤⠰⠨⠭⠭⠭⣼⣿⣏
⠀⠈⠓⠦⣠⠒⠒⠾⡶⠠⠒⠂⠀⠠⡄⠀⡀⠀⢂⠉⢀⣀⣀⡄⠀⣀⣀⠀⠀⠈⠀⢀⣀⡤⣶⠟⠛⠓⢾⣿⣿⣦
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣴⣶⣶⣤⣤⣤⣤⣤⣄⣤⣤⣤⠤⠤⠤⠤⠤⠤⣶⣖⣚⣩⣵⠞⠁⠀⠀⠀⠀⠙⢮⣋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡏⠀⠀⠀⠀⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠻⠇⣰⠄⠀⣄⢠⡖⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢳⠀⣦⡿⠃⠀⠃⣼⡟⠿⣣⡀⠹⠃⢰⡇⢸⡧⠟⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⠀⠀⠀⠹⣗⣴⣷⣤⢉⣹⢷⡉⠁⣄⣲⠀⢹⠛⠃⣠⡅⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⣼⣦⣤⡉⣽⡷⣴⠚⠛⡄⠟⣂⠀⣽⢿⣷⢣⣌⣵⠄⢠⡆⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠀⠐⠛⠉⣀⠀⠈⠑⢤⡾⣿⠛⣷⡏⢡⠐⣮⣯⡄⠈⠋⠂⠁⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠳⣤⡀⡈⢰⣿⡇⠀⠰⠾⣗⠉⣶⠈⢻⡷⠀⠀⠠⣿⣧⠆⢠⣶⡞⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢻⡀⠀⣣⣤⣹⡆⠒⠀⠒⢹⣆⣿⡷⢸⡁⠐⠒⢲⣉⡳⣄⣴⡀⠿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣑⣾⠋⣿⠟⠙⣷⣾⠶⣬⣅⠛⣡⣤⡸⢧⡾⠻⣿⡷⠈⠿⢙⣶⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⢾⣯⣝⣻⣿⡖⣶⠿⣶⠶⠿⢿⡷⠿⢿⡇⠚⡷⠚⣻⣟⣻⡿⠿⠽⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⢠⡄⡄⠀⠉⠻⠉⠉⡽⢻⡏⠛⠛⠟⠛⠋⠉⠩⠟⢰⡄⠁⡤⠀⠶⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⢀⣤⠀⠀⠐⠄⢀⢀⠁⠈⠀⢀⠀⠘⠁⠀⡄⢲⣄⣁⠘⠀⠀⢠⣴⠀⠚⣧⡀
⠀⠀⠀⢠⡀⠀⠀⡰⢟⣼⣇⠀⠛⣻⣦⡄⠀⡀⠆⠘⠀⠃⢰⡏⡤⠁⣀⠀⠀⠃⢋⠉⠀⠀⣀⡀⠹⠀⠆⣽⠟⣄⠀⠀⢶
⠀⠀⠀⠈⠉⢂⣼⢣⡶⢹⣿⠀⠀⠀⢀⣠⣥⣤⣦⣶⣤⣤⠀⠀⠀⠀⠑⢲⣦⣤⣾⣤⣤⣤⣬⡂⠓⢰⣄⡟⢾⣉⣷⡀
⠀⠀⠀⠀⠀⡞⢣⡞⠡⢈⣿⡇⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⢰⣿⠇⠈⢻⡈⣵⡀
⠀⠀⠀⠀⡘⠣⡟⠀⠀⠀⢹⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣦⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡈⢿⠀⠀⠀⢿⡈⣧
⠀⠀⠀⢠⠓⡾⠘⣷⡄⠀⣸⠁⠸⣿⣿⣿⣿⣿⣿⡼⣿⣿⠇⣾⣿⣇⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⢃⠀⣰⡿⢃⠙⡆
⠀⠀⠀⡻⠰⠇⠀⣿⠛⣄⡋⠀⠀⠙⠛⠻⠿⣿⢻⡟⢻⠁⣼⢿⡛⣿⣇⡀⢘⣛⣙⣛⡙⠛⣉⠈⣤⠠⣶⣟⠾⢻⡇⠸⠈⢣
⠀⠀⠀⡇⣻⠀⠀⢿⣄⠘⡳⡀⠘⠋⢀⠀⠀⠲⣾⣿⡇⢠⣿⣿⡧⣾⣽⡅⢿⡿⠛⠋⠉⢀⠉⠆⠈⣠⣟⠞⣴⠟⠃⢀⡇⣸⡀⢀⡀
⠀⠰⠶⡏⣿⠀⠀⠀⠙⡗⢱⠹⣆⠀⠘⠃⠠⣄⠈⠟⢃⣾⣿⣿⡗⢿⣿⡃⠠⠅⠀⣤⢴⡇⠀⠀⣰⣟⠏⣠⠏⠀⠀⠀⡇⢸⠁⠈⠁
⠀⠀⠀⣇⣸⡀⠀⠀⠀⢸⠀⢷⡘⠷⣤⣦⢤⣟⣶⠀⠈⠐⠍⣿⠾⣿⡟⠁⠠⢶⣦⢘⣿⣿⣷⡺⢡⡏⠀⡟⠀⠀⠀⢠⠇⣼
⠀⠀⠀⢻⣈⡇⠀⠀⠀⠈⡌⠈⠧⠀⠴⡘⢺⣿⣿⣇⠀⣆⣶⣸⡇⡌⠉⢀⣠⣼⣿⣸⣿⠃⢳⠃⣦⠈⣨⠃⠀⠀⠀⣼⠰⡇
⠀⠀⠀⠈⢇⢼⡄⠀⠀⠀⠃⠠⠘⣇⠀⢡⠺⣿⠉⣻⣿⢸⠟⣿⠙⣇⣿⣾⡇⢸⠀⣿⡿⢸⡇⢰⡇⠀⣿⠀⠀⠀⢰⠧⣾
⠀⠀⠀⠀⠘⣤⢳⡀⠀⡠⢺⠀⠂⠙⡄⠀⡄⢹⡀⠹⣿⣼⣄⣷⢠⡿⣿⠿⣿⡄⠐⣿⠇⢸⠁⡏⠀⢰⡏⠐⠄⢠⡟⣶⠃
⠀⠀⠀⠀⠀⠘⣤⣿⡌⠀⠘⡄⠐⠀⢧⠀⡿⢸⡇⠀⠸⣸⡇⢠⢸⡆⣏⢰⣦⠁⢸⣿⠀⣾⣠⠃⠀⣤⠀⠀⢠⠟⢷⠏
⠀⠀⠀⠀⣀⠄⠉⢧⡸⢆⠀⡗⠀⠀⠸⡴⡇⠀⣿⠀⡄⢿⡟⢻⠚⠛⣟⢻⡏⣀⢸⡱⡿⢙⡿⠀⢈⡏⢀⣠⢁⡜⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠑⣼⠓⣾⡀⠃⠀⢳⠙⢦⣼⡆⠰⠊⢷⡈⡄⠀⠑⣾⠀⠁⣼⢃⣽⠋⡇⠂⢸⡥⣏⣬⠛⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡅⠚⠁⠈⣧⡀⢈⣽⠀⠀⠈⠻⠷⠶⠾⠃⠀⠀⣿⣏⣁⡼⢃⠀⢾⡧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⢡⡤⠶⠚⣚⠖⠰⣿⣏⣉⣹⣏⣉⡉⠙⠙⢳⡶⠒⠤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠄⠀⠚⠀⠈⠉⠀⠀⠿⣤⣤⣤⣤⠟⠀⠀⠈⠉⠉⠉⠐⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⠀⡇⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡤⠀⢀⡀⠀⢻⣦⣼⡇⠀⣧⡐⣿⠁⢀⣀⠠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣉⡉⣧⡈⡿⣿⠏⢰⡿⠉⡠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⣉⣀⣹⣿⣇⣨⡍⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⠛⢿⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⢻⣧⠈⠉⠻⢷⣦⡀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⣴⣦⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠛⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⣴⡟⠁⣀⣤⣾⠟⠋⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⢸⣿⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣴⡾⠟⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⣠⣴⡿⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡬⠍⠉⠩⡩⠉
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡔⣿⢯⣽⣿⣿⡟⣲⣶⣶⣶⣖⡲⢲⣶⣶⣶⣶⣶⣤⣤⣖⣵⣚⣠⢤⡀
⠀⠀⠀⠀⢱⣿⣿⢿⡇⠉⠻⣿⠻⣿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣖⣦⡀
⡆⠀⠀⠀⣾⠟⢉⣾⡇⠀⠀⠀⠳⣄⠙⠿⣯⣿⣿⣏⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡷
⣇⣠⣶⠚⠁⠀⠛⢹⡧⠀⠀⠀⠀⠘⣷⣤⠃⡟⠃⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⡟⠛⠉⢀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⢴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣾⡿⠋⠉⠹⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⠀⢐⠁⠀⠓⠉⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠑⠋⠋⠻⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢀⡞⠁⠀⠀⠘⠁⠀⠀⠀⢠⣤⡀⠀⠀⢀⣀⠀⠀⠀⡠⠊⡀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠅⠘⢠⠆⠀⠅⠀⠀⠀⢀⣠⣿⣿⣿⢷⠀⠀⠈⠟⠃⠘⠠⠎⡠⢂
⠀⠀⠀⠀⠀⡆⠘⠈⢀⠄⠀⠀⠠⠀⠐⣤⣹⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⣠⠄⢁⡀⠀⠀⠀⠀⠀⠄⣻⢿⣿⣿⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠠⠀⢊⠀⠋⠀⠀⠐⠀⠐⠢⠀⢀⣨⣽⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠁
⠀⠀⠀⠀⠀⠀⠋⠑⠄⠀⠀⠀⠀⠀⠘⠀⢨⠛⠈⢿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁

⠀⠀⠀⠀⠀⠀⠒⢶⠶⠒⠢⠤⣄⡀⠀⣀⡤⠤⠒⠒⠛⠉⠉⠉⠉⠉⣩⠗⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠲⣶⡒⠋⠀⠀⠀⣀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⣤⣤⡀⠉⠓⠲⠤⠤⠤⣤⣄
⠀⠀⠠⣤⡔⠃⠀⠀⢀⡴⠊⠁⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⣿⢸⣿⣷⠀⠀⠀⠀⣠⠖⠉⡟
⠀⠀⠀⠈⡇⠀⢀⡴⠋⠀⠀⠀⠈⢀⡢⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⢠⠞⠁⠀⡜
⠀⠀⢰⡛⠀⣠⠎⠀⠀⠀⠀⠀⠁⠀⠁⠘⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣞⠀⠀⠀⠃
⠀⠀⠀⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⣸⣀⠀⠀⠳⢦⣀
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠤⠭⢽⣛⣒⣛
⠀⠀⢠⡇⠀⠀⠀⠀⣤⠞⠉⠉⠀⠀⠀⠀⠀⠀⠑⢄⡈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡤⠔⠊⠁
⠀⠀⣸⠀⠀⠀⠀⢸⡗⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠻⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣲⠶⠒⠚⠉⠁
⠀⠀⣯⠀⠀⠠⠈⡄⢄⠀⠉⠛⠒⠒⠒⠊⠁⠀⠀⠀⠀⢀⣤⢾⢫⠃
⠀⢰⢿⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠘⠀⠘⡆
⠀⡼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠸⡄⠀⠀⠀⢳
⢀⡇⠀⢳⠀⠀⠀⠀⠀⠀⠠⠡⠐⠀⠀⢀⡴⠋⠀⠀⠀⠙⠦⣀⡀⠈⣇
⣸⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⡏⠀⠀⠀⠸⣆⠀⠀⠀⠀⡀⠀⢠⠞
⠻⣄⠀⠀⠀⠸⣆⠀⠀⢰⠁⠀⣿⠀⠀⠀⢶⠤⢤⣀
⠀⠈⠙⠒⠶⠴⠋⢧⡀⣾⠀⠀⢹⠀⠀⠀⢸⡄⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⡀⠀⠣⣀⣀⣸⢃⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠈⠉⠙⢮⢖⡡⢠⣼⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⢸⣿⠦⠤⠀⠀⠀⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⢍⣑⠒⠀⠀⠀⠀⠀⠀⠈⢣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢌⣆⠤⢆⡄⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠰⠀⠀⠀⠌⠢⠈⠢⡈⠂⠂⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡵⡇⠀⠄⣀⠂⢄⠢⠀⢈⠀⣀⠠⠄⠤⢱⣄⣣⡀
⠚⠁⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠳⢄⡀
⠀⠀⠀⠀⠓⠤⢄⡀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠘⣠⠴⠊⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣁⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠿⠘⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⡠⠖⠀⡠⠒⠀⠀⣏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡠⢹⠀⠀⠀⠀⠀⠨⠭⠄⡠⠾⢑⠟
⠀⠀⠀⠀⠀⠀⢸⠤⠒⡉⡀⢀⠌⠀⠀⠀⠀⢣⠄⠀⡀⠀⠄⠈⠙⢁⡘⠤⠐⠈⠁⠀⠀⠑⢌⠠⠀⠐⠄⡴⠊⣠⠔⠁
⠀⠀⠀⠀⠀⠀⣁⡠⢊⠊⢠⠊⠀⠀⠀⠀⠀⢹⡒⠁⢀⣀⣀⠴⠈⣡⢀⠄⠠⠠⠴⠄⢀⣀⢀⡉⠀⡄⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡝⡰⢁⡴⠁⠀⠀⠀⠀⠀⠐⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⣠⣟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⡈⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠂⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⠀⠀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢀⠀⠀⠈⠈⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣦⠤⣀⠀⠿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠙⣿⣿⣿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠙⣷⣌⠑⢄⠀⠀⠀⠃⢀⠆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠘⠛⠣⣨⠂⠀⠀⢀⠞⣀⣿⣿⣿⣿⣷⡶⠀⠀⠙⢿⣧⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⣀⣲⢦⡤⠗⠋⠀⠉⠛⠋⠛⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡑⠂⠤⣤⣀⠀⠀⢰⣶⣶⣿⣦⣄⠈⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠦⠀⢹⣿⣷⠀⠀⠈⡙⠻⢿⣿⣧⡀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣿⡿⠀⠀⢠⣿⡀⠀⠈⠉⠉⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠉⠀⣀⣴⣿⣿⠃⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣃⣀⠀⢀⡤⠚⢡⡿⠟⠋⠀⠀⠀⢀⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠭⠭⠵⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠐⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣽⣿⣿⣿⢖⣦⣾⣿⡿⣿⣿⡟⢸⢞⢲⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣤⣭⣽⣿⡿⢋⣵⣾⣿⣿⣿⢳⣿⡿⡇⠹⣙⡢⠋⠁⡬⠛⠁
⠀⠀⠀⠀⠀⠤⠶⣾⡿⢋⣴⣿⣿⣿⣿⣿⠃⠙⡛⠀⠀⠠⡀⠀⠀⠈⠉⢙⠇
⠀⠀⠀⠀⠄⠲⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⡀⠀⠀⠨⡀⠀⢁⠤⠃
⠀⠀⠀⠠⠶⢾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡀⠀⠂⠈⠳⠤⠝⠁
⠀⠀⠀⢀⣴⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋
⠀⠀⠀⢠⣼⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⢀⡠⠄
⠀⠀⠀⢀⣼⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣴⣤⣶⡿⠋
⠀⠀⠀⠁⣽⠨⣿⣿⣿⣿⣿⣿⣝⡛⣿⢕⣣⢴⠟⠛⠉
⠀⠀⠀⠀⢰⡆⣿⣿⣿⣿⣿⣿⡀⠅⠈⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠐⠀⠱⠹⣿⣿⣿⣿⣯⢢⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠸⣄⢣⢽⣿⣿⣿⣿⣫⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢯⠙⢿⣧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠹⡆⣬⡻⣿⣿⣿⣷⣷⣄⢀⣠⠾⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠉⣿⠑⣮⠿⣿⣿⣿⣿⣀⠉⠋⠉⠙⠶⠆⠀⠀⠀⠀⠀⠀⠐⣘⣧⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⡓⢮⡛⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠀⢤⣄⣉⡛⠿⠿⣿⣶⣶⣶⣶⣾⣷⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣦
⠀⠀⢻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠘⣿⣿⣿⣿⣆⣀⣀⣀⣰⣶⣶⣶⣶⣾⡿⠿⢿⡟⠛⣿⠛⢛⠉⠻⠏⠁⠉⠙⠟⠛⠛⠿⠻⢿⢿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠿⠀⠠⠾⠀⠀⠒⠒⠒⠂⠀⠙⠀⠀⠀⠉⠀⠀⠴⠀⠐⠇⠚⠛⠛⠿⠻⠿⠿⠩⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣷⡤⠄⠀⠀⠀⢀⣀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⠠⠄⠀⠤⣤⠀⣈⠉⠹⠯⠄
⠀⠀⣸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠿⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢈⣿⠞⠋⢉⠀⠀⢀⣠⣤⣨⣅⣤⣄⡀⠀⠀⡉⠙⠳⣿⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⣀⣴⠟⢁⣴⢿⣷⠖⠛⠉⢞⣡⢭⡭⣌⡳⠉⠛⠲⣾⡿⣦⡈⠻⣦⡀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢋⡴⢻⡇⠀⠀⢀⣘⠛⠛⣩⠤⠤⢍⠛⠛⣃⡀⠀⠀⢸⡟⢦⡙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠞⠉⣰⣿⣇⡀⠀⠊⢁⡠⠖⠒⠒⠒⠒⠲⢄⡈⠓⠀⢀⣸⣿⣆⠉⠳⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡄⠉⣽⡏⠻⡄⠀⠺⠀⠀⣠⠔⠢⣄⠀⠀⠗⠀⢠⠟⢹⣯⠉⢠⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⢇⣾⠁⠀⠗⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠲⠀⠈⣷⡼⣷⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡏⠈⠙⠲⠄⠀⠙⢺⣿⣏⣀⠀⠀⠀⠀⣀⣹⣿⡗⠋⠀⠠⠖⠋⠁⢹⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣇⣠⡤⡠⢞⣣⣤⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠤⢤⣜⡳⢄⢤⣄⣸⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣠⣴⡟⣇⠀⢰⣿⣦⡻⢄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢟⣴⣿⡆⠀⣘⢻⣦⣄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣧⠻⠆⢸⣿⣏⠻⢦⡀⠀⢠⡄⠀⠀⢠⡄⠀⢀⡴⠟⣹⣿⡗⠰⢟⣼⣥⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣍⡛⠛⣻⣿⣓⠦⠀⠀⠈⠻⣤⣤⠟⠁⠀⠀⠴⣚⣿⣟⠛⢛⣩⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠹⣿⣿⠿⠳⠶⡶⢤⣴⣦⡴⢶⠶⠞⠻⣿⣿⠏⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣀⣿⣿⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣿⣿⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡏⠖⡀⠀⠀⠀⠀⢀⠲⢹⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣟⣉⣉⣉⣙⣩⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡤⠠⣤⠠⣤⠀⠀⣠⠔⠐⢤⠀⠀⣤⠀⠠⡄⠠⣤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢻⡄⠀⠀⢻⠃⠀⣿⠀⠈⠀⠀⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠐⠛⠀⠀⠙⠒⠄⠚⠀⠀⠛⠒⠒⠃⠐⠛⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡾⠿⠛⠛⠋⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣶⣤⠀⠀⠀⣠⣤⣴⣤⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⣾⠟⠉⠁⠙⣿⡄⠀⠀⣴⡿⠛⠉⠈⠹⣿⡀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣼⡿⠁⠀⠀⠀⠀⣿⡇⠀⣼⡟⠁⠀⠀⠀⢀⣿⠇⠀⠀⢰⣿⠃
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⢠⣿⠃⠀⠀⠀⠀⣼⡿⠀⢰⣿⠁⠀⠀⠀⠀⣼⡿⠀⠀⠀⠈⠁
⠀⢀⣼⡿⠁⠀⢀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠀⠀⠈⣿⣆⠀⣀⣤⣾⠟⠁⠀⠘⣿⣄⠀⣀⣤⣾⠟⠀⠀⢀⣾⣿⡆
⠘⠻⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠁⠀⠀⠀⠀⠙⠻⠿⠟⠋⠁⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠄⠀⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣄⡀⠀⠀⣠⡤⣤⣤⣤⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠩⢶⣶⣤⣽⣿⣿⣿⡿⠏⣳⣾⣿⣿⡞⣿⣿⣿⡟⢈⡿⢛⡛⣿⣿⣿⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠤⢤⣭⣭⣿⣿⣿⠿⣉⣴⣿⣿⣿⣿⣿⡇⣯⣿⣿⡷⠘⣇⠻⢆⡟⠁⠀⣌⠿⠋
⠀⠀⠀⠀⠀⠀⣀⣉⣭⣭⣽⣿⠟⢡⣼⣿⣿⣿⣿⣿⣿⢟⣲⠽⠟⠀⠑⢄⠈⠉⠁⠀⠀⠘⠑⠶⢴⠄
⠀⠀⠀⠀⠀⠀⠊⣉⣩⣾⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠲⠄⠀⠀⠀⠉⠦⠀⠀⢠⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠚⠍⢿⣿⡟⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠄⠀⠀⢀⠰⢄⣀⢀⡠⠞⠁
⠀⠀⠀⠀⢀⢴⢾⣿⠏⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢙⣄⠀⠀⠐⠍⠉⠲⣦⠼⠋⠁
⠀⠀⠀⠀⠀⢀⣼⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠈⠖⠀⠠
⠀⠀⠀⠀⠐⠹⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣦⠄
⠀⠀⠀⠀⠠⢾⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⣄⣤⠖⠒
⠀⠀⠀⠀⢀⣴⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿⡆⢠⣖⣰⣶⣿⡿⠃
⠀⠀⠀⠀⠉⢙⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⡽⡼⣾⣟⠯⢮⣰⣼⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠋⢣⢸⣿⣿⣿⣿⣿⣿⣿⣯⠘⠆⠘⠷⠻⠶⠽⠋
⠀⠀⠀⠀⠐⠀⢿⡀⣿⣿⣿⣿⣿⣿⣿⡷⢁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠘⠃⠀⠷⠹⣿⣿⣿⣿⣿⣿⣜⢠⠌⠀⠀⠀⠀⠀⠀⣰⠾⠛⠋⠉⠉⠛⠻⢮⡳⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠘⣏⡐⢇⣹⣿⣿⣿⣿⣿⣿⢫⣖⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠫⠻⣽⣆⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠸⣧⡘⢁⢻⣿⣿⣿⣿⣿⣿⣎⠄⠀⠀⠀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠏⣧⣇⠀⠀⠀⠀⠀⢠⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣸⣯⠙⢿⣿⣿⣿⣿⣿⣳⡂⠰⣾⣟⣟⣡⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⣿⠀⠀⢀⢀⣴⡏⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⡯⠸⣧⠹⢿⣿⣿⣿⣿⣟⣄⠀⠉⠀⠀⠙⠛⠚⠀⠀⠀⠀⠀⠀⠀⠂⠀⢋⣌⣿⣶⣶⣿⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⡻⢳⢪⣭⠻⢿⣿⣿⣿⣦⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣼⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⠪⣟⠃⡀⠉⠻⠿⢿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣤⣶⣥⣤⣠⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠷⠶⠦⣭⣩⣭⣿⣻⣿⣟⣿⡿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣶⣲⣾⣿⣭⣭⡭⠭⣭⣴⣶⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣤⢶⣻⣽⣶⣿⣿⣿⣿⡿⢷⣦⣬⡝⣿⢙⣽⡏⣼⡏⡇
⠀⠀⠀⠀⠀⣠⣾⣛⣷⣾⣿⣿⣿⣾⣿⣿⣿⢟⣿⠾⡟⠁⠀⢾⣿⣿⢷⣿⣿⣿⢹⡟⣿
⠀⠀⠀⣀⢼⣉⣙⣻⣷⣿⣿⣿⢿⣿⣿⣿⣵⡿⣿⡦⠀⠀⠀⠰⣾⣇⣾⣿⢹⣿⣿⡇⣿⡀
⠀⠀⠀⣸⣀⣨⣿⣿⣿⣿⡿⣸⣧⡝⠏⣼⣿⠀⣾⠀⠀⠀⠀⠀⠹⢸⣿⣿⣼⡇⣇⠃⢘⣧⡀⠀⢀⣀⣤
⠀⠀⠀⢩⣭⣿⣿⣿⡿⣟⠇⢿⣿⣿⡗⣻⡟⠀⡿⠀⠀⠀⠀⠀⠀⠸⡏⠀⣿⡇⢸⣟⣾⣟⣿⣿⣿⣟⣫
⠀⠀⠀⡸⣺⣿⣿⢿⢭⣿⢷⣿⣿⣿⡟⢸⡵⠀⣷⣄⠀⠀⠀⠀⢠⠶⠿⠀⣿⣧⣸⣿⢟⣛⣹⣷⡿⠋⠁
⠀⢠⣮⠾⣿⣿⣟⣶⣾⠻⣿⣿⣿⡟⠃⠀⢣⠳⡹⣽⣧⡀⠀⢀⡿⠀⠈⠻⠛⣉⡥⠖⠛⠋⣡⠾⠁
⢠⢳⢿⣿⣿⡿⠹⣿⣿⡧⠀⠉⠛⠃⠀⠀⠘⣷⣍⡛⠓⠛⠛⣄⠀⠘⢋⣭⣭⣥⣶⣂⠴⠚⠁
⡏⡿⣟⣽⣿⡇⠀⠘⠿⠷⠄⣠⣶⣾⣿⢿⣿⢟⣟⡹⢷⣶⣶⣷⣮⣿⣿⣿⣿⣿⠋
⡷⢋⣾⣿⣯⠁⠀⠀⠀⢰⣾⣟⠿⣯⣾⡯⢗⡿⣾⣿⠼⠀⠀⠉⠉⠉⣉⠽⠋⠁
⠁⠙⠿⣿⣟⠀⠀⠀⠀⠈⠙⢿⣷⣄⣋⣖⣣⣴⠞⠁⠀⣀⣤⣶⣶⣾⠁
⠀⠀⠈⢹⣿⣤⡀⠀⠀⠀⠀⣀⣠⣤⡿⠿⠛⠉⣠⣶⣿⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢻⣟⣷⣄⠀⠀⢿⠛⠉⠁⠀⠀⣀⣼⣟⠟⣷⣹⢿⣿⢿⡟⡿
⠀⠀⠀⠀⠈⢻⣯⣟⠇⢀⣼⠆⠀⠀⣴⢋⡴⠉⠘⢦⠱⠋⣿⣿⡞⠃⡇
⠀⠀⠀⠀⠀⠀⠹⣿⡃⢰⡟⠀⢀⠜⡵⣿⣷⣖⣚⣩⣵⣮⣙⢿⡗⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣇⢀⡼⠃⠀⢋⣷⣾⣯⡭⠽⠿⣾⣟⣛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡟⢰⣴⣿⣟⠙⠿⣿⣻⠿⠯⣭⣭⣭⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠛⠃⢿⣶⡬⣶⡲⠯⠭⠭⢭⠭⠥⠴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣉⣁⡈⠙⢃⣀⣀⣀⣤⣤⠶⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⢛⣉⣥⣤⣤⠶⠶⠦⠤⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣾⣟⣥⡶⢚⣋⣩⣥⣴⠶⠶⠦⠤⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠉⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠛⠛⠻⠿⣶⣶⣶⡒⠒⢒⣠⣾⠏⠀⠀⠀⠀⠀⠀⣠⡾⠛⣡⡴⠚⠁⣰⡿⠁
⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣭⡀⠀⠀⠀⠀⠀⣠⡾⢋⡴⠟⣁⣤⠀⣼⡟⠁
⠀⠀⠀⣠⡿⠋⣠⠤⠤⣄⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⠾⠋⠀⣩⠶⠛⣉⣀⣼⡟
⠀⠀⣼⠟⠀⣼⣥⣈⣷⡌⡆⠀⠀⠘⣆⠀⠀⠀⣀⣴⣶⣶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⠁⣿⡇
⣠⣾⠏⠀⠀⠹⣌⣛⣛⡴⠃⠀⠀⠀⣿⠀⣠⠾⡋⠁⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠛⠛⠓⠻⣿⡄
⢻⣷⣤⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣿⠘⢑⣪⠀⣾⠏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⡛⠶⢤⣰⡿⠃
⠹⢷⣬⡛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠰⠶⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣄⠀⠀⢮⣙⠦⣄⣿⡇
⠀⠀⠈⠙⠷⣦⣀⡀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⡾⠋⠁⠀⠀⠀⠙⢷⣌⠢⣍⠳⣄⠻⣧
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣟⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣾⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⠳⣌⡙⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⢿⣿⠿⣿⠷⠶⠶⠶⠿⠿⣿⡻⢟⡻⢟⡦⠭⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⡙⠢⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡻⣕⡘⣷⠀⠀⠀⠀⠀⠘⢿⣦⣙⠲⢭⣙⡒⠂⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⣸⡧⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣬⣉⣛⣓⣨⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠐⠋⠙⠓⠶⣤⠀⣠⠆⠀⠀⠀⠀⢀⣺⣼⣶⣶⣶⣄⢰⠂⠀⠀⠀⠀⠀⣤⡤⢴⣦⣤⡀⣠⡄
⠀⠀⢀⠀⠂⠀⣀⡠⠒⣀⠀⣿⡀⠀⠀⠀⢴⣿⡿⣿⠟⠽⠋⠀⠛⠀⠀⠀⠀⡄⠄⠀⣤⠴⢒⣲⠄⠹⠇
⠀⠀⠈⠀⠀⢤⣌⠁⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⠦⠤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⢈⣀⠀⠻⢠⣆⠀⠀⠀⠀⠀⡊⠁⠹⣿⣿⢿⣿⠉⠻⠇⠀⠀⠀⠀⠀⠄⠀⠿⣿⣿⣯⣤⣦
⠀⠀⠀⠀⠀⠈⠉⡠⠆⠘⠇⠀⠀⠀⠀⠀⠉⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣘⠓⠿⠇⠈⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⣠⣶⣤⡄⡴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢶⣴⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣄⢹⡄⡇⡇⡧⠠⣼⠀⠀⠀⠀⠀⠰⣟⢰⢷⣼⢱⠶⣇⠀⠀⠀⠀⠀⠀⢤⠀⣸⡷⢮⣵⣦⣀⣤
⠀⠀⠤⣾⡘⣧⠗⣧⣏⠈⠻⠀⠀⠀⠀⠀⠙⠿⢿⡟⠛⠻⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠥⠇⠰⠗⠂⠈⠋
⠀⠀⠀⠀⠈⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⣄⡀
⠀⠀⠀⠀⢨⣿⣿⠿⠿⠓⢺⡇⠀⠀⠀⠀⢀⠠⠃⢀⠩⠙⠦⠀⡀⠀⠀⠀⠀⢀⣴⡟⢻⢻⣿⣿⣧⣶⣦
⠀⠀⠄⠈⠀⣙⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡴⠀⠀⠉⠀⠀⠀⠀⠸⠆⠀⡀⣸⠯⠿⠆⠈⠉
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⣿⠀⡝⠣⣴⡆⠀⠀⠀⠀⠀⠀⠀⡩⡮⣿⣿⠂⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡉⠺⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠠⠶⠐⠂⢀⡿⠿⢿⣷⠆
⠀⠀⠀⠀⡀⢰⡈⢀⡀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠉⠀⠀⠀⣹⠶⠴⠾⠋⠀⠀⣠⠁⢀⡤⢄⠀⢀⣤⣖⣒⣳⣢⡄
⠀⡀⢀⠀⠀⠀⠷⣶⠣⠀⠀⣠⠄⠛⠉⠢⠀⢀⡀⠂⠀⢈⣠⡼⠃⠀⠀⣠⣤⣾⣿⡛⡄⠛⠁⣶⢇⣼⢉⣽⠉⢉⡉⠉
⠀⠈⠡⣴⡘⠳⣄⣺⣿⢷⡞⠁⠀⠖⢀⡀⠀⠀⠀⢀⡾⠉⠉⣅⢀⣠⣾⣿⣯⡄⠻⣷⣴⡀⢀⣿⠸⣏⢻⡵⠂⠋⠁
⠀⠀⠐⠈⠛⠓⣺⡿⠛⠀⢀⠀⠀⣀⠈⠋⠀⠀⢀⡼⡁⠀⣾⣿⣿⣯⣅⢘⣿⣷⣿⣿⣦⡝⣣⣏⣴⣿⣾⣷⡄⣀⣤⣀
⠀⠀⠀⠀⠁⡠⠋⠰⢁⠀⠀⢰⡁⠈⠾⠂⣠⠔⠋⠁⢿⣸⣿⣇⣨⡟⠃⠁⢠⣴⣿⡟⢩⣿⣷⣿⠟⣰⡟⢠⣮⠌⠁⠉⣿
⠀⠀⠀⠀⣰⠆⢀⠀⠀⠀⠆⠀⠁⢀⣰⠏⠀⠀⢀⠀⢸⡯⠽⠏⠁⠀⠀⢀⣠⣿⣿⣷⣸⣿⡿⢁⣾⠻⣄⣌⠇
⠀⠀⠀⢠⡇⠠⠉⠀⢠⠀⠀⢀⡠⠞⠓⠀⠀⣄⢠⡿⠋⠁⠀⠀⢰⣦⠐⠿⣿⢿⣿⣿⣿⠋⣱⣿⣿⣇⠛⡉
⠀⠀⢀⠏⠀⠀⠂⠀⢠⠞⠉⠉⠀⠀⠀⣀⡴⠟⢁⣄⠀⠈⢆⣀⠀⠀⠀⠀⡀⣾⡿⢿⢇⠸⣿⣿⣷⣽⢿⡇
⠀⠀⣦⡄⠀⢀⣀⡶⠉⠁⢀⣀⡾⠗⠚⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡴⠊⠀⠰⢿⣿⣿⡗⢸⡇
⠀⠰⠁⢀⣴⣏⠀⢀⣵⣶⡟⠁⠠⠀⠀⠈⠐⢦⠀⣀⡘⠃⠀⣠⣬⡄⠌⠀⠀⠀⠀⢆⠀⠙⠈⠋⣘⠃⣿⣧
⠀⡀⡄⠉⢀⣾⠿⣿⠃⠋⢀⣠⣶⣄⣀⡀⢈⣠⡄⠀⠀⣀⢀⡟⠁⠁⠀⡀⠀⢀⣠⣤⣶⣶⣶⣄⠁⢸⣇⠹
⠀⣿⡧⠔⠁⢰⡀⣿⠀⢓⣸⣿⣿⣿⣿⣿⣿⣷⣷⣾⣧⡉⠘⢠⠏⠼⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡄⣿⠙⠆
⠀⠀⠀⠀⠀⠈⣧⢸⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡆⡈⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠸⢈⡄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠁⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⢿⡇
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠃⠀⠛⠛⠛⠿⠟⠟⢻⠙⢻⠀⣰⣿⣿⣷⡀⠴⠃⣠⣍⡉⢩⡉⠉⠘⠀⣰⡌⠣
⠀⠀⠀⠀⠀⢠⠃⢸⡄⠈⢠⡤⠀⣠⠀⣴⡄⠊⡀⣅⣠⣿⣿⣿⣿⡇⢆⣠⣿⣿⠇⣀⡀⠀⠠⣦⡈⠁⢰⠃
⠀⠀⠀⠀⠀⠀⢳⣄⠰⠀⠀⠀⠀⠁⣤⠘⡿⠊⣉⢛⣼⣿⣿⣿⣿⣷⠿⠏⠻⠅⢰⣿⣿⣠⣀⡉⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⢇⣤⣿⠀⢶⣖⠉⠀⠐⠃⠿⣿⡿⣿⢿⢻⣿⣿⡆⢸⠀⡈⠁⣿⣾⣿⠃⣹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⢶⢄⡾⣀⡖⢘⠂⡄⢀⠈⡏⠀⢧⠈⢇⢹⡀⣾⢿⡏⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⡀⡜⢀⠏⢁⡞⢢⡷⠺⡷⠛⣤⠚⢷⠌⡿⢣⠸⣾⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⣵⣇⣾⠀⣾⠀⢸⡇⠀⡇⠀⢹⠀⣸⠀⣿⣼⣿⣏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⡄⣿⣿⡏⢱⣿⣤⢼⣧⢨⣷⣤⠾⣤⣟⣶⢻⣿⣿⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣾⢿⣼⢻⣿⡟⢻⣫⣾⣿⣧⡼⢿⣠⣿⠿⢟⣛⣿⢯⢱⡏⣆
⠀⠀⠀⠀⠀⠀⣶⢀⡼⠋⣸⠋⠾⠘⣷⣭⢿⣷⣿⣽⣧⢿⣿⡷⣶⣭⠿⡟⢿⣽⠋⢦⣼⡇⢙⠂
⠀⠀⠀⠹⡃⠚⠲⣏⣀⡞⢁⣀⢷⡀⠰⢙⣠⢹⢿⣿⣟⣴⣷⣾⣽⣷⣾⣷⣾⡏⢃⡼⠋⢡⠏
⠀⠀⠀⠀⢙⣦⣿⣿⠛⣧⠘⠁⠀⠳⡐⠆⢻⣼⠀⠻⠙⢋⠈⡟⠉⡟⠙⣯⠀⣧⢼⠅⣱⠏
⠀⠀⢀⠞⠋⠳⣿⡿⠠⢤⣳⡄⠀⠀⠙⣆⠀⠛⠢⠷⢠⠾⠀⠷⠠⢷⠐⣿⠀⠇⠈⣰⠏
⠀⠀⠈⢧⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠐⠀⠀⠀⠉⠀⠈⡠⠀⠀⠀⣀⠏
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠒⠤⠖⠶⠘⠀⠠⠶⠔⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣦⡀⠀⠀⠀⠀⠀⢀⣤⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣭⣭⣭⣭⣽⣷⣶⣤⣤⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣤⣔⣄⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠃
⠀⠀⠀⠀⣠⠴⢟⢭⣍⠉⠉⠉⠀⠀⠀⠀⣤⣤⣶⣶⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠶⢶⣤⣤⣀⡀⠀⠀⡼⠃⢸⠇
⠀⠀⠀⠀⠣⡀⠐⠿⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣶⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣙⣛⣓⠢⢸
⠀⠀⠀⠘⢍⡉⠁⠀⠀⠀⢀⡆⠀⡼⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠛⠉⠉⠸⡄⠀⢸
⠀⠀⠀⠀⠀⠈⠛⠫⢝⣒⣋⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣫⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠙⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣠⣶⣿⢶⡦⣄⠀⠀⠀⠀⠀⠴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠷⠾⠁
⠀⣀⠀⡀⢻⠸⣿⢸⠃⠘⠓⠒⠄⢀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠀⠈⠁⢸⠀⡇⠘⢸⡇⠀⠀⠀⠘⠉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣈⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⣶⠀⣆⢸⡧⠀⠊⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⡄⠰⠘⣿⣁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠙⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⢩⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣉⣀⡤
⠀⠀⠀⠀⠀⠀⢠⣶⠤⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⣠⣤⠴⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠿⠛⠻⠿⢶⡆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠆⠀⠀⠠⣶⣛⠉⠠⠄⠐⠒⠒⠛⢛⣫⣇⠘⠀⠀⠀⠀⠀⢰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣛⡛⠿⠿⠿⢛⡛⠉⠁⠀⠀⠀⠀⣠⣦⠈⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢠⠘⣿⡷⠂⢤⡆⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠄⠁⠀⢹⣇
⠀⡤⠤⢤⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢰⠀⠞⠆⢰⠀⡆⢸⠋⢠⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢃⢴⣿⣶⠀⠹
⠸⠁⢠⣶⣿⣯⠈⠉⠀⠀⡀⠈⠁⠀⠀⠀⠒⠤⢄⡀⠀⠀⢸⡀⢸⠀⢆⠀⢸⠀⡇⢸⠄⠘⣄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣾⣿⠑⣠⣦⡀
⠈⠰⣿⣷⣿⣿⠀⠀⠀⠀⠡⠈⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⢸⠀⢸⠀⠀⢣⢸⡤⡇⢸⠀⠀⠈⠆⠀⠀⠈⠁⣀⠄⠀⠀⠀⠀⠀⢀⣉⣼⣿⣿⠏
⠀⠀⠈⠛⠿⠃⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⠀⢸⠀⠀⠈⢸⠀⡇⢸⠀⠀⠀⢸⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠀⠸⠀⠦⠜⠸⠀⠇⠸⠄⠘⠤⠎
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣄
⠀⠀⠀⠀⠀⡀⠘⢿⣿⡌⠙⢿⣿⡄⠐⣦⣤⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣤⠀⠀⣠⠄⢸⡿⠁⣼⠋⢈⡷⠂
⠀⠀⠀⠀⠈⠋⠀⢸⣿⡿⠀⢸⣿⡇⠀⣼⣿⡧⠙⠿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡷⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠐
⠀⠀⠀⠀⠀⠀⠰⠿⣿⣁⣴⣿⡟⠁⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⡾⠟⠀⠀⠻⠀⢸⣧⠀⢿⡄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠁⠀⠀⠀⠀⠀⢀⡀⠆⠠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢀⣴⣇⠀⠀⢻⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⠀⠀⠸⡟⢂⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⠻⢉⣾⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⠀⠰⣷⢼⣷⡀
⠀⠀⠀⠀⠚⢿⣷⡄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⢿⡶⠖⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⣧⢼⡿⠁
⠀⠀⠀⠀⢤⠄⢹⡇⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⣴⣦⠉⠝⠃⣄⡀⠀⣠⣶⠇⠀⠀⠀⠀⠛⠃⠤⠐⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙⢁⠆⠿⢟⣻⠡⣾⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡿⠶⡗⠘⣿⠇⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣠⣤⣬⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⠟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡄⣻⣶⣤⣀⣀⡀⠀⠀⠀⣀⣤⠞⢉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣦⣤⡎⢀⣀⣤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠇
⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣻⣿⣿⣟⣻⣯⣿⣿⣟⣾⣻⡿⣻⡟⣻⡿⣻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣷⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠏
⢀⣠⡴⠾⠟⠛⣿⣶⡛⠀⠙⠻⣮⣉⣠⣤⣤⣥⣿⡬⢥⣬⣧⣭⣠⣟⣘⠛⠁⠋⠘⠋⠾⠇⠸⠟⠹⡿⢻⣿⣿⣿⡿⣶⣦⣤⣤⣾⣷⣶⡟
⠀⠙⢣⡀⣀⠻⠿⠟⠀⠀⢀⠀⣻⡟⡟⠓⠉⠉⠈⠉⠉⣁⣀⣨⠭⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠾⠿⢿⣿⣿⣻⡅
⠙⠻⢦⣥⣞⡃⠀⠀⠀⠀⣴⡀⣿⣓⣳⣤⣤⠖⠈⠉⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠉⠁⠉⠀⠀⠀⠀⠻⣝⠛⢻⡀
⠀⠀⠀⠀⠉⠛⠻⠶⣶⣮⣿⣿⡟⣛⣛⣿⣿⣿⣭⣭⣭⣵⣶⣶⠶⠶⢷⣴⣤⣶⣴⣶⣶⣾⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣿⣿⣯⣭⣭⣭⡭⠽⠿⠿⠿⠟⠛⠛⠛⠻⣿⡻⠏⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⣿⠁⠘⢠⠄⣴⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣴⣄⣺⣣⣖⣷⡀⠈⠀⢴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠄⠀⠈⠉⠑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⣆⣠⣤⢦⡤⠀⠀⠀⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢂⡀⠚⠳⢾⠋⠀⠀⠀⠀⢸⣿⠎⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡘⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣠⠐⠉⠙⣿⣷⡄⠀⠀⡞⠧⣀⠘⢦⠀⠀⠬⠚⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⡹⠀⠀⠈⠀⢻⣷⠀⢀⠀⠀⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠳⠢⠆⢁⠀⢺⡟⢠⡏⠀⠀⠀⠀⣡⠋⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠆⡂⠀⠐⠲⠶⣫⢠⠏⠀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠁⠀⢀⠀⠀⠀⠠⠄⠀⠀⡰⠋⠀⠀⠀⡠⠂⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡔⢀⢇⠀⠀⠀⠘⢀⠎⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠝⠒⠀⠈⠁⠸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠂⠁⠀⠠⢤⣾⣞⡁⠀⢀⣀⡤⠔⠀⢀⡴⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠛⠛⢅⠀⠀⠀⠀⠀⠒⠋⠉⠙⠛⠛⠯⠥⠤⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⡀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⢀⠘⠋⣉⡝⠁⠤⣤⡤⡄⢤⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⠈⠙⠐⠊⠁⠀⠀⠀⠀⣀⠴⠀⣤⠖⠃⢰⣴⣶⣤⠴⡾⣷⠒⠈⣭⣤⠂⠀⠀⢀⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⣀⡠⠶⣿⠉⠉⣹⠛⠙⣿⠦⣀⣉⣳⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⡤⠤⠚⠉⠉⠉⠉⠀⢀⠈⠁⠀⠀⢸⠳⠀⣻⠦⣼⣇⠀⠈⢉⡟⠛⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢨⣿⣧⠀⣿⣤⣈⡏⠁⠀⣼⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⠾⠿⠿⠀⠈⠛⠻⠿⠶⠚⠙⠓⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣾⣿⣹⣷⠾⣶⡆⢒⡾⣶⠆⢀⣰⣶⣿⣿⠀⠐⡂⠦⣴⢆⢸⡯⢿⢸⣴⣷⡟⠟⣿⣟⣿⢷⣆⣠⡀⠀⣤⡤⢀⣀⢀⠀⠀⠀⢢⢨⣄⣦⡀
⠀⠀⠀⠀⠛⠻⠿⠛⠛⠛⠓⠛⠛⠛⠛⠛⠸⢰⣿⣬⣴⣿⡿⣦⣞⠁⢼⡧⢸⡏⢻⡔⣿⢻⣿⣏⣿⡆⣧⢸⠌⠜⠟⢺⡣⣽⡿⣿⢿⣶⢠⣄⡿⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡑⠉⠀⠙⣷⣶⣄⡐⠓⠻⣿⣼⢿⡻⠿⢿⡶⣯⢿⠀⠁⠄⣉⠦⣤⣷⡏⠸⠁⣸⣋⣮⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠏⠈⣤⣦⣶⣵⣿⣿⣿⣷⣤⣀⡓⠒⠻⠶⠂⠘⠋⠉⠉⠀⠶⠶⠀⠹⠽⠿⠃⢀⣾⢿⠝⢸⡗⢫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣵⡂⠂⣿⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡿⡯⣽⢿⣿⣿⣿⡿⣻⣿⣿⣿⣶⣭⣻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣽⣿⠽⢏⣻⣅⠎⢹⡇⠼⠛⠛⠛⢯⣹⢒⠿⢉⢿⣿⣟⠯⡛⠿⣿⣗⣤⡀⠀⠀⠀⠀⠀⠀⠛⠾⠿⣿⣠⡆
⠀⠀⠀⣀⡤⡾⣧⡾⢾⣿⣿⣿⣿⣿⣿⡿⠛⡃⠀⠉⠈⠀⠀⠀⠀⠠⠀⠀⠀⣏⠇⠀⢀⠾⠿⠉⠛⠒⠠⠍⡛⠻⣿⡦⠀⠀⠀⠀⢠⡔⣠⡞⢿⠁
⠀⠀⠀⢻⡘⣶⠟⣤⣾⣿⣿⢃⣿⣷⣿⠗⠈⠀⠀⠀⠀⣶⣤⠈⢀⣀⠀⠀⣸⡍⡄⠀⡆⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⠹⠦⣄⠀⠀⠀⠈⠙⠃⠈⠃⠃
⠀⢠⠶⢦⣷⡼⣶⣿⣿⣿⣿⣸⣿⣿⠁⠀⠀⢀⣀⣠⠒⡙⢿⣧⡀⢻⣷⣤⡿⢷⡘⠒⢢⡀⠀⢠⠤⡎⠁⣠⣤⠶⠚⢉⡱⠾⠗
⠀⢸⣧⡘⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⠀⠀⠀⠈⠉⣁⣀⠀⠀⠙⠛⠀⠛⠿⠯⠄⠻⣤⠠⠙⠶⠴⠂⣀⠀⢈⠛⢠⣾⢛⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⢿⣿⣆⠙⣿⣺⣿⣿⣿⣿⣿⣿⣤⡤⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣼⣿⣶⣆⣀⣤⣬⣵⣦⣌⣀⣉⣛⢛⣦⣴⠀⠀⠀⠀⣸⣿⡟⠛
⠀⠀⢸⣿⣿⠀⢿⣿⣿⡿⣿⣿⣿⣝⣿⡁⢰⣿⣿⣿⣿⣿⣿⣿⢷⣯⣿⣿⣿⣿⣿⣟⣿⡿⢛⣋⣹⣿⣿⣯⡉⠛⠻⠿⢻⣧⠟⠀⠀⠀⠀⠈⠛⠟
⠀⠠⠾⣿⣿⡄⠀⠈⠻⣷⣈⠿⢿⣟⣿⣷⣾⣿⡟⢻⣿⣿⣿⣇⣺⡿⢟⣽⣿⣿⣿⣧⢶⣿⡿⠿⣿⣯⡜⠉⠉⠒⠀⠉⠛⠁
⠀⠀⠸⣿⣿⣿⡐⠲⣦⠠⣻⣆⡈⠀⠙⠿⣿⣿⡇⠺⣯⣿⡿⣉⣭⡵⢿⣿⣿⡻⠿⠛⠛⣉⣀⣤⣿⡿⠇⠀⠀⠀⠀⠉
⠀⠀⠀⠛⣽⣿⣿⣄⠀⢤⣌⠙⢿⣦⡀⠀⠀⠹⡇⠀⣿⣿⣷⠟⠋⡴⢿⣿⢏⣠⡾⢿⣿⠟⠀⣠⠋
⠀⠀⠀⠀⠿⣛⡟⠻⠻⣦⣹⡄⠀⠈⠉⢧⣀⣴⣧⢰⣿⡿⠁⣀⣠⣷⣟⡿⠟⢁⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠨⣿⠟⠹⠚⢆⠰⣤⣤⣽⣿⣧⢸⣯⠀⠀⣠⣤⣤⣤⣀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢳⣝⠶⠬⠞⠳⣸⣿⠀⠀⠉⠉⢉⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠙⠄⠀⠀⠀⠉⠁

⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣷⢦⣄⡀
⠀⠀⠀⣿⡟⠻⢷⣬⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⣿⡇⠀⠀⠈⠙⠷⣯⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠻⣿⣷⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠹⣿⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡄⠈⠻⣿⣿⡆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣄⠀⢸⣷⡀⢘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣸⡇⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣿⣆⠈⢿⣧⠀⠻⣿⣿⣷⣤⣀⣀⣀⣀⣤⣸⣿⣠⡟⠁⠀⢳⡀⠀⢹⣿
⠀⠀⠀⠀⠀⢻⣦⣄⢠⣼⣿⣿⡄⢈⣿⣆⠀⢹⣿⣿⣏⠻⣿⣿⠿⢿⣿⣿⣿⠿⠿⠷⣾⣷⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⢯⡻⢿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠙⠿⢷⡈⢻⣶⡿⠶⡷⠶⢦⣴⣶⣤⣄⣀⢉⡉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⠉⠛⠛⢦⣠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣶⠶⠿⠿⠿⢛⣷⣶⠖⠀⠀⠀⠀⠀⣸⡟⠋⠙⠋⠉⠂⠈⠲⣤⣄⣀⠀⠉⠛⢦⣀⠙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣥⡄⠀⠒⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⢤⣽⣿⢠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⢻⣧⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⠈⢉⡀⢿⣶⣄⡀⢠⡀⠀⠀⠀⠈⢻⣿⣆⠀⠛⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡧⠀⠾⢻⡟⠁⠀⠀⠀⢀⣴⣿⠃⠒⠺⢿⣿⣿⡿⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠐⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠋⣿⠟⠁⣠⣾⣿⠋⠀⣠⠄⠀⣼⣿⡋⠀⠀⠀⢀⡈⠀⠀⠀⢻⣿⣿⠉⠛⢿⣷⣄⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⢀⣼⠏⠀⡰⢫⡿⠁⣠⣾⡏⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⣿⡄⢀⣈⣿⣿⣷⣦⣀⣼⣧⣤⣄⣀⣠⡆⢰⠀⣾
⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⡿⠀⠘⣫⣿⡇⠀⢸⡛⣿⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⠋⠙⠻⣿⠄⠀⠉⠉⢳⣿⣤⡆
⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁⠀⠐⣿⣿⡀⠀⠘⠃⠙⠻⣿⣦⣤⣤⣀⣀⠉⠙⠓⠶⠾⠿⠿⠿⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣃⡇
⠀⠀⠀⠀⠀⠋⠁⠀⢸⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠠⠄⠚⠛⠛⣛⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠠⣄⡀⠠⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⠀⠀⠛⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⡀⠀⢠⠈⠹⠟⠳⠄⠀⠉⠳⣤⣈⠙⠦⠈⠳⠆⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⢻⣇⢸⣿⡆⠀⢀⣿⣧⡖⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⢧⠀⠀⡀⠀⠀⠀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡎⣿⣷⠀⠸⣿⡟⠀⣰⢃⣾⣿⣿⣿⣿⣿⣿⣏⠙⢿⡎⠙⣿⣿⡇⢸⠁⠀⣀⠀⠻⣆⠀⠟⠀⢰⣶⣦⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠈⠙⣇⢹⣿⡀⠀⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣿⠀⠈⠙⣿⡐⣦⠀⠙⢦⠀⠙⠀⠀⠀⠘⠛⠻⣿⣿⣷⣶⡶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡇⠀⠋⠀⣿⣿⣿⣿⣿⠿⠋⠙⣿⠁⣿⠇⠀⠈⠳⡀⠀⠛⣷⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠰⠃⢀⠟⠀⠀⠀⠀⠣⢄⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⢸⣿⣿⣿⡧⠖⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠛⠛⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣿⡟⠁⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡀⠐⠋⠉⠛⠿⣷⣶⣶⣾⣿⣧⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡄⠀⢿⣦⠀⠀⢰⣿⣆⢈⡄⠀⢰⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢿⡀⠘⣿⣧⣤⡟⠀⠉⠉⡹⢁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⢷⡀⠈⠻⠿⣿⣷⣶⣾⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠁

⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠍⠁⠀⢀⣀⣬⣿⣿⢦⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠃⢤⣀⣤⣾⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⢟⠟⣴⣾⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠟⠉⠻⡻⣙⣸⡿⣿⢃⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⠀⠀⠀⠀⠈⠉⠉⡰⢁⣾⠿⠿⠿⣿⡦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠭⠒⣒⠢⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢸⣻⡄⠀⠀⠀⠀⠢⠄⠀⠀⠨⠔⠁⠀⠚⢁⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣷⡙⢦⣀⣄⠓⠦⡤⠄⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⡟⠁⠀⠀⣤⣶⡿⢟⡛⠋⢉⣿⡿⠛⠲⠤⣭⣍⣃⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⣿⣿⣿⣿⠁⠀⠀⠀⠉⠻⢿⣿⣷⣶⣿⠾⠁⠀⠀⣀⣤⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣤⠶⠚⠙⣿⠿⠟⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠀⠀⠀⠀⠘⣿⡄⡀⣘⣧
⠀⠀⠀⡀⠀⠀⢠⣾⠖⠶⠦⡤⠘⣿⣷⣄⠀⠀⠀⠀⢠⡟⠁⢀⣠⣤⣤⣶⣶⣶⡎⠻⣿⣿⣿⠠⠶⢄⡀
⠀⠀⣾⠀⣀⢠⣮⠓⢶⣂⠀⢀⣉⣉⠛⢿⣶⣀⠀⠀⠘⢷⣶⣿⣿⣿⡭⣭⣽⣏⣀⣠⣶⣿⠶⠒⠉⢁⢣⢀⠇⢠⠷
⠀⠀⠘⢇⢽⡾⣟⡳⣶⣿⣟⠲⠾⢭⣉⡉⢙⣿⣿⣷⣤⣾⣿⣿⣿⣶⣾⣿⠟⣿⣥⡶⠾⠋⢅⠤⢒⣫⢏⡞⠜⠁
⠀⠀⠀⠀⠙⠟⣾⣿⣿⢿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣟⣭⣵⣶⣶⣶⣶⣾⠻⣵⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠛⠻⠟⠛⠻⠿⢿⣿⣟⣛⣛⣛⣻⣿⣿⡿⣿⣿⣟⣿⡿⠛⠿⠛⠙⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢚⣋⣉⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⣶⠶⠶⠾⠃⢠⣿⣏⠀⠉⠝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⢉⠠⡙⠓⢁⠀⠀⢀⣿⣿⣿⣯⡓⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠠⣿⢷⠄⠀⠁⠀⠀⣼⣿⡿⠻⡏⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢫⡄⠀⠀⡾⠀⠞⠁⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⡟⠀⠀⢸⠇⠀⠀⠀⠀⠀⡀⠀⢰⢠⣿⣿⠀⠀⢸⠀⢸⡏
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡿⢁⣰⣿⣿⠃⠀⠀⢿⡆⠀⠀⠀⢀⡾⠀⠀⡿⢸⣿⡿⠟⣛⣺⣧⣼⣷⣄⣀⣀⡀
⠀⢀⣠⣴⣞⣛⣉⣥⣿⠟⣠⣿⠛⢹⣿⠀⠀⠀⠘⢿⣤⣀⠀⣼⡇⠀⣠⢇⡾⠿⠛⠋⠉⠉⢀⣀⣉⣉⣉⣉⣽
⠈⢙⣿⣿⣿⣭⣽⣿⠏⣼⠟⠁⣄⠈⢿⣦⣀⣀⡀⠀⠉⠉⠁⠙⢷⣶⠶⠋⠀⠀⠀⠀⠲⣛⢛⡋⠉⢉⣽⠟⠁
⠀⢻⣿⣿⡷⢶⣿⢏⣾⠟⢠⣼⡟⣆⠀⣍⠉⠉⠉⠁⠀⣠⡴⠀⠀⢀⡤⠀⢀⠀⠀⠀⠘⠡⠋⢀⣴⠛⠁
⠀⢸⣿⣿⣿⣿⢏⣾⠏⢠⣼⣟⣧⣠⣤⡬⢧⡀⠒⠚⣛⣉⣀⣤⠶⠋⣀⡴⠋⣠⠆⠀⠀⠀⢠⣾⠃
⠀⣼⣿⣯⣽⠏⣼⠏⠐⣾⣿⡟⠋⠉⠀⢰⢾⣿⢓⡾⠛⠋⠩⣄⣀⡉⠩⢶⡛⠁⢀⡤⠂⣴⡿⠁
⠀⢸⣿⣿⡟⣸⡟⠀⢶⣿⡙⢿⣄⠀⠀⠀⠉⣠⡟⠁⠀⠀⠀⠀⠈⠙⠋⠉⠙⠛⢛⣵⣿⠟
⠀⠘⢿⣿⢃⣿⠁⠀⣿⡿⠀⠀⠙⠻⠶⢶⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁
⠀⠀⣾⡟⢸⡇⢰⡄⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣰⣿⠃
⠀⠀⣿⡇⣿⠁⢀⣿⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣰⠇⣰⣿⣿⣆
⠀⣸⣿⠀⣿⠀⣘⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⢡⣦⣿⣾⡿⣿⣿⣿⡀
⢀⣿⣿⠀⣿⢀⣹⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣿⣿⣿⠟⢰⣿⣿⣿⣇
⢸⣿⢿⠀⣿⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⢉⣽⣿⠋⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣤⣴⣄
⣿⡿⢸⡇⢿⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠉⠻⣿⠏⠀⠀⠀⣠⠞⣡⣿⣿⣿⡆
⣿⡇⢸⣷⢸⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡾⢟⣿⣿⣿⡇
⣿⡇⣾⡿⣆⠋⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⣩⠶⣛⣿⣿⣿⡇
⢻⣿⣽⡇⣿⣆⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⢰⡿⠀⠙⢿⣶⣤⣀⣀⣀⣀⣠⣤⣾⡅⠀⢈⠴⢞⣫⣽⣿⣿⡇
⠸⣿⣿⣿⣹⣿⣦⡙⢷⡀⠀⠀⠀⠀⠀⢠⡿⣿⣼⣧⡄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠈⠻⣦⠀⠈⡭⠴⢾⣿⣿⣿
⠀⠹⣿⣿⣿⣿⠿⢷⣄⠉⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣾⣧⣶⣴⡀⣄⣀⣀⡀⠀⠀⠀⢹⣇⠀⢒⣩⣭⣽⣿⣿⡆
⠀⠀⠈⠉⠉⠁⠀⠀⠙⢷⣦⣀⠀⠀⠀⠈⣿⣿⣿⠈⠃⠙⠛⠛⠟⠿⠿⠿⡿⠿⣧⠀⠀⢸⣿⠀⠠⠤⢶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣀⡘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣏⠀⠈⢛⣛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⢶⣤⣤⣤⣶⡶⠶⠶⠾⠿⠛⠛⠛⠻⣧⣜⢷⣮⣝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣝⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠈⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠄⠀⠀⠁⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠐⠀⢠⠁⣔⠀⠀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢠⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢟⢾⣿⣟⠟⠳⣾⣷⣶⣿⡀⠠⠀⡀⠀⠈⠁⡀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡞⢏⣌⠀⠀⠀⠄⢃⠴⠒⠝⢍⠱⠈⠭⠻⣟⢤⣀⣀⡄⠀⠀⠀⣠⠏⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⢀⣤⠄⠀⠀⣼⡿⣝⢛⣭⠎⣀⣀⠈⢄⠀⠀⠈⠀⡳⡞⠀⠠⠃⡈⠯⠃⢀⠀⢀⣰⣯⣾⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⢀⢀⠀⠀⢀⣴⣿⣿⢽⣾⣶⡁⢀⡿⣧⡀⠈⠀⠀⠀⢀⠄⠆⡠⢀⠔⠈⠐⠀⠀⣽⠿⣿⣿⣧⠊⠀⠀⠀⠀⠀⣀⠔⠁
⠀⡾⠋⢀⢔⡫⢭⡛⢽⡆⠀⢈⡉⠉⠠⠘⠁⠀⠠⠧⠶⢫⠀⠘⢤⡏⠀⠀⠀⠀⡀⡉⢂⣿⣿⠷⠂⠀⠀⠀⢠⠋
⡴⣅⠀⠀⡏⣶⡂⢸⡌⠠⠀⠚⠁⠀⠀⡀⠀⠈⠁⠠⠅⠎⠠⠀⡩⠁⠀⢀⣀⣀⣁⣁⣾⣿⠿⠂⠀⠀⠀⠀⠘⡄
⢆⠘⣆⠀⠳⢭⡥⠞⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠈⢈⡀⠤⣴⢁⠶⣵⢾⡁⠀⠘⣿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠘⡀
⠀⠂⣈⠀⠐⠂⠒⠈⠀⠀⠀⠀⣠⠆⣠⡎⠀⠀⠐⠂⠘⢰⣷⣼⣶⣿⠇⠀⠣⠀⠀⣸⣿⡟⠂⠈⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⠁⢤⣔⡃⠀⠀⠀⠂⡁⠀⣰⣿⠉⢠⠘⠀⠈⣀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠈⠑⠠⢤⣶⣷⣶⣮⡻⣿⣷⣤⣤⡤⣴⣏⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⠟⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠻⡻⣿⣿⣿⣿⣽⣿⣿⠟⠉⠀⠀⠀⡠⠁⠀⡞⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠙⠿⣭⣉⠉⠉⠑⠈⠛⠆⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⢙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠠⣀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠢⠀⠈⠀⡀⠈⠓⠲⠄⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠀⠀⠀⠀⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⣄⡀⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠛⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢾⡇⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠈⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⣿⡆⢸⣿⠇⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠛⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⣷⠀⢀⣀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠀⠻⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄⣤⠆⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⠀⣴⡄⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠟⠀⠛⠃⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⣿⣶⣶⡄⠀⠀⠀⠀⠀⣠⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡉⠉⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⣄⡀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⡴⠖⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⣿
⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⣿⡇⠀⡿⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠱⣦⣄⣀⣀⣤⣿⠃⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⣤⡀
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣿⡆⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠴⠉⢻⣆
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢴⣿⠁⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠻⡇⠀⠘⠀⢸⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⠃⠀⠘⠀⠀⣿⣷⡀⠀⠀⠀⠀⣸⡷⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣯⣽⠿⠂⡀⠘⠛⢃⡀⠃⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣄⢀⠀⠀⣠⣼⠛⠢⠀⠁⠲⠆⠀⠀⠀⠀⠀⠀⢸⠀⠸⠇
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠛⠛⢿⣿⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡿⠃
⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠃
⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡟⠛⠻⠛⠛⢿⡿⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠛⠲⠿⠿⠿⠿⠿⠿⠿⠚⠛⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⡀
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⣧
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢾⠆⢰⣿⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣧⡀
⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢷⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠃⢀⠀⠀⣠⠿⠻⣤⡔⠀⠈⢳⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠻⣄⣠⡿⠃⠐⠂⠀⠹⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠉⣷⡶⠶⣾⡇⠀⠙
⠉⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣿⡆⣤⡞⠻⡄
⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⠛⠛⠛⠛⠛⠿⠂
⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⣶⣀⡀⠀⠀⠀⠀⠀⠀⠉⠷⣶⣶⣶⣶⣶⣶⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠦⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠻⠛
⣤⣄⣀⣀⣠⣄⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣠⣤⣤⣤⣄⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠉⠈⠙⠋⠉⠉⠉⠙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣤⣤⠄⣀⣠⣤⠴⣶⣶⠶⡒⠒⠒⠶⠆⠀⣀⣤⣴⣶⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⢖⠛⡩⢶⠃⠿⢟⣫⣽⣶⣶⡜⢟⡴⣛⠽⠻⣟⣭⢾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢏⣴⠇⣠⡴⢖⣒⣛⠉⢷⡁⠀⣹⠗⡞⠀⠁⠠⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢻⣿⡾⢸⢏⡶⢋⣽⣧⠶⠿⢃⡤⢂⣤⣦⣄⠀⢀⣶⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠈⡏⡼⠋⠘⣠⣴⣯⣵⣦⣤⣬⡽⠞⠫⠵⢦⣌⣧⠸⣿⢡⣟⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡷⣰⠃⠀⣸⣿⢻⣤⡥⣋⠕⡊⡲⣶⣦⡴⢣⡿⠀⠀⠃⣸⣿⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⢰⠋⠀⢀⣤⣬⣤⠄⣽⣇⡴⢂⠀⣯⣜⠏⡀⠀⣿⣿⣹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡇⠀⠀⠀⠸⢦⣤⠄⠀⠀⢒⣭⢿⣯⡶⣻⠏⠋⣙⣴⠃⠠⢿⣿⠉⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣻⡿⠃⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⠿⠟⢁⠄⠀⠸⣿⣾⠃⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢿⡁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣲⣶⠶⠞⢉⡀⠂⠆⢻⡏⡇⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠳⣤⣀⣙⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⡿⣾⢁⡀⣴⣃⣾⠀⣾⣿⢸⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣼⣷⢾⠋⠀⠈⣇⣿⡆⢻⡇⣿⢋⡼⣻⣫⡏⠀⣿⡟⣾⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠁⠙⠅⠈⠀⠀⢀⣸⢇⡏⣸⣿⣷⠏⣸⣩⠟⢀⣾⣟⣽⣫⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣼⢡⣟⡾⠁⢠⡟⣡⣼⠿⣫⣾⢁⣥⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⡼⣡⠟⣋⣤⣴⠿⣛⣩⣴⠟⠫⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣠⠟⣛⡵⣟⣯⣷⠾⠟⢛⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡡⠂⣴⢃⡾⣫⠾⠛⡉⠀⢀⣶⢛⡍⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⢺⢇⡞⡽⢁⡞⠋⠀⢸⠟⣙⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡌⢸⢸⠣⠇⠈⢧⣀⣀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⢸⢸⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⢸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣀⡾⠿⠿⢳⣦
⠀⠀⠀⠀⠀⠀⢠⣿⠀⣧⣼⡟⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣀⣿⡀⢳⣿⣧⣄⣀⡀⠀⣼⡇⠀⡀
⠀⠀⠀⢤⡀⠉⠁⣽⣇⠀⠹⢟⡻⠿⠿⠛⠁⠀⢸⡿⠀⢀
⠀⠀⠀⣬⡁⠀⠀⠹⣿⣦⡀⢸⣿⣶⣶⠀⠀⢠⣿⡅⠀⣸⡿⠂
⠀⠠⠀⠛⠑⠀⠀⠀⠀⠛⠻⣿⣿⣉⣿⣆⣀⠾⡆⠀⢠⡟⠁⢀⡀
⠀⠀⠢⠀⢁⡀⠀⡐⠆⠀⠚⠉⠉⡙⠛⠛⠉⠈⣿⣿⣿⣄⢀⣿⠷⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢹⡆⠁⠀⠀⠀⣀⡾⠧⣤⣀⠀⠀⠀⢹⣿⣿⣿⣏⠀⣠⣤⠆⠈⠉⠉⠉⠙⢷⡄
⠀⠀⠀⠀⣸⣷⣿⣧⣤⡴⠒⠉⠀⠀⠀⠀⠀⠀⣴⠿⣹⣿⣿⣿⠛⠛⠁⠀⠀⣀⢰⣶⣄⡸⠃⠀⡄
⠀⠀⠀⢀⣾⣿⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢧⡙⣿⣿⣧⣸⣷⡀⠀⠀⠀⠈⠫⠾⠃⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣧⣴⣿⣿⣆⠀⠀⠀⠀⠀⢹⣮⢠⣿⣿⣟⢹⡦⢿⡄⠀⠸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠄⢰⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⢯⡈⢻⣦⠀⠀⡄⠀⢸⣿⣾⣿⣿⣿⠀⢡⣀⣷⠀⠀⠁⠀⠀⠀⠀⣸⣿⠀⣶⠆⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⢿⣀⠽⣷⠀⣠⣤⣾⣿⣿⣿⣿⣿⣷⣦⣿⣿⡀⠀⠀⠘⠀⠀⢰⣿⠋⣾⡏⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣄⣠⣤⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⡄⠀⠀⠀⠐⣿⠇⢰⡟⠀⠀⢠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢃⡀⠀⠀⢻⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣥⡿⣿⠿⠃⠀⣠⠟⣁⢹⣇⠀⠀⠀⠀⡄⠀⠀⠀⣿⣯⢀⡼⠀⠀⢈⠿⠃
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠛⢻⣦⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣯⣤⡿⠛⣿⣦⠀⠀⠀⠀⠀⠀⢠⣿⣇⡿⠃⠀⠄⠁
⠀⣤⠀⠀⠀⠘⢿⣾⠁⢠⠀⠀⣾⣡⣿⣿⣿⣿⣿⣡⣴⣿⡟⢀⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣩⣷⡄⠉⢻⣿⣷⣤⣴⣦⣤⣾⡿⠋⠀⠊
⠈⣿⣆⠀⠀⢰⡟⢻⣧⣈⠳⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⠘⣿⣧⢈⣉⠻⠧⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠘⢿⣦⣀⠀⠉⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⣧⣰⣿⣏⠈⠁⠀⠀⠀⠀⠀⠔⠋⠁
⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠃⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢠⣿⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⡏
⠀⠀⢀⠀⣤⣿⣷⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⡇⠀⡀
⠀⠀⠾⠿⣿⣿⣿⣿⣼⣿⡀⠀⠀⣠⣴⣶⣶⣶⣦⣤⣤⣤⠀⠀⢀⣴⣿⣿⠿⠿⢿⣿⣷⣦⣬⣿⣿⣿⣿⣁⣿⣶⠟⠀⠀⠠
⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣷⣴⣿⣿⡿⠋⠉⠀⠉⠙⢿⣿⡇⠀⢸⣿⡟⠃⠀⠀⠀⡿⢻⣿⣿⢻⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⢠⡟⢹⣿⣿⣿⣿⡁⠀⠀⠀⠀⣀⣠⣾⣟⠀⠀⢸⣿⣷⣶⣶⣶⣷⣶⣿⣿⡟⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣾⠁⣾⣿⣿⡟⢿⣿⣿⣿⣿⣿⡿⠟⠛⠟⠀⠀⠈⠛⠋⠉⠉⠙⠛⠉⠀⠀⠀⠀⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠏⠐⣿⠿⡿⠁⠀⠙⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠘⠀⢿⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠷⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣄⢀⣠⣴⣿⣿⣿⣿⣿⠿⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠙⣿⣿⣧⣈⣿⡏⠉⠀⠀⠀⠀⠀⠀⡿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠙⣿⣿⣿⣿⣇⡀⠀⣀⣠⠞⠁⠞⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣙⣶⣾⣟⣡⣍⣿⡿⠛⢻⡟⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣀⣉⣀⠉⠉⠉⠙⢻⣤⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠟⠉⠈⠀⠈⢠⡄⠹⠃

⠀⠀⠀⠀⠀⠀⠀⣾⣷⠶⣤⣤⣄⣀⣤⣤⣄⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⣀⠔⢒⣠⠤
⠀⠀⠀⠀⠀⠀⠰⡍⢹⠂⣽⠻⣿⣌⣡⣿⣿⣿⣿⣿⣦⣄⡀⠰⠷⣿⣿⣿⣿⣿⣿⣿⣶⣾⣧⣶⣏⣤⠴⣚⣡⠄
⠀⠀⠀⠀⠀⠀⡄⠀⢸⠃⣾⠀⠁⠹⢛⣿⠛⢷⣴⣿⣿⣿⡿⡀⠀⢻⣿⣿⣿⣿⡋⢾⣿⣿⡿⣿⣿⣿⣿⣯⡤⣖⣩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠈⠆⡀⠙⠻⣿⣽⠉⢏⠑⡮⠝⣿⢿⣿⣿⣿⣿⣿⣴⣿⣿⣟⣿⣿⣿⣿⣿⣟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠿⢾⡀⠀⠀⠀⠀⠀⠙⣠⡀⠈⢹⠇⣿⣿⠓⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⢀⣴⣆⠀⠀⠀⠀⠀⡁⠀⡿⠇⠀⠀⠀⠀⠀⠀⠩⣙⡶⠎⠀⠘⠁⠀⠠⠟⢛⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣄⣀
⠈⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⣀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠓⢄
⠀⠀⠀⠀⠉⠳⢂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣟⣛⡛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣉⣂⠀⠀⠀⠘⠟⠀⢡⡛⢿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⢩⠿⣿⣯⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠤⠛⡝⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡵⠛⠁⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠌⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠎⠀⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣦⡄⠀⠀⠙⠻⠶⠄⠀⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢂⠈⠀⡜⠀⠀⠀⡀⡀⣀⣀⣀⣠⣴⣾⣿⣿⣿⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠊
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠠⠂⠀⠀⠀⠰⠀⠀⠐⠣⠬⣽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠐⠀⠀⠔⠀⠀⡂⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠂⠀⡘⠀⠠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠞⢋⣿⠿⣳⣦⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣴⠾⢗⣵⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣈⠁⠀⠡⢪⢞⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣤⣤⣠⣀⣀⠀⢣⠀⠀⣀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⢿⠿⡿⠳⡿⣿⠛⠛⣿⣿⡳⣾⣷⣶⣶⣿⣿⠂⠀⠀⠀⢀⡠⠔⣲⡇
⠀⠀⠀⠰⢾⠟⠿⠛⠟⣽⡿⣆⣻⣤⣴⠿⣿⠀⢩⢿⡏⠀⢻⣿⡿⣿⡿⠝⢿⠟⠿⣿⣶⣦⡋
⠀⠀⠀⠙⠪⢍⠐⠚⢥⡍⠞⠠⠬⠛⠿⠒⠛⠀⠈⠿⠄⠀⠑⠋⠀⣈⡀⠀⠀⠀⠀⠛⢺⣄⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠄⢀⣀⣄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠂⠨⠄⠈⠄⠀⠀⠀⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⣀⣀⣤⣄⣄⣀⣀⣀
⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⢉⣉⡀⢀⣈⣉⣉⠛⠛⣉⣤⣤⠀⢩⣍⣙⠛⠶⣦⣄
⠀⠀⠀⣾⠃⢀⣀⡈⠉⠛⠛⠶⢶⣦⡴⠞⠋⠁⠀⢀⡖⠋⠉⠀⣴⠏⣉⠉⠉⣽⣛⣋⣴⠖⡲⠀⠀⢹⡇⠀⠀⠙⢿⣦⣀⣤⣴⠶⠶⠟⠛⣷⡄
⠀⠀⠀⢿⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⢠⣀⠀⠀⠀⠊⢀⣠⣶⠟⠀⣿⡁⠀⠀⣼⠀⠘⠃⠞⠇⠀⢀⡌⠳⢦⣴⣇⣀⠙⠏⣀⣤⣶⣿⡷⠂⣿⠁
⠀⠀⠀⠸⡇⠀⠀⢰⣿⣿⣿⣿⡏⠀⠠⣬⠳⣄⠀⠙⠛⣩⡿⠷⢿⡟⠃⠀⣈⣵⣀⣀⢀⣐⣶⣾⣋⣭⡙⣾⡷⣄⠸⡆⢀⡉⡻⠟⠉⠀⣸⡏
⠀⠀⠀⠀⢻⡀⠀⠀⣙⣿⡿⠁⠀⠀⣀⠩⣤⣌⠳⡝⣿⣏⣼⣿⣶⣿⡟⠛⣫⡽⠿⠭⠿⠶⠶⢿⡟⠻⠟⢛⣿⢈⡣⠶⣏⠤⠈⠃⠀⣴⡟
⠀⠀⠀⠀⠀⠙⣦⡀⢉⣿⠍⠉⢛⣒⡭⢤⣀⠙⢷⣝⣮⠻⠿⢿⣶⣤⠁⠘⢁⣠⣤⣤⣤⣤⣤⣤⣉⠃⠐⠫⢖⣩⣴⢺⡅⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⠷⣾⣯⠥⢤⣘⣛⠃⠐⠾⣿⠶⢭⣝⡓⠀⠴⢤⡀⠀⠀⢻⣿⣭⣤⣤⣤⣤⣼⡟⠀⠀⠪⢭⣤⣖⣲⣶⢒⡂⢺⣷⡀
⠀⠀⠀⠀⠀⢠⠏⣞⠟⢻⡧⣤⣙⣛⣉⣻⠷⠒⠒⠦⢬⣽⣓⠂⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠟⠉⠀⠀⠀⠀⢁⣉⣀⠡⢽⡭⠁⢺⣿⠃
⠀⠀⠀⠀⠀⣿⡈⠐⠀⣸⡧⢬⣭⣥⠤⢽⣯⠭⠭⢍⣛⣒⡦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⠿⢶⣦⣤⣤⡀⠀⣐⠭⣈⠉⢹⣇⠀⠺⣿⡆
⠀⠀⠀⠀⢠⣯⠔⡢⡀⣸⣿⡀⢻⣿⡀⠘⣿⠀⠀⠄⠶⣖⣻⡅⢀⣤⣶⣋⣉⣁⣠⣤⣤⣤⣤⣀⡤⢍⣻⡾⠋⣑⣈⠁⢸⣇⠀⠙⣮⣷
⠀⠀⠀⠀⣼⠡⡪⢞⠁⣼⣿⡆⠸⣿⡇⠀⠻⣦⣄⠀⠉⢻⣿⣿⣿⣿⠀⠀⣿⣄⣧⣦⣦⣤⣤⣼⠀⠀⣿⠇⠀⣹⠃⠀⣿⡗⠀⠀⠩⣻⣧
⠀⠀⠀⠀⣿⢠⢖⡡⠀⣾⣿⠃⢐⣿⣿⣦⡀⠘⢧⠀⠀⠈⢿⣿⠯⣿⡄⠀⡟⡿⡟⡟⣿⢻⣿⢿⡀⢸⣧⣤⠾⠃⠀⢀⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⡶⣫⠖⢀⣿⡇⠀⠈⠸⣿⣿⣷⡀⠈⡇⠀⠀⠼⣿⣢⠙⣿⣄⣿⠉⠧⡇⠟⠈⠸⣸⣷⡏⣼⠁⠀⢀⣦⣿⠿⠃⠀⡀⠀⣜⣿⡇
⠀⠀⠀⠀⢸⡆⠁⠀⢠⣿⡆⣸⡟⡀⠉⢻⣿⡇⠀⢻⡆⠀⢸⣿⣿⣁⢻⣿⣿⠀⠀⠀⠀⠀⢠⢿⡏⣷⠿⢿⣿⣿⡏⠈⠀⠀⢱⣱⣦⣼⣿⠃
⠀⠀⠀⠀⠈⣿⣴⡄⡄⠻⡿⣿⣿⡅⠀⠘⠛⣗⠀⠈⢳⣄⠸⣿⣿⣯⣾⡇⠸⣦⣤⣤⣤⣴⠏⢸⣇⣿⠟⠀⠀⠈⠀⠀⠀⣠⣣⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣧⣷⢠⢠⠡⢻⣦⠀⠀⠀⠀⠀⠀⣈⠛⣶⣾⣿⣿⠏⢧⣀⣉⣥⣤⣭⣏⣀⣸⢻⣿⡆⠀⠀⠀⢠⣀⣰⡟⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣷⣤⣿⣬⢠⢰⠀⢠⢃⢅⣧⣉⢹⡵⠛⣯⣷⣶⠶⠶⣦⣤⣤⡤⣬⣠⢟⡀⢦⢦⢰⣸⢻⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣧⣦⣧⣮⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣼⢸⡞⡄⢾⣸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠋⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢠⡀⣄⠀⠀⠀⠀⢀⣠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣦⣼⣶⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠤⣟⡯⡙⣝⣦⠴⠒⠋⠉⠉⡅⠀⠀⣡⣤⣄⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⡙⠒⠆⠙⢋⢝⠀⣰⣧⠀⠀⠀⠁⠀⠸⡟⢏⣹⠂⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⣤⣛⣒⠒⣬⠶⠤⠔⢉⣡⠞⠁⠸⡆⠀⢠⣄⡀⠀⠈⠉⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⡺⠵⠒⢢⡽⠁⠠⣴⣿⡍⠀⠀⠀⢸⣇⡰⠁⢈⠛⢆⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⢨⡟⠩⢭⣟⣀⠀⠈⠁⠈⠙⣦⠀⣠⢛⠟⠛⡄⠀⠀⠈⣆⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠉⠙⢳⣤⣀⠈⣖⠁⠁⠀⠀⣫⠞⠁⢰⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⣀⣀⣀⡀⠹⡌⢣⣸⠀⣠⠔⠋⡝⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⡋⠋⠙⢯⡁⠀⠑⢧⠀⣿⠞⠁⠀⠀⠸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣀⢱⠀⠀⢸⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⢉⣽⣏⠢⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡿⣿⣿⣽⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡏⣇⢣⠙⢿⢿⠩⣑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠈⢧⠑⢄⠑⠌⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠃⠐⢠⠋⠙⠣⢄⣉⡐⠚⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣧⠃⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⡇⠀⠀⣶⠀⠀⣾⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠇⠀⠀⠟⠀⠀⠺⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠆⠀⠀⠤⠀⠀⠰⠀⠀⢀⡼⠃⠀⠀⢀⣤⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⠀⠀⠀⠀⠈⠙⠶⠶⠶⠶⠶⠶⣄⡀⢰⡶⠖⠋⠁⠀⠀⢰⡏⢠⣄⠀⣤⠌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢻⣄⡾⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠀⠀⠀⠀⠀⣀⢺⡀⢀⣤⣠⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣿⠀⠀⠉⠀⠀⣿⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⠳⣌⣀⠁⣈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠉⢩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠴⠞⠛⠉⠁⠀⣠⠀⠀⠈⠛⠶⢤⣴⠞⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠼⢷⣄⡀⠀⢀⣠⡾⠧⠤⣤⣀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⢠⡞⢡⡞⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⣡⡟⠀⠀⠀⠀⣠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏⠀⠀⠀⢀⡾⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿⠀⣀⣀⣸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⠀⣴⠏⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣼⠋⠉⠉⣿⠛⠲⠶⠦⣼⣧⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⠙⠷⣤⣤⣟⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠁⠀⢠⡾⠋⠉⣷⠀⢀⣀⣤⠴⠞⠛⣯⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣠⣭⠿⠟⠲⠦⣤⣄⣀⣀⣼⠇⠀⠀⠀⠘⢧⣀⣤⠼⠟⠋⠉⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⡆⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠙⢿⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠸⠇⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡄⠈⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⡀⠘⣀⣠⣦⣤⢀⣤⠄⠠⠶⣆⢼⡟⠘⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢑⠤⣖⣿⡿⣛⣋⣽⣿⡷⢀⢾⡀⡉⠘⠙⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣪⡵⢚⣩⣶⣾⣿⣻⡅⢻⠇⠀⠀⠂⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣼⣞⣹⣶⣿⣿⣿⣟⣷⠣⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠦⢄⣠⡄
⠀⠀⠀⠀⠚⢁⡈⣻⣻⣿⣡⡾⢿⣯⡜⠙⠎⠥⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠉⠁⢀⠤⠊
⠀⠀⠀⠀⠀⠈⣽⣽⢏⣾⣿⣶⡌⠈⠈⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⣸⣿⢃⣾⣯⣄⠉⠀⠐⠀⠀⠀⠀⠘⠿⠂⠀⠱⣄⡤⠂⠀⠐
⠀⠀⠀⠀⠀⣿⣏⣾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁
⠀⠀⠀⠀⠀⣿⣿⣍⡁
⠀⠀⠀⠀⠀⠸⣿⣏⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠹⣿⡁⠀⠀⠐⠋⠀⠁⠇⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠘⢖⠀⠀⠀⠀⠀⠀⠂⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠳⢦⣄⣀⣠⡾⠟⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣯⣿⣿⣿⣦⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⠿⠟⠋⠛⠋⠉⠉⠈⠉⠁⠉⣁⣈⣉⡈⠉⢡⡾⠉⠉⠙⠛⠿⣷
⠰⣶⣶⣶⣿⠿⣿⡟⠉⢉⣉⣁⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠉⠙⠛⠉⠁⢸⣇⣀⣠⣶⣿⡿⠟
⠀⠈⠻⣿⣿⣷⣹⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣩⣽⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⣿⠏



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⠋⢉⠙⠒⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⢸⠋⠹⣆⠀⠀⠘⠛⠻⠻⠋⡡⣤⠈⢻⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⢀⣄
⠀⠀⠀⠀⢸⠀⠀⢸⡄⠀⠀⣙⠒⠢⠰⢝⡋⠀⢨⣿⣏⠻⠛⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠘⠛⠶⢶⡶⠤⣀⠈⠇⢸⡖⣷⠀⠠⡆⢲⣯⡜⣿⢻⠿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⡀⠀⠀⠀⠺⠗⣧⠀⣤⡁⠀⡀⠁⠯⠻⠇⢱⡾⠰⢻⣏⣿⢿⣿⣿⣿⣶⣤⣀
⣠⠤⠤⠤⠄⠀⣠⠴⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠙⠋⠻⣷⠾⣿⢠⣤⠀⢀⠛⠀⠁⠛⠉⠽⠇⡯⢻⣿⣿⡋⠉
⢳⡒⠶⠖⢚⣋⠁⠀⠀⠀⠀⠀⠀⠸⡀⠀⠱⡀⠀⠀⠀⠀⠀⢰⣀⠈⠙⢳⡿⣸⣷⠖⠀⠀⠀⠉⠛⣯⡻⣿⣧
⠀⠙⠲⢤⣀⠈⠑⠖⠊⠁⠀⠀⠀⠀⠃⠀⠀⠈⠂⠀⠂⠀⢠⣿⡯⠀⠀⠘⠃⣿⠁⠀⠀⠀⠀⠀⠀⣿⡱⣿⣿⡆
⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⣠⡴⠋⠛⠉⠀⠀⠀⠀⠛⢀⠀⠀⠀⠀⠀⡸⣙⡻⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡠⠊⢸⠿⠷⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣰⣔⣄⣄⡴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣤⣶⡆⠰⠟⣃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠉⠉⠩⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣯⠁⠴⠷⢽⣿⣿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠉⠳⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣌⣻⢷⡄⢰⣜⢳⣽⣿⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⢙⣿⣷⣄⠀⠀⠀⠀⡀⠀⠈⢀⡄⠀⠀⠀⠀⡀⢈⡿⡧⠀⠶⢎⣃⣿⣿⠇⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⣿⣿⣿⣧⠀⠀⠀⢳⡇⢀⣼⣿⣄⠀⢀⣰⣾⢻⡇⢰⣶⣶⣾⣿⣿⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⢋⣷⣤⣠⣤⣽⣿⣿⣿⣿⣶⣾⣿⠏⢨⣴⣿⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠛⠿⣿⠿⢯⣽⡿⡟⠟⠙⠉⠀⠀⠈⣿⣭⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⣿⣿⣿⣿⡿⠦⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣨⣿⣿⡿⠋⠀⠀⠀⠈⠉⠓⠒⠲⠶⠖⠒⠾⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⠉⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠙⠻⠻⠿⠏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⢄⣀⠈⠗⠶⣬⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣦⡀
⠀⢀⣈⠉⠛⠶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⢟⠷⣙⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⣆
⠀⢀⠋⠙⢢⣼⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⢀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢹⣷⣄⣀
⠀⡘⠛⢲⣾⣿⣿⣿⣿⣿⣿⠯⠂⠀⠁⠀⣠⡴⢆⣾⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣿⣿⣆
⠀⠉⢓⣿⣿⣿⣿⣿⡿⠓⣳⣀⡤⢀⡀⣐⣿⢱⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣾⣿⣧⡀
⣈⠓⣾⣿⣿⣿⣿⠛⠉⠀⢰⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⢿⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣵⣾⣶⣷⣶⣶⣍
⠈⢙⣿⡿⣿⣿⡯⢠⣴⣿⣶⡿⢿⣿⣷⣾⡿⡿⠘⡿⠙⢰⠁⣾⣿⡿⣿⢿⠟⠁⠛⠛⢛⠟⣳⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢻⣿⣿⣿⣿⠃⠘⠋⠁⢀⠴⠋⠛⢟⡏⠀⠀⠣⡘⢦⣸⠸⠝⢛⠛⠌⠢⠁⠀⠀⣰⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⢼⠃⠀⠀⠀⠀⢀⣠⠌⠀⠀⠀⠀⢻⣕⡟⢦⡑⠤⠀⠀⠀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⢻⣹⣿⡿
⠀⢸⣿⣿⣿⣿⣜⠷⣶⣶⣶⣿⣿⡦⠀⠀⠀⢀⡰⡌⣷⡧⣄⠉⠛⠒⣏⡚⢯⣈⠈⢸⣧⣉⠋⠛⠿⣿⣿⠆⠇⢹⠗
⠀⣾⣿⣿⠻⣿⣿⡧⠁⢙⣿⣿⣋⠀⠀⣀⠊⠉⠷⣷⣾⣷⠞⠻⢶⡿⠵⢿⡶⢭⣛⣦⣭⡛⢛⠦⣌⠒⠧⠤⠃⠘
⣸⡿⢻⣿⣿⣿⣿⡗⢰⣾⣿⣿⡿⡜⠓⢇⠀⠀⠃⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣤⣄⣀⣠⣤⠞
⢸⠃⠏⣿⣾⣻⣿⣿⠂⢠⣭⣾⣿⣿⡟⠉⢻⣶⠋
⣸⢰⢰⠸⢿⣯⣿⣿⣷⣆⠉⠙⠻⣿⡷⣤⠜⣿⡀
⢸⡇⠀⡀⡈⣿⣿⣿⣿⣿⣷⣢⡰⠿⢿⣿⢀⣆⠍⠐⠤⡀
⠘⡇⡀⡇⡇⡇⠟⠿⣿⣿⣿⣿⣷⣴⣂⠻⢿⣿⣗⠢⢄⡈⡆
⠀⠸⡇⠀⠁⠇⠠⠰⣨⠟⢿⣿⣿⣿⣿⣧⣌⣻⣿⣧⣤⠝⠃
⠀⠀⠘⠢⠦⠶⠛⠚⠁⠀⠀⠈⠙⠿⣿⣿⣿⣿⣯⢿⣿⣷⠠⣄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣾⣿⣿⣠⣿⣕⠀⠈⠀⠀⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢟⠿⢮⡑⠨⡐⠀⠄⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⣏⠏⢎⠀⠱⡀⠑⠀⣁⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡂⠀⡈⢆⠣⠐⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢠⢀⠱⠘⡄⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡈⡄⠄⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢀⠎

⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⣸⣿⣿⡿⣻⢦⣀⠀⠸⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣦⣤⣤⣤⣤⣀⣀
⠀⠀⠀⣾⣿⣿⣿⡇⣉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⢻⣫⣿⣿⣿⢳⣶⡆
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠈⠉⠛⠛⠛⠗⠟⠻⠋⠻⠇⠁⣻⢆⡋⠨⠛⠙⠉⢉⢺⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⠀⠀⠀⠉⡩⣿⣯⡃⠀⠄⠀⠀⠀⢀⡘⠀⠬⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠄⢤⣠⣼⠟⠀⠂⣤⣄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⡶⠒⠒⠒⠒⣖⠒⠒⠒⠒⡒⠲⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢚⠉⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⡉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠲⡄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠉⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠒⠹⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡌⠀⠈⣆⢀⣀⠀⡴⠒⠲⡂⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢅⠄⠈⠉⠀⠈⠉⠀⠍⠁⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢸⠀⠀⠀⠀⠀⠀⢀⡜⢀⡀⠀⠀⠀⠀⡀⠀⣤⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⡀⠀⣀⣤⠴⠚⢹⠀⡾⠈⡒⠀⢢⠊⢹⡄⠹⠑⠤⡤⡤⢤⠲⢤⠶⠟⠚⠓⠲⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣸⡷⢚⣩⡒⠒⠛⣁⣤⣭⣿⣥⣄⣘⣓⠚⠁⠀⠰⣄⡀⠈⠢⡙⣌⠳⡀⠀⠀⠀⠀⠈⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⣠⡍⠁⠀⠸⣿⣿⣿⣿⠛⠻⢿⣿⣷⣦⡀⠀⠈⠻⢄⠀⠱⡈⢆⢳⠀⠀⠄⠀⠀⡸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⡟⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢳⡀⠱⡈⡄⡇⠀⡤⠀⡴⢁⠀⢰⡁
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⠀⠀⣠⣆⠐⠦⣀⠄⠀⠈⠉⠉⠉⢁⠀⠀⠤⠰⠒⣶⢠⡇⠀⢇⢱⡁⣰⡧⠊⣠⠊⢠⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⡏⢸⠐⠏⢓⠶⠤⠤⠤⠤⠒⠛⠂⠤⠴⠒⠉⣸⠁⠀⢸⣼⡶⣋⣡⠖⢁⣴⡿⠟⡇
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠓⠒⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾⠏⠻⣥⠤⠶⠋⠁⠀⢸⡇
⠀⠀⠀⢠⣾⣗⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠑⠠⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠤⠒⡿⠀⠀⠀⠀⠀⠘⣦⠀⢱⠀⠀⠀⠀⠀⢸⠁⣀⣤⣄
⠀⠀⣠⣯⡤⠤⣤⣀⠈⠐⢲⡄⠀⢠⠂⠀⠀⠀⢠⣴⡷⡀⠀⠀⠉⠈⣰⠯⣤⡄⠈⢹⠀⣄⠀⠀⠀⠀⢻⢨⠏⠀⠀⠀⠐⠀⠋⠉⠉⠉⠈⠑
⡰⢊⠵⣂⡤⠤⠤⢶⣿⣦⣀⣉⣤⠧⠤⢄⢠⢴⠟⠉⢺⡕⠦⠀⢠⠴⡃⡴⠛⣹⣆⣂⠀⢸⡖⠈⠁⠀⠸⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀
⣷⢡⠞⠁⠀⢰⡏⠁⠀⠈⠉⠁⢠⠀⠀⠀⡇⠀⠀⠀⠀⣿⠛⢉⠻⡎⠈⠀⠀⠘⠋⠁⠀⠀⠓⠀⢀⠀⠀⢀⠈⠉⠁⣦⠀⠀⠀⣇⠀⠀⠄⠀⢱
⣿⢸⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⣤⢀⠄⢨⡇⠀⠀⠀⠀⢸⡩⠲⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⢄⠀⢂⣺⠀⣤⠤⠶⠋
⢸⣼⡄⠀⠀⢇⠀⠀⠀⠉⠉⠉⠙⢦⣒⡾⠥⠤⠤⠤⠠⣜⠀⠁⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣤⣤⣭⣉⠉⠁
⠀⠉⠛⠓⠶⢠⠁⠀⠀⠒⠒⠒⠚⠙⡇⠀⠀⠀⠀⠀⣠⢧⣀⣀⡉⠥⡀
⠀⠀⠀⠀⠀⠘⠶⠤⠤⠤⢤⣀⠠⠜⠁⠀⠀⠀⢠⠜⠁⠀⠀⢀⡀⠁⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣷⣤⣤⣤⣶⣶⣶⣶⣴⠤⢤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⠁⣾⣶⣿⠿⢛⢭⢿⡃⠀⠈⠃⠀⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⣻⣿⣄⠈⠻⢿⣦⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠝⠿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⠿⠛⢋⣠⡴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⣀⡤⠖⠒⠺⣽⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣋⠀⣠⠖⠋⠁⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡯⠚⣡⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠁⠰⠉⠟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⣅⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁