Clock Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⢈⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⠤⣖⣲⡭⢿⣛⣋⣁⣀⣠⡶⠦⣤⠀⠈⢙⣁⣀⣀⣀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⣶⡭⠷⢚⣻⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠓⠿⣄⡀⠸⣏⢆⠀⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡴⠞⠉⢀⡠⠤⣒⣺⣭⡭⢭⣭⣙⣒⠤⣄⠀⠙⠳⣜⢯⣣⡀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⡠⣖⡽⠞⡋⠁⠀⠀⡆⠀⢀⡆⢭⡛⢦⣑⢄⠀⠈⢿⣷⡵⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⣪⠞⡉⠀⠀⠛⠀⠀⠀⠀⠀⣨⡕⠾⠃⠀⠙⢦⡳⡀⠀⢻⡽⣞⢆⠀⣼⠰⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡜⡴⢃⣺⣿⠂⠀⠀⠀⠀⠀⠀⣼⢫⠇⠀⠀⠀⢠⠌⢳⡙⡄⠀⢷⠘⢮⡾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⡜⡼⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⣷⠏⠀⠀⠀⠀⠀⠀⠀⢷⢱⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠃⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠦⢤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⡇
⠀⠀⠀⠀⠀⢹⠀⠘⣿⡇⠚⠁⠀⠀⠀⠘⢧⡴⠞⠛⠻⡀⠀⠀⠀⠀⠀⠀⠘⠃⣸⢣⠃⢀⡇⢰
⠀⠀⠀⠀⠀⠘⣇⠀⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⡏⡜⠀⣼⠁⠏
⠀⠀⠀⠀⠘⠄⢹⡄⠀⢿⣧⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⡾⢣⣞⡜⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠻⣷⣄⠀⠀⣴⣶⠆⠀⠀⠀⠀⢀⠀⠀⠀⢉⡵⣫⠎⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠾⣝⣦⡀⠈⠻⣷⣤⣛⡛⠀⠀⠋⠀⠀⢈⣁⡤⢖⡫⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠴⠞⠉⠙⢦⣄⠈⠙⠻⠭⠭⠽⣛⣛⣛⠭⠕⠚⠉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡟⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠙⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⠛⠛⠛⡟⠋⠉⠻⠛⠉⠙⣿⡇
⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇⢹⠀⢸⠈⣹⠂⡧⡎⢹⠤⢰⠉⡧⣽⡇
⠀⣸⠃⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⡇⠈⠀⣸⠘⠓⢀⣇⠙⠊⢀⡈⠓⠁⣾⡇
⠀⢹⡄⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣧⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⠟⠡⠀⠐⠀⠠⠉⠻⣆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⠁⠂⠀⠀⠀⡔⠀⠀⠂⠘⣇
⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣠⣤⡴⠞⠋⠀⠀⠀⠀⣿⠐⠀⠀⠀⠚⠁⠀⠀⠐⠂⣿⠇⠀⠀⢀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡀⠀⠀⣈⣠⣾⠋⠀⠀⣨⣿⣿⣿⣷⣦⠀⢤⡤⠀⣠⣶⣶⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠳⢴⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⢟⣩⣤⣼⣧⣤⣝⡻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠟⠋⣡⣿⠟⡋⠉⠉⠉⠉⢙⠿⣿⣍⠻⢿⠇
⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣸⡛⣅⠀⠉⠀⠀⢀⡴⠋⠀⣘⣿⣧
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠁⠘⣿⡇
⠀⠀⣿⠇⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠁⠐⠒⠒⠚⠁⠀⠀⠀⠈⠁⣿⡇
⠀⠀⢿⡇⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣸⡏⠀⣴⣶⣶⢰⡶⣶⡄⣶⢶⣦⣴⡶⣶⠀⠀⠘⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡿
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⣾⣿⢸⣿⣼⡇⡗⣿⣿⣟⣿⢺⠀⠀⠀⠘⢿⣦⣀⠂⠀⢠⠀⠀⢣⣠⣾⠟⠁
⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠿⠿⠿⠸⠶⠾⠃⠿⠾⠟⠻⠷⠿⠀⠀⠀⢀⡼⠋⠻⠿⢶⣶⣶⡾⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠉⠀⢀⣀⣤⡴⠶⠶⠶⠶⠖⠶⠶⠶⢤⣤⣀⡀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⢀⣠⡶⠟⠋⠉⠀⠀⠀⠀⢠⡄⢠⣤⠀⠀⠀⠀⠉⠙⠳⢶⣄⡀⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠰⢿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣠⡾⠋⠀⠀⠀⢸⠇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠙⢷⣄⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣦⡀⠈⢻⣧
⠀⠀⠀⠀⠀⣾⡟⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⣧
⠀⠀⠀⠀⣼⡟⠀⢀⡿⠁⣶⠀⣿⣷⠀⠀⠀⠀⠀⠀⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠈⣷⡀⠀⢿⣇
⠀⠀⠀⢰⣿⠃⠀⣼⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣧⠀⠘⣿⡄
⠀⠀⠀⣼⡟⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⡇
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⠐⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇⠀⢸⡿
⠀⠀⠀⢹⣷⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠘⣿⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢰⣿⠁
⠀⠀⠀⠀⠹⣷⡀⠀⢿⡄⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢠⡟⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢻⣷⡀⠈⢻⡄⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⡟⠁⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣦⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⠳⣄⡀⠀⠀⣭⣿⠇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣯⡉⠀⠀⠀⢀⣴⠟⠁⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⢦⣄⠀⠋⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠚⠃⠀⣠⡴⠟⠁⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠉⠀⣀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⣀⠀⠉⠉⠛⠛⠳⠶⠶⠶⠶⠛⠛⠛⠉⠁⠀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠒⠒⠶⠶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣤⠶⠛⠛⠛⠛⠛⠲⢶⣤⡉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠉⠛⠳⣦⠀⡏⢨⡇⢠⡶⠛⠉⠉⠉⠛⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠉⠀⠀⠀⠀⣸⣇⣇⣀⣇⡟⠛⠲⠶⢤⣄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣉⣉⣭⣽⣶⣦⣄⠀⠈⠛⢦⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣤⣤⠶⢞⣫⠶⢚⣋⣭⣤⣤⣤⣤⣀⠉⠻⣦⣀⠀⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⡾⢋⣴⠞⠉⠁⢠⠤⡄⠀⠀⠉⠛⢶⣄⠙⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠋⡴⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⢷⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⡿⢡⡞⠁⠀⡠⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣾⠁⡞⠀⠀⠀⠑⠦⣄⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⢸⡇⣠⣀⠀⠀⠀⠀⠉⢿⡉⣹⡀⠀⠀⠀⠸⣭⠇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠸⡇⢸⡇⠓⠋⠀⠀⠀⠀⠀⠀⠉⢧⠹⣄⠀⠀⠀⠀⠀⣼⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢷⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣳⠀⠀⠀⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⢣⡘⢷⣀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠈⠁⠀⣠⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡀⠉⠛⠶⣤⣄⣀⣀⣙⣁⣠⣤⡴⠞⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢙⣆⠀⠀⠀⠀⠉⠉⠉⣀⣀⣤⡤⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⠋⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⢢⣼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⠒⠒⠒⠀⠉⠁⠐⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠁⠀⡀⠀⠠⡀⠂⠲⠀⠀⠶⠀⢐⡒⠀⢠⠄⠀⣀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⡀⠄⠲⠡⠀⠃⠈⠀⠀⠀⢰⡆⢀⡶⢶⣦⠀⠀⠈⠉⠐⠫⠌⣐⠤⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⡠⠐⠈⢛⡀⢀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠈⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⠐⢟⠢⢄⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⡀⠔⠁⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡼⠧⢤⡿⣴⡶⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠣⣝⢦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡠⠊⢀⠴⡥⠊⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠑⣌⢦⡀⠑⣄
⠀⠀⠀⠀⢀⠜⠀⡠⢤⠈⠀⠀⠀⠀⠀⢤⡉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠈⠣⡑⡄⠈⢦
⠀⠀⠀⢠⠏⠀⣴⡀⣀⠀⣠⣤⡀⠀⠀⠈⢿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠘⠌⢆⠀⢣
⠀⠀⢠⠃⠀⡄⠈⠀⣿⢰⠃⠀⢹⡄⠀⠀⠘⣿⣿⣏⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣹⠇⠈⠋⢂⠀⢣
⠀⢠⠃⠀⠌⡘⠀⠀⣿⢸⡄⠀⣼⠀⠀⠀⠀⠘⠴⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⠄⠀⡆⠀⠀⢃
⠀⡎⠀⠈⠱⠀⠀⠈⠉⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠐⡠⠀⠈⠄
⢀⠁⠀⢦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⠀⢀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠸
⠀⠀⠀⠀⠀⣼⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡖⠀⠀⠒
⠀⠀⠘⠃⠀⠻⠷⠆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣶⠄⠀⡤
⢀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟
⢸⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠈⠙⠀⠰
⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆
⠀⠂⠀⠐⠣⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠰⠀⠀⢠⠃
⠀⠈⡄⠀⢀⠄⠀⠀⢻⣶⡋⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡇⠀⠀⢀⢀⠀⠀⠆
⠀⠀⠘⠀⠀⢀⣀⢰⡏⠙⣷⠀⢀⣤⢤⢶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠥⢼⡧⠖⠀⢀⠃⠀⠌
⠀⠀⠀⠐⡀⠈⠁⠀⠙⠚⠁⢀⣟⡾⣯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠛⠃⢀⠌
⠀⠀⠀⠀⠈⢆⠀⠁⠀⠄⣠⣿⠿⠒⠛⠁⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠈⠐⠁⢠⠊
⠀⠀⠀⠀⠀⠀⠣⡀⠈⠼⠛⡁⠀⠀⠀⠀⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⡋⠀⠀⠀⠀⢀⠔⠡⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠠⠐⢄⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡗⠀⠀⡀⠀⠡⠊⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠁⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢷⡄⠀⠀⠀⠀⠀⠀⣀⠴⠛⢠⡔⠀⠐⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠁⠀⡤⢀⡀⠀⠀⠀⠘⢷⣤⡴⠃⠀⠀⠀⢀⠀⠄⠀⠠⠄⠀⠁⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⣀⠀⠀⠀⠐⠀⠁⠄⠀⠀⠿⠀⠐⠀⠤⠙⠀⠀⠁⠀⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠙⣧⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡈⢷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣾⠏⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠱⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠁⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠘⠀⠞⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢀⠎⠀⠀⣀⠶⠋⠁⠀⡈⠂⠀⠀⠀⠨⡀⠀⠉⠳⢎⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⠀⠀⠁⠀⣀⢄⣈⠤⠐⠒⠒⠒⠠⠌⣐⢄⡀⠀⠈⢂⠀⢳⠀⡇⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢧⠀⠇⠀⡀⠂⠂⣡⠞⠁⠀⠀⠀⢀⢀⠀⠀⠠⠀⠑⠺⡀⠀⠈⢀⡞⢰⠃⠀⡰⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣄⠟⠁⠀⠀⠠⠶⠀⠘⠚⠀⠀⢰⠀⠀⠀⠘⢦⣀⡈⠀⠀⠀⠈⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⢀⣠⡏⠀⠄⢰⢴⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠈⡆⠉⠉⠉⠉⠉⠉⠀⠋
⠀⠀⠀⠀⠀⡸⠀⠀⠠⡀⠙⣄⠀⠉⠉⢠⠁⠘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⢀⠔⠁⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠉⠑⡴⠁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⠛⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⡆⠈⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢤⠀⡀⠀⡌⠀⠀⠀⠉⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠖⢊⣠⣤⡴⠶
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢒⠀⢀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠁⠀⠀⠓⠀⢀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⡅⢀⠀⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣠⡴⠋⣠⠊⠀⢀⣀⠀⠀⢂⠜⠓⠤⠄⣀⣀⣁⣀⣀⠤⠔⠛⢄⢂⠀⠀⠀⢀⡀⠀⠑⢄⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠁⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠄⢰⡎⠈⢀⠀⢀⠈⠑⣄⠀⣠⠐⢚⣁⣠⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣮⠀⠐⢸⡇⠀⠀⢸⡇⢰⡏⣄⠀⣤⠀⠜⣴⠛⠄⠈⠉⡁⠀⢠⡆⠀⢀⡄⠁⢀⠀⢀⠀⠀⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠸⡇⢠⡀⠈⠇⣼⢁⣛⡀⠘⠀⠾⠁⣾⠀⢘⠛⠁⠀⢸⡇⠀⢸⡇⢠⡟⠀⣶⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠁⠘⣷⡀⢰⣇⡜⠸⣧⣀⡄⣲⡀⠹⠶⠶⠶⠞⠀⢸⡇⠀⢸⡇⢸⠇⢀⣍⣠⣼⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣠⠈⠷⠴⠞⠀⠀⠀⠿⠤⠞⠷⠤⠖⠀⠀⠀⠀⠈⠻⠶⠤⠖⣿⣄⡀⠀⢠⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡆⠀⠀⣼⣿⠛⣿⡇⠀⠀⢰⣿⠛⣿⣷⠀⠀⢰⣿⠟⢻⣷⠀⠀⢀⣿⡟⢻⣿⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇⠀⣿⣧⣤⣤⣿⣿⠀⣿⣷⣤⣤⣾⣿⠀⣿⣿⣤⣤⣼⣿⠀⢸⣿⣤⣤⣼⣿⡇⢸⣿⣦⣤⡀
⠀⠀⠀⠀⣴⣿⠟⠉⣿⡇⠀⣿⡏⠉⠉⣿⣿⠀⣿⡟⠉⠉⢹⣿⠀⣿⣿⠉⠉⢹⣿⡀⢸⣿⠉⠉⢹⣿⡇⢸⣿⡋⠛⢿⣷⡀
⠀⠀⠀⢸⣿⠃⠀⠀⠙⠿⠿⠟⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⠈⠿⠿⠿⠋⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠻⠿⠿⠟⠀⠀⠈⣿⡇
⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠃⠀⠀⠸⠿⠿⠟⠀⠀⠀⠿⠿⠿⠃⠀⠀⠺⠿⠿⠟⠀⠀⠐⠿⠿⠿⠂⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣶⣶⣦⣿⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⣶⣶⡄⠀⠀⢰⣶⣶⣦⠀⠀⠀⣶⣶⣶⡄⠀⠀⢴⣶⣶⣶⣾⠿⠛⠉⠁⠀⣴⣦⠀⠈⠉⠙⠿⣿⣦⣀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣽⣿⠟⠁⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠘⠛⠛⠋⠀⠀⠀⠛⠛⠛⠁⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠸⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠸⠿⠿⠷⠀⠀⠀⠿⠿⠿⠆⠸⣿⡇⠻⠿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣤⡀⠀⠀⠀⠀⠻⠿⠟⢀⣿⡇
⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡷⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣬⣥⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣘⣾⣿⣇⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣙⠻⠈⡀⠈⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣿⣿⡏⠉⢧⠓⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⢀⠀⠐⣶⡌⢦⣙⣛⣵⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢸⣿⡇⢀⣈⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣙⣥⣿⠀⠀⡿⠿⣮⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⡇⣿⣿⢸⣿⡇⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⡇⠀⢸⠀⢸⡟⠛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⣿⣧⣿⣿⣼⣿⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⠀⠀⠃⠀⠘⠀⠸⠇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣠⠶⠞⠛⣛⣛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣤⡴⠶⠶⠶⢮⣍⠛⠂⢀⣤⠶⠛⢛⣉⣉⣙⠛⠶⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠳⢶⣤⡀⠀⢈⣁⡴⠚⠋⠉⠉⠙⠛⠷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⢠⡾⣰⠏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠋⠙⠻⣆⣀⣀⣀⠀⠀⠀⠈⢻⡄⠈⣷
⠀⠀⠀⠀⣾⢡⡟⢠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢿⣷⣶⠞⣩⣭⣭⣉⡛⢶⣄⠀⠀⢻⠀⣿
⠀⠀⠀⠀⣿⢸⡃⣿⠀⣤⡤⠴⠶⢚⣩⣤⡀⠀⠀⠴⠾⠿⠶⠿⠿⠿⢿⣿⣦⡙⣧⠀⠘⠀⣿
⠀⠀⠀⠀⠻⠸⣇⠸⣦⠀⠀⠀⠘⣆⢻⣿⠷⡠⢂⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣭⣙⡻⢶⣤⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢃⡸⠿⣦⣄⣀⣈⡤⢖⣫⠞⢋⡭⠶⠛⠉⢉⡉⣉⣉⠉⠛⠻⢿⣿⣷⣮⣙⢷⣄⠙⣦⡀
⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢁⠴⠯⢍⠝⠉⠠⡆⢠⡆⢸⡇⣿⣽⠀⠀⢠⡄⠈⠻⣿⣿⣷⡝⢷⡈⢷⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⡇⠸⢿⠸⣷⠖⠀⢸⡇⠀⢀⡈⠻⣮⢻⣆⢻⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣰⠶⡆⠃⠀⠁⠀⣾⠀⣷⠀⠀⠘⠁⠀⠏⣿⠀⠙⣧⢻⣆⢻⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡐⠀⠀⡇⢿⣀⡟⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⢀⣴⣧⡆⠀⠸⣇⢻⡄⣧
⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡇⠀⠈⠁⠈⠛⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⢿⠘⣧⢸
⠛⢠⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⢰⠛⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⢸⡆⣿⠘
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠸⠶⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆⢀⡼⠁⠀⡀⠀⠀⠀⠀⠀⠘⣧⠅⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⢸⠀⡿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠀⡟⣻⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠏⡇⠀⠀⠆⢀⠇⠘
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡄⠸⣧⠀⢠⡟⢳⠀⠀⣤⣤⣤⠀⠀⣠⠀⠀⢀⠶⠆⠘⠛⡟⠀⠀⠀⠘
⠀⠀⢀⠘⣯⣉⣉⣉⣉⣉⣉⡈⢿⣆⠹⣧⡈⠳⠞⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠸⣆⠀⠀⠀⠃⡀
⠀⠀⠘⣧⠈⢻⣭⣭⣭⣭⣭⣥⣤⠻⣷⣌⢿⣦⡀⠀⠀⢀⡏⠀⠀⡏⢻⠀⢠⣤⠟⠀⢀⡠⠮⡭⠋
⠀⠀⠀⠈⠳⣄⠙⢷⣶⠶⠶⠚⠛⠶⣾⠿⢷⣯⣻⣶⣤⣈⠀⠀⠀⠻⠞⠀⠀⣀⣠⠖⣋⡴⢊⡕
⠀⠀⠀⠀⠀⠉⠳⠦⠉⠻⠃⠀⠀⠀⠘⣧⣀⣉⠛⠿⢿⣿⣿⣿⣷⣶⣾⣿⣿⠿⠞⣋⣥⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠤⠾⣷⣶⣒⣒⡂⠀⠀⠀⠉⣉⣉⣀⣤⠴⠖⠋⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄
⠀⠀⠀⢸⢿⡧⠶⠶⢼⡿⡇⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠓⣠⣶⣖⢢⡐⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆
⠀⠀⠀⠀⡏⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⢸⢘⡃⢘⡃⢘⡃⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢧⡀⠠⠅⢀⡼⠀⠀⠀⠀⠀⢸⢨⡅⢨⡅⢨⠗⠒⢄⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠸⠛⠒⠒⠛⠃⠀⠀⠀⠀⠀⠘⠒⠒⠒⠂⠠⡀⣁⡜⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠃⠈

⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⣾⠿⠽⠿⠿⠿⢿⡀⠀⠀⠀⠀⢠⠞⠉⠁⡄⠉⠙⢦⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣤⣄⠀⠀⠀⠀⣿⣯⠖⠒⠒⢿⡻
⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⣤⣾⣿⣿⡇⢿⣇⣿⠂⠀⠀⠀⢰⠇⠀⢠⡄⠀⢹⠀⠁
⠀⠀⠀⣿⠀⠀⢘⠀⡤⣼⠁⠀⠀⠀⠀⢸⠆⢀⡀⠃⣀⡤⠎⠀⠀⠀⠀⠀⠉⠻⢿⣿⡆⠉⠁⠀⠀⠀⠠⡈⢧⡀⠀⠀⣀⡞
⠀⠀⠀⠙⠒⠒⠒⠒⠛⠛⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠊⠃⠀⠀⠀⠀⠀⠀⠁⠐⠋⠛⠋⠁⠉

⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⠋⠻⡟⣷⡄⠀⠀⠀⠀⣾⠿⠽⠿⠿⠿⢿⡀⠀⠀⠀⠀⡜⠔⣠⣾⡷⡠⠄⢄
⠀⠀⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣧⡇⠀⠀⢡⣼⣿⠀⠀⠀⠀⣿⢘⡃⢚⠃⣛⢸⠇⠀⠀⠀⠀⠃⠀⣿⣿⡇⣅⠈⢀⠂
⠀⠀⠀⠀⢀⣿⣿⣿⢷⠀⠀⠀⠀⠀⠈⢿⣟⣀⣀⡀⢈⣿⠏⠀⠀⠀⠀⣿⢨⡅⣩⡁⣡⣶⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣌
⠀⠀⠀⠂⠙⠛⠛⢿⠟⠃⠐⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠙⠒⠒⠒⢂⣿⣿⣧⠀⠀⠀⠀⠀⠉⠉⠹⠯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢸⣟⠂⠀⠀⠀⠀⠀⠀⠀⣠⣦⢀⣸⣇⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣧⠠⣢⠀⠀⠀⠀⠀⣠⣴⣶⢤⡀
⠀⠀⠀⠀⢤⣘⣿⣿⣷⡆⠀⠀⠀⠀⠀⠉⡝⠉⠀⠀⠙⢮⠁⠀⠀⠀⠀⣴⠒⠚⣯⣾⣿⣿⡆⠇⡇⠀⠀⠀⢰⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠂⠀⣇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠸⣄⣀⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⠶⠤⠤⠾⠅⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠂

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

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣿⡟⠁⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠈⢻⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⣿⣤⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣿⡟⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣿⡟
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⡁⠰⣨⠃⢠⣄⡟⠀⠛⠀⡏⠀⡄⠘⠃⢠⡄⢹⠀⣾⣿⣿⣿⡦⠀⣶⠇⢸⠀⠘⠏⠀⡟⠀⠲⣾⣦⠀⣶⣿⠃⠀⢻⡇⠀⠂⣽⠀⣾⣿⠀⠐⢶⣿⣿
⣿⣿⣿⣯⣉⣀⣸⣆⣈⣩⣇⣀⣧⣀⣧⣀⣁⣼⣄⣈⣠⣾⣀⣉⣹⣿⣿⣧⣀⣿⣄⣸⣀⣶⣦⣀⣇⣀⣉⣹⣿⣀⣿⣇⣠⣄⣈⣇⣘⣀⣼⣀⣉⣹⣀⣈⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣯⣭⣭⣽⣭⣿⣿⣿⣿⣿⣯⣭⣯⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣭⣯⣭⣭⣯⣽⣿⣿⣿⣿⣭⣯⣭⣽⣿⣿⣿⣿
⣿⣟⣟⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿
⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣯⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣏⣿⣿⣧⣀⡀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣀⠀⠀⠉⠻⣶⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠋⠀⢹⣷⣮⣢⡀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠉⠀⡨⠜⠃⠀⠈⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⢨⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠾⠀⢰⣾⣿⡢⣤⣤⣤⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⡄⠘⢿⣦⣶⣿⣿⠟⣋⣥⣶⠾⠿⠟⠛⠛⠳⠶⢶⣤⣉⠻⢿⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⣿⣿⠟⣡⡾⢛⠉⠂⠀⠁⢸⣶⡹⠈⠈⢐⠨⠙⠻⣄⠉⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠠⢀⣾⢋⠔⠁⢸⡾⡇⠀⠈⣿⡇⠀⠀⠨⠇⠀⠁⠌⠳⣆⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠂⢀⣀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠠⢶⠇⡙⣧⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⠇⠀⠸⠘⠜⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⢀⠀⠀⠛⠂⢰⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⠀⣠⡀⠀⠀⠀⠀⠀⠀⡸⠧⡀⠀⠀⠀⠀⠀⠀⠠⡄⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠐⡀⠷⠃⠀⠈⠀⠀⠀⠐⣦⣠⡶⠀⠀⠀⠀⠀⠀⠀⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠇⢠⠀⢀⣄⠀⠂⠀⣀⣴⠉⠁⠀⠀⠀⠈⠀⠀⢀⡀⢀⢁⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡈⣄⢢⢼⡻⠀⠀⢠⡿⠋⠀⠄⠀⠀⠁⠀⠀⠀⠚⠃⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣌⢦⡁⡀⠀⠠⣮⠀⠀⠀⢀⠀⠀⠀⠰⣆⠀⠀⢂⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣝⢦⣔⠠⠃⠀⠀⠐⢯⠆⠀⠀⡐⠇⣡⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⣽⣲⠄⠀⠀⠀⢤⣴⣶⣶⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠋⠉⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢱⣀⣤⠮⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠀⠀⠴⣤⣀⡎⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠟⠉⠀⠀⠀⠀⠀⠈⠳⡀⠀⠨⢥⡬⠇⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠛⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⣸⡇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠠⣔⠩⠔⠐⠂⠉⠉⠁⠀⠀⠈⠉⠉⠐⠒⠤⠍⣲⠄⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠄⢊⠗⠊⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠺⣑⠠⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠊⡠⠞⠁⠀⠀⢀⡠⠔⠂⠉⠀⠀⢀⠈⢁⡀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠈⠳⢄⠁⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠁⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠈⠢⣄⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠳⡀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⠄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠃⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⡀⠁⠀⠘⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠘⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⣀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠰⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡹⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣠⠞⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠓⢄⡠⠀⠀⠀⠀⠀⠀⠏⠰⠀⠀⠀⠀⠀⠀⢄⡠⠚⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠠⠄⠀⣀⣀⠠⠄⠊⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠢⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠔⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⠁⠈⠁⠒⠠⠤⠄⣀⣀⣀⡀⠠⠤⠄⠒⠈⠁⠈⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⢷⠾⠷⠾⠷⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢣⣰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠞⣱⠃⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠈⢳⠦⡀⠀⠀⠻⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⣀⠀⠠⡧⠴⢤⠇⣠⣒⠛⢁⠎
⠀⠀⠀⠀⠀⠈⠫⣀⣨⠆⠀⠀⠀⠙⠃⠹⣸⠀⠀⠀⣠⣠⢄⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠚⠉⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠔⠊⠙⠲⢍⠀⣴⣒⠃⠁⢈⣍⡷⠀⡠⠚⠉⠉⠉⢢⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⡇⠀⠀⠀⠀⣨⠇⠈⠈⡇⠀⢸⠀⠀⠀⠧⣀⠀⠀⠀⠀⡇⠀⠘⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠧⠤⠴⡂⢫⡤⠴⠒⢲⡃⣠⣼⠋⣱⠖⣲⠏⢨⡓⠒⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⢀⠠⢀⣀⢀⠀⠀⠀⢀⠀⠀⣘⡆⠈⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠊⠁⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⠁⠀⢀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⠀⠀⠓⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠀⠀⠀⢢⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠏⡢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠄⠀⢠⠀⢀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠉⠂⢄⣀⠀⠀⠀⠆⠀⠀⠀⢘⠁⠐⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠤⠶⠤⣄⣉⣉⣉⣛⡤⠬⠥⠦⣄⡄⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠆

⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇
⣿⠛⠛⠛⠿⡿⠟⠟⠛⠛⠟⢿⡿⠻⠟⠛⠻⠻⣿⠟⠛⠛⠟⠟⠻⡿⠻⠿⠛⠛⠻⢿⡿⠛⠛⠛⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢀⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠛⠀⠀⠀⢸⡏⠀⠉⠀⢹⣿⠈⣿⠁⠉⠉⠈⣿⠀⣿⡏⠈⠉⠁⢹⡇⢸⡏⠀⠀⠈⢸⣿⠀⠀⠀⣿⡇
⣀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠸⠿⠿⠿⠿⠾⠟⠐⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⠠⣶⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⣶⡆⢰⣶⣶⡆⢀⣀⣤⣤⣴⣶⣿⣧⣤⣄⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⣇⣤⡶⠟⠋⠉⢀⣀⣀⣀⣀⠈⠉⠛⠷⣦⣀
⣿⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠘⠁⣼⡿⠋⢀⣤⡶⠟⠛⠋⠿⠋⠛⠻⠷⣦⣄⠈⠻⣷⡄
⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠃⢀⣾⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠈⢿⣆
⣿⠀⠀⠀⢰⡶⠶⠶⠶⢶⣶⠀⣶⠶⠶⠶⠶⣶⠀⣶⡶⠶⠶⠶⣾⠇⢀⣾⠏⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠈⢿⡆⠀⢿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⢸⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸
⣿⠀⠀⠀⢸⣇⣀⣀⣀⣸⣿⠀⣿⣀⣀⣀⣀⣿⠀⣿⣇⣀⣀⣸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠲⣿⡇
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢠⣿⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣾⠃⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣾⠏
⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣷⣄⠈⠙⠿⣶⣤⣤⣶⣦⣤⣴⡾⠟⠉⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⡈⠉⠉⠉⠁⣀⣠⣤⡾⠗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢀⢀⣤⣤⣤⣤⣤⣄⣀⡁⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⣪⠒⠉⠉⠀⣄⠀⠉⠉⠙⠺⢕⢦⠀⠋⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣁⠀⣸⠇⠸⠀⠀⠀⠏⠀⠀⢰⠀⠀⠀⣧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣯⣅⠢⡀⠀⡶⢦⢄⣈⡤⣖⣒⣿⣦⣦⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢠⢫⣾⣿⣿⣿⣿⣯⡵⠃⣀⣈⣷⣾⣇⠾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢆⠻⣛⣽⣿⣽⣿⠿⠿⣿⣿⣿⣯⣭⣙⣳⣾⢿⣿⣿⣿⣿⡽⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠈⢛⣽⠟⣫⣵⠞⠛⠋⠉⢹⠛⠛⢿⣟⠿⣷⣽⣿⣯⣟⡻⢃⠃⠀⠀⣀⣠⣤⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠂⢠⡞⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡝⣿⣿⡅⠀⠛⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡟⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣷⡈⣿⣿⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⣼⢡⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡇⢸⡇⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⢸⠀⡇⠁⠀⠀⠀⠀⠸⡋⡀⠀⠀⠀⠀⠀⠀⡀⣿⢸⣿⢸⡇⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉⠀⠀⢻⠸⡆⣇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⢸⡇⡇⠀⠀⠉⠙⠛⠲⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⢃⣾⡟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠘⣦⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢃⣾⠟⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⠄⠈⠳⣿⣷⣭⣒⠤⠀⠰⠀⠀⢀⣠⡴⢟⣥⣾⣷⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣈⣿⢿⣯⣭⣽⣿⣿⣿⣿⣿⣿⢿⣟⠉⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⡉⠉⠉⠉⠉⠉⠀⠀⠹⣻⡄⠀⠙⣆⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⠋⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠁⠀⠀⠀⠉⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠋⠉⠙⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢠⣾⣿⣠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣶⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣤⣌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡘⠀⠄
⠀⠀⠀⠀⠘⡀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠉⠁⠀⠈⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠐⢡⠞
⠀⠀⠀⠀⢆⠹⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠤⠤⠤⠔⠒⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠃⠘⠦⠈⠙⢦⣤⣀⣀⠀⢀⣀⣀⣤⠖⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠉⢻⣆
⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⣴⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠋⠉⠀⣠⡞⠁⠀⠀⠉⠛⢷⣄⠀⠀⢿
⠀⠀⠀⡼⠃⠀⠀⣠⡾⠋⠁⠀⠀⠀⠈⠙⢿⠶⣦⣤⣀⣀⣀⣀⣀⣤⣤⠶⠶⠛⠉⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠘
⠀⠀⠸⠁⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣭⣉⣉⣉⡁⠀⢀⣀⣀⣀⣤⣤⡴⠶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠿⠓⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⡟
⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⢀⣠⣶⡾⠿⠛⠛⠉⠉⠉⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠃
⠀⠀⠀⠀⠀⢿⣿⣦⣤⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠶⠶⠾⠟⠋

⠀⠀⠀⠲⠊⣻⣧⠀⠀⠀⠀⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣿⠻⣧⠀⠀⠀⠀⣿⡇⠀⢀⣴⢶⣄⣀⣤⣤⣀⣠⣄⠀⠘⠛⣿⣴⠞⠻⣦⣤⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡆
⠀⠀⠀⠀⢸⡏⠀⢻⡆⠀⠀⠀⢻⡇⠀⠈⠁⣀⣹⡏⠙⣿⠟⠉⠙⠗⠀⠀⣿⡃⠀⠀⢸⡏⠀⠀⠻⣧⠀⠀⣴⠶⠶⢶⣄⢀⡶⠟⠛⢿⡇
⠀⠀⣀⣀⣼⠟⠋⠉⢿⣄⣀⠀⢸⡇⠀⣴⠟⠉⢻⡇⠀⣿⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⢿⡄⢰⣿⣶⣤⣤⣿⣿⠃⠀⠀⢸⣧
⠀⠀⠉⠉⠉⠁⠀⠠⡾⠛⠉⢀⣸⣇⠀⢿⣤⣀⣼⣇⣠⣿⣦⡄⠀⠀⠀⢴⣾⣷⡆⢤⣾⣷⡦⠀⢠⣾⣷⣾⢿⣄⣀⣠⡄⢻⣆⣀⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠈⠉⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡴⠶⠾⣇
⠀⠀⠀⠀⠀⠀⠀⣶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡴⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠛⠳⣶⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠂⠀⠀⠀⠀⢀⣀⣸⡇⠀⣀⣀⠀⢠⣄⣴⠆⠀⠀⢸⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⡴⠾⠋⣾⣯⣽⡷⠘⣿⢻⣶⣦⠀⢰⡆⢰⣶⡿⣧⠀⣾⢉⣿⡇⢸⣯⣽⠆⠈⣿⠁⠀⠀⠀⢸⡷⣌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⢷⣄⠀⠻⣤⣀⡀⠀⣿⠸⠇⢸⡇⢸⡇⠸⣿⠀⢿⠀⠻⠾⠙⣧⠀⠛⠶⠀⠀⠛⠀⠀⠀⠀⣼⣇⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣇⠈⠳⣄⠓
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠛⠳⡄⠈⢣⡀⠈
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⡽⠛⣏⠈⢳⡀⠙⢦⠀⠙⠦
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢿⡁⠙⣄⠈⢧⡀⠹⣄⠀⠳⣄
⠀⠂⠀⠀⠀⠉⠉⠛⠛⠛⣷⠶⣶⠶⠶⡶⠶⢶⠶⠶⢶⡶⠶⣶⠶⢶⡶⠶⢶⡶⠶⠶⡶⠟⠛⣏⠉⢧⡀⠹⣄⠈⢳⡀⠱⣄⠘⢦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⢧⠈⢧⠀⠹⣄⠈⢧⠀⠈⢧⡀⠈⢦⠀⠹⡄⠀⠳⡄⠀⠙⣆⠀⠘⣆⠀⢳⡀⠘⢦⡀⠙⢦⡈⢧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠳⠈⠧⠈⢧⠀⠈⠦⠈⢳⡀⠀⢳⡄⠈⢧⡀⠹⡄⠀⠹⡄⠀⠘⣆⠀⠈⢦⡀⠳⡄⠈⠳⠀⠀⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠀⠀⠘⠀⠀⠈⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠔⠊⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣿⡿⠤⠔⠒⣒⠲
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠊⢁⡀
⠀⠀⠀⠐⠿⣯⣥⣶⠾⠟⠛⠋⠉⠀⠀⠀⠀⣀⣤⣄⠀⠀⢰⠿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣆⠀⠀⠀⠀⠀⣼⡿⠛⠻⣀⠤⡎⢀⡀⢱⠀⠀⠀⠀⢄⠀⠀⠀⢹⠀⠀⠀⡜⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣬⠇⣠⠄⠀⠀⠀⠙⠀⠠⠄⢸⠀⠳⣀⢀⡌⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⢿⡄⢀⡀⠀⠇⠀⠀⠀⠀⠳⢄⡠⠎⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⡏⠐⠀⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⢠⠀⠀⠀⠀⠀⠀⢠⠎⡇⠀⣀⠤⠒⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⢀⡆⠀⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⠀⣸⠀⣧⣊⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢧⡀⠀⠀⠀⠀⠁⠈⠒⠒⠒⠂⠀⢀⡾⠀⠀⠀⢀⢔⠚⢻⠉⡏⠢⣀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⠁⠀⠀⡇⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠺⣍⠱⢀⡖⠚⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⠤⠄⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢻⢰⠊⣉⡓⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⠀⠀⠘⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢻⣸⠀⡄⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣴⡳⣼⣿⣧⠀⠀⠀⠀⠀⠀⢸⠀⠘⠘⠒⠒⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣌⡑⠚⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠋⠁⠙⠋⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣙⣉⣉⣉⣙⣳⠀⠀⠀⠀⢸⣙⣉⣉⣉⣛⡃⠀⠀⠀⠀⢘⣛⣉⣉⣉⣋⡇⠀⠀⠀⠀⣞⣋⣉⣉⣉⣋⡆
⠀⠀⠀⢸⠀⠴⡆⣲⠂⢸⠀⠀⠀⠀⢸⢰⣲⣖⣖⡆⠁⠀⠀⠀⠀⠈⠠⠄⠤⠠⠄⡇⠀⠀⠀⠀⡇⠀⠠⠤⠤⠀⡇
⠀⠀⠀⠸⣀⣀⣁⣉⣀⡼⠀⠀⠀⠀⠸⣘⣚⣓⣓⣃⡄⠀⠀⠀⠀⢠⣘⣁⣃⣘⣀⠇⠀⠀⠀⠀⢇⣀⣈⣉⣉⣀⠇


⠀⠀⠀⢠⠴⠤⠤⠤⠴⢤⠀⠀⠀⠀⢠⠴⠤⠤⠴⢤⠀⠀⠀⠀⠀⢠⠤⠤⠤⠴⢤⠀⠀⠀⠀⠀⡤⠦⠤⠤⠤⠦⡄
⠀⠀⠀⢸⠒⣒⣒⣒⠒⢺⠀⠀⠀⠀⢸⠒⠒⠒⣒⣺⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⣻⠀⠀⠀⠀⠀⡗⢒⠒⢒⠒⠒⡇
⠀⠀⠀⢸⠀⠣⢌⠴⠁⢸⠀⠀⠀⠀⢸⣀⣀⣸⠰⠆⡃⠀⠀⠀⠀⢂⣀⣀⣎⠀⠀⡇⠀⠀⠀⠀⡇⠐⠐⠐⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⢸⣉⣉⣉⣉⣉⣽⠀⠀⠀⠀⢸⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⢨⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⣯⠭⠭⠭⠭⣽
⠀⠀⠀⢸⠀⠀⠆⡖⢲⠞⠀⠀⠀⠀⢸⠀⠀⢠⠊⡙⠆⠀⠀⠀⠀⡀⠀⠀⡰⠊⡙⡄⠀⠀⠀⠀⡇⠀⠀⡴⢊⡛⡄
⠀⠀⠀⠸⠤⠥⠥⠗⠁⠀⠀⠀⠀⠀⠘⠒⠒⠚⠤⠥⠃⠀⠀⠀⠀⠘⠒⠒⠓⠬⠵⠃⠀⠀⠀⠀⠑⠒⠒⠳⠬⠴⠁


⠀⠀⠀⢀⣰⣒⣒⣒⣒⣢⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⡴⠶⠶⠶⠶⠦
⠀⠀⠀⢸⢺⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠘⠀⢀⢀⡀⠀⡇⠀⠀⠀⠀⡏⠉⠉⠉⡡⣶⠄
⠀⠀⠀⢸⠘⠤⠤⠤⢤⠜⠀⠀⠀⠀⢸⠀⠘⠲⠃⠀⡀⠀⠀⠀⠀⢀⠀⠰⠛⠄⠀⡇⠀⠀⠀⠀⢧⡀⢠⣪⠖⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠄⣀
⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢆
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠈⠓⠶⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠤⠤⠤⠤⢤⡀⢠⠤⠤⠤⠔⠒⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢀⡿⠛⠻⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⡛⠂⠀⢤⡀⠈⠻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠔⡽⢋⠃⣀⠀⠀⠀⠀⢀⠔⠉⠁⠀⡤⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠞⣀⠀⠁⠀⣀⠖⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡤⠄⡲⠹⡇⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠊⣀⠀⢀⠄⢧⡀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠄⣀⣤⣤⣄⣁⡏⠀⢸⣠⢾⡇⠀⠀⠀⠀⠀⠀⢸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⣰⡷⠟⣻⣿⡿⠋⣽⠁⠀⠈⡇⠈⠃⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣴⠏⠀⢠⣿⣿⠃⠀⡿⡄⠀⠀⠓⡙⠀⣴⡿⠛⠓⠒⠒⠻⢿⣄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⣾⠋⠀⠀⣸⣿⡏⠀⠀⢡⠀⠀⠀⠀⣧⠞⠀⠉⠈⠀⠀⠀⠀⠀⠈⠑⠲⣤⡀
⠀⠀⠠⠊⠀⠀⠀⠀⢀⣾⠃⠀⢀⢰⣿⡿⠀⠀⠀⠀⢂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠛⣦
⠀⠰⠁⠀⠀⠀⠀⠀⣾⠃⠀⠀⠁⣸⣿⡇⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⢸⠆
⢀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢀⣤⣾⣿⣿⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⣤⣀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⡾
⢸⠀⠀⠀⠀⣠⣾⣷⣶⣾⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⣼⠳⣀⣴⣾⣿⣿⣿⣧⣀⡾⠋⠀⠀⠀⠀⠀⡼⠃
⠘⣶⣶⣶⣿⣿⣿⣿⠿⠛⠋⠁⠀⣿⣯⠀⠀⠀⠀⠀⢠⡇⠈⣿⣿⣿⠟⠋⣰⢿⣿⢧⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠏⡀⠀⠀⠀⠀⢸⠀⠀⠈⢯⡀⠀⠀⠁⠚⢸⠃⠁⠀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠀⣠⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠙⠶⣤⣴⠾⠋⢰⠇
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠴⠶⠚⠛⠳⠶⠶⣦⣄⡉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⠲⠶⣦⡄⠀⡞⠉⡇⠀⣠⡶⠞⠛⠛⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⠴⠞⠁⠀⠀⠀⢹⡆⣇⠀⡇⣼⠧⣤⣤⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⠒⠳⠿⣤⣀⠀⠉⠛⠶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣤⠾⣋⣤⠴⠶⠚⠛⠛⠛⠛⠻⢦⣄⠀⠈⢳⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠚⠛⠋⣩⡶⢛⣩⠴⠖⠛⠛⠛⠛⠛⠶⣦⣄⠙⢷⣦⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠟⢁⡴⠋⠁⠀⠀⢰⣉⡷⠀⠀⠀⠀⠉⠻⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣾⠋⣰⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠃⡼⠁⠀⢠⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣸⠃⠀⠀⠀⠈⠓⠦⣄⣙⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⢠⡴⡄⠀⠀⠀⠀⠀⠻⣄⡼⣄⠀⠀⠀⠀⠳⠞⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡀⠀⠀⠀⠀⢰⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⣠⠏⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢆⠙⢦⣀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠁⠀⢀⡼⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⠀⠙⠳⠶⣤⣤⣀⣀⣉⣉⣠⣤⣴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢻⡄⠀⠀⠀⠀⠉⠉⠉⠁⣀⣠⣤⡶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠳⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠈⣤⡟⠁⠀⢀⣾⢿⡧⠀⠀⠀⠸⣿⡎⡀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣠⡿⣿⡇⠀⠀⠀⢸⣿⡇
⠤⣀⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣿⠋⣼⠃⠀⠀⠀⢈⠟⠿⠦⢄⣀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⡿⢀⡼⠁⠀⠀⠀⢸⠛⠷
⢀⠉⣿⣷⡦⢄⢸⡁⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⠸⠁⠀⠀⣀⠹⣿⣷⡦⣀⢺⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢽⣿⠟⠁⠀⠀⠀⠀⡾
⠀⠀⠚⣽⡏⠀⠘⢦⡀⠀⠀⠀⣿⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠊⣿⡃⠀⠙⢄⡀⠀⠀⠀⡿⠀⠀⢀⡰⠛⠁⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠠⣿⠀⠀⠀⠀⠱⡄⠀⣦⣿⣄⠴⠋⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠈⠱⡀⠀⣦⣧⣀⠖⠋⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⣴⣿⠀⠀⠀⠀⠀⢹⡄⢙⠟⠋⠀⠀⠀⠀⠀⠐⠀⡴⢏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠀⠀⠀⠀⠀⣷⡄⢻⠟⠁⠀⠀⠀⠀⠀⠀⢀⠴⡇
⠚⠉⢿⣿⠀⠀⠀⠐⢿⣿⣿⡞⠀⡀⠀⠀⠀⠀⠀⣠⠏⠀⠘⢧⠀⠀⠀⣠⠴⠛⠉⢿⣷⠀⠀⠀⠸⣿⣿⣷⠏⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠘⣆⠀⠀⢀⣠
⠀⠀⡈⣿⡀⠀⠀⠠⠛⢿⣿⡟⠁⠀⠀⠀⠀⣀⣾⡅⢀⠀⠀⠘⣇⠰⠊⠁⠀⠀⠀⠎⣿⠀⠀⠀⠐⠻⣿⣿⠗⠀⠀⠀⠀⠀⣰⣎⠀⠀⠀⠀⠸⡄⡴⠋
⠀⠈⠀⢻⣆⠀⠀⠀⠀⣿⠛⠋⠥⠤⢄⣀⣰⣿⣿⡶⠀⠀⠀⠀⢹⠆⠀⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⢨⣿⠛⠃⠤⠦⣄⣀⣴⣿⣷⠄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⣿⡄⠀⠀⠀⡿⠀⠀⠀⠈⠀⠉⢻⣿⡧⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢸⣧⠀⠀⠀⠈⡯⠀⠀⠀⠉⠀⠉⢻⣿⣦⡀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠠⢁⡿⣄⠀⠀⣻⠀⠀⠀⠀⠀⠀⣩⣿⠁⠙⣦⡀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠄⢡⣷⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢹⣟⠈⠙⣄⠀⠀⠀⡽
⠀⠀⠀⠀⣸⠃⠈⢳⡀⢸⠀⠀⢀⠀⣀⡤⠺⣿⠇⠀⠈⠳⡀⢠⣏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⢦⡀⣿⠀⠀⠀⢀⣀⠴⢿⡿⠄⠀⠨⢳⡀⣼⡃
⠀⠀⢀⡴⠃⠀⠀⠀⠈⢻⡧⠒⠈⠉⠀⠀⠀⣿⠀⠀⠀⠀⣸⠇⢱⡄⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⢻⠴⠒⠉⠉⠀⠀⠀⣷⠀⠀⠀⠀⡼⠀⢳
⡠⠞⠋⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢠⡼⠁⠀⠀⣳⣀⣠⣀⡤⠞⠃⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣠⡿⠁⠀⠈⣧⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⢀⣶⠋⠊⠀⣰⣿⡇⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢨⡾⠑⠁⢀⣾⣿⡇⠀⠀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⣼⠟⢹⡇⠀⠀⠀⠀⣿⣷⣂⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⣼⠟⠀⠀⢀⡾⠃⣻⠃⠀⠀⠀⢸⣿⣗
⠀⠛⣶⣤⣀⠀⢨⠆⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣿⣶⠟⠀⠀⠀⠀⢸⠂⠀⠉⠑⠻⣶⣤⣀⠀⣸⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢺⣧⡾⠃⠀⠀⠀⠀⣸⠀⠈
⠀⠈⠹⢿⡏⠉⠺⡀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠈⡿⣿⠇⠑⢾⡀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣼⠋⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⣿⠃⠀⠀⠈⢣⡀⠀⣀⣿⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠙⢆⠀⡀⣀⣏⠀⣠⠜⠃⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢈⣾⠀⠀⠀⠀⠀⢷⡀⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⣦⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⡤⠴⣿⣿⠀⠀⠀⢀⣦⣽⣷⣰⠁⠀⠀⠀⠀⠀⠀⠀⡴⠁⢛⡄⠀⠀⠀⠀⣀⡤⢾⣿⣟⠀⠀⠀⢀⣤⣿⣧⡾⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢣⡀⠀⠀⠀⢀
⠀⠀⠙⣿⠀⠀⠀⢀⣼⣿⣿⡧⠄⠀⠀⠀⠀⠀⣠⢋⠀⠀⠀⠸⡄⢀⡴⠊⠁⠀⠀⡝⣿⠀⠀⠀⢀⡾⣿⣿⡥⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢻⠀⢀⡴⠋
⠀⠀⠁⣿⡅⠀⠀⠀⠀⣿⡿⠧⢄⣀⠀⠀⢀⣾⣿⡡⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⠀⢡⣿⡀⠀⠀⠁⢊⣿⠿⠦⢀⡀⠀⠀⢀⣾⣟⡀⠀⠀⠀⠀⢧⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠳⣾⣿⣟⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡘⡿⠀⠀⠀⠀⠙⠲⣿⣿⡛⠉⠀⠀⠀⠀⢹⠀⠠
⡀⠀⠀⢀⠟⣧⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠘⣿⠟⠷⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠘⠙⣦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⡿⠙⢦⡀⠀⠀⠀⢼
⠀⠀⠀⠀⢰⠏⠻⣄⠀⢻⠀⠀⠀⠀⠀⢀⣽⣿⡂⠀⠙⣆⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠣⡄⠀⣳⠀⠀⠀⠀⠄⢀⣽⣿⠀⠀⠻⣄⠀⢠⠇
⠀⠀⠀⣠⠎⠀⠀⠈⠣⣼⣆⢤⠰⠖⠚⠉⠈⣯⠂⠀⠀⠠⢣⡜⢧⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠉⠢⣿⣀⡤⠰⠒⠙⠁⠘⣏⠱⠀⡄⠘⢢⡛⢆
⢀⣠⠖⠉⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣠⠟⠀⠈⣇⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣴⠋⠀⠸⣆
⠉⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠞⠑⠀⠀⠾⠧⠀⠀⠀⠹⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠠⠎⠃⠀⠠⠾⠇⠀⠀⠀⠻⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡷⠖⠚⠛⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠳⠶⢤⡞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⡊⠉⡆⠀⠠⡊⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠔⠊⢁⣀⣠⠠⠬⠯⠀⣀⣀⡈⠑⠢⣀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⢫⠷⠒⠉⠁⣀⠀⣤⣤⢄⣤⢀⣀⠈⠉⠒⠴⣹⠂⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⡠⠚⠁⣀⣔⠶⠙⠋⠉⠉⠁⣄⠉⠉⠘⠻⢷⣠⣀⠈⠓⢄⠈⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⡪⠋⠁⢰⡆⠀⠀⠠⡆⠙⡦⠀⡄⠀⣤⠈⠙⢕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠊⠀⠀⠀⢸⡇⢹⠀⠀⠃⠲⠆⡰⠀⢻⠀⠀⠀⠀⠑⣷⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣯⠃⠰⢆⠀⡠⡈⠁⠈⠀⠀⠀⠀⢠⠁⠀⠉⠀⠀⠰⡄⣤⠘⣵⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣾⠃⠀⠀⢸⠸⣇⠇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠮⠄⠀⠀⠘⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣜⣀⣀⣀⣀⣀⣀⡀⠠⡄⠀⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣇⠀⠋⠈⠞⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠋⠈⠇⠛⠀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⢠⠠⠀⢰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠱⣧⠀⣠⠸⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢀⡀⠀⣼⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠹⣧⡀⠀⠀⠀⠀⣩⠁⠀⠀⢀⠀⠀⠀⢰⡂⠀⠀⠀⠀⢁⣼⠏⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢝⢄⠀⠀⣤⠁⠀⠀⠀⡟⡦⠀⠀⠀⢋⣄⠀⠀⡠⡮⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠯⣦⣄⡀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣡⣴⠟⠊⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⢄⡀⠁⠚⠳⠶⠦⠶⠦⠴⠶⠚⠓⠉⠀⣠⡔⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⠔⠁⠀⠉⠑⠒⠤⠤⢀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠑⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⢀⡀⠀⢀⣀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⢀⣀⣸⣧⣀⣀⣿⣀⣀⣸⣇⣀⣸⣿⣀⣀⣿⣇⣀⣸⣇⣀⣀⣿⣀⣀⣼⣇⣀⡀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠹⣿⣿⣇
⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡋⠉⠉⢹⡏⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣀⣀⣠⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣄⣀⣀⣸⣅⣀⣀⣸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⠋⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣨⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⡀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢙⡏⠉⠉⢹⡏⠉⠉⢹⡏⠈⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⠋⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣰⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣆⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡏⠉⠉⢉⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡉⠉⠉⢹⡏⠉⠉⢹⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⢸⣇⣀⣀⣠⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣄⣀⣀⣸⡇⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⢸⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⢸⣇⣀⣀⣘⣇⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠉⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠀⠀⠀⢀⣀⣀⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣰⡿⢾⣿⣿⣿⣆⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣄⣠⣬⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠭⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡋⣩⣤⡶⠶⠲⠾⠒⠲⠶⠴⣶⣥⣂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⣿⠿⡋⠉⠀⠀⠀⠀⠙⠃⠃⠀⠀⠀⠈⢙⠷⣄⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢠⡿⠋⠙⠴⠚⢀⠈⠀⠀⢲⡀⡇⡆⠀⠀⠀⢀⠋⠀⠀⠙⢄⢜⣷⡀
⠀⠀⠀⠀⠀⠀⢨⣾⠞⠀⠀⠀⠡⢮⡈⢦⠀⠀⣨⢧⠇⡇⠀⠀⢠⠎⠀⠀⠀⠀⠀⢑⣻⣷⡄
⠀⠀⠀⠀⠀⠀⣸⠃⡐⠀⢀⠀⠀⠀⣩⠒⠁⠀⠀⠀⠀⠠⠼⡀⠋⠀⠀⠀⢀⣌⠘⣭⡤⢹⣽⡄
⠀⠀⠀⠀⡀⣿⠏⠉⠃⠠⠶⠦⢤⠄⠀⠀⠀⠀⠀⠀⠤⣬⡁⠀⠀⠀⠰⣪⠕⠊⠀⢀⠀⠀⢻⣷
⠀⠀⠀⠀⠱⣾⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢰⠐⣿⠂⡂⠀⡥⠶⠞⠃⠀⠀⠀⠀⠀⠀⢴⣧⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡂⠰⠐⠆⣿⡇
⠀⠀⠀⠈⢀⣿⠈⠁⠀⠓⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠙⡆⠀⠀⠀⠋⠉⠉⠁⠈⠉⢁⣧⡇
⠀⠀⠀⠀⢩⣾⡆⠀⠀⢠⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢱⣶⢄⡀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠐⡛⢿⣤⠶⡀⠙⢋⡴⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣷⡞⢁⠄⠀⢠⢯⠏
⠀⠀⠀⠀⠀⠘⡴⡷⡜⠃⠀⠈⠀⢀⢜⢝⡕⠀⠀⡄⡄⠀⠀⢦⡀⠀⠀⠀⠉⠀⠲⠊⡠⢮⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⡀⠀⣀⠈⠛⠠⠇⠀⠀⡇⢻⠀⠀⠀⠙⠂⠀⠀⠀⠀⢀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⠅⣐⠂⠀⠀⠀⠈⠁⠈⠀⠀⠀⢠⠺⡀⠀⣀⢔⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣝⡐⢤⣠⣄⣀⠈⠈⠧⠀⢀⢠⣀⡡⣖⣾⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠛⠶⠬⠭⠷⠭⠵⠖⠒⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⣄⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠀⠀⠀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⠀⠀⠀⣠⡇⢰⠾⠓⠒⠋⠉⠉⠁⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠤⠄⢴⡋⢙⡒⠊⠡⣀⣿⠀⠀⠀⠀⣀⣠⠤⠔⠒⠉⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠤⠤⣄⣀⣀⣀⣀⣠⡞⠋⠀⡈⣷⠖⠛⠻⣤⣶⠆⠉⠁⠀⠀⠀⠀⢾⢱⡁⣏⠉⠙⢦⠀⠀⠀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢹⣄⠀⢀⡽⠁⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢈⡁⠉⢛⠛⢦⢸⣤⣤⠤⠦⢤⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢄⠐⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⢃⡀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣠⣴⣚⡯⠴⣚⣩⠴⠛⠀⠁⠀⠀⠛⠲⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣴⠞⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠁⠀⠀⠀⠀⣠⠃⣠⣺⣯⣽⠾⠷⢚⣉⡥⠴⣒⣋⣍⡳⢄⠘⢳⡶⣄⡽
⠀⡀⢀⡠⢖⣠⠴⠶⠋⠘⠦⠄⠶⠲⠶⠶⠶⠶⠶⠶⠶⠓⠒⣻⣦⣄⡤⠶⢛⡷⠚⣛⣉⠥⠶⠚⣋⠭⠵⢚⣩⠥⠒⠀⠙⢆⠳⣄⣻⠀⠐⢂
⡞⣰⢋⡖⣡⠄⡀⢀⡴⠒⠒⠒⠒⢒⣒⣒⣒⡒⠒⠬⠽⠉⠉⣻⠀⠉⠀⠀⠈⡇⠈⠙⢦⡀⠠⠤⠤⠒⠊⠉⠀⠀⠀⠀⠈⢮⡳⢬⠅⠀⠀⠀⠀⢀⣠⣤⡀
⠳⠷⡘⡿⢡⡞⠀⡼⢠⠞⠩⠭⠤⠒⠒⠒⠒⣒⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⢶⣤⣄⣙⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠁⠀⠀⣀⡤⠖⢻⣿⣿⠿
⠀⢠⠠⣓⠚⠳⠜⣡⢏⠄⠀⠀⠀⠀⢀⡴⣻⣿⣷⣿⣀⣀⣠⣯⣀⡀⠀⠀⠀⡼⠦⠤⣉⡉⠉⠻⣍⠉⠉⠉⠈⠉⠓⠺⢍⣦⡠⠤⣼⠋⠁⠀⠀⣾⠛⢳⡀
⠀⠈⠑⠉⢦⣀⣈⡵⠋⠀⠀⠀⠀⠀⢸⣘⣿⡞⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⡼⢰⠛⠑⡦⣙⢳⡄⠈⢦⡀⠀⠀⠀⠀⠀⢻⣤⣶⣶⣯⠀⠀⠀⠀⣿⠉⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢚⣓⣫⡴⠃⠀⠀⠱⣄⠀⠀⢀⣠⣴⡫⠟⠛⠛⢷⡐⠂⠲⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣀⣀⣤⣤⡴⠶⢼⣷⡿⠟⠛⠉⠀⣀⠀⠀⠀⠀⠀⢛⣓⡺⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⡽⠉⣳⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⠁⠀⠀⠀⠀⠱⡄⠀⠀⡼⠉⣇⠀⠀⠀⣰⣿⠟⠛⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢧⣀⡰⠋⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⣄⡼⣡⣄⠹⣄⣠⢾⣇⡿⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠓⢒⣒⣒⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡴⠁⢸⡄⠸⠃⠈⢹⡇⠀⢀⡠⠞
⠀⠀⠹⡏⠙⠲⢾⡶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣈⠙⠓⠠⠤⠼⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣷⠀⠀⢀⣘⣷⠞⠉
⠀⠀⠀⢧⠀⠀⠀⠁⠹⣤⡄⠀⠀⢀⡀⠀⠀⠀⠀⠉⠙⠓⠒⣶⠶⠶⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢹⣀⡴⠋
⠀⠀⠀⠘⣦⡀⠀⠀⠀⠉⠙⠖⣋⢁⣹⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠹⢥⣤⣤⣤⣤⠴⠋⠁⠀⣯⠆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣠⠼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠶⠤⠤⠖⣚⠭⠒⠉⠈⠳⢤⡤⢠⠤⠴⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⢰⠃⡇⠀⠀⠀⣀⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠸⢖⣒⣂⠉⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡠⠔⣹⠋⡀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣚⡿⠶⡖⠒⠲⣮⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠿⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠻⠶⠚⠛⠋⠤⠞⠛⠷⠶⡛⠶⢦⡶⢶⣿⠶⣂⠴⢶⣤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡄⣿⠀⣤⣤⣤⣤⣤⣤⢸⣿⢠⣄⡀⠀⠀⠀⢸⣿⣿⣿⡿⠟⣿⠀⣿⠛⠛⠛⠛⠛⠛⡇⠀⡟⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣧⣭⣼⣿⣿⣿⣿⣿⣿⣦⣭⣼⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠛⠶⠃⠀⢸⣿⣿⣿
⠀⠀⠀⢸⡏⠉⠉⢩⣭⡍⣭⣭⢩⣭⡍⣭⣭⢹⡇⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⡟⠛⣻⠛⠛⡟⠛⢻⠛⠛⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣀⣀⢘⣛⡃⣛⣛⢘⣛⡃⣛⣛⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠿⢿⡷⠤⣿⠦⠴⡷⠤⢾⠦⠴⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⠿⠿⠸⠿⠇⠿⠿⠸⠿⠇⠿⠿⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⣀⣸⣇⣀⣽⣀⣠⣇⣀⣼⣀⣠⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⢸⡇⠀⣻⠀⢀⡇⠀⣸⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⢸⣿⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⣾⠃⠀⠀⢳⣿⡟⠀⣦⠀⠀⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⠀⠀⡀⣠⣞⣧⡄⣴⡟⠀⠀⠀⠸⡹⣷⣄⣷⠃⠀⠀⠸⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠠⡀⠙⠛⠾⣿⣿⣏⠀⠀⢀⠀⠀⠰⠉⢻⣿⠒⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢀⣤⡶⠊⠈⠛⠷⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⡀⠀⠀⢀⠀⠀⠾⢿⣦⡀⠀⠀⠈⠻⠄⠀⠀⠀⠀⣀⡀⡤⠀⠀⣴
⠀⠀⠠⢶⣯⣿⡇⠀⠀⠀⠀⠈⠛⠷⣽⡀⡀⠀⠀⠀⠈⠙⠀⠀⠈⠀⠀⢠⡿⢹⣱⣄⡀⠀⠂⠀⠀⠀⣦⡀⠈⠋⠀⢀⣾⡟
⠀⠀⠀⠀⠀⢉⠻⣄⡀⠀⠀⠀⠀⠀⠈⠛⠿⣴⣄⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⣽⢟⡟⣦⠀⠀⠀⠀⠉⠻⠛⠒⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⣄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠁⢈⣿⠃⢎⡆⠀⠠⢄⠀⠀⡀⠴⠄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠁⠀⠀⠀⠀⣾⠀⠈⠁⠴⠿⠃⠀⠀⠀⠀⢀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠀⠌⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣌⡀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⣤⠀⠀⠀⢀⣆⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⢾⠏⠀⣘⣴⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢠⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠛⠲⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⣁⠔⠁⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⢶⣼⣿⣷⠀⠀⣹⣿⡇⠀⢴⣧⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⣋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣇⣿⣿⡇⠀⣿⣷⢠⣿⣇
⠀⠀⠀⠀⠤⢀⣀⣀⡈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢥⣸⣿⣿⢺⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠻⢷⣦⣀⣀⣤⣤⣤⣤⣀⣀⣠⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣷⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠘⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣶⣶⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠻⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠋⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⢐⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⣻⣾⣶⢠⠀⠀⠀⠤⡀⣠⠀⠀⣒⢢⡀⠀⢀⡀⠀⠀⠉⠉⠙⠻⣗⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠘⠀⠀⠀⠅⢊⠎⠀⠙⣿⠀⠀⠰⡁⠈⠀⠀⠀⢀⠄⠠⠨⠕⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠐⣈⠔⠈⠀⣀⣀⣀⣸⡀⠀⠀⠈⠁⠀⠨⣀⠁⠀⢀⠄⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡿⠓⠂⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠒⠛⠀⣁⠐⠠⠍⠈
⠀⠀⠀⠀⠀⠈⠀⠄⠊⠁⡴⠙⠈⠀⣀⣤⡶⣾⠿⣟⢻⣿⢿⢿⢿⣶⣦⣄⣀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠾⢋⣠⢶⡛⠉⠀⠀⢇⠀⠀⠀⠉⠙⢻⣍⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡾⢋⣴⠋⠀⠈⠡⠀⠀⢨⠴⠆⠀⠀⠀⡐⠛⠉⠢⡄⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡟⠀⡞⠳⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡽⢆⠘⢃⠂
⠀⠀⠀⠀⠀⢠⠀⠁⠀⣾⠀⡸⠀⠀⠀⠁⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣷⠘⡆
⠀⠀⠀⠀⢠⠸⢠⠀⠀⣧⢀⡧⡄⠰⠀⠀⠀⠀⠀⡀⣦⢦⡿⠶⠶⠶⠖⠒⠒⠲⡗⠶⢾⡀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⣇⠈⡄⠀⠀⡀⠀⠀⠀⠀⠈⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢀⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣄⢱⣤⠄⠻⠄⠁⠀⠀⠀⣼⠀⠀⡀⠀⠀⠀⠀⠲⠠⣤⡰⠁⡼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⣀⠀⠀⠰⠀⠀⣤⠏⠀⠀⠀⠀⡂⠀⠀⠀⢀⡾⠃⣨⡾⡁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠘⢧⣌⢦⣴⠂⠀⠀⠀⢘⠀⠀⠀⠀⢰⡄⢀⡤⠌⠁⣼⢟⠵⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⢈⠳⣬⣙⠳⠦⠤⠼⠧⠔⠒⠮⠓⠋⢠⣠⣕⣻⠏⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢽⠶⠄⡊⠉⠉⠓⠚⠶⠶⢶⠤⠡⠒⢘⠋⠁⣊⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠃⠀⠀⠀⠁⠀⠀⠀⠀⠤⠥⠀⠍⠄⠐⠀⠀⢹⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠂⠐⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠤⡀⠀⢀⠤⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠘⣶⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠖⠀⣿⠀⠀⠀⠛⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⡴⠋⠳⣤⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠒⠚⠉⠀⠀⠀⡄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠦⣀⣀⡠⠔⠊⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⣠⣤⣄⣀⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢀⡻⣿⡿⠟
⠀⠀⠀⠀⢰⣿⣿⣿⡋⠉⠀⠀⠀⠀⠈⢙⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣀⣠⣴⣶⣿⠿⠛⠹⡇⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠛⠛⠛⠋⠉⠀⠀⠀⠀⣷⠀⠀⠀⣩⠽⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠯⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣶⣶⣶⣶⣦⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣽⣿
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣤⡶⠞⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠳⠶⣤⣀
⠀⣰⠟⠁⢀⣠⠖⠉⢀⣤⠤⠶⣶⣶⣦⣀⠀⣤⣤⣤⡄⠀⠀⢀⣠⡴⠒⠂⠀⢤⣤⠀⠒⢤⡀⠈⠹⣧⡀
⠀⡟⠀⢠⡞⠁⠀⠀⠀⠁⠀⣠⣾⣿⡿⠛⠀⠉⢹⡅⠀⠀⠀⠘⠻⣿⣶⣤⣀⠀⠘⣦⠀⠀⠙⣦⠀⠘⡇
⠀⠁⠀⢸⠀⠀⠀⣜⠀⣠⣾⡿⢟⠁⡠⢔⣂⣭⣭⣽⣿⣭⣽⣶⣦⣄⢙⡻⣿⣿⣶⣿⡇⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠈⠂⠀⠀⢻⣿⠟⠋⢀⠝⣡⣶⠟⠋⠉⠀⣶⡆⢀⠉⢙⠻⣿⣿⣧⡈⠙⠿⣿⠇⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⢣⣾⠋⠀⠈⠃⠀⠀⠈⠀⣾⠀⠉⠀⠈⠻⣿⣷⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⣿⠁⠦⠀⠀⠢⡀⠀⠀⢸⡏⠀⠀⠀⠐⠂⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⣿⡇⠀⠀⠀⠀⠀⠈⠢⠀⡿⠀⠀⠀⠀⠀⠀⡀⢸⣿⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣇⢸⠀⣷⣿⡇⠿⠇⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠙⠛⠈⣿⡇⠀⡄⡷
⠀⠀⠀⠀⠀⠀⢹⡘⡄⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⡀⠀⢸⣿⡇⢰⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠑⠑⠈⢿⣿⣆⠚⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢉⢀⣿⡟⠀⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠖⠀⠀⢡⡄⠀⠀⠰⠀⢔⣱⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣦⣄⣀⣈⣁⣀⣀⣤⣶⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠂⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠒⠚⠉⠉⠉⠉⠉⠒⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡝⠶⠛⠻⠿⣷⣶⣄⠀⣤⣤⣀⡠⣴⡬⠾⠟⠚⠳⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⣀⣤⡶⠾⢿⣿⠊⠛⣿⠿⢿⠿⠷⣦⣤⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣏⢻⡀⣨⣟⣁⡀⣴⣿⠄⣈⣳⣄⣼⡿⢛⣿⣶⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠛⡿⣿⣿⣦⣿⠿⠖⢀⣀⣁⣀⣀⡀⠐⠿⣿⣥⣿⣿⠋⠛⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⡹⣫⣿⣋⡴⠞⠋⠉⠉⠀⠄⠉⠉⠙⠲⢦⣝⢻⡙⢯⠒⠺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣩⡾⠋⠀⠀⠀⠀⠀⣰⠺⠀⠀⢠⡞⠂⠈⠳⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⠀⠀⠀⠀⠆⡇⠀⠈⠉⠀⢠⠽⠇⠀⠀⠀⡈⢧⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠃⠀⠀⣾⣷⣀⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢸⡋⠑⠈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡟⠀⡀⣀⠉⠁⠙⠦⠦⣀⣀⢠⡏⠀⠀⠀⠀⠀⢀⡀⣆⢹⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⢰⠂⡿⠀⠀⠀⠀⠀⠈⣷⣾⠀⠀⠀⠀⠀⠀⢈⡇⠛⢸⣷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢿⣿⠞⠀⠀⣀⠀⠀⠀⠀⠠⠋⠈⠣⡀⠀⠀⠀⢀⡄⠀⠂⣼⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⠠⠄⠿⠀⠀⠀⡰⠁⠀⠀⠀⠑⣄⡀⠠⠺⣧⣄⡖⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠃⠐⢆⠀⠈⡅⠀⠀⢠⡆⠀⠀⢨⣦⣀⣶⣿⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⣦⡀⠼⣦⣤⣀⣀⣘⡋⢀⣠⣢⣾⡿⢟⠕⡊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣭⣓⣻⣿⣿⣛⣛⣩⣵⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⣿⡟⢿⣿⣿⣿⣿⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠙⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣷⣶⣭⣉⣛⡛⠻⠿⠻⠿⢿⢶⡶⠦⠤
⣿⣿⣿⣿⣿⣿⣿⡿⢷⣶⣦⣤⣤⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⠏⢃⣀⠀⠀⠀⣼⠀⣶⠂⡀⠀⠀⠀⠠⣤⣴⣾⡏⠉
⣿⡏⠈⠙⠀⠙⠆⠘⠍⠉⠉⠙⠉⠉⠛⠛⠛⢣⡀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⣿⣿⡇⠀⢰⣆⣁⡀
⣿⡇⣤⣠⣾⡆⣥⣔⣀⡄⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠱⠍⠴⠦
⣿⣿⣿⣿⣿⣿⣶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⢀⣀⣀⣶⣶⣤⣶⣶⣬⣦⣶
⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⡀⠀⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢿⣶⣾⣿⣧⣤
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⣂
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣓⠒
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠁⢈⡁⣟⣿⣿⣿⣿⣿⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠏⠋⠉
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦
⣿⡇
⣿⡇
⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣄⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣷⣶⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⣶⣼⣿⣿⣿⣿⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⠏⠀⢀⣠⣄⡙⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⡟⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⣿⣿⣿⣿⣧⣐⣶⣿⣿⠟⠁⠀⣾⣿⣿⣯⠟⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⢹⡇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⣤⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢋⣀⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣙⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠲⠶⠶⠶⠒⠛⠛⠛⠋⠉⠉⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⡴⠶⠛⠉⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡴⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⢦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⣄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢈⡷⠟⠚⠋⠉⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡖⠛⠋⠹⢒⡻⣾⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢫⠂⠁⠀⠀⠀⢀⣠⠤⠾⠃⣀⣀⣀⣀⣀⣀⠘⠧⢤⣄⡀⠀⠀⠀⠈⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⣀⡴⠚⣉⡤⠖⠚⣉⠉⠤⠄⠠⠤⠉⣉⠓⠲⢤⣉⠓⣦⡀⠀⠀⠀⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⢁⡴⠋⡡⠴⠈⠁⠀⠀⢰⠀⡐⠀⠀⠀⠀⢱⠢⢈⠛⢤⡋⠳⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠁⡴⠋⡠⠊⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢆⠈⢣⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡞⢡⠚⠤⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⡈⢧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠃⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⡠⢄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⠀⠀⡀⢀⠐⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⠢⢀⡐⠀⠀⠀⠀⠀⠙⠤⠁⠀⠀⠀⡀⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢉⡷⢤⣉⡐⠀⠀⠤⠤⠤⠄⠠⠐⢂⣁⠴⢾⣁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣡⠎⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⠳⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠾⠋⠁⢀⣤⠶⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣉⠀⠀⠀⠉⠉⠙⠓⠲⢤⣄⠀⠀⠀⠀⢀⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣆⠀⢠⣀⣀⠀⠀⠀⠈⠙⢦⡀⠀⠈⠻⣦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣟⣯⣭⣥⣤⣤⣿⣉⠁⢠⣶⣾⣷⣶⣽⡀⠀⠀⠘⣷⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣴⣾⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣦⣝⢿⣿⣿⣿⣷⠀⠀⠀⠸⠇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⣶⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣝⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠸⡆⠀⣸⡇⠀⠀⠀⠀⠈⢻⣿⣦⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢿⢀⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⢠⡟⠀⢠⡄⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⢸⡇⠀⢸⡇⠀⣿⣿⠀⠀⠀⢀⣀⣠⠤⠶⠚⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⡇⠀⢸⡇⠀⢻⣿⡆⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⢿⡀⠀⢿⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⢀
⠀⠀⠘⠃⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⢀⡿⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢿⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠋⠀⣠⡾⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⢿⣿⠛⠁⠀⠀⠾⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣴⣦
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

⢀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⢿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡾⣯⣿⣳⣽⡾⣷⣿⣿⢌⠻⣾⣿⣼⡎⣳⢿⣾⣿⡟⢿⣌⠳⡌⠳⡌⢓⢜⣧⣷⠟⡈⠅⠐⠒⢈⡘⣷⣭⡅⣻⠋⠁⠤⠀⠐⠂⠒⠒
⣿⡿⣷⢞⣿⣷⣷⣟⠫⣝⡷⠌⠻⣿⡅⣿⣿⡻⣦⡙⢦⡉⠣⢜⢯⡾⣋⣕⡞⠂⠄⠀⢀⠀⠀⠀⣳⠈⠿⡌⠘⣤⠀⠀⠀⢒
⣿⣷⡿⠾⣛⡛⠻⢿⠿⣟⣽⣷⢦⡈⢡⣿⣓⠿⣮⡙⠢⢜⣵⣟⣵⣛⣸⣄⡸⠀⣤⠀⠄⠠⠀⢠⠺⠦⣯⠗⠳⢾⡇⢀⠀⠐
⠾⠛⠉⢈⡿⠿⠒⢶⠈⠻⡿⣧⣻⣧⡾⣯⣙⡳⠤⡉⣧⡼⣇⠀⠀⠀⣸⠃⢹⡣⠈⣄⠀⠀⣁⢸⣂⡠⣯⣵⠦⠊⢀⠒⢀⡂⠀⠀⠚
⠏⠀⠔⠁⠀⠀⠀⠸⢑⡀⢈⡘⣿⣿⣿⠶⢬⡉⢲⣴⣟⡿⠉⣳⢆⡖⠁⠀⠀⠈⠻⣿⣦⣶⠟⣿⢉⡿⠋⠀⠀⠈⠁⡖⠉⠀⠈⠄
⣤⣼⠀⠀⠀⠀⢀⠰⠊⠈⣭⠷⣿⣟⣯⣛⠲⢤⣿⡿⣸⣱⣿⢥⠽⠆⡇⠀⠀⠀⠀⢈⠙⠁⠘⠯⠋⠄⠀⠀⠀⠀⠀⠈⡖⠏⠉⠀⠐
⠗⢻⡅⠀⠀⢠⠀⡼⠀⠀⠣⣄⣿⡷⣬⣍⣉⣹⣿⣷⣿⢿⣿⣏⣀⣽⡃⢀⠀⣠⢤⡀⠀⠀⣆⣠⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢠⣰
⣱⠊⢞⣀⠀⢠⡄⡦⠤⠤⠤⣾⡏⠃⠴⠶⠤⡇⠂⣡⡏⢿⢭⡿⣶⠆⠀⠀⢙⠒⠘⠷⠤⢰⠛⠀⢍⠀⠀⠀⠀⠀⠀⡆⠀⠰⠼⡘⠑
⣯⣯⠮⣥⣢⠇⠷⣥⣐⣄⡁⢹⣿⣉⣉⣓⠞⠀⠀⠀⣹⡿⠤⢷⣣⠀⠀⠈⣿⠀⠀⠀⠀⠈⢁⠀⠀⠳⠀⠀⢀⠔⡇⠀⠀⡀⣷⣮⣛
⠈⠀⠀⠈⠁⠀⠀⠀⠉⠻⣿⡾⣿⠭⣉⡱⠈⠉⢲⣎⠡⠄⠒⢚⠂⠑⠀⠈⠄⠘⡁⢢⣤⡤⠈⣀⠠⣒⡥⠀⠀⠀⣿⠀⠀⠃⢰⡅⢌
⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠓⠖⢪⡁⠈⢂⠀⠀⢷⠘⠖⠨⠾⡇⠂⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⣶⠀⠀⠠⠘⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⠀⠀⠀⡀⢸⢦⠰⠒⠉⣐⠒⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡃⠀⠀⠐⠀⣿⠀⠀⠀⠀⢳⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣶⡻⣎⢧⠡⠌⣰⠾⠃⠀⢄⠀⠀⠀⠀⠀⠀⣆⠀⡇⠀⠀⠀⢀⣿⠀⠀⢀⠀⢸⢄
⠀⠀⠀⠀⡀⠠⠓⣦⣀⡀⠀⠀⠀⠀⠈⢀⡼⢋⡤⠚⡻⣦⡳⡐⠁⠀⠀⠈⠠⠄⠀⠀⠀⡄⡠⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠀⢸⢋
⠀⠀⠀⠀⠀⢂⠞⠁⡇⠉⠁⢲⣤⣴⣾⣿⡿⢫⡴⢋⡸⠎⢷⣌⠠⣀⠀⠀⠀⠀⠀⠀⠈⡰⠁⠀⠀⠀⡔⠀⢀⡎⢸⠀⠀⠀⠀⢸⢝
⠶⣀⣤⣀⣄⡌⠀⢸⡇⠀⠀⣸⣿⣿⣿⣿⢿⣫⣶⢏⡴⢋⡔⢉⡿⣮⣑⠠⡀⠀⠀⠇⡸⠁⠀⠀⠀⢰⠃⠀⡼⠀⢸⠀⠀⠀⠀⠸⣌
⠀⠈⢀⠀⠹⡃⠀⡸⡇⠀⡼⢹⣿⣭⣿⣿⡇⠹⣷⡟⣴⠏⣴⠃⡜⢣⠝⡻⢶⣤⣔⣰⠃⠀⠀⠀⠀⠌⠀⡰⠁⠀⢹⡄⠀⠀⠀⡠⣌
⠀⠀⠀⠀⣢⢷⠀⡅⣁⠞⠁⢸⣿⣿⢿⣿⣿⣦⠈⠛⢷⣾⣡⡾⣰⠏⡴⢁⡏⡚⢳⢯⠀⠀⠀⠀⠀⠀⣰⡃⠀⠀⠀⡇⠀⠀⠀⣷⢮
⠃⠀⠀⠀⡿⡼⣾⠇⣧⣀⠀⢸⢽⣷⠿⣿⣿⡾⠀⠀⢻⠙⠿⣵⢯⣼⢇⣾⢰⡧⠊⠁⣀⡂⡀⠀⣠⣜⣥⣿⡀⢀⠀⠻⡀⠀⢘
⠘⡀⠀⢀⡿⣅⢿⠀⡇⠈⢳⣾⣿⣛⣿⣯⡛⠀⠀⠀⢾⠇⡇⣿⡿⠞⠛⠬⠶⠶⠶⣖⣬⠄⣠⠞⢁⠔⠾⠛⠓⣆⡀⠀⠙⢦⡀
⠀⢡⡤⠘⠀⠇⠐⡄⢧⠎⢹⣿⣿⣿⣿⣿⣿⠀⠀⡜⠀⠀⣧⡋⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠃⠀⠀⡀⠁⠀⠀⠉⢀⡢⡌⠳⠤
⠀⠀⢣⠇⠀⠈⠄⢸⠳⡉⢺⠛⠛⢛⠛⢛⠟⠀⠘⡀⠀⢢⡈⠙⢶⣤⣾⠤⣀⣐⣶⣄⡀⠀⠀⢰⡀⡀⠀⢀⠀⠀⠀⣀⡾⠭⠛⠷⣀
⠀⠄⠘⡄⠀⠀⠘⣋⣁⠉⣋⣈⠭⠋⠉⠇⠀⢀⡴⠶⠷⠶⠒⡛⢶⣈⣩⣧⠴⠾⠟⠛⠛⠒⠚⠉⠉⠉⠉⠛⠲⠦⠤⣤⠤⠶⣿⣥⡀
⡀⠀⠀⠱⡄⠀⠀⠹⡗⠒⢾⠟⠖⣒⣊⣩⣯⣷⣦⣤⣴⣼⣟⢋⣉⠁⠀⣀⠀⠀⢀⣀⣀⣀⡈⠂⠀⠀⠀⢀⡀⣀⣀⣻⣆⣀⣠⡽⢿
⠙⣄⠀⠀⢇⠁⡀⠀⣿⣷⣿⣿⡯⠽⠛⠋⠉⠀⠀⠀⠀⠙⠛⠹⢍⠁⠉⠉⡋⠉⠁⠈⠉⠉⡄⠈⠈⠁⠈⡏⠉⠉⠀⠈⡿⠉⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠋⠉⢷⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣤⣀⠀⠐⢧⣀⣀⣀⢠⡤⠞⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢬⡇
⠀⠀⠀⠀⠀⢸⠅⠀⠉⠀⠀⠈⣿⠀⠀⠀⣀⣽⠜⠦⠤⠤⠤⢄⣀⠈⠑⡦⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡤⠞⢁⡠⠖⠋⠁⠀⠀⣀⣀⣀⣀⠀⠀⠙⠋⣰⠋⠙⠦⠼⠁
⠀⠀⠀⠀⠀⢧⣀⠤⠞⣏⠳⠊⢁⣠⡤⠶⠚⠛⠉⣉⣉⠉⠉⠛⠶⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣠⠞⠉⠀⠀⠀⠀⢾⡀⣤⣞⣀⠀⣀⡀⠈⠻⣦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⠎⢠⠞⠁⠀⠀⠀⠃⠀⠀⠀⣁⡉⠉⠉⠁⠈⠐⠃⠀⠈⢳⡄⠈⣧
⠀⠀⠀⠀⠀⢀⡎⢠⡏⢀⡀⠙⠀⠀⠀⠀⠀⢮⠁⡇⠀⠀⡄⠀⠀⠀⢤⠀⠀⢻⡀⠘⣆
⠀⠀⠀⠀⠀⣼⠀⣼⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⡆⢳⠀⠞⠁⠀⠀⢀⣠⣤⠀⠈⣇⠀⠸⡆
⠀⠀⠀⠀⠀⡏⠀⣯⣴⣚⣆⠀⠀⠀⠀⠀⠀⠀⢃⢸⡮⠤⠲⡄⠀⠀⣈⠉⣳⠀⣿⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⣿⢈⣁⠞⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠉⠉⠁⠀⠀⠈⠉⠁⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⢹⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢳⠀⢻⡄⠀⠀⡀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠃⠀⢀⡟⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠻⣄⡀⠀⠀⠶⠀⠀⠰⣿⣛⡷⠀⠀⠛⠀⠀⠀⣠⠟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠞⠉⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢦⣄⡀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⣀⡤⠞⠋⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠈⠁⠀⢤⣄⣀⣀⣀⡀⣀⣠⡤⠶⢿⣉⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⡶⠞⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢰⣿⡆⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢠⣶⡆⠀⠀⠀⢠⣶⡄⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⠶⠶⠶⠶⡿⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⣿⡿⠿⣷⣄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣿⣿⣇⠀⣿⣿⡄⠀⠀⣼⣟⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⣿⣿⣿⠀⣿⣿⣧⣤⣾⣿⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⣦⣀⠀⢀⣤⣶⠀⢸⣿⡿⠶⣿⣿⣿⠶⣿⣿⡿⣿⣿⣿⣿⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⠃⠀⠀⠻⣷⣶⣿⣿⣷⣶⣿⠟⠀⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⣿⣿⣿⣧⢀⡀⠀⠀⠈⠛⢿⣿⡿⠟⢉⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠛⠿⣿⣷⣦⣄⡀⢸⣿⣇⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢸⣿⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⠿⠿⠛⠁⠉⠓⠒⠀⠉

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

⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛
⣿
⣿⠀⠀⠀⠀⣀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⡀
⣿⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢧⡤⢧⣶⣤⣤⡤⠄⣤⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣤⣤⠄⢀⣠⣤⣤⣶⡆
⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⡌⠿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠛⠀⠀⠀⠈⠙⠛⠛⠉⢹⣿⣿⡇
⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⢺⣷⣾⠗⠀⢠⡄⢲⣼⣿⣿⣿⡇
⣿⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣷⡿⢿⣿⣿⣿⣿⡿⠏⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⣐⠈⠉⠈⠙⢻⣿⡇
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⠀⢈⣶⣏⣱⠈⠁⠀⠀⢤⣀⣈⡁⠀⠀⠀⣼⣿⣇
⣿⠀⠀⣿⣿⣿⣿⢿⠿⠟⠿⠛⢉⡉⠉⠉⠙⢿⣿⣿⣿⡿⢿⣿⣿⡟⠛⠛⠺⠿⠛⠛⠛⠟⠛⠙⣷⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋
⣿⠀⠀⣿⣿⣿⣿⣷⣶⠟⠀⢀⠀⠀⠀⢸⣆⣈⣿⣿⣿⣷⣾⣿⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⡀⡀
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣿⣾⣶⡤⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠲⠶⡿⣾⣟⠻⠓⠋⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⠉⠁
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠤⠙⢿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠉⠉⠀⢐⣠⣼⡿⡍⢀⣀⣸⣧⣤⡆⢀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠠⠆
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣄⡻⣿⣿⣿⠿⠿⠿⠷⠖⠒⠒⠀⠘⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣰⣦⡄
⣿⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠛⡛⠙⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣤⣤⣤⠀⠀⠀⠀⠀⣀⡀⣿⣿⣿⣿⡁⠄
⣿⠀⠀⣴⣿⣿⣿⣿⡟⣿⣾⣿⢿⣿⡂⣦⣡⡏⠉⢡⣄⡀⠀⣠⣤⣀⠀⠀⠀⠀⡀⣴⣦⣾⣿⠿⣿⣿⣶⣶⣦⡤⠀⠙⠻⢿⣷⠿⠟⣇⡁⠀⡁
⣿⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣁⠀⠛⠛⠳⢦⣤⣀⣠⣿⣿⣿⣿⣏⣤⡀⢠⣽⣿⣷⣤⣤⣾⣿⣿⣿⣠⣦⣥⣤⣄⣠⣄⡀
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣆⣠⠀⠀⠀⠾⠷⠿⠶⠾⣶⠦
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠙⠋⠀⠀⠨⣤⣭⠉⠀⠀⠤⠤⡤⣤⡀⠀⠀⠀⠀⠀⠀⠉
⣿⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡻⢿⣶⣤⣀⡀⠉⣀⣠⡄⠀⠠⡄⠠⠼⠗⠐⠀⠀⠀⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣅
⣿⡀⢀⡈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⡏⠙⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣿⣿⢀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡅⢿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢋⣉⣉⣉⢻⣿⣿⣿⠋⣉⣉⣉⡙⣿⣿⣿⡟⢉⣉⣉⣉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣘⣛⣛⣋⣸⣿⣿⣿⣄⣛⣛⣛⣁⣿⣿⡿⢃⣤⣤⣤⣉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢠⣤⣤⣤⢹⣿⣿⣿⠁⣤⣤⣤⡌⣿⡇⣾⣿⣿⣿⣿⣿⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⡇⣿⠇⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣬⣭⣭⣥⣾⣿⣿⣿⣦⣭⣭⣭⡴⠟⣰⣿⣿⣿⣿⣿⣿⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣭⣭⣭⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣿⣿⣱⠟⠀⠈⠓⢦⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠖⠃⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠴⠻⠁⡍⢉⠋⠁⠀⠀⠀⠀⠈⠓⠦⣤⡼⠃⢠⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⣠⡾⠟⠉⠙⣦⣄⠀⢀⠠⢞⣾⠋⠁⢀⠆⠀⠉⠀⠀⣸⣧⠀⠀⠀⠀⢠⣔⣉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠙⢧⣈⠐⠀⠓⠚⠑⠀⠀⠠⣿⡆⢀⣿⣿⡀⠀⡠⢐⠿⠋⡠⠀⡤⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡀⢀⠀⠀⠀⠀⠀⢀⡙⣷⣤⠖⠁⡠⠞⠀⢀⣿⣧⣼⣿⣿⣷⠉⠀⠁⠀⢀⠀⠚⣡⣶⡿⠗
⠀⠀⠀⠀⢠⣷⣀⡀⠀⠊⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣡⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠉⠛⠿⢷⣼⣱⣷⣄⣀⠀⠀⠀⠀⢀⣼⣯⠄⠀⠀⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣴⣧⢈⣴⣄⡀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣈⠛⠟⠋⠈⠈⠿⣦⣠⣰⣿⡟⠁⠀⢠⣿⣆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣸⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠢⣤⡀⠀⠀⠉⠛⠿⠿⠴⣦⣴⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠂⠔⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢁⣀⣤⡤⠂⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠿⠦⣀⣀⠀⠈⠙⢶⡄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠛⠉⠀⠀⠀⠀⢀⡜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠶⣶⣠⣄⢘⣻⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡴⠯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠙⠿⠛⣿⣿⣿⣿⣿⣿⠉⠁⠁⠀⠁⠀⠀⠀⣠⣴⣿⡶⠞⠁⠀⠀⠜⠉⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⠉⠁⠁⠀⠀⠀⠀⠀⠁⠀⠚⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡏⢇⣀⣀⣠⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⢹⠛⢷⣭⣭⠟⢹⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡗⢺⡿⣷⠲⢾⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠟⠀⠈⠳⢾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡏⠀⠀⠉⠳⣤⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠹⣆⠀⠀⠈⢷⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠸⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠻⠀⠀⠀⠘⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠷⠶⠾⠷⠷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢟⡝⡩⢋⠝⢉⠟⢉⠝⢉⡽⢩⠝⣩⢋⡽⢋⡽⢩⠟⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢋⠞⡴⢃⡴⢁⡴⢋⡴⢋⡴⢡⠞⣡⠞⣠⠎⡴⢁⡜⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠊⠀⠏⣠⠋⢠⢎⡴⢋⡴⢁⠞⢁⡴⢡⠞⡰⠋⡴⢃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠁⠐⠧⠊⢠⠎⠐⠋⠀⠚⠀⠀⠀⠁⠚⠴⢣⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⠶⠟⠛⠛⠛⠛⠛⠛⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⢦⡀⣋⣀⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣳⣄⣠⠆⠀⠀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⢽⣻⣽⡷⠂⠁⡄⠀⠸⣿⣋⡀⣿⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢾⣿⣯⣾⡙⣟⣥⣴⡶⠶⠿⠶⢶⣶⣭⣛⡿⣯⣄⡨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢋⣵⡿⢛⡉⠀⠀⠀⠁⠀⠀⠀⢈⠙⠿⣦⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⡟⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠂⠹⣧⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡎⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢿⡇⠘⡇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⢸⣿⠠⠀⠀⠀⠀⠀⠀⠀⢠⣞⡇⠀⠀⠀⠀⠀⠀⠀⠄⢸⡧⠀⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡀⢠⣿⠁⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠻⠷⣶⣤⣤⣬⣤⣤⣴⡾⠟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣻⣄⣀⣠⣞⣉⣳⣄⣀⣀⣴⣛⣶⣀⣀⣠⣖⣛⣲⣀⣀⣠⠖⠒⣆
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣏⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⠙⣯⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⢻⣛⠛⠛⠛⣷⣦
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣉⣀⣀⡀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡤⠒⠒⢢⠠⠒⠒⢢⠠⠒⠒⢢⢰⠒⠒⢢⡔⠒⠒⢢⠄⠀⠒⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⢣⣤⣤⡜⢰⢤⣤⡜⢰⣤⣤⡜⢸⣤⣤⣜⢣⣤⣤⡜⣢⣤⣤⡇⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⣇⡀⠀⡀⢀⠀⠀⣸⢀⠀⠀⢸⢸⠀⠀⢸⣇⠀⠀⣸⡁⠀⠀⡇⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠒⠋⠱⡔⠒⠒⠂⠐⠒⠚⠂⠐⠓⠒⢢⠐⠛⠒⢢⡔⠒⠒⢢⠒⠒⠚⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⠴⠤⣔⡳⠤⢤⠆⡰⠤⢤⠄⡠⠤⢤⠎⡰⠤⠤⡜⡳⠤⢤⢚⠤⠤⡤⠃⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣀⣀⡿⣀⣀⢀⠀⡀⣀⣀⠇⣀⣀⢀⠇⡇⠀⣀⢮⣀⣀⣀⢇⡀⠀⡘⠀⣸⣿⣿⡇
⠀⠀⠀⠀⢀⣼⠃⠀⡎⠉⠉⢁⠆⠉⠉⢷⠂⠉⠉⣲⠂⠈⠉⡳⠂⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⠃
⠀⠀⠀⠠⠾⣥⣄⣀⣉⣉⣉⠁⠑⠒⠒⠈⠒⠒⠒⠉⠒⠒⠒⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠞⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣛⣛⣛⣟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⢠⣾⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⢿⣟⣛⣿⢿⣿⡇⠀⠀⣠⣶⣶⣦⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠁⠀⠀⠀⠙⠛⠧⠄⠙⠃⠀⡴⣿⢾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⣴⣾⣿⣦⣦⡀⢀⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠹⢿⣟⢻⡿⢣⣿⣿⣿⣿⣯⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⢰⣿⣿⣿⣿⡿⢿⣦⣀⠀⠈⠉⠁⣰⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⠉⠛⠛⠶⠶⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠋⠉⠉⠉⠉⠛⠻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠿⠿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⣀⠔⢊⠩⠔⠒⠃⠈⠘⠒⢄⡈⣧⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠈⠀⠰⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⡆⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⣠⣡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣷⡀⠀⠤⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠁⠀⠀⠀⠈⠢⡀⠀⠀⢀⠤⢤⠞⠉⠁⠀⠀⠉⠲⢤⡌⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⡇⠀⣀⣠⣤⢶⣶⣾⣿⡷⠀⠀⠀⠑⢲⡗⢤⣀⠀⠀⠀⠀⠀⠈⢑⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣆⢹⡘⣿⡽⠟⢶⢲⠲⠶⢶⣿⣶⣤⣬⣱⣦⣄⡀⠀⠀⣔⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⡿⢿⣛⡭⠖⠛⠋⠉⢉⡍⠛⠒⠯⣽⢿⣷⣿⣿⠛⠻⢿⡒⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢻⣅⠀⠀⠀⠙⢦⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡴⠣⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠃⠈⢣⡙⣆⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡧⠄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢑⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⢠⡄⠀⠀⠀⢀⣠⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡇⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢠⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢷⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⣠⡎⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠴⢿⣷⣋⡄⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⣌⣒⣤⠼⣀⣀⣀⠠⠤⢤⠐⣐⣩⣶⠞⣋⣉⠁⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⣮⠟⠉⠛⠶⢿⣶⣾⣾⣿⣿⡻⠟⠛⠒⣰⠁⠿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⡄⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⡟⠁⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠒⠒⢦⡀⠀⠤⠀⡜⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠓⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣾⡣⢄⠀⡀⠀⠀⣀⣹⣄⠀⡀⠀⠐
⠀⠀⠀⠀⠀⣤⣤⣼⡧⣄⢀⠀⠀⡠⠼⢦⣠⣤⠀⠀⠀⠀⠀⠘⣰⢡⢢⣿⣿⣿⠿⢓⣁⣈⣿⠛⢤⣶⣶⣿⣿⡅⢀
⠀⠀⠀⠀⠄⣿⣿⣿⠿⣊⣉⣿⣉⡓⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠃⢸⡸⢟⡭⣛⣽⠿⠿⠿⠿⢿⣷⣾⡿⢿⣿⢿⠘⡆⡀
⠀⠀⠀⠘⣀⠟⡫⣟⡽⠿⠛⡛⠻⠿⢿⣿⣝⠿⣿⠀⠀⠀⠀⠀⠀⠀⢠⢫⡾⠋⠐⠀⠀⣿⠀⠀⠈⠻⣿⣦⠙⠚⠈
⠀⠀⠀⠀⢀⢎⡾⠋⠀⠀⢻⠁⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⢰⢀⣧⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⣞⡿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣾⢸⣿⡇⠆⠀⠀⠀⢠⣿⠀⠀⠀⠀⣀⣿⣿⢀
⠀⠀⠀⠀⣿⡇⠦⠀⠀⠀⢰⡿⠀⠀⠀⠀⠦⢸⣿⠀⠀⠀⠀⢠⢸⠈⣿⣇⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢁⣿⡟⣸
⠀⠀⠀⠀⢿⣧⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⢿⡟⠀⠀⠀⠀⠀⠀⢣⠘⣿⣦⠀⡀⠈⠀⠀⠀⠀⠀⡠⣾⡟⣡⢃⠄
⠀⠀⠀⠀⠘⢿⣷⣀⠀⠀⠀⡀⠀⠀⠀⣠⢪⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣄⣘⣀⣀⢤⣺⡴⠋⠀⠁⠋
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣦⣤⣥⣤⣖⣯⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠙⠛⠛⠛⠛⠋⠻⠆
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⠉⠉⠉⠉⠀⠈⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣤⠶⣦⡀⠘⣧
⠀⠀⠀⠀⠀⠀⣾⠁⢀⡴⠶⢦⣄⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⡟⠀⠀⣾⠇⠀⣹
⠀⠀⠀⠀⠀⠀⣿⠀⠘⣷⡄⠀⢹⡆⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢷⣄⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢧⣄⣀⣀⣠⡾⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠳⠶⢦⣄⠀⠀⠀⠀⢀⣴⠿⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠘⣯⠀⠀⣶⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠳⣄⡀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⡀⠀⠀⣀⣠⡶⠋⠀⠀⠀⠈⠛⠛⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⡏⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣴⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⡇⠀⠀⠙⠦⢤⣀⣀⣠⠤⠶⠛⠁⠀⠀⣾⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⡇⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⣰⡟⠋⠁⠀⠀⠉⠻⣦⠀⠀⢰⣿⣟
⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⣿⡇⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣠⡾⠋⠀⣾⣿⠉⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⢰⣿⡟
⠀⠀⠀⢠⣴⣶⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⡀
⠀⠀⠀⠀⠈⠉⠉⣿⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣴⠟⠛⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣼⡇⢸⡏⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣿⠀⠸⣧⡀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠰⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⠻⠶⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⣴⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿
⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠉⠉⠈⠉⠉⠛⢷⣤⠀⠀⠀⠀⣿
⠀⠉⠙⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠹⣧⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⣿
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⢀⣴⠟⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣀⣀⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢦⡼⢩⣽⣗⣛⡽⠻⡿⣏⡽⣁⣹⣄⠤⡼⠉⡙⡆⣠⡰⣬⣉⠘⡍⣛⣬⣳⢯⢘⣼⢂⡀⠈⣠⡴⣖⣒⡲⢤⡈⠈⠗⣤⠸⠾⠀⣦⡀⠈⣛⢓⠚
⢰⡷⠓⢰⣯⡋⠿⠳⠾⣿⣯⡵⣗⣓⣤⢸⣋⡾⠊⠠⠟⢀⢀⢀⣠⡀⠡⣣⡙⠃⠜⣼⠃⣀⣤⢾⣡⢎⢕⡴⣮⢂⣷⢸⡀⠠⠐⠂⠀⡏⠙⢉⡦⢀⠁
⠈⠟⢮⠿⢍⡉⢶⡆⠙⡳⠿⠟⣴⠥⣎⡭⡀⢟⡛⠲⠼⠀⠎⢋⡰⡷⡦⢩⣿⣿⡛⠉⡏⣍⡼⢞⡵⣻⣋⠀⠻⠼⢏⠼⠀⣑⡀⢀⠈⡄⠰⠀⡀⣭⠅
⠀⠀⣤⠃⢢⠤⣽⠮⠯⠯⣕⠒⠣⠟⢐⠰⠇⢘⠃⢃⠐⢣⡞⢶⣍⢢⢛⡫⡿⣿⠽⢻⠏⡽⡌⠙⢎⠹⡮⠁⣈⡡⠁⣤⣀⠀⠂⢙⠄⢷⠆⠐⡏
⠀⠀⠀⠐⠋⠀⠘⠈⠀⠀⠙⢦⣤⡤⢆⠲⠅⡵⠨⠬⠔⠀⠙⠚⠀⠉⠙⠷⠦⠾⠓⠛⠉⠊⠀⠀⠑⠓⠒⠊⠙⠃⠑⠛⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠺⠛⠟⠛⠺⠊⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠾⠺⠳⠗⠗⠺⠛⠃⠀⠀⠀⠀⠀⠘⠾⠺⠸⠓⠛⠺⠸⠓⠗⠟⠛⠃










⠀⠀⠀⠀⠀⠀⠐⠂⠶⠶⠶⠦⠶⠶⠶⠖⠆⠀⠀⠀⠀⠀⠀⠀⠰⠇⠶⠰⠶⠶⠲⠆⠀⠀⠀⠀⠀⠀⠀⠰⠦⠖⠒⠲⠶⠆⠶⠶⠲⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠋⠛⠛⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⣠⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠸⠃⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠋⠀⠀⠛⠀⢀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠖⠛⠛⣓⠲⣾⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣷⣿⣿⣷⣶⣤⣍⠛⣦⠸⣧⣤⣼⠇⢾⣅⣠⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢠⠄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣀⣿⣆⣀⣀⣈⡙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡟⠀⡞⠀⣿⣿⣿⣿⣿⣟⣡⡴⠞⣛⣯⣭⣿⣿⣿⣿⣿⣿⣿⣶⣾⣟⠻⢿⣿⣿⡇⣹⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⠟⢋⡽⢋⣡⣶⣿⠿⠛⠋⢉⣍⠉⠉⠙⠛⠿⣿⣿⣿⣷⣄⠈⠛⠶⠟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⡶⠛⠁⣴⠋⣴⣿⠟⠫⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠃⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⡿⠃⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡙⢿⣿⣿⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢸⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠄⠀⠀⠀⠀⠀⠀⠲⠷⢸⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡟⣸⣿⠁⢰⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣧⠀⠹⣿⣿⣧⡀⠆⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⣿⠃⢀⡟⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⣧⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣴⠟⣴⡿⠃⠠⠏⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣿⣿⣿⣦⣌⡁⠀⠀⠀⠿⠀⠀⢀⣠⡴⢟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡻⣿⣿⣿⣿⣿⣶⣶⣶⣶⣛⣫⣵⣶⠿⢿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣱⣿⠋⠙⠛⠻⠿⠿⠿⠿⠟⠛⠉⣁⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⠶⠶⠒⠛⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⣻⠗⣖⠓⠒⠛⠒⠒⡟⣷⢲⣿⠖⣶⣲⢶⠶⠖⠒⠒⠒⠒⠒⠲⠒⠒⠒⢲⣖⡒⠒⠒⠒⠚⠛⠻⣷
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡹⢅⢸⠄⠀⠀⠀⠇⡷⣻⣼⡇⢾⡡⢞⢸⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⠀⣩⣙⠓⠶⢤⣀⡀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⠀⠀⠀⠘⣆⢀⠉⠀⠀⠀⠀⡀⠿⠭⠺⠗⠿⣵⣿⢸⠀⠀⠀⣴⣿⠿⡿⢿⣷⡆⢸⠀⣿⠈⠙⠲⠶⢤⣩⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⢈⡴⡄⠀⠀⡀⡀⠈⠈⠉⠉⠀⠀⠀⠊⠀⠀⠀⢹⠿⠒⠐⠰⢿⠇⢸⠀⣿⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣭⣀⣿⠁⣀⣤⣠⣔⠣⠗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⠒⠁⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢤⢾⣿⢶⠛⢤⢿⣤⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣾⣷⣦⡀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣶⣿⣿⣿⣦⣈⣿⣿⣧⣀⣀⣀⣸⣥⣀⣀⣀⣀⣀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⠀⠈⠩⣗⡀⠀⠁
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇⠀⠀⠈⢹⠀⢤⢀⠄⠀⣀⣠⣴⡿⠟⠿⣿⣟⣿⣿⣿⠋⠉⠻⣿⣿⣿⠃⠎⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢉⡝⠋⣀⠀⠀⠴⡸⠄⠀⠈⠀⠄⣿⢿⣿⣧⠀⠸⣿⣷⣿⣟⣿⢦⡀⠀⡟⠛⢿⣷⡕⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣶⣌⠑⠀⠚⠀⢸⣃⣀⡉⠀⠀⠀⠀⠀⢀⣠⠄⢹⡦⢼⣿⢿⢻⡧⢼⣧⠸⡄⢚⣹⣿⣶⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣿⣼⣷⣶⣿⣿⣿⡿⣷⡕⢦⡛⠒⠒⠖⠃⠀⠀⠀⠀⠀⢠⠏⢠⠀⠨⠄⢈⠋⠈⢚⡇⠈⠉⠷⠯⠭⠝⡿⠟⠀⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⡀⠈⠉⣿⠛⣟⠛⠓⠒⠒⠒⡏⠠⢞⣧⡀⣄⢼⣿⣷⣿⣷⣶⣤⣼⡆⠀⠀⣿⣶⣾⣧⠀⣷
⠀⠀⢀⡀⠀⣀⣀⢠⣿⢫⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣧⠤⠤⣿⠀⣿⣤⣤⡤⣤⠤⠌⢲⠄⠀⠻⠹⢎⢿⣿⣿⣿⣿⣿⣿⠗⢀⡼⡻⣿⣿⣿⡇⣿
⠀⠀⣿⠉⡹⠋⠉⢙⣷⣿⣿⣿⣿⣵⡟⠿⣿⣿⣿⡀⠸⢿⣿⣄⠀⣿⠀⢹⣀⣇⠀⢸⡄⢀⡞⡄⡇⠀⠁⠀⠈⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠹⣿⣿⡇⣿
⠀⠀⣿⠾⠖⠒⢒⠞⠈⠉⠛⠟⠉⠧⡇⠘⡇⠀⢸⡂⠀⡞⠳⣍⠢⣼⢀⣾⠤⡧⠤⠤⡄⢰⠃⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⡂⠒⢲⠛⠒⠒⢲⠞⠒⠂⢠⠆⠀⠐⠐⡲⢷⠀⠉⠉⡽⠑⢾⠠⣿⠄⠶⠤⠤⠧⣾⠀⠀⠀⠀⡇⠀⠀⣿⣿⣏⣿⡇⠀⠀⠠⡄⠀⠀⢸⢿⠟⣇
⠀⠀⣿⠦⣴⣃⣀⣀⣴⣁⣀⣀⡰⠃⣤⣤⠤⡴⠥⢼⣤⡶⡴⠤⠤⠾⡶⠿⠭⣿⠭⠍⢽⠛⠓⡖⡷⠒⠛⡖⡶⢿⣿⣿⠿⣧⣀⣠⢲⣿⣄⣀⣈⣿⡄⣿
⠀⠀⣿⣞⣁⣀⣠⡎⠀⠀⢀⡞⢁⣾⠛⢁⣴⣁⣀⣀⣉⣻⣷⣀⣀⣠⣇⣀⣀⣿⣀⣀⣸⣀⣀⢻⡷⣄⣀⣛⣳⣄⡀⢱⡀⣤⣿⠇⠹⡉⢩⠟⢫⡀⠀⠿
⠀⠀⣿⣇⣀⣴⣍⣀⣀⣠⣏⣀⣀⣀⣠⣏⣀⣀⣀⣀⣰⠃⠀⠀⠀⢹⣀⣀⣀⣇⣀⣀⣸⣀⣀⣀⣣⣀⣀⣘⣄⣀⣉⣓⣷⣀⣀⣀⣀⣘⣄⣀⣀⣱⣄⣴
⠀⠀⠈⠉⠉⠉⠉⠉⠈⠁⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⡤⠔⣲⠠⠯⢍⢁⢀⡠⠠⡄⡠⠀⢀
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠸⡿⠐⠚⢃⢀⣶⢈⢇⠌⢂⡀⠉⠨⠄⢌⡛⢒⠀⠀⠼⠏⢈⣩⡄⠀⠂
⠀⠀⠀⠀⠀⠈⠁⠀⢀⡈⢠⣿⡇⠰⠨⠙⣽⣯⣤⡜⢿⡐⠒⡦⠈⠖⠉⠔⣀⠀⠛⣋⠛⠒⠊⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂⢀⠔⢂⣰⣷⡳⡏⠀⠀⣯⠾⠚⠁⠀⣀⠠⠴⢲⣠⡤⠀⠈⠝⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡷⠉⢻⡟⣀⣀⢌⠀⢦⣴⣿⡶⣒⠚⢙⣿⣥⠴⠒⠞⠀⠤⢌⣉⣋⠙⠐⠉
⠀⠀⠀⠀⠀⠀⠠⠤⠶⠚⠁⠀⠈⠀⣬⣇⡴⣀⠻⠜⠛⠛⡽⡂⣽⣳⠬⠲⢤⣤⠄⠀⠠⠐⢦⡤⠥⠤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢤⢄⣂⣈⡨⠌⠿⢟⣻⣯⣈⣥⡉⠀⢀⡉⢉⡛⢛⠒⠀⠠⠄⠒⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⠒⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⢀⣠⠤⠒⠒⠉⠉⠉⠉⠉⠉⠉⠓⠒⠤⢄⣐⠨⢙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⢀⡤⣤⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⢝⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡵⠊⠀⠀⠀⣠⡄⣠⡄⠀⠀⠀⣿⠀⣠⠟⠀⠀⠀⠀⢠⣄⠀⠀⠀⠙⠢⡁⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⠋⠀⠀⠀⠀⠀⣸⠇⣸⠇⠀⠀⠀⠋⠛⠙⠛⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⢢⡑⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢂⡞⠁⠀⢀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠎⠀⠀⠀⢤⡕⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⠀⠀⠀⠘⣆⠂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡞⠀⠀⠀⠀⣾⢠⡇⢸⠗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢀⣴⣋⠀⠀⠀⠀⠘⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⠁⠀⠀⠀⠀⠋⠀⠙⠉⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢱⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⢀⣠⣶⣿⠅⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡇⠀⠀⠀⢰⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡟⠀⠀⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⢀⣩⡟⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠟⠀⠀⠀⢠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡆⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡞⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢳⡀⠀⠄⠀⠀⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡏⠀⠀⠀⡰⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⢳⡀⠀⠀⠘⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠀⠀⡰⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠠⠱⣄⠀⠀⠀⠀⠀⠀⠀⠛⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠀⠀⠈⢀⡜⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢌⠢⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⣈⢹⡆⠀⠀⠀⢀⡴⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡑⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢲⡄⠀⠀⠀⠀⠉⠉⠀⠀⣀⠴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡂⢉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠂⣀⡠⠔⠊⠅⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠟⠓⠦⣀⡀⠉⠙⢒⠒⠶⠤⠤⠤⠴⠖⠒⠛⠉⠁⣀⡤⠖⠛⠙⠒⠲⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⣀⣤⠤⠞⠛⠒⠲⠶⠤⠬⠤⠤⠥⠴⠖⠒⠚⠛⠦⢤⣀⡀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣀⡤⠔⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣀⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉

⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠾⢶⡤⠴⠿⢦⠤⠴⠷⣦⠤⠶⠶⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠲⠚⠀⠀⠲⠊⠀⠐⠖⠃⢀⣲⣞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⡭⠇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⢠⠤⡄⣤⢤⠀⡖⢦⢰⠒⡄⣖⢲⠀⡖⣆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⣒⡃⢘⣚⠀⣉⢉⢈⡭⡅⢨⠤⡀⡬⢤⠀⣇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠧⠼⠘⠤⠇⠙⢚⠂⢓⣛⢘⣒⡃⢙⣚⡀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣇⣀⣳⠀⢻⣹⠄⣏⣳⠸⠭⠇⠯⠼⠀⠧⠗⠘⠤⠇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⠀⢰⡒⡆⢴⢲⡀⣖⣲⢰⣚⡆⢯⣹⠀⣏⡧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣩⡥⠬⠭⠤⠬⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣴⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠹⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣋⣭⣍⠀⠀⠈⢿⣆⠀⠻⣶⡟⣄⣴⣾⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣠⣀⣤⣴⣷⣿⣿⣿⣿⣿⣿⡌⠑⠦⠀⠀⠀⠀⠀⠘⠿⣮⡁⠈⠉⠙⢦⣤⣴⣿⠿⠷⣶⢤⡀⣀⡼⠛⠛⢷⠄
⠀⠀⠀⠙⠋⠳⠈⠻⠿⢉⣤⣿⣿⣷⠀⠀⠀⠀⢀⣠⣤⣴⡆⣄⡀⠁⠐⠂⢠⣌⣀⡁⠀⠀⣠⡶⣿⣿⠙⠄⠀⠀⠀⠙⠓⠤⠴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⢴⣶⠿⣆⠈⠻⢿⡏⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡊⠛⠷⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⢀⣾⠀⠀⠘⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠙⠷⣶⣶⣿⣿⣿⣿⢛⠽⢿⣷⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠈⠁⠎⠀⠼⢻⣿⠿⢻⡃⠀⠀⠀⠀⠠⣴⣿⣿⡙⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠰⡚⠀⠈
⠀⠀⠀⣶⣤⣄⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠶⠀⠀⡴⠃⠀⣤⣀⣠⣄⠀⠀⠀⠀⠀⠙⢫⣷⣤⣄⠀⠀⣿⣿⣿⣿⡞⠉⠈⠁
⠀⠀⠀⢿⡿⠿⠷⣿⣶⣦⣀⣄⠀⠀⠀⣼⣧⠀⠀⠀⠈⠀⠠⢖⣼⣿⣿⢯⡿⠷⡀⠀⠀⠀⠀⠀⠉⠙⠀⠀⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢄⠀⣿⣿⣿⣿⣿⣷⣾⣦⣿⣿⣆⣾⣦⠀⠀⠀⠀⡈⠙⠻⠧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⠆⣀
⠀⠀⠀⢤⣤⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡴⣄⠀⣾⣿⣴⣆⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠿⢿⣿⡿⠿⣿⣿⣿⡿⢿⣿⣿⣦⣿⣿⣿⣿⣴⣦⠀⢠⣄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣯⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠁⠈⠙⠛⠛⠛⠶⠿⠿⠅⠀⠀⠰⢷⣴⠿⠟⠋⠙⠛⠻⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢿⣤⣿⣷⣾⣻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣟⣻⣾⣶⣿⡟⢿⠛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣿⣿⣿⣿⣿⣿⣷⣾⢷⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠉⠙⠻⢿⣿⣿⣷⣦⣼⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠸⢿⣾⣷⣷⣶⣶⡆
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣤⣤⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠹⢿⣇⡄
⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⣶⢸⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⠀⣰⢷⠠⠊⠀⣤⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⣀⠀⢀⡠⠄⡼⠿⠏⠀⠀⠀⠀⣄⣠⣴⣿⣿⡳⢠⡾⠁
⠀⠀⠀⠀⠀⣠⠝⠶⣤⡀⠀⠻⣿⣿⡿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⢋⢡⡄⢀
⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⣦⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣋⠵⣾⣿⣼⣧⣸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣼⣧⢺⡟⢾⣿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⠘⢻⣿⣽⣿⣿⡿⣿⣿⣟⢩⠁⠀⠉⠙⠻⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⠀⠈⣿⣿⣿⣽⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣧⣄⣿⣿⡏⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣧⠄⠀⠀⠀⠀⠀⠈⣻⣿⣿⡿⠀⡸⢁⣀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡋⠁⠀⠀⠀⠀⠀⢠⣿⣽⣿⡇⠀⠠⠋⡽⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠹⣿⣿⡗⠀⠐⣌⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠻⣷⠀⠀⢸⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣧⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠻⣜⣧⠈⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⠒⠛⡋⠉⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠄⠀⠀⠂⠈⠉⠉⠁⠒⠦⣄⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⠔⠈⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⣁⣠⣴⡾⠿⠛⠛⠉⠉⠉⠉⠉⠀⠀⠉⠛⢷⣮⡽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠏⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢦⣱⠀⠀⠴⠚⠛⠛⠛⢂⠒⠀⠀⣙⣛⣉⣉⠀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠉⠀⠀⠀⠘⠛⠛⠀⢴⠁⠀⢏⠁⠍⠉⠉⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⢠⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⡆⠀⠀⠀⢀⡴⠀⠠⢍⠀⠀⢀⠀⢠⡀⠀⢸⡾⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡴⠟⠀⢀⡀⠀⠡⠖⠁⠀⢀⡉⣳⣸⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡆⠀⠙⢦⣍⣉⣛⠳⠞⠫⠵⠋⠀⠀⣿⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣆⠀⠀⠉⠛⣒⡓⠂⠂⠀⠀⠀⢰⡏⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⡃⠀⠐⠀⠀⠀⠀⠀⠀⠀⡴⠋⠻⣄⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⢦⡀⠀⠀⠀⠀⠀⢀⡾⠉⠲⣄⡿⣋⣘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣄⠀⠙⠳⠶⠶⠒⠒⣻⠑⠶⠦⡤⢿⠢⣌⠣⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⠙⢦⣀⣀⠀⢀⣴⠏⡀⠀⠀⠀⠉⡇⡏⢗⠮⣳⡶⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣜⡁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢳⡀⠀⢀⡖⠉⠙⢢⢂⡿⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣧⠀⡘⡂⠙⢦⠀⠀⠀⠀⢀⠀⠀⢀⠐⣆⢈⣷⠠⡾⠙⠓⠿⠕⠊⢱⡜⠋⡙
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠬⠶⠼⢀⣟⡄⢈⡷⠻⡄⠠⢀⣰⠃⠈⠋⠘⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⠠⡤⠤⠏⠐⠁⠀⠀⢻⡎⠀⠀⠀⠀⠀⠀⢀⣠⠞⠒⠒⠛⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣇⣀⣀⣀⣀⣀⢠⣤⡓⠀⠀⠤⠴⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠂⣾⠏⠙⠉⠉⡟⢓⠉⠉⠉⠉⡏⢩⠀⣧⠀⠀⠀⠀⣠⡴⠒⠛⠋⢹⠃⠀⠀⠀⠰⡋
⠀⠀⢀⡴⠶⠁⠂⠀⠰⢦⡄⠾⠁⠒⣄⠀⠄⠈⠉⢥⡀⠀⢀⡾⠃⠀⠀⠀⢐⡭⠶⠶⣦⠔⠋⠁
⢰⠞⠋⡠⠀⣉⠟⡿⠦⣤⣽⣼⡿⣧⣸⡇⢸⣿⣷⡀⡳⢠⡟⠀⠀⠀⠀⠀⣬⠀⣖⠼⡁
⠈⠉⠀⠋⠀⠈⠙⠶⠒⠃⠐⠓⠚⠶⠞⠃⠩⠿⠥⠔⠐⠉⠀⠀⠀⠀⠀⠰⣯⡬⠍⣊⣉⣊⡽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣦⣾⡁⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢀⣴⣶⣦⡄⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⣀⣀⣠⣤⣄⣸⠁⣀
⠀⠀⠀⠀⢸⣿⡀⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⡈⠹⣿⣿⣿⣿⣿⡋
⠀⠰⠿⠋⠀⢻⣷⣼⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠀⠀⠀⠀⢉⣴⣿⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣶⣬⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣾⣿⣟⣻⣿⣿⣿⣛⣿⣷⣿⠀⠀⠀⣰⣿⢿⣿⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣰⣿⣿⣻⣿⣿⣌⠿⠛⠿⢿⣻⣿⣿⡟⣿⣿⣿⣧
⠀⠶⠀⠀⠀⢀⣴⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠏⠶⠀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠃⢀
⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣽⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⠀⣠
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡏⣿⣿⠯⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⣀⡉⠉⣀⡈⠻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠛⠛⣷⡆⢿⣇⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⡾⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠾⠁⠘⠻⠟⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠛⠋⠛⠛⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⣶⣤⠶⠚⠋⠉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⣆⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠛⠓⢦⡀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⢀⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣴⡦⡀⣻⣤⣄⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢠⡾⣿⠃⠀⠀⣟⠀⠀⠀⠀⠀⠀⠘⠿⢿⡟⠁⠀⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣸⣇⣤⠶⠚⢻⡄⠀⠀⠀⠀⠀⠀⣠⣾⣶⡄⠀⠀⣸⠃⠉⠙⠛⠒⢲⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠋⢠⡴⠶⠤⣍⠓⠶⠶⠶⠚⠋⠁⠙⢿⡴⠶⠚⠛⠀⠀⠐⠲⣤⡈⢷⣀⣀⡴⠖⠲⠶⣄⠀⠀⠀⠀⢻⡄⠀⢻
⠀⠀⠀⠀⠀⠀⢠⡟⠁⢻⠁⢾⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁⠑⠀⠙⢁⣤⠶⣦⣤⠈⡇⠀⠀⠀⠀⢿⡟⠉
⠀⣀⠀⠀⠀⠀⣿⠁⠀⠘⣇⠈⠳⢦⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠚⠉⠉⠁⣠⡾⠁⠀⠀⠀⠀⠸⣧
⢾⣉⡛⢮⡀⠀⠘⣧⠀⠀⠈⢧⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁⠀⠀⠀⠀⣠⠾⠛⠒⠒⠚⠋⠁⠀⢀⠀⠀⠀⠀⠀⣿
⢸⡉⠙⣶⠛⠉⡷⠈⠳⣤⡀⠀⠉⠳⢦⣄⡀⠀⠀⠀⢀⣀⣠⡴⠞⠋⠁⠀⠀⠀⢀⣠⢴⣟⠁⠀⠀⠀⠀⠀⠀⢀⡴⡿⠀⡄⠀⠀⠀⣿
⢯⣉⣉⡟⠋⠘⠧⣄⣀⡬⠽⠶⣤⣀⠀⠈⠉⠉⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⡴⠚⠉⠀⠀⠈⠳⢶⡾⠁⠀⢀⡴⠋⣼⢃⡼⠓⠒⠛⠚⠃
⠀⠈⠉⠑⠛⠦⣄⣀⠀⣷⠀⠀⠀⠉⡻⠒⠶⠦⠤⠤⠤⠶⠶⠒⠛⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⠤⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⣧⣤⡤⣼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⣧⠀⠀⡟⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠷⢚⠗⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣶⣶⣶⣦⣤⡤⢤⣤⣴⣚⣡⣄⠀⢈⣉⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⠤⢼⡇⢸⣭⣭⣭⣭⣭⡷⢤⡽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢍⣉⡻⠤⠤⣤⡤⠷⢾⡇⠀⠀⠀⣀⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠛⠶⣄⠀⠀⠀⣴⡶⢦⡀⠀⣠⡴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢠⣾⡟⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠂⢀⣠⡴⣞⣩⡤⠶⠒⠚⠛⠛⠓⠒⠶⢤⣄⡈⠳⣤⡀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠚⣩⡵⠛⣩⡤⠶⠚⠛⠉⠉⠉⠉⠙⠓⠲⢤⣍⠓⢦⣝⠲⣄⡈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⠳⣄⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠹⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⡿⠟⣿⠟⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣷⣤⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡀⠹⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢦⣙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⣌⡙⠲⠦⢤⣤⣤⣤⣤⠤⠴⠖⢋⣡⢴⣞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠛⠉⠉⠛⠒⠶⠶⠦⠴⠶⠶⠒⠛⠉⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡞⢛⠏⠉⠉⠉⣈⠉⠉⠋⠭⡷⡲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡟⢀⣀⡤⠄⠀⠈⠳⠤⠄⠀⠀⠈⢌⣒⣛⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣹⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣾⣿⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⢼⣾⣧⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣰⣷⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠁⢀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⢸⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⡇⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣳⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣫⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡃⣾⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡴⢚⣿⡿⠿⠛⢉⡉⣰⣟⣶⣼⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⠀⠀⣠⡼⠁⣹⠏⡝⠀⠀⣾⢁⣿⣿⠛⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠀⣽⣿⣿⡿⠠⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣼⡿⠃⠀⠀⠀⠀⠀⢿⣿⣿⣟⣿⣿⡿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡟⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡧⣯⣿⣿⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆⡻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣅⣀⡀⠀⠀⠀⠀⢠⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣼⢿⣿⣿⣷⣤⣼⣿⣷⣭⣿⠛⠗⠁⠀⠀⠀⢀⣨⣉⣺⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠝⠛⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣷⢆⣤⡄⠁⠻⠋⡶⢂⣤⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⣌⠀⣨⣉⠀⠸⠤⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣤⣤⣶⣄⢈⠛⠁⡈⢉⢻⣿⣿⣿⣿⡿⣿⣿⣾⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⠿⠿⢿⠆⢤⠠⠚⣺⣿⣿⣿⣯⢽⣿⣿⢣⣽⠿⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠕⠈⠉⠛⠛⠛⠛⠛⠋⠀⢀⣤⣀⣒⡶⣦⣶⣿⣿⣿⡟⠻⠞⠿⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣿⣿⠿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⣀⠉⢻⡅⠁⠀⢂⠀⣁⣾⠋⠀⠉⠑⡍⠈⢹⣦⡾⢛⣿⠛⠛⠋⢿⡀⡸⠏⠛⠛⢻⣟⠿⣖⣾⠉⠉⡕⠉⠁⠈⢹⣆⡁⠀⠐⠀⠁⣽⠋⢁⡀
⠏⠁⠐⠚⠓⠒⠁⠈⢫⠈⠙⠃⠘⠃⠀⢹⠀⠒⠓⠛⠈⠛⢀⡀⠀⠒⠛⠙⠒⠀⢠⠀⠈⠚⠓⠀⠀⣹⠁⠑⠋⠒⠛⠋⢨⠋⠀⠑⠒⠛⠂⠀⠉
⠠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠦
⠲⡆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡖
⠑⠃⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠓
⢐⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢒
⣿⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢼
⣾⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢹
⡶⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠴
⠲⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⢝⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡴
⠳⠄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠴
⢸⡅⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿
⠒⣳⣒⣶⢲⡶⣒⣶⣶⢆⠐⢲⠐⠒⠀⢛⣷⠂⠒⢂⠀⠀⢺⣅⠒⠒⠀⠐⠒⢀⣼⠂⠀⢀⠒⠂⢲⣞⠃⠐⠒⢰⠒⢀⢶⢶⣖⡲⡶⣲⣖⣲⡛
⠈⠉⠉⠉⠁⠉⠉⠉⠈⠉⠀⠁⠉⠀⠀⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠈⠉⠈⠉⠉⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛