Pag Emojis & Symbols

⠀⠀⠀⠀⣠⡶⠶⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡿⡇⠀⡔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⢄⣀⡀⠀⡠⢀⡒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠈⠉⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠀⠀⢀⠀⢐⠀⠄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠒⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⢣⠀⠀⠉⢄⡀⠁⠖⢄⡀⠀⡆⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⢣⡐⠂⢄⡀⢀⠀⡆
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠐⠂⣀⢰⠄⣀⡀⠀⠀⢀⠀⠀⠈⣅⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠂⠄⠀⠈⣄⠀⠀⢀⡴⠁⠐⡀⠀⠀⠀⢘⠀⠄⠉⠁⠂⠌⠃⠔⠉⠣⠄⠀⠀⠄⣀⡤⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡴⠖⡁⠀⠀⠀⠀⡋⠐⠐⠄⣀⣀⡈⡀
⠀⠀⠚⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡆⠀⠐⠃⠈⠁⠉⠂⠷⠒⢊⠁⠈⠳⠄⠁⣲⠖⡩⠤⡶⠛⠀⠁⢠
⠀⢀⠀⠰⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⡄⠈⠆⡴⠂⠐⢤⢒⡡⢌⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠙⠤⢤⡀⠄⠐⠂⠉⠀⠀⠀⠀⠀⡄⠀⠀⡀⢀⠎⠀⠀⠈⠉⠀⠀⠀⠈⡉⠕⠤⠁⠀⠀⠀⠀⠀⡐
⠁⢄⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠠⡅⠐⠂⠈⠸⠀⠀⠈⠂⣄⢈⡀⠤⠔⠀⠁⡐⢂⠒⢢
⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣀⠌⡅⠀⠀⠀⠊⠐⢁⠔⠁
⠀⠀⠀⠐⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠈⠉⠛⠉⠣⠀⠐⠀⠃⢂⠁⠂
⠀⠀⠀⠀⠘⠹⢤⣀⡀⠀⠀⡠⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⣀⡄
⠀⠠⠀⠀⠀⢀⠀⠀⠠⠄⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠂⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠊⠁⠦⠤⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⠀⠄⣀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢄⠀⣀⣠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠁⢂⠄⡀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠤⢨⠐⣂⠀⠀⠀⠀⠈⠠⠄
⣶⣖⢲⡶⢶⣶⠶⡶⢶⠶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡿⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡿⢃⡤⠀⠲⢮⡻⣿⣿⣿⡍⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⣟⣷⣿⣵⠄⠒⢢⡟⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⡇⠀⠀⠀⢹⣮⢻⣿⣿⡄⠈⠻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠟⠋⢠⣿⣿⡿⣭⡟⠂⠀⠀⢸⡗⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣇⠀⠀⠀⠀⠈⠁⠹⣿⣷⠀⠀⠀⠉⠋⢸⡇⢀⣄⣁⣉⣈⣁⢸⡇⢋⡉⠀⠀⠀⣾⣿⠏⠘⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢀⣠⣤⣭⣧⡛⢍⠙⢭⣽⣍⣮⣥⣬⣀⡀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡔⢆⡀⠀⡀⠀⠀⣀⠻⣤⢃⡀⠀⠆⣸⢇⡙⠀⠀⠚⠁⢿⡔⣋⠘⠛⠋⡘⣿⠏⠠⠀⠀⠀⠀⢀⣤⣶⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⣿⣿⣤⣉⣴⣇⡀⢀⡥⠀⠹⣆⠀⠈⠰⢉⣼⠁⠀⠀⠀⠀⠈⣷⡍⢆⠑⠤⣱⠏⠠⠀⠀⠀⣠⣄⣉⣤⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⠟⠛⣉⢉⡙⠛⡙⠶⢤⡙⢦⡰⡕⠉⠻⢷⣆⡀⢀⠰⢿⠋⠱⣎⢦⡴⠋⠴⠶⠞⠛⡛⢉⣉⣛⣛⣿⣿⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣥⣌⠙⡁⠀⢀⠴⣡⠎⣷⠈⠀⠀⠈⢅⠀⠈⠀⠂⠀⠀⢈⣿⡱⠄⢀⠀⠀⠈⠈⠉⢭⣽⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠫⠈⠂⠐⠀⣠⠞⠋⠀⠻⣧⡀⣤⠄⢨⣥⣤⡦⢠⣄⣠⣿⠏⠀⠙⢷⣄⠀⠄⠂⠀⢎⡙⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠋⠄⠀⠈⢀⣀⣤⡴⠞⠁⠀⠀⢠⡾⠛⠁⠀⠰⠂⠀⠀⠠⠀⠀⠈⠉⢿⡄⠀⠀⠉⠳⢦⣤⣀⡀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣸⣿⣿⢟⣡⣤⣶⠶⠶⠛⠛⠋⠁⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡂⠀⠈⠠⠀⠈⠛⠛⠻⠶⠶⣦⣤⣌⡹⣿⣿⣧
⠀⠀⠀⢠⣿⣿⡵⠋⠉⠀⠀⢀⣀⠠⠤⠀⣈⣉⣀⣐⣒⣛⡁⠄⠀⡚⠀⢠⣄⠀⡄⠀⡀⠀⢀⠀⠶⢘⣛⣒⣖⣒⣙⣓⠀⠦⠔⠀⡀⠀⠀⠉⠙⠾⢿⣿⡆
⠀⠀⠀⣾⢟⡀⠀⣀⣤⠾⠛⠉⣡⣶⠾⠿⠿⠿⠿⣿⣿⣿⣿⣷⣦⡉⢤⣼⣯⠀⠀⠠⣅⠠⣀⣴⣾⣿⣿⣿⣿⢿⠿⠟⢿⠿⣶⣄⡈⠙⠲⣤⣀⠄⠀⠹⣿
⠀⠀⢠⡿⢈⣤⡾⠋⠡⢄⣤⣚⣍⣀⠲⠄⢴⣶⣾⣿⢻⡿⢻⣿⣿⣿⠘⡏⢁⣄⠰⠷⣉⠃⣿⣿⣿⡿⣿⡟⣻⣷⣴⣤⠀⠐⢂⣉⣻⣦⠀⡈⠹⢷⣤⡀⢻⡇
⠀⠀⣼⡟⡟⠩⠄⢈⣡⠶⢶⣭⠶⢝⣳⢦⡀⠈⠻⢿⣷⣶⣾⣿⣿⣿⡀⠃⠈⠛⢷⠀⠀⢂⣿⣿⣿⣿⣶⣾⡿⠟⠁⢀⣠⢾⡳⠾⢯⡶⠶⣤⡒⠠⡉⠛⢻⣧
⠀⠀⣿⠁⣹⢀⡴⢋⣄⠛⠛⢛⡻⠷⣌⠻⣽⠦⡀⠀⠙⢿⣤⠉⠉⢻⣷⡀⠀⠈⠀⠀⠀⣼⡿⠋⠉⣩⡿⠋⠀⠀⣰⣿⠟⣫⠾⢟⣛⠛⠛⢠⡙⢶⣤⡜⣼⣿
⠀⢠⣿⢸⢀⡿⢠⠎⠀⠁⠄⣁⠄⠀⠈⢳⣌⢷⣿⣆⠀⠀⠹⣷⡀⠈⣿⣧⠀⠀⠀⠀⢸⣿⠁⠀⣸⠟⠀⠀⣠⣿⣿⢣⡾⠁⠈⠠⠀⠀⠃⠄⠙⡆⢹⡄⡇⣿
⠀⢸⡇⢸⢿⣇⠈⠀⠞⠀⠠⣄⠀⠁⠀⡀⢻⣮⣷⢙⢿⣦⡀⠈⡿⠁⢸⣿⠀⠀⠁⠀⢾⣿⠀⣰⠋⠀⣠⣾⠟⣹⣧⣿⠀⢀⠀⠀⠀⢀⠀⠂⠀⠃⢨⣇⠁⣿⡆
⠀⢸⡇⢸⠀⣿⣄⠀⠀⠀⠀⠘⣦⣬⣀⠀⣸⣷⡿⣸⠃⠙⣿⣄⠃⠀⠾⡏⠃⠀⠀⠉⠸⠏⠀⢾⢀⣾⠏⠱⠀⣿⢿⡏⠴⣊⣤⣴⠂⠀⠀⠀⠀⢠⣿⠁⡆⣿⡇
⠀⣸⡟⢸⡄⠘⢿⠇⠀⠀⠉⠈⠉⠙⠻⢷⣤⣛⣽⠏⣠⠀⠈⣿⡄⠀⠀⠁⠀⠀⠀⠈⠁⡆⠀⢠⣿⠃⠀⠀⢱⡟⣞⣡⣾⠟⠉⠉⠀⠀⠈⠁⠀⠸⠏⠀⠀⣿⡇
⠀⣿⡇⣼⠃⠀⠁⠀⠀⠀⠈⠀⠀⠠⠄⠀⠉⠻⢷⣦⣘⠀⠃⢸⡗⠀⢰⠟⠀⠀⠀⠀⠐⡇⠀⣾⡏⠀⠀⢈⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢻⣧
⠀⣿⣧⡟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⣸⡇⠀⢸⠀⠀⠀⠀⠀⠀⡃⠀⣿⣇⠆⣴⡿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢸⣿
⠀⣿⣿⠀⣿⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣿⣧⠙⠐⠀⠘⠀⠀⠂⠀⠀⠀⠁⠀⠈⠿⢰⣿⠁⠀⠀⠀⠀⡀⠀⠀⠀⠒⠂⠐⠂⠀⠀⠀⠀⢸⠎⣿
⠀⣿⡏⠘⣿⣛⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣼⡿⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⣾⣿⠀⠀⠀⣀⠄⠁⠀⠈⠉⠛⠀⠁⠀⠀⠀⠀⠀⣼⢰⣿
⠀⣿⡇⢸⢸⠅⠀⠀⠀⠀⠉⢀⡄⠈⠁⠀⠄⠀⠀⠀⢹⣿⠃⡀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢹⣿⢀⣴⡾⠋⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⢀⡀⠘⣮⣿
⠀⣿⡇⢸⠀⠀⠀⠂⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⠀⠀⣄⡟⠀⠀⠀⠀⠛⢴⡄⠀⠀⠈⢻⣾⠏⠀⢀⠀⠀⠤⠀⠠⠄⠀⠐⠀⠀⠁⠄⠀⡆⢸⣿
⠀⣿⡧⢸⣆⠀⠉⠠⡄⠀⠠⣀⠀⠂⠄⠀⠀⠀⠻⣶⣼⡿⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠐⣇⠐⠀⠀⡄⢺⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠙⠁⠀⢠⡿⢸⣿
⠀⣿⡇⢈⢻⣆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡃⢻⣿
⠀⣿⣧⠈⠀⠹⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⠞⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣾⣿
⠀⢿⡟⢠⠀⣤⡈⠻⣶⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠀⠀⠀⠀⠶⠄⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣴⠟⢀⣤⠼⡇⣿⣿
⠀⢸⣇⣸⠀⠉⢻⣦⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠃⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⣠⡟⠁⠸⣇⣸⡏
⠀⠈⣿⠹⢀⠃⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⠐⠂⠑⢀⣼⠏⠆⠀⢀⡿⣿⠇
⠀⠀⢿⡄⠉⠀⢳⠀⡌⢿⡄⠆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠁⣰⣿⠀⠀⠀⠀⠀⠘⠁⠀⢰⢆⣾⠋⠀⡇⢠⡋⢠⣿
⠀⠀⢸⡇⠉⠃⠼⢆⠀⠈⢷⣎⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠈⠙⠛⠻⠿⠛⠋⠁⠀⠀⠀⣴⣿⡟⠀⠀⠠⠀⠀⠀⠀⠀⢹⡾⠃⠀⣀⠛⠀⠙⢸⡏
⠀⠀⠘⣿⠀⠃⡀⠀⠑⠀⠘⢻⡄⠰⠚⠘⠓⠒⠒⡀⣿⡏⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠀⠀⠲⠖⠐⠶⠆⢡⡾⠃⢠⢀⠉⢠⠀⠀⣿⡇
⠀⠀⠀⣿⣦⠀⠁⠀⠀⢃⡀⠀⢿⣦⠀⠀⠘⠛⡈⠃⣸⣧⢻⣿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣾⣿⣿⣿⡏⣿⣧⠘⠋⣻⠃⠀⠀⣰⡿⡉⠄⠉⣀⠀⡁⠀⣸⣿
⠀⠀⠀⠸⣿⠈⠀⠀⣀⣈⡑⠂⠈⢻⣏⠲⠈⢀⣹⠰⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣿⣿⡆⠀⠀⠐⠀⣰⠏⠀⠀⠔⢡⣄⠐⠄⠀⣿⡏
⠀⠀⠀⠀⢻⣇⠀⠀⣄⠉⠉⠳⡀⠀⠹⣷⣤⣜⢻⢸⣿⡝⣿⣄⠻⢮⡍⣻⡿⠿⠿⢟⣟⣩⡵⠛⣰⣿⢿⣿⡄⡸⢃⣤⢾⠏⠀⠐⠀⠋⠁⠀⡀⠀⣸⡟
⠀⠀⠀⠀⠈⢿⣧⠀⢘⣃⡀⠀⠙⠀⠀⠘⣄⠙⣷⠈⣿⡇⣹⣿⡆⢀⠀⠸⠇⠘⠃⠹⠃⠀⠀⣰⣿⡟⢸⣿⠀⣴⠏⠀⢆⡀⠀⠀⠁⠀⣀⣤⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣷⡀⠉⠃⠀⣀⡾⠀⠀⢹⡆⠸⡇⢸⡧⠘⣿⣿⡈⡆⠀⠐⠁⢀⠂⢀⠦⢰⣿⣿⠁⢸⣿⢃⡟⢀⡚⠀⠠⢂⣀⡀⠀⠉⢁⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠂⠀⠈⠁⠀⠀⠀⠙⡄⢹⣾⡇⠀⢻⣿⣧⠘⠀⠀⠄⢠⠀⠃⠀⣾⣿⡏⠰⡈⣿⠘⢃⠎⠁⠀⠀⠌⠁⠀⠁⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠘⣧⠄⠀⠀⠀⡀⠘⣧⢸⡇⠒⡨⣿⣿⣦⡀⠐⣤⣼⠗⢀⣼⣿⡿⠃⢀⢰⡏⣰⢏⠀⢀⠠⠋⠀⣈⠁⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡈⢻⡇⠀⠀⠀⠀⠈⢎⡱⠆⠁⡈⠻⢿⣿⣷⣶⣶⣾⣿⡿⠟⠁⠰⠁⡞⢱⠁⠀⠀⠀⠀⠀⡼⢋⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠹⢦⣄⠀⠀⠀⠈⣷⠆⠀⠀⠀⠀⢸⣯⡉⠉⣽⡏⠀⠀⠀⠀⣿⡀⠇⠀⠰⢄⣉⠶⠈⣠⣾⠟⠁

⣿⠉⠙⠋⠉⠛⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢉⣯⣏⡉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠌⢛⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠛⠟⠻⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣰⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠹⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⡶⡶⡶⢶⢴⣶⢶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⢀⢀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⣀⣀
⣿⣿⡿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠷⡄
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⣤⡀⠀⣤⡄⣤⡄⢀⣤
⡀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣤⣿⡇⣿⣷⣿⠁
⣊⠀⠀⠀⠈⠂⠱⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⠀⠻⣿⠇⣿⡏⠻⣷⡀⢴⡦
⠉⡀⠀⠀⠰⡀⠀⢱⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⢠⡤⢀⣤⠄⣤⣠⡤⡄⣤⣀⡄
⢠⡏⠁⠀⢀⢷⡀⠀⠇⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠟⠻⠯⠤⢽⠗⠿⠻⠴⠷⠿⠻⠇
⡘⢰⠏⡃⠐⡈⣷⠀⠘⡈⠀⡆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤
⢁⠏⡜⢉⢄⠁⠸⡇⠀⢁⠀⠃⣿⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠁
⠎⡜⡀⣉⠋⠀⠀⠻⠀⠈⠀⢠⢿⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠎⠔⠀⣉⠣⠀⠀⠀⠀⢀⠀⠀⠘⠈
⡢⠀⢀⡉⠀⠀⠀⠀⠀⠘
⠀⠐⣉⠙
⠀⣈⡉⠃⡇⠀⠆
⠊⠉⢻⠀⡇
⠛⣷⠘⠃
⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢢⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢎⢣⡑
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⢱⡘⣎⢳⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⢹⡘⣄⢳⡘⢦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢎⢧⠱⡘⣆⠱⣄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠀⠈⠈⠌⢆⠣⡐⠌⠣⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⢹⡄⢂⠀⠀⠀⠢⡑⢝⢦⡑⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣼⡇⠀⢣⠀⡀⢀⠈⠂⠑⢌⠒⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠋⠀⠼⠂⣱⠀⣠⠀⣦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠉⠀⠀⠀⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⣠⣀⣀⣠⣤⣾⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡄
⡁⢨⡿⢻⡿⣿⣿⣷⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢗⠀⣧⠀⢀⣀⣠⠔
⣤⢾⠇⢸⡄⣸⣿⣿⣿⣇⠱⣦⣀⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⢿⣶⡰⢁⠚⣤⡿⡖⣯⡿⢃⡞
⡏⠜⠀⣸⢠⢻⣿⣿⣿⣿⣦⠀⠹⣿⣷⣿⣣⣀⠀⠀⠀⠀⠀⠀⠀⣬⠭⣧⡀⠀⠀⢠⣯⠏⣸⣱⠋⣤⡿⡟⡼⢹⣷⢛⡷
⠃⠁⣰⠏⢘⣟⣿⡿⢏⢻⢇⠙⢦⣄⠈⠻⣿⣿⡷⣶⣴⢦⣄⣀⣼⠜⡄⠘⠙⡦⣤⠟⠃⣸⠿⣁⡾⣱⢂⠟⣵⠏⣥⠏
⠇⣴⠋⢀⣼⣿⡟⠀⠘⢸⡇⠑⡀⠻⡕⠀⠙⠿⣿⣿⣮⡓⣨⠏⠁⣼⠃⡀⠀⡙⣾⣦⣞⡧⢞⣽⠟⠰⢫⡾⢁⣾⡟⠀⢀
⡞⢁⣶⣿⣷⣿⠃⠀⢀⡏⠁⡀⢸⡲⠙⣆⠘⠂⠈⢻⣿⣿⣷⡟⠻⣁⡴⠃⠀⠁⠙⢯⡣⠴⠋⠀⢀⡾⣡⣶⣿⡿⢁⠀⣄
⣶⣿⠟⠉⣿⡏⠀⢸⣟⠀⡆⠡⡈⠃⠡⠸⣧⡀⢀⠀⠙⢿⣿⣿⣿⣯⣷⣤⣤⠃⢀⣨⠈⡇⣧⡴⢋⣴⣿⠟⠉⣸⢹⢠⠙
⣹⢈⠀⠀⢻⣧⢠⡟⢸⠀⠃⠀⠁⠀⡀⠀⠁⠙⢪⣧⠀⠰⡎⠛⢿⣿⣿⣿⣾⣿⢻⣤⣷⠛⠻⣤⡾⠟⢉⠠⠆⢠⡆⠀⣰
⠛⣾⣴⣶⡀⢿⢨⠀⣾⡄⠀⠀⡄⠀⠇⠀⠀⠸⠆⠹⢧⠀⠙⠀⠀⠙⢯⢥⣍⣍⣿⣿⡿⢶⢶⡟⡀⢺⠀⡀⠀⣄⠓⠀⢻
⠛⠟⣳⣗⣦⣿⣻⢼⢿⣧⡜⢢⡅⠀⡀⠀⠀⠀⠀⠀⠀⠓⢆⠀⠀⠀⣀⠐⢤⡙⢿⣿⣷⡶⢋⠘⠁⠘⡀⠁⠀⠿⡄⢠⣿
⢧⣵⣳⢚⣟⡾⣌⢿⣿⠷⣽⣻⣼⡛⢷⣾⣧⣤⡀⠀⠀⡀⠀⠄⠦⠀⠌⠐⢂⣡⣾⣿⡿⠀⢁⠀⡠⡟⠃⠠⠁⢠⠄⠹⠃
⣿⢼⠯⣏⣢⣷⡞⠙⢬⣯⠿⣋⠓⡏⢫⣞⡶⢛⠀⢀⣠⡴⡶⢾⣛⡾⢿⣿⢿⣿⡿⡛⠷⠀⠸⠏⠀⠀⡘⠁⠰⠃⠀⠀⢀
⢶⣿⢇⣶⡓⣶⢜⢉⡡⢭⡓⠯⠫⡟⠙⠤⠂⡈⣰⣿⡭⠷⠶⠶⠶⠦⠼⠦⠶⠯⠜⠁⠀⢀⢸⠀⢀⠏⡀⠀⠀⠀⠀⣠⡶
⣺⠅⣐⣯⠗⠀⠐⠒⠀⡴⢉⣉⠀⠊⡀⠁⠋⠑⠿⣿⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿
⣀⡠⠈⡂⠨⠒⠄⠉⡒⠐⡡⢠⠥⠈⠀⠀⣁⣀⡀⠈⠉⠉⠉⣛⣻⣿⣶⣀⣄⣤⣀⣄⣤⣴⣦⠤⣤⣶⣶⣶⠾⠟⢛⡉
⠉⠉⠉⠙⠛⠛⠶⠶⣆⣀⣤⣶⣁⣠⣁⣘⣿⣯⠷⠶⠖⠓⠛⣋⣉⡉⠉⠁⢀⣤⣤⡄⠀⣰⣆⠀⠻⢿⣿⣟⣀⣀⣾⣷⡤
⣤⣶⣤⢀⣀⠀⠀⠀⢠⣤⣭⡍⠉⠛⠉⠁⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣤⡤⣼⣿⣿⡿⠶⠟⠛⠒⠚⠛⠉⠉⠉⠁⠀⣠⡄
⠛⠛⠻⠿⠿⢧⣤⣀⣻⣿⡿⠟⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠁⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⣰⣿⡿⢷⡀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠤⣤⣼⣧⣀⣤⡴⠶⠖⠒⠒⠒⠚⠛⠒⠒⠒⠒⠒⠶⠤⠤⢤⣙⣛⡚⠛⢓⠂
⣀⣀⣤⠤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠦⠤⠴⠶⠇⠔⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠞⢷⣶⣾
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠒⠒⠒⠛⠛⠒⠒⠲⠶⠤⣄⣀⠀⠀⣠⠤⠦⠖⠋⠉⠉⠙⠦⠤⠶⠦⣄⠀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⢁⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦⠀⢸⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠢⠀⠀⠀⡼⠀⠀⣀⡼⠋⠀⠀⠉⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣏⣀⡀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠔⠛⠉⠁⠀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⣀⣤⣤⣤⠶⠿⠷⣦⡄⠀⠀⠉⠳⡄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⣷⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠈⠳⣀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡞⠉⠀⠀⣀⠖⠉⠙⠚⠲⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢸⡀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢰⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⡔⠚⠷⣾⡁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⡴⠞⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡤⠾⡀⠀⠀⠈⣇⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠲⠦⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠻⠦⣄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⠄⠀⣠⠃⠰⠁⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⡏⠀⠀⠀⠀⠀⢀⣀⣀⠀⢧⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠘⠆⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢁⣀⣀⡤⠴⠚⠁⠀⠈⠳⡈⡇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠸⠀⠀⠈⠣⠀⠀⠀⠳⡀⠀⠘⢆⠀⠀⢀⣀⡤⠴⠞⠓⠚⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠹⡇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠞⠉⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⡠⠊⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠐⠃⠀⠀⠀⠀⠃⠀⠀⠀⠨⠄⠀⠠⠤⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⢉⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⣀⣀⣀⣈⠦⣤⣀⣀⡀⣀⣀⣀⣀⣤⠤⠴⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⢰⢸⡿⢋⣈⠠⣘⡽⣆⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣳⣿⣽⣶⡄⣉⢻⣿⢡
⠀⠀⠀⠀⠀⡟⣾⠁⠌⢣⠻⡑⠀⣿⣧⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⣿⠍⠽⢝⡗⠘⠆⣻⢸⡆
⠀⠀⠀⠀⢀⡇⣣⡶⠊⠁⣊⠐⣠⡘⢿⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⣿⠃⣠⡀⠈⠨⣗⣶⡿⠈⡇
⠀⠀⠀⠀⢸⡇⢻⣎⣤⣌⣛⣻⣿⣿⣌⢿⣧⢻⠷⠶⢦⣤⣤⣤⣤⣤⠶⠶⠟⣰⡿⣁⣾⣿⣿⣿⣛⣷⣿⡧⢰⣷
⠀⠀⠀⠀⢸⡷⣼⣟⣻⠮⣭⣿⡽⠯⣽⠛⢛⣷⡙⠄⠀⠈⢹⡯⠉⠀⢀⢎⣼⣋⠓⣋⠭⢭⣽⣿⠿⣿⣿⣧⠸⣿
⠀⠀⠀⠀⣿⠃⠊⠻⣿⡻⠦⣖⣀⣂⣊⣼⡋⢩⠙⣰⠀⠈⣸⣇⠀⢀⡈⢾⣉⢝⣷⣈⣇⠀⠐⠶⠛⣿⣟⢥⡒⣿⡀
⠀⠀⠀⠀⡯⠀⢁⣺⡹⣷⡑⠌⢀⣬⡿⠷⠶⠶⣬⣍⣠⣴⡿⠛⢷⣀⣤⣬⠶⠶⠶⢭⣜⠂⣀⠀⣾⢋⣼⠀⣱⢸⡇
⠀⠀⠀⢰⡗⡸⢣⣈⠱⠋⣣⡶⢛⣁⣤⣍⣉⠙⠒⣐⣠⣦⢠⠋⢀⣦⣁⡀⠚⣉⣉⣤⣍⠛⢶⣅⠋⡙⣃⡴⣡⠸⣇
⠀⠀⠀⣼⢃⣟⣸⣫⡴⢟⣡⣞⣛⣛⣏⣹⡍⠛⢦⡉⠄⠙⠐⠀⠘⠃⠆⣱⠞⢉⣵⣭⣞⣛⣓⣌⡻⢦⣙⢇⡤⣿⣿
⠀⠀⠀⣿⣿⣿⠟⢋⣤⡿⠿⠛⠛⠯⣍⠩⣍⠀⠀⠹⡄⠒⣤⣩⡞⢂⡾⠁⠀⢈⡉⢉⡩⠟⠛⠻⣿⣄⡙⠻⣟⣿⣿⡆
⠀⠀⢈⡸⢿⣡⡞⢉⠒⢾⣻⣷⣌⠳⠀⡤⠈⠃⠀⠀⢹⡖⢹⣿⠀⣾⠁⠀⠀⢈⠥⠈⠰⢋⣟⣛⡭⣌⠽⠷⣌⠿⣟⡀
⠀⠀⢏⡴⣾⠏⣀⣠⣴⠶⠏⠉⠐⠶⠴⠖⣀⠀⠀⠀⢨⢱⣌⣏⣼⠛⠀⠀⠀⢀⡀⠶⠶⠆⠈⠙⠳⣦⣀⢠⡙⢷⣦⡙⡄
⠀⢠⡿⠷⠾⠋⢈⡝⠛⠀⠀⠒⠒⢛⣓⣒⡠⠔⠞⠲⣿⣾⣿⣹⡿⣼⡶⠦⠲⢦⣀⣒⣂⠀⠈⠋⠐⠒⠋⡉⠙⠷⠾⠻⡆
⠀⢨⡶⡶⢿⣖⠁⢀⣈⣙⠁⠴⣻⢿⣿⡙⣛⣧⠸⣦⡿⠛⡜⢿⡱⡿⣷⡴⣰⣿⡛⣹⣿⢿⡷⠄⠺⠅⠀⠈⢲⡶⢶⡶⣿
⠀⣿⡷⠞⠛⢃⠦⠛⠻⠂⡀⢀⡸⣷⡙⠻⠿⣿⡇⢸⡇⠁⢻⣾⠃⠃⣿⢀⣿⡿⠟⠋⣱⡯⣠⠀⡀⢹⠿⠷⠬⠹⡶⢶⣝
⠀⢋⣴⠿⠛⠃⠀⠀⣿⠀⢻⣦⡑⠺⢦⣀⣁⠜⣇⢸⠇⠀⠺⣿⢓⠂⣿⣨⠿⢐⣀⣴⠿⠂⣠⠞⠁⢀⡌⠀⠉⠛⠻⣦⡙
⠀⠞⣡⡾⢛⡵⠂⠀⣟⠀⢀⠀⠀⣱⠐⢻⣯⠀⡻⠉⠀⠈⠠⢣⡀⠀⠈⠟⡰⣸⡟⢡⠠⠃⠀⠀⠄⠠⣿⠀⠐⣿⢶⣤⡹
⠀⡼⢋⡴⠁⣩⠀⠀⡿⠀⠀⣤⠄⠛⢠⡇⢹⡆⠁⢀⡾⠉⠉⠉⠉⠱⣆⠀⠇⣿⠘⣃⡀⡨⠈⠜⠀⠀⣇⠈⣿⠛⠳⣌⢷
⠀⣰⠏⣠⣾⣧⠀⣰⡇⢀⠐⢄⠀⢙⠃⣷⠇⡇⠀⡼⠃⠀⠀⠀⠀⠀⠈⢤⠃⢻⠀⡿⣠⠄⢀⠄⠀⠀⢸⣆⠼⡷⣶⣌⠳
⠀⢡⡾⣋⣵⡢⣔⢽⡁⠄⠑⠤⣆⠈⣁⠀⣸⠃⠀⠀⢀⣀⡀⠐⢄⣀⣠⠈⡆⡈⡦⠱⠀⣰⠷⠊⠀⡀⣸⡏⠢⣠⣤⡙⢷
⠠⢋⡼⠋⣁⣈⢡⡾⣷⠒⠀⠀⢈⡀⠨⠐⠃⡴⠃⠈⠁⠈⠀⠀⠀⠁⠀⠁⠘⢣⠀⠀⠀⠀⠀⠀⢄⣈⡗⠈⢀⡉⠉⠻⣦
⠀⡾⢡⡾⠋⡡⠉⠀⡿⠛⠁⠀⠀⠀⠀⡇⠈⠓⠀⠀⣀⣠⣤⣤⣤⣀⣀⠀⠀⡎⠀⡄⢀⠁⠀⠀⣀⣹⡅⠀⠈⠙⢷⣄⠙
⠀⢰⡟⠁⢊⠀⠀⣼⣧⣤⠆⠀⠀⣾⢠⡅⠀⠂⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠻⠀⡧⣸⠆⠀⠀⠈⣻⢇⠀⠐⢦⣄⠹⣧
⠀⡟⠀⣴⣯⣄⣯⠞⣝⣡⡀⠀⠐⣹⡎⣧⠈⠧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢰⣧⠿⠁⠀⠀⠙⣿⢦⢻⣦⣜⡿⣷⡈
⠀⢀⣾⠋⣿⣿⣿⠶⢻⡇⠀⠀⠊⠁⠻⠻⣦⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡿⠋⠑⠀⠀⠀⢠⣿⢸⢿⣿⣿⣇⠘⣷
⠀⡾⢣⠀⠙⢯⣿⣶⢸⣧⠀⢀⡀⣼⠃⠀⠹⣆⠠⠹⣿⣿⣿⣿⣿⣿⡿⠃⢠⡿⠁⠐⣦⠀⣖⠀⢸⡇⢰⣿⣬⠔⠉⢦⠘
⠀⢰⡇⠀⠀⠾⢻⡿⣇⢻⡄⢸⡇⠋⢰⡏⠀⠘⣷⣤⣴⣿⣿⣿⣿⣷⣤⣶⠋⠀⠘⣦⠁⠀⡿⢠⣿⢰⣿⢻⣿⠀⠀⢸⣇
⠀⠻⣧⠀⠀⠀⣸⡀⢻⡻⣿⡘⣇⠀⠘⠇⢀⠀⣟⢳⠈⢹⣿⠿⡽⠉⠰⢻⡇⠐⠠⠋⠀⢠⡇⣾⡟⣿⠁⣿⠀⠀⠀⣸⡿
⠀⠀⠹⡄⠀⠀⣿⠀⠈⠹⣟⣳⣿⣷⡄⡀⠀⠀⠋⢠⡀⠀⣯⣼⠃⠀⣀⢸⡇⠀⢀⢀⣴⣿⣼⣿⠏⠁⠀⣿⡀⠀⢠⡟
⠀⠀⠀⠈⠀⠀⣿⡀⠀⠀⠈⢿⣷⣿⣷⣿⣦⣠⡀⠘⣧⠀⠘⠃⠀⢠⡏⣈⢀⡴⢿⣿⣿⣿⠿⠁⠀⠀⠀⣿⡇⠀⠊
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⣴⠀⠉⠹⡿⣿⢷⣟⣧⣹⣆⠀⠀⢀⣾⣼⣿⣌⢶⡿⣿⠉⠀⣤⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢆⠀⡐⣿⡄⠀⠀⢠⡏⠘⠛⣿⣿⣿⣆⠀⣾⣿⣿⡟⠏⢸⣇⠀⠀⢀⣿⢀⢀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣽⣷⠀⠀⣼⡇⠀⠀⣼⡏⠻⠛⡼⠿⢃⣿⡄⠀⠀⣿⠀⠀⣾⣯⠸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⣿⡇⠀⢠⣿⣿⠀⠀⠁⠀⣼⣿⡇⠀⢀⣿⡆⣼⠿⠃

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⢸⣿⡆⠉⠀⠀⠒⠚⣛⣿⣿⣿⣿⣿⣟⣫⣭⣿⣿⡭⠿⠿⠿⠿⣭⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣽⠋⠥⠈⠁⠉⠀⣿⡇
⠀⠀⠀⢸⣿⡄⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣤⡤⣀⣀⣤⡄⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⢀⣿⡅
⠀⠀⠀⢸⣿⡟⢀⣿⣿⣿⣿⣿⣿⣍⢻⣿⣿⠛⠻⠿⣿⣷⣍⡿⣿⣽⢿⣌⣼⡿⠿⠟⠛⣿⣿⡏⣹⣿⢏⡽⢍⣿⣿⠀⡇⣿
⠀⠀⠀⢸⣿⡐⣸⣿⣿⡿⣿⣿⣷⣿⣶⣿⠏⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁⠉⠀⠀⠀⠀⠀⠘⣿⣼⡟⣡⣾⣁⣿⣿⣿⡙⠃⣿
⠀⠀⠀⢸⣿⠁⣼⣿⣿⣾⢿⣻⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣿⢟⣿⣿⣿⣿⢷⡜⣿⡁
⠀⠀⠀⢸⣿⣼⠸⣿⡿⣿⣷⣟⣽⢿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣳⣯⣿⣿⣿⣿⠈⡟⣿⠃
⠀⠀⠀⢸⣿⣾⠀⣿⠁⣿⣿⣿⣿⡷⡽⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢋⣾⣿⣻⣿⣿⠟⣿⢸⡇⣿
⠀⠀⠀⢸⣿⠘⣰⣿⣾⣿⣿⣿⢿⡿⣿⣎⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣱⣿⢿⣽⣟⣿⣿⣧⣿⡄⣇⣿
⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⢟⣣⠷⣼⣇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠯⣴⡜⢋⣛⣿⣽⣿⣿⠿⣿⣿⡇⣿
⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣿⣿⣿⣿⣧⣿⣶⣾⣿⣾⣿⣿⡷⣸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⢿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡸⣿⡇
⠀⠀⠀⢸⣿⠈⣿⡟⣱⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣯⣿⣯⠱⣿⡇
⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣀⣈⣳⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣥⣭⣤⣁⣸⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣟⣮⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⢯⣾⣷⣹⠆⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠛⠝⣷⣾⣾⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣼⣿⣧⣿⣿⠯⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣿⣿⣿⡏⠐⣴⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⢸⡏⡙⣿⣿⣿⣿⣿⣧⠙⣿⣿⣧⢾⣿⣿⣟⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⡿⢁⣾⣿⣿⣿⣿⡏⣿⣿⡇
⠀⠀⠀⢸⡗⣇⣿⣿⣿⣿⣿⣿⣷⡌⢻⣿⣌⢿⣿⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⡟⡰⣹⣿⣿⣿⣿⣿⣷⢿⣿⡇
⠀⠀⠀⢸⣿⡘⣿⣿⣿⣿⣿⣿⣿⡸⣆⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⡰⢣⣿⣿⣿⣿⣿⣿⡆⡄⣿⡇
⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣇⢫⢦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠁⣾⣿⣿⣿⣿⠛⣿⣻⡆⣿⡇
⠀⠀⠀⢸⣿⣤⠻⣿⣷⣼⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢶⣿⣿⣿⡿⣃⣰⣟⡷⣰⣿⡇
⠀⠀⠀⢸⣿⣿⣯⣿⣿⣏⠈⠻⣿⣿⣟⢤⢻⣿⠿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣿⣿⣿⠛⠀⢡⣿⡏⣠⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠈⠻⣯⣞⠣⣿⣇⠈⠹⢿⣽⣿⣿⣿⣿⣻⠿⠋⢀⣺⡧⣟⣳⣿⡟⠽⡛⢠⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠓⣿⣿⡿⣧⣈⠣⣄⡀⠈⠛⣿⣟⢋⠀⣀⡤⢞⣡⣿⣿⣿⣟⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣏⠙⢿⣿⣿⣦⡀⠈⣽⢶⠻⣿⣷⣮⣛⡛⣛⣛⠛⡛⣛⣿⣷⠾⣿⣿⢻⣯⡉⣀⣴⣿⣿⡿⠋⢹⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣻⣿⣿⡄⠀⠉⠻⣟⠿⣷⣶⣀⠀⠻⣯⠀⠸⡟⠋⠉⡩⠛⡁⠙⣧⣾⡏⠜⣻⣿⣿⠿⣻⠟⠉⠀⢠⣿⣿⣟⣿⡇
⠀⠀⠀⢸⣟⢩⣉⣉⡁⠀⠀⠀⠈⠣⡈⠙⠻⣿⣶⣦⡈⠰⣧⢄⡱⣆⡐⢧⠎⣡⣿⣿⡿⢛⠟⢁⠞⠁⠀⠀⠀⣈⣉⣉⣀⣿⡇
⠀⠀⠀⢸⡏⢸⣿⣿⣿⣿⣶⣶⣤⣄⡈⢆⢢⡈⠻⣿⣟⢷⣿⣤⣤⣥⣼⡷⠾⣿⣿⠋⡠⢁⡴⢁⣠⣴⣶⣾⣿⣿⣿⣿⣟⣻⡇
⠀⠀⠀⢸⡧⣼⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣷⡝⣦⠘⣿⣆⠈⢿⡟⢩⡟⠀⣼⡿⢁⡞⣰⣿⣾⣿⣿⡿⣿⣿⣿⣿⣟⣋⡁⣿⡇
⠀⠀⠀⢸⣟⠈⠉⠉⠉⠉⠙⣷⣿⣿⡻⣿⣿⣿⣿⣷⡘⣿⡤⠈⣷⣾⠁⢶⡿⢡⣞⣼⣿⡿⡛⢿⣬⣿⣿⠏⠉⠉⠉⠉⠁⢹⡇
⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡻⣛⣿⣷⡘⣧⠆⢸⡇⢘⣿⢁⣿⣿⢳⢏⣤⣿⣿⡻⢿⠋⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⢸⣿⡿⣦⠀⠀⠀⠀⠀⠘⢿⣯⣿⣿⣿⣦⣿⣿⣧⢹⡄⢿⡿⢸⡇⣾⢯⣟⣡⣾⣯⣟⢷⣽⠋⠀⠀⠀⠀⠀⣰⠏⣿⡇
⠀⠀⠀⢸⣿⠀⣿⠳⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⣿⣿⣿⣻⣿⣿⣿⣯⡿⠟⠁⠀⠀⠀⠀⠀⠈⠀⡇⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠂⠦⠄⠠⠀⡄⣄⣀⣈⠙⠛⠿⢿⣿⣿⣿⡆⣹⣿⣿⣿⡿⠿⠛⠋⢁⡀⣀⣄⠐⣂⣁⢀⡀⠀⠀⠁⣿⡇
⠀⠀⠀⠸⣯⣤⣄⣠⣤⣤⣤⣤⣭⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣬⣭⣥⣬⣥⣤⣀⣠⣤⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠴⠦⡀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢰⢔⢥⠪⢁⣠⣄⠌⠕⡵⡰⡶⢡
⠀⠀⠀⠀⠀⠀⠠⣦⡤⠖⠒⠒⠒⠛⠯⣷⣧⣵⢟⢹⡍⠻⣦⣼⣾⠽⠛⠒⢒⠒⠲⢤⢴⠄
⠀⠀⠀⠀⠀⠀⢠⡏⡏⣯⣭⣭⣍⣌⣍⣙⣿⠟⢁⠘⠇⡈⠻⣿⣋⣉⣡⣩⣭⣭⣽⢹⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠐⡇⣿⠱⣶⠢⠠⠤⢽⣹⠰⠵⡣⢌⢮⠄⡏⡯⠤⠴⠔⣶⡎⣿⢸⠈⡇
⠀⠀⠀⠀⡀⠄⣼⠘⡁⢽⠨⡂⢕⣧⣤⣀⡀⢈⡶⢽⡎⣶⡱⠀⣐⣤⣤⠪⢰⠅⡯⢘⠀⣧⠠⢀
⠀⠀⠀⢈⠐⣖⢯⣯⣇⢹⣀⣇⠠⢻⡰⣮⡻⣷⣗⣼⣇⣺⣾⢟⣽⢦⡟⠄⣸⣠⡏⣸⣼⡵⡺⡂⡁
⠀⠀⠀⠀⡢⡓⢉⣿⣿⠟⢓⡲⡠⢈⢿⣮⠴⣯⡡⠜⠳⢍⣽⢦⣵⡿⡡⢄⢒⠚⠻⣿⣿⡉⢚⢄
⠀⠀⢀⡜⠁⣢⠟⣑⣁⠐⠵⡫⣜⣛⣝⣽⣁⠎⡿⢏⣽⢯⠡⣜⣯⣫⣛⣣⢜⠮⢂⣌⣊⢻⣖⠈⢢⡀
⠀⠀⠈⢣⡀⠹⣯⡩⡙⠠⠲⣕⢫⣭⣩⣿⡉⢆⣿⣌⣩⣞⡰⢩⣟⣍⣭⡝⢨⢖⠌⢉⢍⣼⠏⢂⠜⠁
⠀⠀⠀⠀⢑⡥⣌⣿⣿⣦⡬⠥⠑⢈⣾⢟⠲⣟⡁⢦⡔⣊⣻⠞⡻⣷⡑⠈⠬⢥⣴⣿⣿⣁⢬⢊
⠀⠀⠀⢈⠠⠯⣞⣟⡏⣸⠋⠏⠀⣼⠱⢟⡵⡿⡯⢻⡟⢽⢿⣮⡻⠞⣧⠂⢹⠙⣇⢹⢻⣳⢵⠅⡀
⠀⠀⠀⠀⠁⠂⢻⠠⡅⣺⢐⠅⡲⡟⠛⠉⠁⢊⠷⣸⣗⠿⡑⠈⠩⠛⢛⢖⢨⡀⣗⢨⠀⡟⠐⠈
⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⡰⠾⠔⠔⠒⣺⣸⠐⡲⣴⢪⢖⠀⣇⣗⠒⠢⠢⠿⢆⣿⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠘⣆⣇⣟⣛⣛⢋⢋⣋⣩⣿⣦⡈⢠⡄⢁⣴⣿⣍⣙⡙⣛⣛⣛⣳⣺⣸⠃
⠀⠀⠀⠀⠀⠀⠐⠟⠓⠦⠤⠤⠬⣤⣖⡿⡟⡻⣮⣸⣃⣵⠟⣻⢿⣲⣤⠤⠤⠤⠴⠚⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠼⠊⡚⢔⡈⠙⠋⢂⡢⡳⠕⠗⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠲⠖⠁⠀⠀⠁⠈

⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⡷⠄⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠒⠛⢄⢉⣆⣤⠚⠓⢤⡀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠚⠂⡤⠀⢸⠁⠀⠀⠣⠀⢨⡀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⡀⠁⠀⠀⡄⠀⠀⠚⠀⠀⠀⠀⢋⡄⡔⠀⠈⠤⠰⡂⠀⠀⠀⠒⢤⠀⢃⠀⣀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⡠⠈⠀⠀⠀⠀⠰⠆⠀⠑⢄⣄⠀⠀⠙⠂⠀⠀⠀⢠⠒⣄⠁
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠀⢀⣤⠀⠀⠀⠀⠛⠀⠤⡆⠀⠀⠐⣀⠀⠀⠆⠀⠀⠄⡄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠢⢤⡤⠀⠠⠄⠐⢀⠀⠀⡠⠂⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂⠀⠈⠁⠈
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠄⡀
⠀⠀⠀⠀⠉⠒⠠⠄⢀⡀⠀⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠒⢠⣀⢀⠀⠙⢀⢀⡔⣃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⡠⠤⠀⠉⡦⠤⠤⠂⠒⠁⠙⠖⠒⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⣐⠁⠈⠀⠀⡨⠈⠄⠀⠀⡁⠀⠠⠔⠉⠐⡄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠚⠁
⠀⠐⠁⠀⠀⠀⢀⠠⡄⠀⠀⠓⠊⠀⢀⠀⡥⣀⡀⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⢐⠦⠔⠈⠀⠇⠀⠀⠙⠀⠀⠀⠀⠀⠥⢀⠀⠠⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢃
⠀⢇⠀⢠⣀⡠⠈⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⡀
⠀⠈⢻⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⢤⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠈⠲⠴⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⢈⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢣⡄⡄⠀⠀⠀⠀⠀⠀⠘⡀⠀⠉⠖⠁⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⡀⠀⣈⠀⠀⠀⠀⠀⠈⠀⠀⠴⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⠐⡀⠀⠀⠀⢄⣀⠁⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⡆⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢹⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡙⠈⠃⣀⡀⠆⠀⠃⡀⠀⠀⠀⠀⠀⠙⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠐⠣⣀⠰⢠⠔⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠦⠀⠤⢁⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠖⠒⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠆⠰⠀⠻⠀⠆⠰⠄⠘⠁⠰⠀⠠⠀⠆⢀⠆⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⣧⣴⣦⣶⣦⣶⣴⣧⣴⣷⣾⣤⣾⣶⣿⣶⣧⣴⣿⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠄⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⢠⠠⣤⠀⠀⣤⡤⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠈⠛⠁⣤⣶⣦⣴⣶⡄⠘⠃⢀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⣶⠀⠰⠀⠀⠀⠠⠀⠀⠀⢰⠟⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠂⠀⠀⠰
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⡶⠾⠟⠋⢶⡆⠀⣤⠀⠹⡟⠈⠛⠳⢶⣶⣿⣀⡀⣠⡄
⠀⠀⠀⠠⠀⠀⠀⢀⠀⢤⣤⣦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠛⠛⢶⣤⡀
⠀⠀⠀⠀⣠⡶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠚⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢹
⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠸⠿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⣸
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢿⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⡇⠀⣻⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢸⣿⣧⣠⠛⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⡿⣿⡇⠈⢷
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣸⣿⠀⢻⣷⡀⠸⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⣿⣿⠀⢸⣿⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢀⢸⣿⠀⠀⣿⣿⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘
⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⣀⡾⢸⡿⠀⠀⣿⣿⣆⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣿⠇⣼⡇⠀⠀⢹⣿⣿⡆⠀⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⣸⡏⠀⣿⠁⠀⠀⢸⣿⣿⣷⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣿⠁⢰⣿⠀⠀⠀⠸⣿⣏⢻⣧⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣿⣿⠃⠀⣿⣿⠀⠀⠀⠀⠻⣿⡈⣿⣷⠀⠰⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄
⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⠀⠀⠛⠁⢀⣾⣿⠏⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣧⠸⣿⣆⠀⠘⣧⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⠀⠀⢠⣿⣿⡟⠀⠀⣾⡏⣿⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⣆⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠘⣤⠀⠀⠀⠀⢀⡛⠀⠁⣿⣿⣿⠀⣀⠀⣿⡇⠉⠀⠀⠀⠀⠀⠀⠘⣷⡀⢿⣿⣷
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠸⠃⠀⣰⣿⣿⣿⣴⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⣿⣿⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⢀⣿⣿⣿⣿⡿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⣿⣧⡀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⣠⠈⢻⡏⠉⠉⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡈⣹⡿⢀⣀⠀⢀⣄⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠙⣿⣿⣶⣤⣽⣦⣤⣶⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣶⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⡄⢀⡘⠁⠀⠀⠈⠉⠉⠉⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠟⠀⠀⣀⣤⣤⣤⣤⣤⣀⠓⠄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣤⠞⠃⠈⠻⣶⡀
⠀⠀⠀⠀⠀⠀⡆⢀⣾⠏⠀⠀⠀⠀⠀⢻⡆⠈⠳⡀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡼⠃⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠆⠀⡀⠀⠀⠀⠀⣿⠀⠀⠈⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡀⠀⠠⠀⠰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠉⣀⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠆⠀⠴⠉⠉⠁⠰⠶⠄⠀⠀⠀⠀⠀⠀⠰⠆⠀⠁⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠈⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣴⣤⣤⣴⣦⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⢤⣀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣠⡖⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠙⢦⣬⣉⣛⣓⠓⢠⡀⠀⠀⡚⣒⣘⣡⠶⠛⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠶⠀⠀⠊⠉⠉⠋⠙⢧⠞⠙⠉⠉⠁⠈⠑⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡀⠀⢀⠞⢉⡑⣴⣜⣧⠐⡀⠢⡀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢦⡀⠨⠢⢼⣤⣨⣴⢇⠀⣼⣦⣍⣠⠇⠔⠠⠠⢻⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣈⣷⣀⠀⠉⠶⣀⢁⠀⠉⠀⠈⠞⢏⡉⣈⡁⢀⡁⣴⠣⢺⣅⣄⠠⠠⠒⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⠪⠽⣻⣙⠣⠔⠒⠼⠭⣭⣍⣉⡁⠀⣀⣭⣅⣬⣭⣴⣦⡗⢋⣡⣠⠤⠔⠊⠁
⠀⠀⠀⠀⢀⠄⠂⣀⠤⠶⠇⠀⠀⠈⠉⠑⠒⠂⠤⢤⡰⠿⠙⠎⠅⠙⢉⡠⠔⠾⠛⠋⠉⠁⠀⢲⣄⣁⠀⠐⠂⢀
⠀⠀⠀⠀⠀⢈⡠⠥⢚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠚⠀⠀⠒⠁⠀⠀⠀⠀⠀⠈⢦⠄⣈⡉⠂⠀⡀
⠀⠀⠀⠠⠀⡤⢒⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣈⡁⠀⠁
⠀⠀⠀⠀⢁⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⡆⠠⠤⠀⡀⠀⠄⢀⠀⠀⠀⠀⠂⠀⡀⠤⠔⠒⢉⡽⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⢙⡦⡀⠀⠉⠈⠙⠤⠤⡤⠴⠊⠁⠀⠀⡠⠴⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣬⣴⡦⣐⣦⠤⠤⢀⣀⠀⡀⣀⣠⣤⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠿⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠯⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣤⠤⣤⠤⣤⣤⠤⣤⢤⣖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠈⢳⣿⢃⠎⠀⢀⠀⠑⡄
⠀⣤⡤⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢧⠀⠳⠿⠃⢸⠁⢸⠀⠈⠶⠃⢀⡇⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢤⡄
⠈⢇⠀⠀⠀⠀⠀⠁⠐⠒⠒⠩⠵⠾⢷⣶⣤⣴⣫⣦⠄⢑⣶⣤⠶⠛⠰⠍⠛⠛⠒⠉⠉⠀⠀⠀⠀⠀⡇
⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠘⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠀⠀⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⣸⣻⣿⣟⢒⣤⣴⠒⣿⣏⣹⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣈⣉⣀⣀⣀⣀⣀⣉⣀⢈⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠺⠿⠿⠟⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢍⡉⠉⠉⠁⠠⠀⠀⠀⠈⠉⠛⠒⠦⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠤⠀⠂⠠⣄⠉⠠⡌⡠⣤⡤⠤⡐⠒⣉⡽⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⣴⣶⣤⠈⡀⠀⣿⡑⣁⣤⡄⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⣾⣿⡯⠄⡅⢠⠇⠋⠻⡿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠈⠳⣤⣀⡡⠔⠁⠈⠀⠓⠤⠴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠆⠆⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠠⡄⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡌⡙⠤⣴⡴⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⠆⠈⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⢠⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢻⣽⣯⠟⣶⣲⣶⣶⡶⡐⠶⠛⠉⣿⠙⢤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⣹⠀⢗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣰⡁⢪⢤⣆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠄⣁⠀⠀⠀⠀⣀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠉⠒⠤⣀⠀⣤⠀⠒⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠆⢸⡄⠄⡠⠔⠊⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⠀⠀⠀⠉⠃⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠒⠀⠆⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣠⣶⠿⠛⠋⠉⠛⠻⣶⣄⠀⠀⢀⣶⡾⠛⠋⠉⠙⠻⢿⣧⣈⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠞⠉⣿⡆⠀⠀⠀⠀⡄⢲⣦⠀⢠⣿⣿⣧⠀⢠⡐⣶⡄⠀⠀⠀⢀⣿⡷⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡴⠋⢀⠔⠙⢿⣦⡀⠀⠀⠿⣿⢟⣤⠟⠋⠉⠛⢧⣜⠿⠿⠃⠀⢀⣠⣾⠿⡀⠈⢧⣀
⠀⠀⠀⣀⣀⡷⠄⣏⠉⠛⠷⢤⣎⣀⠀⠀⠙⠻⠿⠶⠶⠾⠿⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠛⠁⠀⠑⡄⢈⣻⡵⠶⣆⣤
⠀⠀⠀⠘⡇⠉⠁⠒⠬⢍⣐⠒⠠⢄⣉⠒⠤⣀⠀⠀⠀⠀⠀⡗⢆⠀⠀⢠⠞⠇⠀⠀⠀⠀⠀⠀⡠⠤⠒⣋⡉⠤⠔⣚⣩⠷⣿⣤
⠀⠀⢸⣛⡳⠶⠒⠈⠙⢢⠀⠉⠓⠦⢄⡉⠲⢌⡙⢦⡀⠀⠀⠘⠢⠉⠎⠡⠊⠀⠀⠀⣠⠔⣂⡥⠖⢂⣉⠤⠖⠒⡝⠉⠉⠙⠳⢼⡀
⢀⠒⠉⠁⠀⠀⠀⠀⢀⠼⠦⢄⣀⠀⠀⠈⠓⢤⡙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠴⢊⣡⠴⠚⢉⣀⡤⠖⢋⣱⣄⠀⠀⠀⠀⠈⠑⠂
⡈⠀⠀⢀⣀⠤⠀⠚⠻⡳⠦⣤⣈⠙⠓⠦⣤⣀⠈⠓⢷⡄⠙⢦⠀⠀⠀⢀⣔⡥⠊⡡⠶⣉⣠⡴⠞⢋⣡⡴⠞⠋⡡⠊⠑⠂⠤⣀⡀
⡗⠊⠉⠉⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠓⠦⢤⣈⣙⠓⠶⢬⣦⣈⣧⡀⣠⠟⣁⢐⣩⠽⢦⠈⢣⡴⠚⠋⠁⠀⠀⠀⠱⣀⠀⠀⠀⠀⠉⠑⠂
⡇⠀⢀⣀⢠⡤⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣦⢸⡛⠒⠶⠶⠚⢿⠉⡤⠖⠋⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣱⠶⢤⣀⡀
⠸⡉⠉⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡧⠤⣀⣀⡤⢼⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢠⣾⣧⡀⠀⠀⠉⠙⣲
⠀⠑⣤⣤⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣤⣄⣀⣤⣼⠀⢸⠀⠀⣤⡀⠀⠀⢸⣿⣿⣇⠀⢸⣿⣀⣽⡲⣤⣄⡰⠃
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⢸⡆⢀⣿⣿⣦⡀⠀⣿⣯⣿⠇⠀⠙⠛⠋⠀⡇⢸⡇
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⡇⢸⡇⠸⠿⠋⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⡇⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢛⣨⠇
⠀⠀⠀⠀⠳⠦⣭⣙⣛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠒⠶⠶⠶⠒⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢚⣫⡥⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢷⡲⠬⣭⣛⡓⠶⠤⣤⣀⡀⢸⠀⣷⠲⠤⠤⠤⠴⡇⠀⣿⠀⣀⣠⡤⠶⢚⣩⣥⠖⢋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣍⡲⢤⡀⠉⠙⠒⠲⠬⣭⣉⡀⣿⠀⠀⠀⠀⠀⡇⠀⢛⣋⣭⠴⠒⣋⡥⢞⣡⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠉⢏⣻⣿⣿⣶⣦⣤⣤⣀⣈⣉⣛⣦⣄⣀⣠⣤⣿⣚⣩⣥⣴⣦⣿⡷⣾⣉⠎⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠒⠋⡯⠤⠤⣙⣿⡔⠉⢙⡏⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠙⣼⣟⡡⠤⠬⡏⠓⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣠⡜⠀⠑⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠁⠘⣤⣀⡤⠃
⣿⡿⠿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣧⣽⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⠛⠂⠀⣤⣄⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠛
⠀⠈⠀⠘⠋⠎⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠰⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢠⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡁
⠠⠤⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⡄⠐⣂⠀⠀⠀⠀⡀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠂⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣭⣥⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣶⣿⣿⠿⣶⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣦⣀⠀⠀⠀⠀⠐⢋⡭⢷⠀⠀⠐⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡹⢃⠻⠂⠀⠀⠀⠀⣀⣀⣸⣀⠀⢠⣴⣴⠞⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠈⠁⠀⠀⠀⠀⠀⠈⠙⢻⣿⣶⣶⠟⢂⣜⣶⡴⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣀⣀⡆⠀⠀⠀⠀⡴⣶⢿⢿⣿⣿⡿⠀⠂⠀⠀⠶⠦⡄⠀⠀⠀⣄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣅⣨⣿⡏⠀⠀⠀⣤⣤⣀⣸⢺⣿⣿⣶⡀⠀⠀⠀⠀⢀⣀⡀⢀⣾⣿⣐⣨⣿⣇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⢿⣿⣿⣿⡇⡄⠀⢠⣶⣴⣤⣼⢸⣿⣿⠀⠀⠈⠀⠀⣤⣤⠄⠀⠘⢿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⡼⠛⢯⣠⣆⠘⢿⣿⣿⠃⡋⠀⠘⠛⠛⠛⠛⢸⣿⣿⠀⠀⠀⡀⠀⠁⠀⡴⣧⠀⠀⠉⡛⠉⠉⣴⡴⣆
⠀⠀⠀⢀⡟⠶⣅⣁⣿⣀⣀⡹⠇⢠⠟⠳⠘⠓⠚⠒⠖⠘⠿⡷⠒⠒⢺⠗⠒⠀⣴⣀⣁⠀⠀⠀⠙⢦⠚⠛⡂⣐⣧
⠀⠀⣠⣶⣷⣴⠋⠉⠙⣤⠀⠀⠀⢨⠗⢺⠀⢴⣶⣶⠀⠀⢀⡀⠀⣀⣤⣤⡀⠀⠉⢿⡋⣀⡀⠶⠶⢾⠶⠶⣿⣟⠉⢅
⠀⠀⣿⣿⣿⠋⠙⠛⠛⡿⠒⠒⠐⢿⡶⣾⠀⠀⣿⠇⠀⠀⣾⣯⡿⣿⣿⣿⠁⠀⠾⣿⣿⡟⠈⠀⠀⣸⠀⢸⣿⣷⣶⡾
⠀⠀⠸⣿⣿⣆⡀⠀⠀⢧⣄⣀⣠⣼⣿⣞⠀⠀⠉⠁⠀⢹⣿⣯⡏⠉⠉⠉⠀⠀⢀⣿⣿⣿⣶⠀⠀⠞⠀⠈⢛⣿⡿
⠀⠀⠀⢽⣿⣿⡿⣿⠀⠀⠻⣿⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⣻⣝⣿⡿⠀⠀⣆⠀⢠⣾⣿⠇
⠀⠀⠀⠈⡈⣿⠦⣞⠀⢈⠀⠈⠛⣟⣫⠅⠀⠀⠸⠿⠿⠿⢹⣏⣿⣿⣿⡟⠀⠀⠀⠛⢟⣿⠀⠀⠀⠈⠻⣿⡋⢛
⠀⠀⠀⠀⠉⡧⣤⡏⠀⠠⠀⠰⠺⣅⣨⡇⠀⠀⢠⣤⣤⣠⢸⡗⣀⣀⣀⡀⠀⠀⠀⢲⡶⢿⠄⠀⠀⢰⡶⡿⣷⣶
⠀⠀⠀⠀⢹⣧⣤⠗⠆⠐⠀⠰⢾⡋⢹⠀⠀⠀⠠⣿⣿⣿⣼⡇⣿⣿⣿⡇⠀⠀⠀⢸⣦⣄⡄⠀⠀⢸⣇⣠⣼⣗
⠀⠀⠀⠀⠸⣅⣨⠷⠂⠐⠀⠤⣔⠉⠓⠀⠀⠀⠀⠛⠛⠛⠓⠓⠛⠛⠛⠁⠀⠀⠀⠀⠫⣿⠁⠀⠀⠈⠙⣿⠅⠉
⠀⠀⠀⠀⠀⢉⣹⠴⠄⠀⢀⣤⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⠀⠀⠀⡟⠟⢷⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠄⠀⠀⢧⣴⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⣄⠠⢤⣑⠢⡄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣲⣾⡿⠿⠿⡻⢟⢿⣆⣀⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣁⣩⡽⢷⡍⠂⠐⣄⣠⣄⠛⠿⠿⡑⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣬⣕⣒⣻⣣⠔⣂⣤⣭⠶⠦⠴⠄⠠⠀⠈⠿⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⠉⠉⠀⡴⠉⠀⠀⠀⠀⠀⠀⢀⡂⠰⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⡍⣩⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡤⢾⣿⢾⠁⠀⠀⠀⠀⠀⠀⢦⡾⠷⠤⠀⠀⠀⠀⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠐⡤⠀⠈⢳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⠃⠀⠀⠂⠀⡀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⢻⡏⠀⢀⣇⣀⢀⠙⠒⣋⠉
⠀⠀⠀⠀⠀⣠⢔⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠈⢧⠐⠋⠈⠉⠀⠀⢠⡾
⠀⠀⠀⠀⣼⠁⣵⣠⣿⡇⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⡀⠡⢀⣦⠀⠑⢊⡞
⠀⠀⠀⠀⢿⣿⣷⣿⣿⡟⠀⠸⣷⣤⣤⣤⣤⣄⣀⠀⣀⣿⡿⠋⠉⠲⢬⠀⡈⢁⠒⠳⠤⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢸⡧⣘
⠀⠀⠀⠀⠸⠋⣿⠉⣿⣿⡄⢀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⢀⡞⠙⠐⠉⢻⣿⣮⣧⠀⠉⠉⢿⣿⣿⡇⠀⠀⣠⣆⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠸⣀⠀⠀⡴⠁⣻⡟⠻⣿⠀⠀⢸⡟⢿⡇⠀⠀⠉⠙⠓⠃⠙⠊
⠀⠀⠀⠀⠀⠻⠂⣤⣧⣰⣿⣇⠀⠀⠀⠀⢸⡷⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡿⢿⣿⠛⠋⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡀⠀⠀⠀⠀⠈⠳⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⢸⣦⣄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⢀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠽⠥⠤⠤⠤⠤⠄⠼⠛⢦⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣘⣃⣠⡆⠤⠤⠤⠤⠀⠀⠀⠐⡶⠀⠀⠀⠀⣤⡆⠀⢀⡀⡀⣀⣀⠀⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣉⣈⣁⣬⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠖⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡤⠀⠀⠀⠨⠱⣤⡀⢀⣀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠋⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⡧⠄⠒⠻⠶⠶⢶⣶⠤⠬⡥⣤⣀⣼⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⢀⣠⡤⠒⠁
⠀⠀⠀⠀⠀⠐⠁⢧⠸⠳⠄⠀⠀⠀⠛⠃⠒⣲⠶⠦⠤⠤⣍⣉⣉⣛⣛⢓⠒⠒⠲⠶⠾⠶⠶⠶⠶⡶⠖⠒⠒⠒⠛⣿⠉⠁⠉⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠨⣗⠛⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⠋⠉⠙⠛⠛⠛⠛⠒⠚⠉⠀⠀⠀⠀⠀⢁⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⡤⣬⣤⣤⡤⠶⣚⣉⣉⣉⠉⠴⠟⠉⠉⠉⠉⠛⠒⠲⠦⢤⣴⣟⡀⠀⠀⠒⠂⠠⠤⠤⠤⠤⠴⠖⠒⠛⠉⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠐⠈⠉⠻⣜⣩⠟⠉⠉⠉⠉⠙⠓⠲⠦⢬⣉⣙⠛⠒⠶⠤⣤⣀⡀⠀⠀⢀⡤⠀⠉⠉⠙⠒⠒⠒⠠⠤⠤⣠⡴⠞⠃⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢤⣬⣉⣛⠒⠦⠤⣄⣀⡀⠀⠾⠃⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠙⠙⠛⠛⠓⠒⠒⠒⣲⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣤⣦⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢫⡶⢶⣶⡄⣻⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⢋⣤⣾⣿⣿⠋⠀⢻⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣴⣾⣿⣿⡿⠆⠀⢸⡟⠀⠈⡇⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡷⠆⠀⣸⡇⠀⠀⠻⣸⡀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣾⣿⡇⠀⠀⣿⡖⠀⠀⣿⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⠀⣿⡏⠀⠀⣿⠀⠀⠐⠀⢀⣀⡰⡿⣿⣷⡆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡉⣿⣿⣆⠀⢰⣿⣇⠀⠀⣿⡀⠀⠀⠀⢀⠛⠛⠉⠉⠩⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣯⡄⢸⣿⠏⠀⠀⣿⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠀⢸⣿⡇⠀⠀⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡞⣿⣿⣷⡇⢸⣿⠇⠀⠀⣿⠀⠀⠀⢸⠙⢶⡄⠀⠐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠌⠻⠂⠀⠀⠙⠀⠀⠀⠸⠦⠈⠋⠀⠆⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡇⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⡇⢠⣜⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠃⠀⠀⢀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡿⠀⢀⢔⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣼⣶⣿⣿⣶⣤⠘⢷⣦⠴⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠈⠂⢸⣿⣟⢻⣿⣻⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠀⡇⠀⠀⠀⢸⢿⣿⣷⣌⣰⢶⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢀⡈⣶⠀⠀⠀⠀⠘⣮⣤⣋⣛⣉⣶⡀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠿⠉⠁⠘⠀⠀⠀⠀⠀⠘⣯⡹⡉⢍⠀⠳⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠳⣟⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⠷⠔⠋⠰⣝⠂⠀⠀⠀⠀⠀⠹⡾⡀
⠀⠀⠀⠀⠀⠐⢄⠀⣿⣈⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣎⠠⠼⢿⡀⢻⡴⣆⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⣀⡀⠀⠉⣿⣿⡧⠀⠀⠒⠀⠞⢾⣿⠛⣿⠏⠀⠀⢄⡙⠿⢇⡟⠀⠀⠀⠃⠘⣇⣤⢜⣂
⠀⠀⢀⠼⠫⠁⠟⠶⠽⡯⠁⠀⠀⠀⠀⠀⢸⡿⠴⢹⠀⠀⠀⠄⠙⢦⠀⠁⠰⠧⠿⠚⢹⠟⡠⠚⠛⢠

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠈⠉⠉⠒⡄⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⢠⠏⠉⠀⠉⢲⡀⢀⢔⢡⠪⠓⢄⠀⠀⡀⠀⠀⠀⡌⠀⠀⠀⠀⢀⡠⠴⠔⣚⣋⡭⠭⠩⢴⡖⠲⡞
⠀⠀⠀⢀⡤⢬⡣⢄⡀⠀⠨⢇⢔⣼⢆⠁⢖⣀⠑⢎⡖⣡⣶⣮⡅⠀⠀⠀⠀⠈⠉⡟⢹⠀⠀⠐⠀⠀⠀⡇⠈⡇
⠀⠀⠀⠙⠓⠚⠁⠀⠈⠹⡕⡹⣫⣷⢟⣷⣌⠁⠀⢈⠳⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⢰⠀⠀⠀⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡫⣪⡿⣋⣼⢦⠀⠐⢆⠘⢏⠉⠉⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣸⡤⠤⠔⠚⠊⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⠙⢯⡪⢛⣵⢟⣳⡀⠈⢳⡈⢆⠀⠀⠀⠀⠀⠀⠰⠏⠭⠭⠄⠠⠒⠒⠊⠉⠹⣶⣶⣦
⠀⠀⢠⠈⠀⠀⠑⠒⠀⠛⢛⡀⠀⡀⠹⣯⡴⢟⣯⡿⣄⠀⠀⠘⣦⣄⣐⣂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁
⡔⠀⠉⠀⠀⠀⠀⡔⠉⠉⠀⠀⠀⣈⠀⢈⣦⣽⣿⡶⣻⣆⠀⣀⠸⠀⠈⠀⣀⢀⢠⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿
⠑⠀⠠⠀⠀⠀⠀⠈⠓⢦⣀⢀⣀⣇⣈⣠⠋⢯⣴⡿⠟⣳⣀⠩⠕⠀⢠⢌⣸⠆⣋⠁⠲⣤⣶⡀⠛⢻⠋⠛⢟⠛⠉⠉
⠀⠀⠀⠉⠑⠤⣀⡀⣠⡪⠛⣆⠸⠀⠢⠋⠈⠘⣷⣾⣿⠉⠀⢀⡠⠔⠉⢮⠳⡀⠀⠀⣠⠏⢻⣿⣄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢨⠛⠃⠀⠲⡦⢰⣿⣿⢋⠥⠊⠁⠀⠀⣀⡠⣆⡗⣶⡉⣳⠄⠀⣿⠃⠀⠀⠀⠀⢀⣀⣑⣄
⠀⠀⠀⠀⢀⡠⠒⠂⠈⠉⠒⠃⠀⠀⠀⠀⡁⡨⠟⠋⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⣏⢼⠀⠀⠀⠀⢸⠉⠉⠁
⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⢀⡠⠴⢏⡀⠀⠀⠀⢠⣾⢿⣿⣷⡄⠀⠀⠈⠑⢁⠄⠀⠀⠀⢨⣀⠀⠿⠂
⠀⠀⠀⡆⠀⢀⠀⢠⡊⠉⢲⠀⠀⠀⠘⢦⡐⠊⠉⠀⠀⠀⡈⠲⠀⠀⢿⣷⢇⢈⣿⡹⠒⢄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⢱⠀⣿⡆⠈⠣⢄⠔⠀⠀⠀⠀⠀⠉⢻⣶⣶⣦⣄⣁⠀⠀⠀⠀⠙⠶⠿⠿⠁⠀⠶⢁⠢⡀⠀⠀⠀⢠⠿⠿⠋
⠀⠀⠀⠀⣇⠹⡽⠀⠀⠀⠀⠀⠀⠔⠉⢹⣿⣿⣿⣿⣿⠋⠉⢻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⢀⠌
⠀⠀⠀⠀⠈⠑⠤⢀⣀⣀⣀⠄⠈⠀⣀⠼⠛⠋⠉⢿⣿⣶⣤⣾⣿⣷⠀⠀⢀⣀⠠⠄⠤⡀⠀⠈⠉⠉⣛⡢⢄⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⢀⠄⠀⠠⣄⠀⢻⣿⣿⣿⣿⣿⣷⡿⠛⠁⢀⡀⠀⣈⠑⢄⡀⡸⠀⠈⠋⠱⡀⠀⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⢀⠀⠀⠀⠀⡇⠀⠻⣿⣿⡿⠟⠁⠐⢏⠉⢠⠁⠀⣿⠟⠃⠸⠓⠤⣀⣀⡠⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠑⠤⠤⠔⠃⠀⠀⠹⡋⣀⠤⠤⠤⡈⣦⠃⠀⠀⠀⠀⠀⠀⠑⠀⠠⡄⠀⠀⢀⠞⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠒⡄⠀⠀⠱⡑⢀⡈⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠈⢈⣁⣤⣴⣾⠁⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⢛⡿⣛⡛⢻⠛⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒
⠰⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⠀⢀⠄⢠⡀⠀⠨⢂⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠀⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠐⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⢘⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⢿⣻⣿⢿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⡧⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡟⢛⡌⠻⢟⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠈
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⠟⢰⠘⢃⡔⢳⢌⠻⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⠀⠩⣿⣿⡟⠀⢸⠀⣼⣇⠻⠈⠈⢿⣿⣟⡝⠁⠈⠻⢿⣿⣿⣿⣿⡀
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⡟⠄⠀⢄⡿⡯⢙⢷⡅⡈⢈⢻⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿
⢸⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣠⢋⡠⢀⣎⣓⣨⣉⣊⣻⡤⣢⠙⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢠
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠀⢥⠃⣿⣿⣿⣿⣿⣿⠘⡤⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢸⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⡿⢿⠿⣿⣿⡿⡫⠸⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣭⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⠀⠀⠀⠀⠀⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀
⢸⠀⠀⠀⣸⣾⣛⠳⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠄⡉⣤⣴⡆
⢸⠀⠀⠀⣹⣿⣿⣿⢧⣤⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀⡀⠸⢗⢼⣿⠋
⢸⠀⠀⣀⠌⠹⠿⠿⠘⠧⠼⢧⣽⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠠⠿⡉⠀⢀⠿⠋
⢸⠀⠺⠧⡄⢣⠀⠀⠐⢫⣭⣀⡀⣸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠶⣯⡅⠂⠀⠰⠋⠀⠞⠳
⢸⠀⠤⠶⣠⣤⡤⠀⠀⠤⠈⡭⢅⠰⠻⣿⣿⣿⣟⡿⠡⣿⣿⢙⡇⢸⣷⣾⣿⣿⣿⣿⣿⡟⠁⠀⠛⠃⠀⠀⢨⣤⣶⣦⣴⠲
⢸⠀⠐⠃⠋⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⣿⣿⡿⡿⢣⣾⣿⣿⠙⠃⣿⣿⣿⣇⢿⣿⣿⡟⠀⠀⠀⠸⠣⠀⠀⠐⠁⠛⠚⠛⠂
⢸⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠋⠀⠀⠄⠀⠻⣿⢡⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣯⢽⠟⠀⠄⠀⠀⠀⠂⠀⡐⢤⣄
⢸⠀⠀⠀⠀⠀⣾⡶⠀⠀⣠⠆⣀⡁⠲⠀⠀⠀⢻⣿⣿⣿⣿⣦⣹⣿⣿⣿⣿⠟⠀⠀⠁⠄⠸⣆⠀⡀⢸⠃⢰⣿⡐
⢸⠀⠀⠀⠀⠀⠛⠀⠠⠄⢹⣦⡎⠁⠆⡀⠂⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠰⢀⠀⠉⠠⣟⡷⢰⡄⠈⠙⠙
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠀⠀⠐⠀⠀⢠⠀⠈⢻⣿⣿⠟⢁⠀⢢⠀⢀⠁⠀⠀⠀⠈⢣⡇

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢆⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠈⡶⠀⡄⢀⠀⠀⠀⠠⠀⠀⠀⠀⠐⣾⣿⣾⠀⠉⠃
⠀⠀⠀⠘⣦⠀⢁⡜⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣇⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⡄⠈⣀⡾⠁⠀⠀⠀⠀⠀⠛⠳⠀⠀⠠⠈⣿⣿⠞⠀⠀⠀⠀⠀⠁⠘⠂⠻
⠀⠀⠀⢄⣠⢹⣦⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠂
⠀⠀⠀⠠⠇⠌⠸⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣹⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡀⠀⠀⢀⣀⡤⠤⠤⠖⠚⠁⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢿⠏⠀⠀⠀⠀⠀⠈⢁⠀⠁⠉⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠈
⠀⠀⢀⠀⡝⠲⠀⠒⠋⠹⠈⣤⠀⠀⢀⣤⠖⡀⠀⠂⠀⠚⠠⠎⠀⠀⠀⠇⡔⠙⠂⠀⠁
⠀⠀⣆⡜⡇⡀⠀⠀⠀⠀⢧⡟⠀⠠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⢎
⠀⠀⠛⠃⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣲⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠟⠛⢻⣿⣿⣿⣿⣧⣠⣴⣶⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⠟⠉⢀⣠⣴⣶⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⣼⣟⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣀⣴⣾⣿⡿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠻⢿⣿⣷⣦⣄⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣼⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣄⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡟⠉⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠈⢿⣧
⠀⠀⢀⡶⡄⠀⠀⠀⣾⡟⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠈⣿⣇
⣠⢤⣴⡇⠀⠀⠀⣸⣿⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠸⣿⡄
⠻⠿⠛⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⡇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣿⣧⣴⣶⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⣿⣿⣇⣀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣀⣨⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⣿⣿⠋⠈⡗⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⡿⠁⠀⢰⡇⠀⠀⠀⠀⠀⣏⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⢸⡇⣿⣿⣿⣶⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠇⠀⢀⡾⠀⠀⠀⠀⢠⠞⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣼⣧⣿⡿⠟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⠀⣠⡾⠁⠀⠀⠀⠀⢸⡘⢫⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡿⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢙⣞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠈⠛⠿⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠟⢋⣴⣿⠟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⢠⣾⠟⠁⠀⠘⣧⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⢿⡿⠀⠀⢰⣿⡟⠉⣿⠀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠀⢸⣷⠀⠀⠈⢿⡅⠀⢻⡆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⢠⡀⠀⠀⠘⣿⣆⠀⠀⠀⠉⠀⠸⣷⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠈⠻⣷⣤⣀⣀⠀⠀⣿⣆⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸⡟⠻⣦⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣴⠞⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣉⣡⣼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⡷⣴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣀⣿⣷⠃⠸⢦⠀⠀⠀⠐⠀⠀⠀⠀⢠⡿⣀⠡⡏⠻⣷⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢖⣿⠒⡒⠶⠍⠉⠀⢰⡇⠀⠀⠀⠀⢀⢰⡆⠀⠑⠺⠛⠿⠃⣰⣄⡟⠗⢿
⠀⠀⠀⠀⠀⠀⠛⠱⣄⣀⠺⠋⠂⠠⣽⡦⠸⢦⣩⠄⠀⡉⠁⠀⠀⠀⠠⡟⠘⠀⠀⠰⣿⢛⠛⠇⣽⠟⠒⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠏⣹⣄⠀⣃⣠⠰⠈⠂⠓⢠⢸⠃⠰⠀⠀⠀⡤⡄⠀⠀⠀⢀⣀⠀⠈⢐⡆⣰⣯⣿⠯⠀⢸
⡄⠀⡀⣀⠀⣠⠄⢸⡏⠀⠸⣿⣿⡍⠀⠀⠀⠈⠈⠀⠀⠀⢠⠈⢀⣇⣰⣤⣠⣤⣼⡤⠈⠋⣠⠃⣘⣻⣿⡷⢸
⠁⠀⠈⠻⡧⠁⢀⣼⣰⣶⣐⣿⠏⠉⠀⡇⠘⢆⠀⠀⠀⠀⣤⠀⣸⣧⣽⣿⣿⣏⠀⡿⢷⣠⠋⢀⡛⢿⣿⣿⢿
⢀⡀⠀⠀⠁⣴⠟⢸⠃⠹⢿⡏⠂⠰⢤⡇⠀⠸⡆⢠⡄⠀⠂⠀⢋⣿⣿⣿⣹⣿⣷⡀⢀⣷⣄⠻⠁⢙⣏⠙⢸
⠃⠑⣂⠀⠰⢿⡀⡋⠀⢀⣫⣅⣀⣴⣾⡇⠀⣶⡶⠛⢳⡇⢀⣡⣾⣿⣿⣿⠉⡿⡟⣧⣾⣿⠁⢘⠃⢨⣷⡇⢸
⠀⠘⡟⡲⢤⣼⡏⣼⡖⢉⣿⠙⠻⣿⣿⠅⠀⠀⣿⡖⢸⣧⡈⠙⡷⡩⠿⠙⣿⠃⢀⣾⣽⣿⣷⣿⣾⣿⣉⣿⣸
⠀⠀⠈⠀⠀⣿⡍⢿⡇⣸⡏⠀⠀⢡⢸⠁⠀⠀⢰⠎⡉⠹⠃⠀⠀⠀⠀⠀⠋⠀⢨⡿⢿⣿⣿⣿⣿⣿⣿⡯⢽
⠀⠀⠀⠈⠚⠉⠑⠈⠓⠙⠃⠐⠀⠘⠙⠂⠒⠠⠒⠜⠳⠦⠴⠰⠆⠰⠤⠄⠠⠴⠟⠤⠎⠥⠼⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠎
⠀⠀⠀⠀⠀⠡⠐⠠⢀⣀⡄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠬⠀⠠⠐⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡕⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⣨⣴⠶⢤⣄⡀⠀⠀⣀⣤⠴⢶⣤⡄⢄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠊
⠀⠀⠀⠀⢀⣀⣀⢀⣿⠋⠀⠀⠀⠈⠻⡄⣼⠋⠀⠀⠀⠈⠹⣇⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠜
⠀⠀⠀⠀⠻⠋⢛⢿⡇⠀⠀⠰⣼⡇⠀⣿⣇⠀⣮⣿⠀⠀⠀⣿⡛⠋⠿⠂⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠔⡵⠁
⠀⠀⠀⠀⣀⡴⠃⠌⠻⣄⡀⠀⠀⣀⠼⠛⠻⢦⣀⠀⠀⣀⣴⢏⠙⣄⢀⠀⠀⠀⠀⡇⠀⠈⢢⡀⡠⡢⠊
⠀⠀⢠⠊⡝⢀⠊⠀⠀⠈⠙⠛⠛⢡⡀⠀⠀⣠⠙⠛⠛⠉⠀⠀⠂⠈⢆⠈⠉⠁⡨⠷⡀⠀⠀⠙⢾
⠀⡰⠁⡜⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡆⠀⠀⡇⠀⠈⠢⡀⠀⠀⠁
⡰⠁⢰⠁⡄⠀⠀⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⣦⡀⠀⠈⠐⠄
⡇⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠇⠝⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⢀⢾⡀⠈⠐⢄⡀
⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⢱⢦⣀⠜
⢃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠂⠈⠉
⠈⢾⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠈⢧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠈⢣⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠋
⠀⠀⠀⠀⠉⣪⣗⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣮⣞⠁
⠀⠀⠀⠀⠸⣀⠰⡞⣛⢿⡗⠶⠶⠶⠶⠶⠶⠶⠶⠶⢺⡟⠛⣳⠆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠌⠑⠒⠁⠀⠀⠀⠀⠀⠈⠒⠊⠣⠤⠜

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠻⠻⣿⣿⣿⣿⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⣿⣿⣿⢰⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣾⣿⣿⣿⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⢜⠮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⢹⣿⣿⣿⠀⠀⠀⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡇
⣿⣿⣿⠈⠀⠈⠪⣻⢿⡿⠿⠿⣿⠿⡻⠟⢁⠀⣿⣿⣿⣿⠀⠀⢸⠈⢄⠈⠢⣀⡀⢀⣀⣀⠀⣀⠴⠊⡀⠃⡇
⣿⣿⣿⡠⡀⣬⠆⠀⠉⠉⠉⠉⠉⠁⠀⠴⡅⢠⢻⣿⣿⣿⠀⠀⠈⢄⢰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣖⢠⠃
⣿⣿⣿⣿⡵⢄⠠⠜⠾⠄⠰⠀⡴⠞⡤⢄⣠⢳⣿⣿⣿⣿⠀⠀⠀⠈⣦⡠⠴⡱⡗⡄⠐⠀⠶⠊⠆⢄⣰⠃
⣿⣿⣿⣿⡹⠁⢀⣈⠺⣟⠀⢀⡿⢛⣀⠀⠙⠞⣿⣿⣿⣿⠀⠀⠀⢠⠏⢄⣀⣙⢿⣇⠤⣔⣿⣋⣨⡠⠘⣇
⣿⣿⣿⣿⡃⠀⠀⢇⠀⠈⢳⡊⠀⢀⡄⠀⠀⢃⣿⣿⣿⣿⠀⠀⠀⢸⠁⠀⠈⣄⠀⠙⣷⠋⠀⢀⡗⠀⠀⣿
⣿⣿⣿⣿⡳⠶⣪⡩⡋⢷⡿⣿⠞⣹⢩⣔⠶⢦⣿⣿⣿⣿⠀⠀⠀⠘⠶⠖⠂⢼⠰⣶⠾⢷⡞⣻⠙⠺⠷⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣮⣖⢌⡡⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣙⡡⠚⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠨⣠⡄⠦⠾⢄⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠈⣧⠁⠀⠀⣠⠠⠀⠀⠀⠀⢀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢉⠑⢀⡘⠀⠀⡄⣴⣄⣐⡀⣀⡀
⠀⢀⠀⠻⠀⠛⠀⠉⠀⢂⡄⠀⠀⠘⠁⠀⣀⠀⠀⠀⡀⠀⠲⠘⠀⠉⠀⠚⠀⠀⠀⠀⠀⠀⢈⢀⠉⡁⠈
⠀⠀⠈⠀⠄⠾⠰⠟⡞⠀⠁⠀⠀⠀⠈⣱⢥⣀⡤⡤⢘⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⠀⠌⢛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠒⠀⠀⠀⢰⡗⣠⠒⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠄⠰
⠀⠀⠀⠀⢀⠠⠄⠀⢄⢐⠀⠀⠀⢄⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⢰⣀⠀⠌⣀
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣁⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠉⠒⢄⣀⠀⢀⡁⠠⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠄⢰⡄⠄⣀⠤⠒⠉⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⠉⠻⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠤⠀⠄⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⣠⣶⡿⠿⠛⠻⠷⣦⣄⡀⠀⠀⠀⣤⣶⠿⠛⠛⠻⢿⣶⣔⠐⣢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣠⠟⢿⣇⠀⠀⠀⠀⠀⣀⣄⠀⠀⢸⣿⣿⡆⠀⠀⣠⣄⠀⠀⠀⠀⠀⣼⣿⡉⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠜⣿⣄⠀⠀⠀⢸⣦⣿⡇⣠⣿⠿⠿⣷⡄⢸⣶⣿⡇⠀⠀⠀⣰⣿⠃⠹⣄
⠀⠀⠀⠀⠀⣄⠀⣿⠲⢾⣟⠀⣠⠃⠀⠈⠻⢷⣦⣤⣤⣭⣽⡾⠉⠀⠀⠀⠀⠙⢷⣯⣭⣤⣤⣶⡾⠟⠉⢢⠀⠘⢷⢤⣀⡀⢀
⠀⠀⠀⠰⡖⠿⢥⣟⣢⠤⢤⣉⣙⠒⠦⢤⣀⠀⠈⠉⠉⠉⠁⢱⣄⠀⠀⠀⠀⣠⡆⠉⠉⠉⠉⠁⠀⠀⢀⣀⡵⠴⠚⣋⣩⠷⣿⣠⡀
⠀⠀⢠⣤⣷⣀⣀⠠⠬⣉⠒⠠⢌⡉⠑⠢⢌⡑⠢⣄⠀⠀⠀⠈⢄⠑⢄⡠⠊⡰⠀⠀⠀⠀⢀⡠⠔⣊⡩⠤⠒⢊⣉⢭⠴⠒⠯⣼⢻⡇
⠀⣀⠸⠖⠊⠉⠀⠀⠀⢨⡆⠀⠀⠈⠙⠲⢄⡉⠳⣌⠳⣄⠀⠀⠀⠉⠋⠃⠉⠀⠀⢀⡴⢚⡡⠖⢋⣡⠴⠒⠋⣁⡤⢾⠀⠀⠀⠀⠉⠓⠤⣀
⢰⠁⠀⠀⠀⠀⠀⣀⣴⣎⡉⠙⠲⢤⣄⡀⠀⠈⠓⢬⣳⡈⠳⣄⠀⠀⠀⠀⠀⢀⠔⣡⠖⢉⠤⠚⠉⣀⡤⠖⠋⣁⡤⠞⣳⣤⣀⠀⠀⠀⠀⠀⠹
⣄⢀⡠⠤⠒⠊⠁⠀⠈⢫⠉⠓⠲⢤⣀⡉⠙⠲⢤⣄⣈⠻⢆⠈⢧⠀⠀⢀⣴⠗⢊⡠⢆⠙⢦⡖⠋⣁⡤⠖⠋⠁⠀⢸⠁⠀⠀⠉⠒⠤⣄⡀
⡏⠀⠀⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣉⡙⠲⣷⣖⡓⠀⠞⣡⣴⡥⠖⢋⣳⣠⡽⠏⠁⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⠀⠈⠙
⢣⣠⡤⠴⠔⠒⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣿⠈⠉⠉⠉⠁⣿⠰⡏⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠓⠲⠤⣄⣀⣰
⠘⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠒⠒⠲⠖⠒⢿⠀⣷⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⢰⣿⠿⢄⡀⠀⠀⠀⢨⠇
⠀⠈⠲⢶⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠒⠶⠶⠴⠶⢾⠀⣿⠀⠀⣷⣄⠀⠀⠸⣿⡿⣷⡄⠸⣿⣦⣶⡏⣷⠦⡔⠃
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡄⢸⣿⣿⣷⣄⠀⢻⣷⠿⠃⠀⠈⠉⠉⠀⢹⠀⣷
⠀⠀⠀⠸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡇⠘⠟⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿
⠀⠀⠀⠀⡇⠿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣛⣥⠿
⠀⠀⠀⠀⠛⠲⠦⣭⣙⣛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠋⠓⠒⠒⠒⠚⡇⢸⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⣛⣩⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠻⣗⠲⠬⣍⣙⡓⠶⠤⣤⣀⣀⣿⢸⡗⠒⠲⠴⠖⠺⡇⠈⣇⢀⣀⣤⠴⠞⣋⡭⣴⠞⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣮⣑⠦⣄⡀⠉⠙⠓⠶⠤⣍⣁⢘⡇⠀⠀⠀⠀⠀⡇⠀⢛⣉⡥⠶⠚⣉⡥⢚⣥⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢹⣉⣶⢿⣷⣶⣤⣤⣤⣀⣈⣉⣳⣤⣄⣀⣠⣴⣷⣪⣭⣤⣴⣶⣯⠷⣮⣉⠎⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠊⣹⠤⠠⢌⣻⣧⠋⠉⣻⠉⠉⠉⠉⠉⠉⠉⠉⠙⡏⠉⠑⣼⣗⡡⠄⠤⣏⠑⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡠⠋⠘⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠃⠹⢄⣀⡤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠺⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡐⠀⠀⠀⠈⠀⠀⢠⣴⠿⡿⣿⣶⣀⣀⠔⠀⡆⠀⠀⡈⢀⡇⠀⠄⠀⠀⣀⣄⡤⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠑⠀⣢⣀⢀⠀⠈⣀⣴⠉⠀⠤⠂⢢⠀⠀⢈⡇⠀⠠⠶⠛⠃⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠹⢀⡀⠀⠠⠀⠂⠠⠘⠪⠼⠁⠂⠈⠀⠐⠁⡇⠠⠀⢘⠃⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠁⣰⠟⠳⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡧⠀⠔⠉⠈⠋⠙⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠈⢸⡇⢠⣄⡀⠀⠀⠀⠂⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢴⣿⣿⣧⣷⡄⠀⠀⠀⠀⠀⠄⠂⠄⠈⠉⠙⠶⠤⠄⢄⠀⠀⢴⠂⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⢷⠀⣀⠀⠉⠁⠀⠀⠀⢀⣿⣿⣿⣷⡖⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠊⠈⠐⠤⠢⠈⠦⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⡀⠀⠀⠨⠾⠇⠤⢳⣍⠂⠄⠀⠀⠁⠀⠉⠉⠁⠁⠀⠀⠠⠄⠈⠀⠀⢢⣾⣾⣮⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⢁⠀⠀⠀⠀⠴⠚⠟⠁⠀⠄⠀⠀⠀⠀⠠⠀⣀⠀⠀⢀⣰⠎⠀⠀⠀⣽⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⢀⢸⡄⠀⠀⠄⠀⠀⠀⠊⠳⣄⠀⠁⠀⠄⡄⠈⠉⠘⠀⠀⠈⠠⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣡⡤⣤⠆⠊⠃⠀⠐⠦⡄⠀⠀⠀⠀⠈⠱⠦⠐⠀⠈⠀⠀⣤⣤⣄⡀⠀⠈⢢⡄⠂⠐⠂⡐⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠻⠦⣀⠀⠀⠀⠀⠱⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠘⠀⠀⣴⠷⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⡿⠃⠀⠀⠈⡄⠠⠁⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣉⣀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠁⠀⠀⠄⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠆⠈⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠸⠿⠟⠉⠉⠛⠛⠻⠛⠃⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢠⠀⡄⠀⠁⠀⠀⠀⣀⣀⡀⠀⢔⣥⠀⠀⠈⠉⠉⠀⠠⣤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡴⣪⡦⢙⠀⠉⠀⠀⠈⠀⠀⠐⠀⠀⠸⠄
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠰⠠⡗⠌⠁⠁⠠⠀⠂⠀⠀⠀⠀⠀⡠⠢⣈⠴⠁⠀⠀⠠⡀⠀⠀⠈⠀⠀⡠
⠀⠀⠀⠀⠀⠊⠀⠐⢰⠎⠠⠀⠀⠀⠀⠔⠀⠅⠀⠀⠈⠃⠀⠀⠀⠀⢀⠔⢈⠄⠉⠀⠀⠀⠀⠀⠀⡀⠐⠂⠔⠒⠈
⠀⠀⠀⠀⠀⡒⠀⠀⠁⠀⠄⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⢢⠄⡢⠖⠁⠀⠀⠀⠀⠀⡀⣂⠀⠃⠰⠶⠀⠐⠀⡄
⠀⠀⠀⠀⣾⣿⣶⣶⣆⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⡀⠌⠀⡠⡂⡡⠑⡇⠀⠀⠀⠀⠠⠀⠀⠑⠚⠁⠉⠡⠠⠢⠤⣒⣠⡤⣄
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠦⠀⠀⠀⠂⠐⠀⠔⠀⠀⣀⣀⠠⠀⠊⠀⡠⠃⠀⡠⢀⠀⠀⠀⠀⡄⠀⠠⠀⠀⢀⠀⢀⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⡇⠀⡄⠈⠀⠀⠀⠀⣾⠑⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠄⠀⢠⣐⡀⠀⠀⠀⠀⢠⣀⠀⠑⢄⠀⠉⠉⠀⡸⠀⠀⠀⠀⠠⣀⡟⠤⠤⠤⠄⢀⣁⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠋⠈⠙⠳⢦⣄⡀⠳⠞⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡀⠉⠁⠀⠒⠀⠽
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠤⠀⠀⢿⠇⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⢀⠁⠐⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠙⠷⡀⠀⠀⣀⠀⠀⠀⣰⠀⠀⠀⠀⠐⠿⠋⠉⠀⠀⠀⠀⠀⠀⢈⠀⠘⠀⠀⠒⠀⠠⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⢀⠀⠈⠠⠀⠀⡀⢤⣾⣿⡄⠀⠀⢀⣤⠀⠀⢛⠁⠀⠀⠀⠠⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⢂⠜⠣⠌⠒⠾⠀⠀⢀
⠀⠀⠀⠀⠁⠀⡸⠂⣉⠙⡄⡦⠡⠀⠄⣥⡍⠀⠀⠀⠀⠘⠃⠀⡐⣴⣷⡄⡄⢁⠀⠀⠁⠀⠀⠀⠀⠘⡎⢀⠀⠀⠀⠀⠀⠀⢂⠁
⠀⠀⠀⢾⠼⣿⡷⠠⠤⠄⠗⠺⡀⠈⠓⠭⠁⠀⠀⠺⠗⠀⠀⠀⢰⣿⡿⠁⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠄⢀⡠⢔⠒⠀⠂⣁⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡁⢀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠇⡐⠀⠀⠀⠐⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢤⣾⡦⠀⣀⣀⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣹⣾⣅⢀
⠀⠀⠀⠚⠉⣡⣾⣿⣿⣷⣤⠶⠞⠋⠁⠀⠀⢀⠀⠂⠀⠂⢠⠀⠁⠀⠈⠀⠀⠀⠈⠀⠀⢠⣦⢺⡷⢷⡿⠏⠋
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⡀⠹⡟⢿⡿⠟⠈⠁⠀⡸
⠀⢻⣿⣿⣿⡿⠛⠛⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣤⠔⠁⠀⠀⠠⠙⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠙⠛⠋⠀⠀⠘⠋⠀⠀⠀⠀⢀⡀⠀⡀⣒⠀⠁⠀⣀⢈⡴⢶⣄⣛⠃⠀⠀⠀⠀⡰⠀⠀⠀⠀⢰⠁⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠁⢰⡟⠻⠟⢹⠋⣿⢻⡟⣉⢻⡟⢻⠏⣉⣻⡄⢀⣤⡞⠀⠀⠀⠀⠀⠠⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⢸⡇⣤⡄⢸⠠⣿⢀⣜⠳⢾⡇⢸⢸⡏⡉⠀⠈⠉⠀⠀⡴⠀⢀⡤⠀⢩⣉⡙
⠀⢶⣾⠇⢀⠰⣄⠀⠀⠀⠁⠀⣼⠀⣿⣇⢸⣇⠀⣸⣟⠗⣨⡇⣼⣌⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠝
⠀⠀⠉⠰⢍⠓⠀⠀⠀⠀⣀⢀⡈⠛⠁⣙⢋⠙⣛⠉⠙⣛⢋⡛⠋⢙⣛⢋⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢚⡓⠲
⠀⠰⡅⠁⠀⠐⠂⠀⠀⠀⣿⢸⡇⠀⠀⣿⢸⠀⣿⠀⠀⣿⢸⡇⠀⠸⡟⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⡙
⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠁⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⣁⣀⣘
⠈⠵⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠐⠒⠮⣡⡔⢮⢂⢀⣀⡠⣿⣿⣿⣿⣿⠃⢼⣶⣀⠐⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠍
⠀⠀⠀⠀⠀⠲⠯⠄⠀⠀⠙⢿⣿⣿⠃⠐⠙⢿⡃⠀⠀⠀⠶⠀⣿⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠐⠀⠤⠤⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣇⡙⢶⣄⣠⡤⠴⠷⠦⢤⣀⣴⠶⣣⢻⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢠⡈⢙⠿⠿⠛⠉⠈⠛⠻⠿⡋⠋⣀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡟⣿⡦⠀⠒⡀⠄⠉⠤⠀⠒⠀⢾⡿⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⠸⣿⠆⠐⠂⠀⠌⠁⠄⡐⠂⠠⣾⡟⢌⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣷⠈⢠⡀⠀⠀⠉⠭⠁⠀⠀⣠⡌⠡⣟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡠⠉⣤⡀⠀⠀⠩⠍⠁⠀⠀⣠⠈⢠⢼⣿⣿⣿⣿
⠀⠀⠀⠠⣾⡇⢰⢀⠉⠓⣴⡞⠀⢳⡄⠈⢉⠀⡄⢺⣦⠄⠀⠀⢸⣿⣿⣿⡛⢵⠂⡄⢈⠉⠀⣴⠃⠘⢦⠄⠋⢁⢰⠀⣶⢛⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣆⠀⢜⠀⠀⣤⣤⣄⠀⠐⡅⢀⣶⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣦⣕⢆⠀⠕⠀⢀⣤⣤⠄⠀⢘⠄⠀⢞⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢳⡄⠇⠀⠶⣰⠆⠀⠟⣠⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣮⡣⠘⠃⠀⢖⡰⠆⠀⠇⢐⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⡖⡴⣿⠦⠾⣴⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣻⣻⣜⠒⣾⣷⡶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠘⢿⣿⡿⢂⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣭⣯⣭⠉⣽⣭⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣩⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣑⡠⡡⣒⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠳⣄⠀⠀⠀⢿⠦⠤⠤⠒⠒⠒⠒⠶⠤⠒⢒⣺⠃⠀⠀⠀⡠⠊⠉⠳⣄
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⢳⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⣴⢻⡇⠀⠀⢠⠋⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⡏⣀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⡇⢺⠷⢤⠀⠀⢀⣰⣾⡅⢸⡇⠀⠀⢸⠀⠀⠀⠀⠘⢶⣄⠹⡄
⠀⠀⠀⢀⡆⠍⠀⠐⠆⠐⣏⣠⠄⠸⢆⠀⠀⡇⠈⠉⢁⠱⣀⠞⡍⣉⠀⢻⠀⠀⢀⠞⡀⢀⠀⠳⠤⠜⠀⠉⠱⡄
⠀⠀⠀⡼⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠘⠦⣼⢀⠀⠀⠀⠀⠉⠀⠀⠀⡀⡄⢇⣠⠋⠀⠑⠈⠋⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⢰⠃⠀⡄⠀⣀⢼⠁⠀⠈⠀⠀⠀⠀⡇⠀⠘⠀⢄⠀⠀⠠⡄⠠⠁⠇⠘⠀⠀⠀⠀⢀⠂⠀⠀⠠⠄⠐⠀⡀⢧
⠀⠀⡏⢠⠛⡇⠀⠀⢸⢰⠆⠁⢀⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⣘⡀⠘⠀⠀⠁⠀⠀⠚⠈⢁⠘⡆
⠀⢰⣠⠏⡆⠀⠀⠀⠈⢰⢠⠀⠼⠀⠀⠀⡧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⢐⠈⠒⠀⢀⠀⣀⠀⠃⠀⡀⠀⠀⠈⠀⢻
⠀⠘⠁⠀⠀⡃⢀⢃⠀⠉⢸⠀⢳⠀⡄⡆⢱⠄⢀⠀⢠⠀⠀⠀⠀⠁⠁⡄⡾⢠⠀⢆⠈⡆⣹⠀⣈⠀⠡⡀⠀⠀⠀⠰⡄
⠀⡇⠀⢸⠀⠈⠁⡌⢠⢃⡞⠋⠈⠂⠀⠈⢹⢀⡘⡇⠘⠄⠀⠀⠀⡴⠀⣼⡷⠞⠈⠈⠃⠀⠈⡆⢸⠀⠀⠈⠀⠀⠀⠀⢧
⢠⠀⠀⠀⠀⠀⢀⠇⡜⠉⠀⠀⠀⠀⠀⠀⠘⡄⠀⢈⠀⠰⠄⠃⠐⠃⠨⢰⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⢠⠀⠀⠀⠀⠀⢸
⢸⠀⢠⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠁⠀⠀⠀⠀⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘
⢰⠀⠘⠀⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠲⠀⠀⢠⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠀⠘
⢈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠄⠀⠀⠐⡀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡆⠀⡇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢈⠊⣧⠠⣁⠰⡄⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡁
⡇⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢠⠆⠺⠬⠤⠾⠷⠧⠦⠇⠼⠼⠤⠄⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⢧⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡰⡿⣀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢯⢩⠁⡉⠉⠉⠉⠉⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⡀⠘⣈⣠⠀⠀⠀⢀⡄⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠁⠒⢲⢉⠟⠒⠒⠛⠛⠋⠉⠉⠉⠛⡑⠉⠉⠉⠙⡩⠽⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠀⠀⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠶⠤⢬⠤⠤⣤⣤⣤⣤⣤⣤⣬⣦⡧⠤⠤⣤⢌⣷⣀⣄⣠⣄⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡀⠠⠀⢿⢇⣼⡕⠂⠐⣀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠠⢈⡇⢝⠀⠀⠍⠉⠀⠀⠘⠷⣰⣤⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠱⠆⡀⠀⠀⠀⠈⠀⠁⠂⠄⡄⠐⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠰⠒⢄⠀⠀⠀⡀⠁⠀⠀⠀⠀⣀⠀⢀⠀⠈⠀⠀⠀⡠⠀⠀⠐
⠀⠀⠀⠀⢀⡠⠤⠤⠠⠄⠀⠀⠐⠥⡀⠀⠀⠀⢀⠀⠘⠡⡤⠀⠀⠀⠀⡀⠀⢈⡀⢀⠔⠀⢀⠀⠀⠠⠂⠀⠂⠀⠀⡀
⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡷⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠰⠀⢀⣺⠋⠁⠀⠠⠁⢀⡀⠀⠀⠀⠐⠄
⠀⢴⣀⠀⠁⠀⠀⢀⣀⣀⠄⢖⡈⠂⡄⠀⠐⠐⠀⠀⠀⠁⠀⠠⠉⠀⠀⢄⡀⠀⠀⠀⢢⠆⠀⠁⠱⠄
⠀⠀⣨⣀⣉⡀⢭⠨⠤⠐⠘⠀⠀⠀⠠⠄⢈⠇⢀⠓⠂⠀⠀⠂⠀⠀⠀⡀⠀⠀⠐⠀⡜⠀⠀⠀⠀⠀⠐⡀
⠀⠠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠉⠁⠀⠀⠒⠄⡀⠀⠐⠀⠀⠀⠀⠈⠁⡀⠠⠔⠀⠈⠀⠀⠀⠀⡀
⠳⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐
⠀⠀⠐⠀⠄⠠⠀⠄⠤⠠⠔⠈⠑⠤⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠠⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⢲⡤⠀⢂⠀⠀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡗⠠⡄⡈⡰⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠄⠀⠀⡠⠊⢀⠁⠀⠀⠀⠀⢃⠈⠀⡀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠄⠈⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⢀⠀⣀⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣴⠞⠋⠁⠀⢀⣤⢞⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣋⣵⡾⠟⠋⡉⠁⠀⠀⣠⣴⡶⠿⠛⠛⠛⣛⣻⡟⠛⠛⠛⠉⠀⢀⠄⢀⣴⢟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⣾⠟⠁⠀⠀⠀⠀⢀⣴⡿⠋⣁⡤⠀⠀⠈⠉⠉⠀⢀⡠⠀⠀⠀⡰⠋⣠⡟⣡⡞
⠀⠀⠀⠀⠀⢀⣾⠏⠀⢠⡟⠁⠀⡀⠀⠀⠀⣠⡿⠉⣠⠞⠁⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡼⠁⣼⠏⣾⡟
⠀⠀⠀⠀⢠⣿⠃⠀⢠⡟⠀⢀⡜⠁⠀⠀⣰⠏⢠⠞⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡼⠁⣼⠏⣸⢸⣇
⠀⠀⠀⠀⣾⠃⠈⠀⣼⠁⠀⡞⠀⠀⠀⠀⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣸⡏⠀⡟⠘⣿⡀
⠀⠀⠀⢸⡿⠀⠇⠀⡏⠀⣸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⡇⠀⢿⣧
⠀⠀⠀⢸⡇⠐⠀⠀⡀⠀⡇⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠃⠀⠀⠃⠀⠸⣿⡄
⠀⠀⠀⢸⣧⠀⣀⠀⡇⠀⠁⠀⢠⡿⠀⠀⢀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠈⣿⡀⢹⡀⢻⡀⠀⠀⢸⡇⡌⢠⡏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣰⠇⠀⠀⠀⠀⠀⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠸⣷⠈⣧⠘⣧⠠⡆⢸⡇⡇⣾⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣼⠏⠀⠀⠀⠀⠀⠀⠁⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠹⣷⡘⣧⠘⣧⣿⡘⣿⣧⡏⠀⢠⡟⠀⠀⠀⣠⠀⠀⣠⠞⣡⣴⠟⠁⠀⠀⠀⠀⠀⢀⡆⠀⣼⢏⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣜⣷⡘⢿⣷⡹⣿⣇⠀⣾⠁⢀⣴⠟⠁⢠⣾⣵⡾⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠀⣸⠟⢸⢱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣿⣿⣿⣿⣼⣇⣴⢟⣡⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣼⠋⠀⠀⣾⢿⡇
⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⢿⣿⣿⣥⡭⠙⣿⢿⣿⣿⣭⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠞⠁⠀⢀⣼⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠾⢫⣿⣇⣾⠛⠇⣈⣿⡿⠿⣶⣤⣤⣤⣤⣤⣤⠤⠀⠐⢉⠀⠀⠀⣠⠞⢁⣴⡿⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⢿⣦⣿⣿⢿⣷⣶⣤⣤⣄⣀⣀⣀⣤⣤⣶⠾⢋⣡⠴⠋⣁⣴⠟⠋⡰⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣹⠿⣿⣾⣿⠳⢦⣭⣉⣉⣉⠉⠉⣀⣀⡀⠁⠀⠀⠛⠉⠀⠀⠊⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⠿⣦⣔⡭⢍⣉⣙⡉⠉⠉⢀⣀⣀⡤⠴⠊⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣬⣭⣍⣭⣤⣤⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⣿⢤⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣁⣀⢀⣤⣶⣿⣿⣿⠇⢸⠳⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⠟⣛⣥⣶⠟⣿⡟⢩⡿⣫⢀⠘⣇⠀⠀⠀⠑⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢢⡤⠖⣪⡅⣰⡿⠟⠫⢀⣠⢾⡿⠐⠛⢱⣿⠘⠄⠻⠣⡄⠀⠀⢀⣙⢮⠁⠐⠦⣄⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠘⠋⢠⠖⠀⢀⣠⢴⣿⠏⡄⠀⣀⣴⣿⣿⡆⠰⣀⡀⠀⣷⢄⡀⠹⠄⢀⡀⠀⢈⣙⠓⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡵⣆⣥⡶⠞⢁⣴⣿⠟⣾⠁⡼⠿⠋⢀⡝⠻⣄⠀⠁⣠⣀⠓⠈⡀⢀⡆⠙⢰⣴⡠⣃⣏⣻⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⠋⠸⢋⣥⡶⠾⣿⡿⠋⠰⢃⠾⠁⢷⠀⢿⠁⠑⡹⠀⠀⠀⠹⠀⠸⡇⠸⡇⡀⣿⣿⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠁⢀⣨⠉⢁⣤⡆⠉⠀⠀⣴⡟⠀⢰⡈⠀⣼⢠⠄⠃⡈⠣⠀⡇⢀⣆⣧⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠶⠾⠛⠛⠛⠓⠻⠛⠛⢿⣄⢠⡀⠁⠘⢠⣾⡷⢐⣿⣸⣠⡀⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢳⢸⡇⣰⢀⣨⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣇⠀⣶⠀⣤⣼⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢿⣇⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⠻⣿⣿⣿⣿⣾⣿⣿⣿⡷⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠹⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⠿⠿⠿⠛⠉⠉⠁⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠁⠀⠀⣠⣶⣶⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⢀⢔⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠀⠀⣀⣿⣇⣀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣦⣀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⣉⡀⣸⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⣙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⢽⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣧⣤⣴⡾⠋⢽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⢰⣿⣿⣿⡟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣿⣿⣿⣿⣿⣏⠹⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠖⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠜⠘⢿⣿⣿⣿⡿⠋⠀⠉⠋⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⡀⠀⢠
⠀⣰⣾⣿⣿⣦⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⣴⡏
⠀⣿⣿⣿⣿⣿⣧⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣟⠉
⠀⠀⣿⣿⣿⣯⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠻⣿⡇
⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢻⣷
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣰⣦⣀⢀⣀⣀⠀⢀⡀⠒⠻⢿⣷⣶⠿⠛⢛⣁⡀⠀⣀⣀⣀⣀⣴⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠙⢻⣿⢟⣽⡷⢋⣴⣾⣿⣿⣷⣶⡀⠘⠃⢰⣷⣿⣿⣿⣷⣎⠙⢷⣮⡻⣿⠟⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⢣⣾⣿⠃⠈⢉⣩⣝⣛⠿⢻⣷⡀⢠⣿⠿⢟⣛⣩⣌⠀⠀⠘⣿⣷⡙⣰⡦⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠁⣾⣿⡁⠀⣐⣚⣿⣿⣿⣿⣦⣹⣷⣿⢣⣾⣿⣿⣿⣿⣶⣂⠀⢘⣿⣧⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡖⣼⣿⠋⠀⣸⠿⣯⣭⣛⣿⡿⣏⣁⣉⣉⣈⡹⢿⣿⣫⣭⣿⠻⣇⠀⠹⣿⣇⢲⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⠏⢈⡭⠅⣀⠔⣡⠄⠻⢌⠉⡟⡞⢉⡉⠋⠛⣁⡉⢧⢯⠉⡱⡇⢠⣜⡢⣀⠲⢭⡁⠹⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢃⣴⠏⠀⢈⣵⣿⣿⠀⠿⢯⣾⡇⠃⠘⡏⠀⠀⣿⠁⠞⢸⣷⡽⠿⢀⣿⣿⣮⡁⠀⢻⣶⡙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢡⣿⡏⠀⠀⢸⣿⣿⣿⣿⣣⣿⣿⣷⡀⠀⠻⣴⡶⠋⠀⣠⣾⣿⣿⢸⣿⣿⣿⣿⡁⠀⠀⢻⣿⡈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣡⣿⣿⡃⠀⠀⠸⢿⣿⣿⣿⢸⣿⣿⠿⠛⠀⢤⡈⢡⡄⠈⠙⢿⣧⣿⠹⣿⣿⣿⠟⠇⠀⠀⢸⣿⣷⣄⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣧⠀⠀⠀⢀⡿⠿⢋⡘⣿⣃⣤⣤⡴⠿⠃⠘⠷⠦⢤⣤⣘⡿⢀⡙⠿⢯⣀⠀⠀⢀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡦⠀⠀⠈⣴⣾⣿⣷⠘⣿⣥⠀⣦⣶⣶⣾⣶⣼⠀⣬⣿⢁⣿⣿⣷⣦⠈⠀⠀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⣸⣿⣿⣿⣿⣄⡟⣿⣦⡏⠀⠸⠁⠈⢹⣼⡟⢿⡈⣿⣿⣿⣿⣆⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⡿⠸⣿⣿⣿⣿⣿⡏⡿⠀⠀⠀⠀⠀⠈⣿⢹⣿⣧⣿⣿⠿⠃⢹⣠⣾⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡙⠿⣿⣇⠀⠀⠈⠻⣿⡟⣇⡇⠀⠀⠀⠀⠀⠀⠹⣼⣹⣿⠟⠁⠀⠀⣸⣿⠟⣃⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣌⠻⣶⡄⠀⠀⢹⡇⣿⣷⠀⠀⠀⠀⠀⢀⡷⣯⣿⡏⠀⠀⢴⣾⠏⣰⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣆⢹⣿⡀⠀⢸⢇⡟⠉⡀⠀⠀⠀⠀⡈⠑⢻⠹⡇⠀⢘⣿⠇⣼⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣝⠛⠄⠈⠘⢷⡀⠟⠶⢾⡶⠞⠛⢠⡟⠀⠀⠐⢛⣣⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⡀⠀⠈⢦⣅⠳⣦⣤⣤⣴⢇⣡⠞⠀⠀⣀⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⢿⣷⣌⣿⣯⣵⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⣧⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡡⠖⠀⢻⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠶⢶⡁⠀⡠⠖⠘⠁⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡄⠀⠀⠀⢷⠀⠀⠀⠀⠀⡴⠏⠀⠀⢤⡀⠀⠈⠋⡠⠒⠒⢶⠄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣤⣀⣀⡀⠀⠀⠀⢰⠀⠀⠀⣸⡇⠀⢀⣰⣾⠀⠀⠀⢀⠴⢂⣀⠄⠢⡀⠃⠀⠀⠈⡃⠀⠀⠃⢸⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⠀⠀⠀⢹⡉⠙⡿⠀⠀⢠⣾⠆⠀⠘⠋⠀⠀⠀⠉⠀⠀⠀⡔⠁⠀⠜⠁⠀⠀⡇⠀⠀⠀⠀⡇⠀⡌⠀⣼⠀⠀⠀⣿
⠀⠀⠀⢠⠶⠛⠉⢁⠀⠀⠰⠿⠃⠀⠀⠀⢣⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠀⡐⠀⠀⠀⢰⠁⠀⠀⠀⠀⠁⡰⠁⣰⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠄⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠽⣋⣙⣶⣒⡴⠃⠀⠀⠀⠀⠀⢸⠳⣤⡏⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢠⠀⢀⣼⠆⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⠀⠀⠀⠀⠀⠀⡁⠈⠛⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠦⢤⣤⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠐⠋⠀⠀⠀⠀⠀⠀⡷⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣯⣤⡔⠉⠙⡶⠛⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣀⣀⣀⣀⣤⡤⠶⠾⠋⠁⠀⢀⡽⠤⠤⠞⠉⠓⢦⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⠤⠤⠶⠞⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⢇⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠐⠶⢤⣄⡀⠀⠀⢀⠴⢤⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠑⢲⡶⠋⠀⢀⡤⠄⠀⠀⠀⠉⠒⣴⠃⠀⠈⡇⠀⠀⠀⠀⠀⡌⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣀⣤⣴⣖⣤⣄⣺⣅⢀⡴⠁⠀⠀⢀⣤⡴⠃⢰⡇⠀⠀⢀⡇⠀⠀⠀⠀⢰⠀⢀⡾⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣇
⠀⠛⠻⢿⣿⡿⠿⠛⠉⢉⡟⠀⠀⠀⣴⠋⢸⠃⠀⠀⣿⡳⠞⢋⡻⠦⠦⣤⠖⠛⠛⡾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠈⠙⠳⢤⣾⠟⣳⣤⡤⠞⡋⠄⢻⣀⠀⢀⣿⣀⠔⠉⢀⡠⢲⡏⠀⠀⢰⡇⠀⠀⢠⠧⣄⠤⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣦⣁⣠⣴⡿⢛⣫⣿⠗⣀⣀⠔⠋⢀⣸⣇⣀⣀⡼⠷⠤⣴⠋⠀⠈⡆⠀⠙⠶⠲⠦⣤⣄⡀⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣿⢋⣤⣾⡿⠋⢀⠔⠋⢀⣈⡽⠋⠀⢀⡠⢾⡀⠀⢀⡿⢲⢶⣇⡀⠀⠀⠈⠙⠳⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣧⣠⡔⠁⣠⣴⣿⠟⢁⣠⣴⠋⠀⡠⠟⠉⢉⠞⠁⢀⡼⠛⠳⣦⣤⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣟⣁⣴⡿⠟⢁⡴⠋⢀⣴⡞⠁⢀⣴⠏⠀⢀⡼⠃⠀⡹⠛⠻⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣤⣴⠋⣠⣶⣿⠟⣡⣾⡿⠁⠀⣠⠟⠀⢠⡞⠁⢀⡔⠉⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣏⣡⣾⡿⠋⢀⣾⣿⠇⢀⡴⠋⠀⣠⠟⠀⠀⣰⠋⠀⢙⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣤⣴⣿⡟⠁⣠⠟⠀⣴⣾⡇⠀⠀⣼⠃⠀⢠⠂⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣤⡞⠁⣠⣾⣿⠟⢁⣴⣾⠁⠀⣰⠃⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣿⡿⠃⣠⣿⣿⠏⠀⡴⠃⠀⣀⡞⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣿⡿⠁⢀⡾⠁⠀⣼⣿⠇⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡎⠀⢀⣾⣿⠃⢠⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣾⣿⠃⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀
⣿⣿⣿⠟⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠀⠉⠉⠛⠷⠀⠀⢠⠒⠉⠀⠀⠈⠛⢷⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⣼⡟⠀⠀⠀⢀⣀⡀⠀⢡⢠⠃⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢈⠝⢻⡇⠀⠀⠀⠻⣾⠇⠀⢸⣹⠀⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢿⣦⡀⠀⠀⠀⠀⠐⠃⠈⠑⢤⡀⠀⠀⠀⣀⣼⢏⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⢀⠛⠻⠟⠛⠋⠁⠀⠡⡀⠹⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠘⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⣝⣴⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⡿⣞⣩⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠒⠘⠓⠉⠉⠋⠉⠉⠉⠉⠉⠉⠙⠉⠑⠚⠁⠒⠚⠋⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⡟⠀⠒⠓⠀⠀⠐⠓⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢩⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠂⡄⠀⠀⢀⠄⠈⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠄⠄⠀⠀⠀⠤⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠠⠳⠀⠀⢠⠛⠛⠀⠀⢠⠋⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠂⠀⢣⢀⢏⠀⠑⢉⢀⠎⠀⠘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠄⠀⢀⠄⠣⡸⢥⠦⠀⠙⢄⠀⠀⣀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⢱⡎⡠⠓⢄⠀⠑⢀⣳⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠂⠀⠠⠜⠁⠀⠀⠀⡀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⣠⣘⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠿⣿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⢀⡻⣿⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣽⡇⣩⣴⢰⡿⢿⠿⠟⣩⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣹⣷⣬⢻⣆⠃⣽⠟⢰⣯⣤⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⣘⡛⢹⣿⠻⣎⢻⣾⡏⣾⠸⣧⣐⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⠻⢃⣝⣯⣵⣿⣦⣽⣵⣿⣷⡌⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⣰⣿⠃⣿⠿⡟⣻⡟⣿⠟⣋⣵⡎⢰⡶⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢠⣿⣿⠀⢻⡇⣷⢽⣇⣿⢸⣯⠈⠁⠛⣿⠀⣾⠟⠹⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⣿⡇⣿⡀⢺⣇⣛⣹⡏⠛⣤⡝⣷⡄⠤⣿⠀⣿⡀⢀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⠈⠛⠘⠛⣣⣮⣙⣛⣻⣿⣂⠈⣛⢛⣁⠰⡿⣇⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣏⠀⢹⣿⣿⣦⠿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣟⠀⠀⢹⣿⣿⣷⣾⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠀⠀⠀⠉⣿⣿⡗⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠷⠶⠦⠀⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡋⠂⡀⣠
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠠⠀⠀⢀⠀⠀⠰⢿⡀⢈⡝⠛⠁⠀⡀
⠀⠀⠀⠛⡀⠀⠀⠤⣀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠈⠀⠀⠀⠀⠑⠊⠁⢀⠀⣰⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⢀⡠⠂⢀⠀⠀⠸⡇⣀⠀⠀⠀⢴⠋⠀⣄⣀⠀⠀⠠⢀⣠⢠⡀⣐⣦⠀⠀⠀⠰⠚⠀⠉⠉⠙⡆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⠃⠈⢁⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡁⠀⢂⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠄⠀⠚⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢁⠈⢀⡀⠀⠀⠀⠈⠉⠑⠀⢹⣠⡤⣤⢴⣶⠲⢶⢲⡶⢄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⢦⣀⠀⠈⣷⣠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠱⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⢀⡀⠀⢦⠀⢠⡀⠀⠈⡿⢷⠰⡆⠀⠙⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠰⢳⡄⠈⠧⠚⢣⠀⠀⠀⠈⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠒⠚⠻⢧⠀⠄⠀⠀⠀⠀⠘⢶⢸⣠⡍
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢨⡄⠀⠁
⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣉⠉⠛⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⢷⠈⠛⠁⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠉⠉⠀⠀⠈⠋⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡸⠶⢶⣾⣿⠿⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⣥⢔⣪⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⠷⣦⠀⠅⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠊⠉⠁⠀⠰⠋⠁
⣴⡄⠤⠤⠶⠚⠋⠉⢉⣭⣵⣾⣿⣿⣶⣌⠀⠀⣀⣀⢀⣸⣦⣄⠀⠀⠀⠀⠀⣠⣴⣿⣶⡄
⢻⣇⠀⠀⠀⠀⠀⣠⣿⣷⣿⣿⣿⣿⣿⣿⣧⠀⣾⣿⣿⣿⣿⣿⣿⣦⣀⣤⣾⣿⣿⠟⠉⡀
⢸⣿⠀⠴⠊⣠⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠛⠋⠉⠛⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁⢀⣼⣿⣀⣾⠄
⢸⣿⡀⠀⣼⣿⣿⣿⣛⣛⣸⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣶⣦⣴⣿⣿⣿⣿⡟
⠈⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣇⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣦⠀⠀⡀⢺⣿⠁⠀⠀⠀⢻⣿⣿⣿⣿⡿⠃
⠀⣿⣿⠈⠛⠛⠛⠛⠋⢉⣄⠉⠉⠀⠀⠀⠘⢿⣦⣀⣠⣼⠟⠀⠀⠀⠀⠘⠻⠿⠿⠋
⠀⢸⣿⡄⡄⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⣀⣠⡾
⠀⢸⣿⡇⣷⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠈⣿⣷⢻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣿⣿⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⢻⣿⡄⢠⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣶⡿⠿⠿⠿⠛⢛⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠛⠋⠁⠐⢒⣂⣉⣡⣾⣿⣿⣿⣿⣿⡟
⠀⠀⢸⣧⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⢿⡟⠛⠛⠉⠉⢻⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠃
⠀⠀⢸⡇⣿⣿⣿⣿⣾⣿⡿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⡇
⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣻⣿⣿⣶⣿⣿⣧⣄
⠀⠀⢸⢁⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣾⣽⣿⣿⣷⣄
⠀⠀⢸⠘⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⠿⢿⣤⣤
⠀⠀⢸⠘⣻⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣨⠟⠹⣦⠞
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⢀⣼⠛⣦
⠀⠀⠀⠸⣿⣛⣻⣿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠗⠚⣿⠃⠀⠘⢧
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠶⠶⠚⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣧⣀⣶⣀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⢸⣿⣯⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠙⠿⠿⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣿⡁⣰⡎⢹⡿⢶⡖⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢯⣤⡿⡇⠛⠋⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣻⣧⡀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢻⣿⣿⣿⣿⡿⢿⣭⣿⣿⣶⣶⣶⣿⣷⠆⠀⠀⠀⠀⣀⣠⣴⣿⡿⠁⢠⣿⡟⠹⣷⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⢻⣿⣿⣿⣿⣿⡿⢿⣧⣘⠉⠁⠀⠀⠀⢶⣿⣿⣿⣿⡟⠁⠀⠈⣿⣷⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠀⣀⠄⠀⠀⠀⠙⣻⠿⠋⠀⠀⠀⠀⢹⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠉⠙⢿⣿⣧⠼⠟⠁⠈⠃⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⣗⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠒⠿⢯⡁⠀⠀⠁⠚⠁⡰⠂⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢻⣶⣆⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣸⣿⡟⢠⣿⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡷⠟⠛⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢹⣟⠃⢀⣾⣿⣿⣿⣿⡟⠛⠃⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢀⣼⣿⡿⠙⢿⣿⣿⡟⠿⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢨⣿⣿⡿⠋⠀⠀⠈⢿⣿⣿⡶⠀⠒⣾⠟⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⠀⠻⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡆⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠛⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⢀⡀⠀⢹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣀⠀⠀⠀⠀⣰⣿⣿⡿⠀⢨⡇⠀⠈⠻⢿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⣶⣶⠀⢠⣿⣿⣿⣧⠀⢹⡃⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣯⣤⣴⣄⣿⣿⣶⣾⣿⣿⣿⣿⣷⣾⣿⣶⣶⣾⣷⣤⣤⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⣿⡿⠟⠛⠻⣿⣿⣿⣿⣿⡛⠛⠛⣛⠛⠛⠿⠿⠿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢀⣿⣷⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢰⡇⠈⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡾⠀⠀⠀⢻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠉⢻⣿⠃⠀⠀⣠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠋⠀⣴⡿⠉⠻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣰⣿⡁⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⢠⣿⣿⡅⣀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⢀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⡄⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣇⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢻⠛⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣄⠀⢈⠙⢿⣶⣾⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⣠⣿⠿⣽⣿⣝⠛⠛⢛⡉⠉⠉⠋⠉⠉⠛⠻⠶⠾⣿⣿⣾⠿⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠿⠛⠉⠛⢻⣿⣶⣿⠛⢣⠀⢠⣶⣄⡀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⣿⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⠀⠘⠻⣿⣿⣶⣤⣀⣀⠀⠀⣀⣤⣶⣿⣿⣦⣠⣾⣇⡌⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⢻⡟⡀⢻⡍⠋⠉⠿⠿⠿⣶⣦⡑⠆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣾⣿⣿⣿⣿⣴⣿⠤⣴⣿⡯⠀⣓⠈⠀⣠⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣟⢷⣤⡿⠃⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠟⠛⠛⠉⠉⠙⠛⠻⢿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣆⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡤⠊⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠰⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡿⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⡆⢀⣷⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢿⡄⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠀⠀⡧⢼⣌⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⠁⠈⢣⠀⠀⠀⠀⡴⢉⣤⣬⡁⠀⠀⠀⠀⠀⠀⠀⣡⣤⡌⠻⡆⠀⠀⡇⠀⠀⢧⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢾⣷⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣾⣿⠀⠁⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠁⠀⠈⡆⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢀⠃⠀⠀⡟⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠜⠁⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠁⠤⣀⣀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⡆⠀⠀⠀⢀⡴⠃⠈⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⣀⠠⠀⠀⢠⠀⠀⢈⠂⢄⠀⠀⠀⠀⠀⠈⠩⠹⠟⠁⠀⣀⡔⠋⠀⢀⡀⠀⠐⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣄⣀⠀⠀⢀⡠⠔⠊⠁⠀⠉⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠈⠠⡀⠀⠀⠀⠀⠀⢀⡠⠉⠁⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⢣⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⣿⠀⢀⡇⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣿⣷⣾⡿⠃⠀⠀⣀⣿⣟
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠻⢷⣄⣠⠾⠛⠁⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡶⠋⠁⠀⠀⢠⡄⢈⠋⠁⢀⣴⣿⣷⡀⠐⠲⡂⠀⢸⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠾⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⡀⠀⠀⢨⣭⣭⣼⣿⣟⣷⣯⡙⢹⢀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠃⠹⣿⡆⠀⠀⠀⣧⣤⣤⣌⣿⣿⣿⣿⣿⡿⡿⢻⢸⠀⢸⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⢸⢸⠀⣸⣿⠈⣿⣧⠀⠀⠀⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⣿⡁⠀⡠⢾⣟⡻⣿⡻⣿⣿⣿⣿⣿⣿⣴⣼⣾⢀⣿⡇⠀⣿⣿⠀⠀⢰⡿⠰⣿
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣿⠀⠀⢹⣿⣀⠀⠘⠳⠈⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣶⣄⠀⢀⣀⡤⠤⠔⠒⠒⠒⠒⢒⣒⠒⠒⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡹⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠉⠙⠻⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢮⠻⣿⣿⣿⣿⣆⠀⠴⢶⣟⣿⣻⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠉⠻⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠉⠀⠀⠳⡘⢿⣿⣿⣿⣷⡛⠋⠍⠉⣾⣿⡟⢉⠘⡟⠉⠉⠉⠛⠻⢯⣕⡀⠀⠙⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠃⠀⠀⡀⢀⣽⣌⢿⣿⣿⣿⣧⠄⠈⡔⣻⣿⠀⠂⠸⣧⡀⠀⠀⠀⠀⠀⡨⢿⣷⣦⣀⠈⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣵⠟⠀⠀⠄⣠⣾⣿⣿⣿⣭⣝⡛⢿⣿⣶⣘⣸⣿⣿⣄⠁⢀⠈⠻⢷⣶⣶⠏⠀⠀⣈⠁⠀⠉⠳⢄⡙⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⢮⣾⠋⠀⡠⢡⣾⣿⣯⣭⣝⡻⢿⣿⣿⣷⣍⠻⣿⣿⡿⠟⣋⣷⣄⡀⠠⢀⠨⢋⠀⢀⢱⣿⡀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣰⣣⣿⠃⠀⠌⣴⣋⣿⣿⣿⣿⣿⣿⣷⣍⠻⣿⣿⡷⢛⠡⠀⠚⠛⠛⠛⠿⣶⣼⣷⣦⣀⠘⠘⢿⣷⣤⣤⣤⣤⠀⠀⣈⣻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣰⣳⣿⠃⢀⢊⡾⣽⣯⣤⣭⣙⡻⢿⣿⣿⣿⣿⠞⡩⣀⣁⠀⠀⠀⠀⠀⣖⣲⠐⠿⣿⣿⣿⣷⣌⡀⢙⠿⣿⠏⡠⠂⢀⠭⠿⠟⢿⣿⣿⣿⣆
⠀⠀⠀⢀⣿⣿⠏⠀⠂⣞⣙⣿⣿⣿⣿⣿⣿⣶⣝⣿⢛⠅⠈⠀⠣⠜⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠻⢿⣿⣿⣿⣦⡑⠤⠈⠀⢠⣶⠃⠀⠀⠀⠹⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣀⡈⡸⣩⠥⣿⣿⣿⣿⣿⣿⡿⢋⠄⣱⣻⠂⠀⠀⢀⡠⠤⠤⠤⠤⣀⠀⠀⠀⠀⡴⢢⡙⢿⣿⣿⡑⢀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⢀⢏⣶⣶⣶⣽⣃⢰⠒⡆⠻⣿⣿⣿⢋⠔⠁⣼⣣⠏⠀⠀⣼⢁⣀⣀⣀⣠⡴⣸⡇⠀⠀⠀⠈⠉⠀⠀⠻⠻⣿⣦⠑⠀⠀⠻⣿⣶⣦⡤⠄⠀⠘⣿⣿⡆
⠀⠀⢸⣨⣭⠭⢟⣿⡭⠘⡶⡡⠡⢛⠟⡡⠀⠀⠀⠈⠉⠀⠀⠀⠻⢷⣯⣭⣭⣷⠾⠛⠁⠀⠀⠀⢰⣾⣭⣽⣶⣶⡌⢻⣷⡌⠂⠐⠌⠩⠐⠀⣈⣉⣴⣾⣿⡇
⠀⠀⢸⣿⡿⠀⠀⣿⠾⢁⢠⣇⡷⢣⠊⠀⣠⢄⠀⠀⠀⠀⣠⡄⠀⠀⠀⠈⣀⠤⠤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠹⣿⣆⠱⡀⠀⡀⢀⣉⣉⣛⠿⣿⣷
⠀⠀⢸⣿⣛⣒⣲⣿⢰⠸⣧⣏⣩⠇⠀⠀⠓⠚⠀⠀⠀⣼⢣⡇⠀⠀⠀⠰⣧⣄⣀⣤⡇⠀⠀⣠⠖⠒⠊⠉⠓⠒⠦⣀⠀⢹⣿⣧⠀⠀⡚⠉⠉⠉⠉⠱⣼⣿
⠀⠀⢸⢸⣿⣿⣿⣿⡌⡷⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣧⡾⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⢸⣇⢤⣀⣀⣀⣀⣠⣖⣽⡆⠘⣿⣿⣦⣎⢀⣤⣶⠒⠢⡄⠘⡇
⠀⠀⠈⣶⣷⠉⠉⢹⡿⡑⠀⢀⡤⠶⠒⠒⠲⢤⡀⠀⠀⠀⣠⣤⣤⣤⡤⣄⠀⠀⣀⠀⠀⠀⠈⠻⢷⣾⣿⣭⣽⣷⠿⠋⢠⣖⣿⣿⣿⣿⣿⣿⣿⣾⡇⣽⢀⠇
⠀⠀⠀⢣⣿⠀⠀⡜⠀⠀⢰⣏⢀⣤⣤⣤⣶⣰⡇⠀⠀⢸⣿⣿⣿⣿⣿⣶⣝⠚⠧⣍⠑⠲⡄⠀⢀⡤⢄⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⢿⡿⠯⠭⠟⣁⡾
⠀⠀⠀⠘⣎⠀⡜⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣤⣙⠺⠤⣀⠈⠓⣋⡀⣴⡟⣱⣶⣾⣿⣿⣿⠟⡵⢋⡷⢶⣶⣾⢣⠇
⠀⠀⠀⠀⠹⡼⠡⣖⢳⠄⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⢀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⡛⠿⠿⠾⠷⢛⣵⣿⣿⣿⢏⢾⡿⢛⣿⣷⣮⣽⠏
⠀⠀⠀⠀⢰⢃⢁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢋⣤⣶⣿⣦⠃⠀⠈⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⡟⣾⣸⡿⢮⣝⢿⣿⠏
⠀⠀⠀⠀⢸⡆⠘⣶⡤⠀⠀⠠⣄⣼⢷⣴⠀⠀⡞⣩⡶⠦⡙⣇⢳⠘⠷⣿⣿⣹⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡿⡟⠀⢘⣿⣷⠏
⠀⠀⠀⠀⢸⡇⢀⣄⠀⠀⣀⠀⠉⡁⢠⣶⡄⠀⢰⣋⣵⣿⣿⡜⠈⣷⣤⣴⣻⢻⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡟⣸⡇⠸⠀⢼⡵⠃
⠀⠀⠀⠀⢸⡇⡸⢸⡇⢸⢿⡆⢸⣿⣿⣿⡇⠀⣾⣤⣿⣿⠦⣙⣄⠘⠿⣿⣿⣧⡻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⠟⣋⣼⣿⢷⣿⠇⢠⡧⠋
⠀⠀⠀⠀⢸⡇⡇⢸⡇⡘⠀⡇⣿⣏⠉⠈⣿⠀⡟⠿⣿⣿⣷⣮⣉⣹⡶⠺⢿⢿⣷⣼⣿⣿⣿⣿⣿⣿⣯⣥⣶⣾⣿⢟⣵⣿⠋⠀⡞
⠀⠀⠀⠀⢸⡇⡇⢸⠁⡇⠀⠷⠃⠈⠓⢤⣿⣼⠇⠀⠈⠋⠉⠛⣿⣿⣿⣿⣿⣧⣝⣿⠿⣿⣿⣿⣿⣿⣿⠿⢟⡫⣷⡿⠋⣵⣶⠞
⠀⠀⠀⠀⠸⡗⢷⡿⢠⠃⠀⠀⠀⠀⠀⠀⠈⠙⠣⢤⣀⡀⠀⢰⢻⣿⣿⢿⠙⠿⢿⣿⣿⣿⣷⣶⣾⡓⠶⠯⠗⣚⣥⣀⡤⠛⠁
⠀⠀⠀⠀⠀⠙⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠃⠬⠭⠼⠤⣀⡀⠤⠤⠬⠭⠟⠛⠋⠓⠒⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣀⠈⠀⠀⠦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⢀⠆⠀⡀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠈⠁⠀⠂⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠁⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⢰⠀⠀⠀⠆⠈⢿⠃⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣿⣏⡿⢵⣞⡣⢄⣈⣀⠓⠀⢄⡀⠐⢀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢀⠔⠂⠛⣆⠀⠀⢀⡴⢏⠻⣾⣿⢻⠟⡟⢿⣾⣧⣯⣽⣒⣬⢆⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⠻⠆⣵⠃⠀⠀⠀⠐⠀⠀⡼⢮⣝⣉⣷⡿⣦⣖⣠⣶⣝⡻⡿⣿⣿⣿⡾⠾⢳⣮⣯⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠆⡶⠃⠀⠀⠀⠀⠀⠀⠀⣋⡄⣠⠨⡬⣡⣴⣎⡁⣟⠁⢱⢧⢬⡿⣿⢥⣄⣀⣷⣾⣷⣾⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⣛⡶⣃⣴⠶⠏⠀⣿⠉⣿⣱⡿⠠⠾⢷⣄⢺⣿⣻⣿⣿⡿⢿⣿⣾⣿⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣾⡤⠀⠀⠀⠀⠀⠀⠸⡖⠷⠛⠁⢠⡄⠟⡇⠘⣯⠉⢠⣤⣦⣼⠇⣷⡜⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡳⡞⠂⢴⠇⢠⡇⢸⡇⠀⢀⡉⠃⠱⣦⣛⣁⠻⣿⣿⣿⣿⣶⣿⣦⣽⣿⣿⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⣿⠀⢾⠀⠈⣇⣸⡃⠀⢸⠃⢠⣾⢌⣽⣏⣹⠿⣷⣽⡛⣿⣅⣿⣿⣿⡿⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⡀⢰⠀⣠⡟⢿⠙⠀⣾⡄⠀⡉⢈⡟⠻⢿⡧⣦⡉⢻⣿⣿⣿⣿⣿⣿⣀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠠⠀⢿⡋⢠⣄⣀⡉⠙⠀⢘⠛⠓⣶⠂⠀⡻⠷⣾⡟⢿⣿⣿⣿⣿⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢠⣾⡃⠠⣧⠀⠸⣷⠂⠨⣷⣦⠻⠀⠿⣧⣄⣤⡿⠘⣿⣿⣿⣿⣿⣿⣿⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣾⡿⣅⢤⣶⣀⣼⣯⣿⣷⣶⣦⣸⣿⠘⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣅⣁⣝⠊⢦⡵⣽⢮⣋⠈⠙⠻⠀⢹⡇⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⡿⢿⣿⣷⣶⣿⣮⣣⣝⡳⣄⠀⠀⣸⣗⠀⣿⣿⣼⣿⣿⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠹⢻⡏⠀⠀⠀⢿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⢿⣯⠻⣧⠹⣿⡀⠨⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡾⠁⠀⠙⢧⣀⢀⢳⡀⣀⡀⣤⢀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠹⣿⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⢻⡆⢻⣇⢹⣷⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠫⡀⠀⠀⠀⠸⣿⠟⠛⣿⣧⡄⠳⠇⠀⠶⠒⠿⠛⠳⠀⠀⠀⠀⢹⡟⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣌⢻⣦⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡠⣴⠋⠀⠀⠈⠀⠀⠀⠀⠀⣀⣴⣹⣟⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣝⣿⣿⢿⣟⢿⣿⣿⣿⣿⣦⠙⣷⣌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠋⠀⠘⢄⠀⠀⠀⡀⠈⠀⣠⣾⠏⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣰⠏⠀⠀⢠⡏⡏⠙⢶⣬⣷⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠆⠀⣠⣾⠟⠁⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠶⣦⣀⣤⠋⠀⠀⠀⠀⡇⡇⠀⠈⠀⠈⢽⣿⠋⠿⣿⣿⣿⣿⣿⣿⡇⣻
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⢌⣉⣻⢛⣿⢷⣴⢶⡇⠀⢈⣿⣿⣤⣤⠀⠀⠸⠇⣷⢀⠀⡀⠀⣾⣿⠀⠀⠙⢷⣌⠛⡿⢻⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⢼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⢠⡀⢱⠆⣿⣾⣾⣿⣶⣿⣿⠀⠀⠀⠀⠙⣦⡀⠀⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠙⠛⠛⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣼⣿⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣿⡄⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣷⣶⡲⢤⣀⡈⠉⠀⠁⠀⠈⣹⣿⣿⣿⣿⣿⢸⡇⢸⣿⣿⡿⠋⣿⣿⠀⠀⠀⠀⠀⠀⠸⣷⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠉⠁⠀⠀⠉⠛⠻⣿⣿⣿⣿⣼⡇⠀⢹⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⡇⠀⠈⣻⣷⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣟⣧⠀⢠⠃⠻⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⢀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠱⡍⠒⠤⣀⠀⣄⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⡎⠀⡠⠔⠊⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠙⣃⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣐⠀⢂⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠉⠉⠉⠛⢷⡄⠀⣰⠟⠋⠉⠉⠉⠛⢷⡔⠊
⠀⠀⠀⠀⠀⣠⣶⣶⣤⣼⡏⠀⠀⠀⢀⣀⡀⠀⢿⣴⡏⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠙⠃⢀⠝⢻⡇⠀⠀⠀⠻⣾⠟⠀⣼⣿⡇⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠀⠛⠁
⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢻⣦⡀⠀⠀⠀⣀⡴⠛⠙⠻⢤⡀⠀⠀⠀⣀⣼⠯⠀⠳⣄⡀
⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠈⠛⠛⠛⠛⢫⡀⠀⠀⠀⣀⠛⠛⠛⠛⠋⠁⠀⠱⡀⠹⡝⢦⡀
⠀⠀⡔⠁⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠢⡀⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠹⡄⠙⣄
⠀⡞⠀⢰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢱⠀⠘⣆
⣸⠀⢀⡏⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⠤⠄⡐⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣇⠀⠸⡄
⡇⠀⣸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠠⠋⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡇⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠂⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠘⡇⠀⡇
⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢠⠇
⠘⡄⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⡜
⠀⠙⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⡼⠁
⠀⠀⠘⣆⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠟
⠀⠀⠀⠈⢦⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⠏
⠀⠀⠀⠀⠀⠙⢮⣓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞⠁
⠀⠀⠀⠀⠀⢠⠊⠙⣍⣿⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣞⣩⠉⢱⡄
⠀⠀⠀⠀⠀⠈⠓⠊⢹⠒⠒⢸⢷⡉⢉⠏⠉⠉⠉⠉⠉⠉⠹⡁⢁⠾⡇⠒⠺⡏⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⢿⣿⣿⣿⣿⣿⣿⡿⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣧⡙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢋⣼⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⡇⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠉⠉⠉⠉⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡄⢠⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣮⣥⣬⣵⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⡙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣝⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⢀⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠻⣿⣿⣦⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠹⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣅⢀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁⢉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣶⣿⣿⠟⠁
⠀⠀⣠⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣠⣶⣿⣿⣿⣅⢀⣼⣿⣿⠟⠁
⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠟⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠹⣿⣿⣿⣶⣶⣴⣶⣾⣿⣿⠟⠀⠀⣼⣿⡿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣧⣀⣰⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄
⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠛⠁⠀⢀⣾⣿⡿⠁⠀⠀⣠⣾⣿⡿⠋⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⠟⠁⣀⣤⣾⣿⣿⣏⠀⠀⣠⣾⣿⣿⣿⣷⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠁⣹⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⠛⠛⠋⠁⠀⣴⣿⣿⠟⠛⢿⣿⣷⣾⣿⡿⠋⠀⣿⣿⡇⠀⠀⣿⣿⣿⠉⠁⠀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿⣿⡿⠋⠀⠀⢀⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⢻⣿⣿
⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣿⣿⡟⠀⠀⢀⣠⣾⣿⡟⠀⠀⠀⣿⣿⣿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⢀⣴⣾⣿⣿⠿⠋⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣼⣿⣿
⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣅⠀⠀⠀⢀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⣼⣿⣿⠟⠋⠉⠙⠻⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⣀⣀⣀⣤⣶⣿⣿⠟⠁⠀⠀⠀⠀⢻⣿⣿⣶⣤⣤⣤⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡪⠕⠚⠉⠉⠉⠒⠮⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⣿⡟⢶⣄⠀⠱⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠁⠀⠀⢀⣀⣠⣿⠁⠀⢸⣿⠀⠘⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠙⠿⠿⠟⠀⠀⣿⡇⠀⠀⡧⠧⢀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠉⠺⡀⠀⠀⢶⡒⠃⠸⢿⣿⣿⠃⡠⠊⢀⣠⣤⣤⣤⣄⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢾⡿⠀⠀⠀⠀⢀⠀⠀⣴⠛⠉⠀⠀⠈⠉⠳⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠠⠄⠀⢀⣀⠠⠔⣎⡢⢄⣀⠀⠀⠀⣀⠄⠀⢸⠃⢀⠀⠀⠤⠄⠤⢀⢹⡄⠈⠢
⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠑⠒⠈⠭⠁⠒⠆⣴⣿⡆⢉⣁⠀⠀⠀⣉⠀⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⡠⠀⠀⠠⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⢿⣿⡇⠀⠁⢀⣀⠀⠉⠀⢿⣿⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠑⠄⠀⠜⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡀⠀⠘⢶⣶⡾⠃⠀⡈⠙⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⣤⣤⡤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⠀⠀⢀⠀⡀⠄⢊⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠑⠒⠄⢁⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠠⠂⠋⠀⠙⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⣸⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠊⠁⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⢂⣀⣀⣀⣀⣀⠀⢀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣥⣤⡏⣻
⠀⠀⠀⠀⠀⠀⠒⠁⠀⣼⣿⣿⣿⣿⣿⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⣩⠀⢀⣥⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⡟⠁⠀⠘⣿⣿⡄
⠀⠀⠸⣟⣻⣿⠀⠀⠀⠀⢿⣿⣿⠀⠻⠃
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⡀⠤⢤⡄
⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⣾⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠈⢿⣿⣇⠀⠀⢰⣿⣿⣿⣿⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠘⣿⣿⡄⢠⣿⣿⣿⣿⣿⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢈⣷⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠻
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠇⠀⡀⢀⠁⣬⠀⠀⠀⠀⠀⠐⠒⠚⠓⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠹⠻⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⢿⣿⣧⠀⠒⠐⠛⠀⠀⠀⠀⠀⠙⠁⠀⠈⠉⠀⣠⣤⣄
⠀⠀⠀⣾⣿⣿⣿⣿⡿⠁⠀⠹⣿⣧⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⢠⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⠘⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⢠⡄
⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⢠⣾⣿⠟⣿⣿⣿⣿⣿⣿⠁⠀⠀⠛⠃
⠀⢰⣿⣿⣛⣿⡏⠀⠀⢠⣿⡿⡎⠂⠈⠻⣿⡛⠉⠙⠓⢤⡀⢠⣤⣾⣧
⠀⠘⣿⣿⡶⢿⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠿⢷⡄⠀⠀⠀⠁⠸⣧⠜⢳⡆⣀⠀⠃
⠀⠀⠙⠙⢣⡈⠀⠀⠀⢸⣿⣿⣟⠀⠀⠄⠀⠀⠁⢠⣶⣿⠀⠀⠹⡷⠾⣿⠟
⠀⠀⠰⣾⣿⣫⡀⠀⠀⡀⢿⣿⣿⣄⠀⠀⠀⠀⠀⡘⠋⠁⠀⠀⣄⡑⠄
⠀⠀⠀⠙⠛⢿⣷⡀⠀⠙⠌⠻⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢣⠈⢷⣄⡀⠀⠀⠈⠙⠛⠛⠛⠂⠀⠀⠀⠀⠀⠛⠻⠟⣠⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣄⣀⠀⠀⠀⠀⣀⣀⣤⣶⣄⠀⠀⠀⠀⡠
⠀⠄⠀⠀⠀⠀⠀⠘⠙⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠐⢠⣤⣌⡧⡄
⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡄⠿⠖
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠰⡄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⣿⣿⣿⣿⣿⠏⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢷⣤⢶⠶⠒⢚⣩⣉⡫⠍⠭⣋⣩⣙⠒⠲⢶⢦⣴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣶⠿⠷⣶⣄⢬⣴⣝⢂⢚⣵⣬⢄⣴⡶⠿⢷⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠕⠉⢈⡀⠈⢧⢽⢸⢯⢿⢸⢥⠏⠀⣈⠈⠑⢽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢶⢋⣉⡑⡜⡾⣿⣿⣿⡿⡞⡔⣉⣉⢳⠆⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⡀⠀⢠⡸⣜⣛⣡⠇⢹⣿⣿⣿⣿⠁⢧⣙⣛⡼⣠⠀⠀⡈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡟⣡⠀⠘⠛⠢⠛⠁⢠⢠⡟⠀⠘⣧⢠⠀⠙⠣⠚⠛⠀⢠⡙⡿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡳⡝⠽⡇⡶⠀⢀⡀⢰⣦⣾⣿⣇⠀⢀⣿⣿⣦⣶⠀⣀⠀⠰⡆⡿⠝⡵⣻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⡘⢧⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣤⠟⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣘⠛⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠛⢛⣠⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡄⠀⠀⠀⢠⣿⣿⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⣿⣿⠀⠀⠀⣼⣿⠟⠁⠀⠸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠄⠀⣼⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠀⠀⠀⠠⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠳⠀⡰⠾⠇⠀⠀⢠⣿⠿⠿⣢⢤⠻⠛⣴⠒⣇⡔⠁⡀
⠀⠀⠀⠀⠀⠀⡀⡀⠄⢲⢀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠸⠁⡀⠀⠀⠀⠀⣀⢿⡸⠗⠈⠁
⠀⠀⠀⠀⠀⠘⠚⢒⠀⠿⠟⠓⠂⠀⠀⠀⠀⠀⠁⠀⡁⢀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡄⠀⠀⠀⠁⠀⠀⠠⡖⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀⠀⠠⣀⣀⡀⠀⠀⠀⠀⠊⠁⠠⣿⣿⣿⣿⣿⠒⠂⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⣤⣼⣿⣿⣷⣤⡄⠀⢀⣤⡀⠀⠙⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⣷⣶⣶⣿⡿⠋⠉⠉⠛⣿⣿⣤⠿⠻⣿⣿⣿⣿⠟⠋⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡄⠀⠰⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠤⢝⣾⣴⠄⠀⠀⠄⠀⠀⢡⡀⠀⠀⠀⡐⠄
⠀⠀⠀⠀⠀⢀⣀⣵⠀⠀⣀⠘⢀⣽⠀⠢⠀⠀⠀⠀⠀⠉⢠⡄⡘⠁⠀⠀⠀⠠
⠀⠀⠀⠐⠂⠐⠘⠉⠧⡀⠹⡃⢬⠁⠀⠠⠐⠘⠀⠀⠀⠀⠀⠃⠃⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠠⢠⠐⠀⠀⠂⠤⠀⠀⠆⠀⠀⠀⠄⠀⠠⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⢄⠀⠀⠈⠀⠀⠠⡀⢀⠀⠂⠀⠀⠀⠀⠀⢐
⣀⠀⣀⠀⠀⠀⠓⣄⡀⠠⠘⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠂⠁⠈⣠⠀⠀⢘⠀⠀⡂⠀⡈⠀⠀⠀⠃⠀⠀⠀⠄⠀⠀⠡⠀⠀⠀⠀⠀⠱⢀⣀
⠀⠀⠀⠀⠈⠃⠄⠊⠠⠀⠈⠖⠀⢃⠄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠱⢛⢄⠂⡤⠀⢠⠤⠀⠠⠄⠀⠀⠀⠀⠙⠄⠀⠀⠈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡀⢀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⡄⢳⡛⣿⣴⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⣘⣀⣀⠀⣀⠀⠀⠀⠀⡀⠀⠈⢉⣠⣤⢴⠶⣦⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠹⣴⡎⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠄⠉⠉⠈⠉⠈⢧⠀⠀⠀⠁⠐⠒⢸⣯⣾⣦⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠁⠀⡜⢣⠆⠀⠀⠈⠀⢀⠂⠁⠀⠀⠀⠈⠆⠀⣛⠙⡉⠉⠩⣉⣈⠁⠀⠀⠀⠀⢀⣾⣡⣍⣁⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡒⠀⢒⠰⠀⢀⣠⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⢁⠈⠉⡁⠀⠀⠀⡠⠉⠟⠉⠀⢹⠁⠸⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠊⠀⠀⠃⠠⠾⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠈⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⠀⠀⠄⠀⠀⠀⠀⡇⠀⠃⢀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠀⡠⠁⠀⡀⣏⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠂⠀⠀⠐⠤⠬⢀⠀⠀⣰⡷⠃

⠀⠰⣄
⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠑⢄⠀⣠⣾⣦
⠀⢠⡀⠀⠀⠀⢱⣿⡋⠙⡆
⠀⠀⠙⢦⡀⠀⢸⠇⠉⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠶⠒⠀⠀⠀⠀⠀⠀⣀⢀⡈⠀⠀⢀
⠀⠀⠀⠀⠙⠢⣸⡀⣠⣿⠇⠉⠒⠠⢄⡀⠀⠀⠀⠀⣠⣤⣴⠶⠶⠾⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠚⠁⠀⣀⡈⠉⠈⠀⣠⠀⠉⠀⠠⠦⠀⠉⠓⠂
⠀⢀⠀⠀⠀⠀⣨⣿⣿⣟⡀⠀⠀⠀⢰⡅⠀⠀⠈⠀⠉⠁⠀⠀⠛⠛⡀⠀⠀⠂⠈⠀⠄⠀⢤⠄⠀⠀⠀⠀⠘⠃⢉⡀⠀⠀⠀⣤⣤⡔⠃⠀⠀⠁⠀⠐
⠀⠈⠳⢤⣠⣾⣿⣿⠋⠀⠈⠉⠒⠂⢰⡆⠠⠤⠀⠀⠰⠆⠀⠀⢠⣤⡄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠛⠀⠀⠀⠉⠁⠠⠴⠖
⠀⠀⠀⣴⣿⡟⠣⢨⣆⣀⠀⠀⠀⠀⠰⠗⢻⡆⠀⠀⠀⡄⠄⠀⠛⠋⠀⠀⣴⣾⠄
⠀⠀⢰⣿⠏⠀⣰⣾⣿⣿⣿⣶⡒⠒⢚⡷⠛⠉⠁⢰⣿⡇⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠰⢾⣿⣀⢰⡿⠁⢸⡄⠈⢻⣧⠀⠀⠀⣀⠠⠄⠂⠉
⠀⠀⠀⢻⡈⠉⠻⣗⠚⣷⠒⢺⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠂
⠀⠀⠀⠀⠙⠲⢤⣀⣀⣽⡴⠋⠀⠀⢀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠸⠛⠁⣴⠀⢀
⠀⢠⣄⣀⣀⣀⣀⣀⣠⠬⡧⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣄⠠⡆⠀⢠⣆⢀⣤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⠀⠀⠀⠐⠛⠀⠀⠛⠁
⠀⠀⠀⠀⠀⣼⣿⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⢻⣷⠰⠆⠘⠋⠀⠀⣀⠀⣀⠀⠛⠁⠜⢛⠀⠤⢈⠁⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠈⢿⣿⠛⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢸⡿⠀⠀⠀⢀⠀⠘⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠁⠲⣄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠐⠒⠒⠶⠿⡷⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣷⡄⠀⠀⠂⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠶⠆⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⡯⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡞⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⢀⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠙⣞⠙⠿⣧⣀⡀⠀⠀⠀⠀⢀⡴⠖⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣻⣟⣗⢀⠀⠀⠀⠀⠀⢰⢤⠀⠀⠀⠀⠀⠀⢠⠆⣸⠀⠀⠀⠀⢀⠀⢤⠜⠿⣾⣟⣻⡁⠁⠀⠀⠀⠹⢫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⣿⣿⣿⣿⣾⣟⣷⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣀⠜⠚⠃⠀⠀⠠⠼⢿⡧⡄⠀⠀⠀⣸⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉⢾⣿⣿⣿⣿⣿⠣⡟⢻⡟⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⢺⠁⠙⢃⠀⠀⠸⠇⢠⢸⡗⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⢏⣈⣩⣀⣞⢀⣷⡇⣿⣿⠃⠠⠀⠀⠀⠀⠰⡆⠀⢸⡇⠀⠙⡀⠀⠀⣀⠀⡁⠀⣸⡀⠷⠀⣴⠛⣢⡰
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠀⣰⣻⣴⢯⡮⠬⡌⡇⣸⣏⣿⢷⡿⠐⣴⠂⠀⠀⠀⢀⡦⠈⠄⢻⡟⠠⣗⠀⠀⠙⡌⠁⡔⣸⣥⣷⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⣤⣭⣾⣟⣩⠭⢲⣵⣷⡿⢾⣫⠿⠁⣰⡯⣧⠀⠀⠀⡾⣷⡄⠐⣌⣡⠀⠉⣇⠀⠉⠑⠀⠻⡍⢝⣇⡀⠀⢈⣤⡴⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢹⣿⣿⣿⣿⡀⣽⣿⠟⢛⠀⣸⠻⠃⠰⣿⠃⠸⠄⠀⢀⠏⣽⠁⠀⡻⣿⣿⣆⠈⠛⡛⠋⣡⣿⠛⢸⡟⠛⠷⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏⠘⠁⠀⠸⠘⠏⠀⡀⣦⠛⠀⢠⣇⠀⢘⠀⠛⡄⠀⠃⠀⠀⠛⠿⡟⠻⢷⠿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠃⠀⠀⣭⣤⡿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠉⠛⠙⠋⠙⠃⠀⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⢣⠀⡄⡄⠀⡄⠀⢀⠀⠀⠀⣆⠠⡀⢄⠱⣄⠀⠐⢄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠃⠂⠜⢠⠃⡄⠀⠀⡌⠀⢀⡜⠘⡄⠑⢤⡀⠈⠣⠀⠈⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣰⣥⣀⣼⣷⠷⠋⠀⠀⠘⢎⢦⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡕⣄⠀⠀⠀⠀⠘⣸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠆⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢣⣴⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠀⠀⡄⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⣆⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⡀⢄⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⢀⣾⣆⣴⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠾⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣠⡾⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠖⣠⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣄⠳⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠩⠤⠖⠚⠉⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⠠⣍⣙⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⠁⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⢄⣀⡀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠸⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⡈⠄⠀⠀⠀⠀⣠⣄⡀⠀⠑⡀⠴⢶⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠉⠀⢀⡉⠈⠉⡀⠀⡐⠀⠀⠛⠋⢹⣇⡀⠀⠀⠀⠀⢠⠂⡀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⢳⣄⣀⠀⠩⢀⢀⡀⢀⣸⣿⠁⠀⠀⡀⣐⠂⠠⠀⡀⠈⠁⠂
⠀⠀⠀⡀⠁⠠⣠⣸⡀⠐⠀⠀⠀⠄⠀⠂⠙⢷⠀⠀⡁⠀⠀⠌⠁⠀⠤⡀⠀⠰⠀⠀⠀⠀⠅
⠀⠀⠀⠀⢀⢰⡙⣿⣷⡈⠄⠀⠀⠀⠁⢠⡷⢨⣦⠂⠉⠀⠀⠀⠀⠀⠀⢘⢡⠃⠈⡀⠀⠀⠃
⠀⠀⠀⠀⠉⣤⠛⣌⢢⡅⢀⣀⣀⠠⡾⢋⡄⠻⣇⣠⡔⢁⠀⠀⠀⠄⠀⢀⠃⣘⠉⣇⠀⠀⠐
⠀⠀⠀⢰⡸⣿⣜⣿⣷⡙⣸⣿⡿⠛⠚⠂⠉⠢⠚⠩⡤⢾⡇⠄⡠⠀⠀⠘⠴⠳⠊⢹⣿⢇⠀⠀⠄
⠀⠀⠀⠀⢷⣝⡛⢛⣋⣴⡙⠏⠀⠀⠀⠀⠀⠁⠀⠠⠤⠞⠰⠂⠀⠀⠀⠁⠀⠀⣠⡾⠃⠡⢁⠀⢀
⠀⠀⠀⠀⠈⠙⠛⠋⣉⡙⡁⠄⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⣎⢛⠀⠀⠀⡠⠂⠠⣤
⠀⠀⠀⠀⠀⣶⠿⠀⠛⠾⢃⣄⠀⠀⠀⠠⠀⠀⠀⠃⢠⡀⠀⢀⠀⠈⠀⠀⠐⣛⡇⡆⢼⣥⠒⡀⢀⣿⠂
⠀⠀⠀⠀⠀⠉⠀⠐⠀⢻⠟⠋⠁⠄⠀⠀⡐⠀⢀⠔⣀⠀⢄⡀⠀⠀⠀⠀⣰⠟⣡⠀⢳⡙⣧⠄⠋
⠀⠀⠀⠀⠈⣶⡦⣀⠠⠀⠀⢠⣄⠀⠁⠖⠈⠀⢰⡔⠀⡁⠢⡉⠙⠆⠘⠚⠒⠊⠀⠀⣲⣛⢰⠀⠀⠄
⠀⠀⠀⠀⠀⠈⡚⡡⠪⡄⠈⠀⠁⠀⢠⣤⠄⠀⠀⠀⠀⢋⠀⠐⠐⠂⠄⡄⠠⠂⡨⠂⠀⠹⠘⠀⠠
⠀⠀⠀⠀⢒⠠⢀⠀⠀⢢⣷⣶⡶⣆⢸⡇⠀⠀⠰⠆⠀⠀⠀⡂⠀⠀⢳⠀⠲⠟⣠⣀⠐⣱⠇
⠀⠀⠀⠀⠁⢘⢘⠀⠀⢘⣿⠋⠙⣿⡘⠿⢆⠀⠀⠀⢀⣤⡄⢸⣶⣄⠁⠀⠀⠂⠩⣵⡾⠁
⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠑⠛⠁⢿⠿⣦⣐⠠⠿⠿⠟⠋⠰⢿⠻⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠂⠀⠀⠀⠀⠛⠋⠉⠀⠁⠀⠀⠠⠀⢁⠀⠉⠙⠛⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠠⠄⠤⠂
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⡆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠁⠉⠉⣀⣀⣠⡄⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣴⠇⠀⠀⢀⠀⢠⠀⠠⣽⣻⣯⡇⠀⠀⠀⠀⠀⠣
⠀⠀⣀⠐⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡂⠀⠀⠀⠀⠾⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⠀⡇⠀⠀⢠⠴⠡⣾⢶
⠀⠀⡇⠀⠀⢀⡀⠀⣀⡀⢈⠀⡀⠀⠀⡀⠀⠒⠉⠋⠁⠈⠁⠀⠑⠀⠀⠀⠀⠸⠣⠀⠇⠀⠘⠂⠠⠀⠀⠀⡿⠃⣁⣿⠀⠀⠀⠀⢠⠁⠀⠀⠀⠨⢲⣿⣯
⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⣋⡠⢐⠀⠤⠤⣤⢄⡄⠀⠀⠀⠀⡀⠤⠤⢠⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠠⠄⢄⠀⠀⠉⠀⠀⠀⠴⠶⢾⡉⠋
⠀⠀⡗⠈⠋⠀⢸⠀⠀⠀⠀⠀⠀⠁⠁⢺⡄⣯⡂⠛⠛⠛⠛⠃⢠⠀⢀⠨⠩⠆⡇⡇⠀⠀⠀⠀⡀⢸⢏⢗⣨⠀⠀⠀⠀⠠⠀⠠⡀⠀⠀⠀⠀⢀⡘⢢⣀
⠀⠀⠁⠀⢀⡀⠸⣄⡤⠀⢠⣐⠒⠥⠠⢙⣶⠀⠀⠀⠀⠀⠀⣀⡀⠆⠚⠐⠂⠀⠄⣀⠀⠀⠀⢠⡀⠈⠉⠉⠉⠁⠀⠴⠛⠀⠀⡼⠁⠀⠀⠀⠀⠀⣴⠁⠒
⠀⢀⣀⠀⠄⠀⠏⠉⠐⠂⠀⣀⠀⠬⠀⠀⠁⢀⣀⣠⡤⠤⠶⠂⠀⠀⢨⣀⡀⡀⠀⠈⠉⠙⠋⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⢠⡇
⠉⠫⠏⠒⠀⠈⠀⠀⡀⢀⣠⣤⠤⠖⠆⠉⢈⡉⠭⠶⡄⠤⢂⡠⠔⣂⠀⠐⠏⠉⠁⠀⠐⠀⠀⠀⢠⠀⡤⠂⠀⠀⠀⠀⠀⠠⢐⠰⠀⠀⠐⠒⠒⠘⠓⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄⠀⢀⠧⠀⠀⢀⣀⠴⠀⠀⢐⠢⠭⠀⢉⠠⠔⠂⣠⠄⠀⠀⠒⠈⠀⠈⠁⠀⠉⠀⠈⠖⡒⡐⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠒⠂⠀⠀⠐⠂⠀⠀⠐⠂⠠⠌⢁⡀⠒⠉⠤⠊⠑⠒⠶⠴⢉⡂⡀⢠⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⡀⠐⠉⣀⠒⠀⢀⠤⠉⣀⠒⠀⠠⠈⡀⢂⡐⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠔⠁⣀⠊⢀⠖⠠⠄⢀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠁⠀⠀⢠⡚⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠈⠑⠢⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣘⣃⠁⡋⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⣠⣴⠿⠿⠷⢶⣤⣀⣀⣀⣀⣠⣴⠶⠿⠿⢶⣦⡄⠑⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢨⣿⠏⠁⠀⠀⠀⠀⠈⢻⣆⢀⣾⠋⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠋⢻⣇⠀⠀⠀⢀⡐⣦⡀⢀⣿⣿⣇⠀⣠⢲⣦⠀⠀⠀⢀⣿⢧⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡞⠁⡰⠋⠻⣦⣀⠀⠘⠿⢿⣣⠞⠋⠉⠛⢦⣙⡿⠟⠀⣀⣠⡾⠯⡀⠙⣤⡀
⠀⠀⠀⠀⠀⠠⣤⢽⡢⠷⣈⣙⠒⠶⠤⣀⡀⠈⠛⠛⠿⠟⠛⣇⠀⠀⠀⠀⢀⡟⠻⠿⠟⠛⠋⠀⠀⢑⣀⡸⠿⢒⣦⡆⡀
⠀⠀⠀⠀⠀⣠⣹⡀⢈⣉⣑⠂⠭⣉⠒⠢⢍⡑⠦⣀⠀⠀⠀⠩⡑⢄⢀⠔⡩⠀⠀⠀⠀⢀⡠⢔⣊⠩⠄⢒⣊⡭⠴⣫⡷⣶
⠀⠀⠀⠀⣀⠿⠜⠋⠁⠀⠀⢱⠀⠀⠈⠑⠠⣌⠳⢌⠳⢄⠀⠀⠈⠒⠓⠈⠀⠀⢀⠔⣪⠵⢚⣩⠤⠒⢈⣡⢼⠀⠀⠀⠉⠛⠤⣀
⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣴⣉⡉⠓⠦⢄⣀⠀⠉⠢⢵⣄⠳⣄⠀⠀⠀⠀⣠⢖⠵⢊⡠⠚⢉⣠⠴⠚⣉⡤⠞⡳⠤⣀⠀⠀⠀⠀⢱
⠀⠀⠀⡧⠤⠔⠚⠉⠀⠀⠈⡆⠉⠛⠲⠤⣌⣉⠓⠦⢤⣙⢢⡈⢧⠀⢀⡶⢛⡡⢐⣍⠙⢞⣉⡤⠖⠋⠁⠀⢸⡀⠀⠀⠉⠓⠢⢄⣀
⠀⠀⠀⡇⠀⠀⠀⣀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢬⡉⡿⠯⠭⠭⠴⡿⢚⡭⠼⢷⠚⠃⠀⠀⠀⠀⠀⠀⠙⢢⣄⣀⠀⠀⠀⢨
⠀⠀⠀⢹⠚⠉⠉⠀⢈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣀⣀⣀⣀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⣏⠀⠀⠉⠑⢲⡎
⠀⠀⠀⠀⠣⣤⡤⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢤⣀⣀⣠⣼⠀⡇⠀⢠⣄⠀⠀⢻⣿⣿⡀⠸⣿⣀⣽⢢⣤⣀⠞
⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⢸⠀⣿⠀⣿⣿⣧⡀⠘⣿⣾⠟⠀⠙⠛⠋⠘⡇⣿
⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⢸⠀⢿⠐⠻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⢹⠸⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣋⡥⠞
⠀⠀⠀⠀⠀⠀⠈⠉⠓⢲⡬⢭⣛⣲⠦⢤⣀⣀⠀⠀⠀⠀⡇⢸⣈⠉⠉⠉⢹⡆⢸⡄⠀⠀⠀⣀⣠⠴⢖⣋⢭⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢬⡉⠓⠲⠭⣭⣛⡒⠶⠇⢸⠉⠉⠉⠉⢹⡇⢸⡧⠶⢚⣫⠭⢒⡿⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢳⠾⣷⣦⣤⣀⣀⠉⠙⠒⠾⣄⡀⠀⠀⣠⣧⠴⢒⣋⣭⣴⣾⡵⢚⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⢻⣋⢍⡻⣿⠙⠛⢻⠛⠛⠛⠛⠛⠛⠛⢻⠟⠛⢍⣿⢋⠍⣹⡟⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⡼⠋⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠹⢄⣀⡤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠉⠉⠡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⡁⠀⠀⠀⠀⠀⠘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢊⠥⠊⠀⠀⠀⠀⠀⠀⠈⠢⣉⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⡡⠒⠁⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠑⠤⡓⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡮⣉⠔⠉⠀⠀⠀⠀⢀⡬⠂⠀⠀⠸⣀⠀⠀⠀⠀⠀⠈⠒⢍⠹⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⡠⠊⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡘⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡀⠄⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡀⠀⠂⢔⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡴⠋⢇⠉⠙⢻⠉⡌⠓⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠜⠀⠀⢸⢀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠾⠏⠀⠀⠘⣌⠒⠋⣀⣬⡀
⠀⠀⠀⣀⡴⠋⠉⠉⠀⠀⢱⡔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠚⠒⠒⠒⠒⠒⠂⢀⣉⣉⣰⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢲⣆⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣶⣿⡻⣿⡿⡜⢹⣿⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣧⠹⣿⣍⠲⢦⠅⢠⣭⣁⡚⢿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄⠱⡽⣷⣌⠐⣼⡊⢙⡛⣓⣀⡙⠛⣿⡶⣶⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣄⠀⠀⣷⢳⡉⠋⣿⡦⣙⣿⣿⣿⣽⡻⢟⣂⡚⠛⣛⣛⡛⠲⣝⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠑⠀⢸⡠⣿⢦⣌⠁⠻⠿⣿⣿⣿⣿⣷⡮⣄⣘⢿⣶⣿⣿⣶⣿⣷⣍⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠉⠀⠀⢁⣥⣶⠿⣿⣷⠦⣤⠌⢙⠻⣿⣿⣿⣿⣾⣿⣿⣿⣶⡼⢿⣿⠷⣆⣣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠹⡟⣧⢸⣷⡐⢵⣂⠹⠺⣵⣦⣈⠙⠛⠿⣿⣶⣍⠛⠉⢿⠉⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⡾⣿⣦⠀⠀⣰⡇⡟⢸⣿⣿⠷⣮⣽⣓⣒⠬⢭⣛⠲⠶⣶⣭⣻⢿⣀⠰⠧⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣛⣫⣾⣿⣿⣿⣝⠷⣬⠻⠿⢟⣛⣛⣿⣿⣷⢦⣌⣻⣿⣿⣿⣿⡤⣤⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⢹⣿⣿⣿⣿⣷⣶⣾⣿⣷⣾⠿⣿⣿⣿⣛⠛⠛⠻⡿⣿⣷⡀⠀⠑⠢⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⢶⣶⣶⠎⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡛⠷⣷⣦⡤⣬⡙⢷⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣬⣙⣻⣿⣯⣿⣯⣙⣛⡶⠿⠶⣾⢍⠻⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠫⠽⠟⠿⠿⠿⠿⡛⠛⠻⢯⣀⠀⠀⠀⠈⠀⠉⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢠⣶⢚⣉⣩⣭⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣖⠛⢉⣭⣽⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠋⠛⠳⢦⣄⠈⡛⢷⣝⢿⣆⠀⠀⠀⠀⠀⣻⠹⠿⣧⣄⡀⢈⠻⣮⡻⣿⡄
⠀⠀⠀⠀⠀⠀⣆⣀⠀⠀⢠⡏⠛⣿⣎⣿⡆⢸⠀⠀⠀⠀⢠⢁⡀⠀⠀⣿⣉⡳⣷⡜⣿⡀⣷
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢈⡿⢟⣻⣿⣼⡷⣼⠀⠀⠀⠀⠀⠹⠂⠀⠀⣯⡽⣟⣿⣷⣿⡇⡏
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠩⡞⢹⣿⣷⠃⠀⠀⠀⠰⣀⣀⣠⣤⣸⣏⠈⢵⠋⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠳⢬⠿⠃⠀⠀⠀⠀⠀⡷⣾⣽⣿⣷⣿⡗⠘⢦⡽⠟⠃
⠀⠀⠀⠀⠀⠐⠀⠀⠴⠒⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡞⠁⠘⢘⣛⣩⠞⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣴⣒⡋⠛⠻⣿⣛⢿⣦⡀⠀⠀⠀⠀⢸⣿⣴⣒⡋⠛⠻⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⢹⣷⣔⣮⢻⣧⠉⣧⠀⠀⠀⠀⡘⠀⠀⠈⠙⣷⣤⣦⡹⣷⡙⢳
⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⢺⣿⣿⣿⣧⣿⡇⣽⠀⠀⠀⠀⠷⡖⠀⠀⢸⣿⣿⣿⣷⣻⣇⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠋⢖⣾⣿⣿⣧⠇⠀⠀⠀⠄⠀⠀⠀⠀⡼⠋⠵⣺⣿⣿⣏⡏
⠀⠀⠀⠀⠀⠀⢱⢄⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠀⠀⠀⠀⢱⣤⣠⠀⠀⠀⠀⠈⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠴⠂⠀⠈⢫⠁⠀⠀⠀⠀⠀⢨⠀⠀⣀⣀⡠⠂⠀⠀⠹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠘⠀⠒⠂⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠲⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠐⠀⠀⠀⠀⠀⣆⠓⠀⠀⠀⢀⣠⠴⢊⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠙⠲⠤⣀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠂⢸⠂⠂⠴⠊⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡆⠀⠀⠀⠈⠃⣢⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣠⣔⡲⠒⠀⠸⠀⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⣠⣶⡿⠟⠛⠛⠛⠛⠿⣶⣤⠠⠤⠤⢀⣴⡾⠟⠛⠋⠛⠛⠿⣿⣮⡀⣩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣼⣿⠃⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢻⣧⣿⠇⠀⠀⡠⣤⣄⠀⠀⠀⠀⠀⢹⣿⣴⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠈⢉⡿⢻⣿⠀⠀⠀⠀⠀⢸⣷⣿⣿⠀⠀⣸⣿⣿⡀⠀⠸⣷⣾⣿⠆⠀⠀⠀⠀⢸⣿⡛⠉⠁⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣸⣿⣆⠀⠀⠀⠀⠀⠙⠛⠁⠀⢠⣿⣿⣿⣧⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣾⡟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⢀⡴⠁⠘⢿⣷⣄⡀⠀⠀⠀⢀⣀⡴⠛⠉⠀⠈⠙⠳⣤⣀⠀⠀⠀⢀⣀⣴⣿⠟⢆⠀⠈⢷⣤⡀
⠀⠀⠀⠀⠀⣠⠖⢉⡞⠁⢀⠎⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⢻⡀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⢣⡀⠈⢳⡉⠲⣄
⠀⠀⠀⢀⡼⠃⢀⡞⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢦⡀⠀⠀⣠⠞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢳⡀⠈⢦⡀
⠀⠀⢠⠞⠀⢀⡾⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡙⠢⢞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢷⡀⠀⠳⡄
⠀⢠⠏⠀⠀⣼⠁⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⣧⠀⠀⠹⡄
⢀⡏⠀⠀⢸⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⢄⠀⠀⠀⠀⢀⠔⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡆⠀⠀⢹
⣸⠀⠀⢀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣉⣺⠉⠐⣆⡀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢿
⡏⠀⠀⢸⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢉⠽⠤⠼⡉⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡆
⡇⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡤⠋⠀⠀⠀⠀⠉⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇
⢇⠀⠀⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀
⠀⢧⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⡼
⠀⠘⣆⣷⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⣰⠃
⠀⠀⠘⢿⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣸⡷⠃
⠀⠀⠀⠈⢷⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡟⠁
⠀⠀⠀⠀⠈⢷⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠻⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢟⠾⢽⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣞⣭⠶⢛⠝⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣸⢶⠟⠋⠻⢟⣿⠛⡿⠿⣷⡶⠶⠶⣶⠶⠶⠶⠶⠶⠶⣾⡿⠿⡟⢻⣟⠟⠋⠛⢶⢮⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣼⠊⠉⠙⢢⡟⣏⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢈⡟⣧⠚⠉⠉⢻⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣋⣸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⣤⣦⣤⣄⣀⠘⣿⣿⣿⠻⣿⣷⣄⠀⠀⠀⢀⣀⣀⣤⡀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠩⠛⢛⣻⣟⡈⢿⣿⣿⣿⣿⣿⣇⠰⠾⠿⡟⠛⠉⢀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⢀⡔⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣶⡄⠀⠐⠀⢠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣦⣤⣄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⡄⠂⠀⠀⣼⣿⠿⠛⢿⣿⣤⣏⣿⣿⣿⣿⣿⡇⠀⠸⣦⣾⡇⣿⣿⣿⣷⣦⣤⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠈⠋⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠐⣿⡿⢿⣿⣿⣿⣿⠿⣿⣿⠟⠉⠀⠀⠈⠉⣩⣽⣿⣿⣿⣿⣿⠛⠛⠉⠁⠠⣿⡿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⠿⠛⣿⣿⣿⣿⣿⠿⣿⣧⣴⡇
⠀⣿⣇⢸⣿⣿⡿⠁⠰⠛⠿⠗⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⠁⣿⣇⣿⣿⣿⣿⣿⡟⠁⠀⠘⢿⣇⠀⠿⠁⠀⠉⢡⣴⣿⡿⠿⠃
⠀⠈⠻⣿⣍⡁⠀⠀⢀⣼⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣿⡟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠈⢿⣷⠀⠀⢀⣩⡇⢀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣼⣿⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⣴⣿⣿⣿⡄
⠀⠀⠀⣄⠀⠉⠀⣰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣶⣤⡞⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⣿⣿⣿⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢻⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⡿⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡇⢻⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠈⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⢸⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⢹⣿⣿⣿⡁
⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⡅⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣧
⠀⠀⠀⣠⣿⡿⠇⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠸⣿⣿⣿
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⣴⣾⣿⣿⣿⠇⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠀⠀⠀⢀⣤⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⢰⣿⠟⠉⠹⡟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠿⠄⠀⢰⠏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⣀⡀⠀⠀⡜⠀⡠⡭⢛⠄⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡉⠀⠀⡀⠀⠈⠁⢒⠻⠖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⢀⣀⠄⠊⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡈⠂
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠃⠠⠀⢈⠉⠉⠀⠀⠐⠀⠈⠉⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⡰⠀⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠀⠀⠒⠊⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠃⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⢀⠴⠂⠉⠀⠈⠈⠑⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⡆⠀⡀⠀⠙⠀⠀⠈⠃⣸⣄⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠠⠂⡠⠀⣀⠒⠀⠤⡄⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠀⠸⡀⠁⠀⠀⠈⠉⠂⠈⠀⠈⠑⢆⠀⠸⠁⠂⢄⢀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠀⠀⠘⠃⢸⠉⠒⠂⠀⠀⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠈⠑⠤⣐⡏⠑⠢⢤⣀⡀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠓⠠⠄⢀⣈⣱⣄⠀⠀⠀⢀⡴⡁⡀⠆⡀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠀⠉⠐⠠⠈⣀⠔⠉⠍⡷⠄⠀⠀⠀⠀⠀⣅⡤⣀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠛⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠴⠋⠀⠀⠀⡀⠂⠺⠋⠱⠈⣦⣀⠀⠀⠀⠈
⠀⠀⠀⣤⠀⠉⠀⠀⠀⠀⠀⢲⠒⠂⠒⠢⠀⠀⠀⠀⠀⠀⢀⡄⠤⠀⠚⠢⢄⣀⣀⣂⣠⠄⠖⠉⢦⡈⠒⠢⠤⣀⣀⡵⠇⠄⢂⣡⡎⠈⡛⠓⡖⠈⠉
⢀⠀⠈⠀⠡⠀⠀⠀⠀⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣩⡈⠑⠀⠁⠀⠀⢨⠢⣀⡠⢾⠩⠔⠛⠛⢏⠃⠁⠀⠁⢀⠁
⢠⠀⠀⠀⠀⠂⠀⠈⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠊⢠⡊⠀⡇⢠⠖⠉⠉⠲⣤⠛⣉⠔⣃⣀⠆⠀⢀⠎⠀⠀⡀⠀⠸
⠀⠸⢃⣀⢀⡇⠀⡀⢀⠤⠔⠒⠁⠀⠀⠀⠇⠀⢀⠁⠀⠀⠀⠀⠀⢠⠚⢄⠡⠈⠐⠓⠁⠀⠀⠀⠀⢹⠚⡝⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠴⡁⠀⠀⠉⠙⢅⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠀⣐⠂⠢⠄⣞⠀⠈⠢⢜⠀⠀⠀⠆⠀⠀⠀⢨⠂⠃⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠋
⠀⠘⢄⠀⠈⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠓⠈⠁⠀⠈⠀⢸⠀⠀⠀⠀⠀⠷⠀⣠⠆⠚⠒⠃⠀⠀⡰⠄⣀⠀⢀⡽
⠀⠀⠈⠲⠀⠈⠀⠒⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡐⠀⠀⠀⢀⡴⢁⡂⠀⠗⠀⠀⡒⢡⠊⢀⠴⠋
⠀⠀⠀⠄⠐⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠙⠲⠶⠚⢯⡀⠀⠡⡄⢠⡄⡤⠑⢥⠔⠁
⠀⠀⠀⠀⠀⠈⡶⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢀⡐⠉⠀⡀⠀⣀
⠀⠀⠐⠁⠀⠀⠀⠈⡳⢦⠤⣀⠀⠤⠤⡖⠊⠉⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠔⠈⠁⠉⠹⣦⣴⡂⠀⠀⠁⠀⠈
⠀⠀⠴⠀⠀⠀⠀⢀⠁⢠⠒⠄⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠤⠒⠀⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉
⠈⢱⠘⠠⢄⣀⡀⠈⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠤⡀⠀⠀⠀⠀⠀⠐⡄
⠀⠄⠀⠀⠈⡂⠏⠑⢁⣀⣤⠀⠀⡔⠀⠠⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠐⠠⡀⠀⠀⠀⣀⣀⠀⡀
⠀⠀⠂⠀⠔⠁⠀⠀⠀⢸⠁⠀⠀⡃⠀⠸⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠰⢄⡀⠁⠠⠄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠉⠢⢤⡀⠀⣌⣠⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⡒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀⠀⢹⡿⢟⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠐⢢⡌⣉⠒⠂⠠⠤⠀⣀⣀⠀⠀⠈⠑⢄⡀⠁

⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠙⢿⣏⠿⣿⣿⣿⣿⠟⠋⠀⠈⠻⢿⣿⣿⣿⡿⠿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠢⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸
⣿⣿⠄⠄⠀⠀⠀⠤⠀⠀⠄⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⢀⣀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠐⠀⠐⠂⠀⠒⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⣿⣿⣿⣿⣿⠋⠁⠂⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣶⣤⢀⡀⠀⡀⢀⣀⢤⣶⣿⣿⣿⣿⣿⠟⠑⠂⠀⠂⠀⠂⠙⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⢈⠐⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⡀⠀⡀⠀⠈⢿⣿⣿⣿⣿⠋⠀⠀⠰⠐⠒⠢⠀⠀⠘⢿⣿⣿
⣿⣿⡿⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠘⠀⠀⠰⠀⠀⠀⠻⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣿
⣿⣿⠀⠀⠀⠀⠈⠓⢴⡤⠠⠉⠀⠀⠀⠀⠟⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⠐⠆⠘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⠀⠀⠀⠠⢀⠠⠒⠒⢄⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣐⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⢠⠐⠐⢄⠀⠂⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡿⠁⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣧⡀⠀⠀⠀⠀⠀⣀⣄⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣶⣶⣦⣄⡀⠀⢀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⢤⡀⣀⣀⣠⣤⣤⡄⠚⣀⠐⠦⠀⠀⠀⣀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣦⠀⠀⠏⠍⠠⡎⠀⠈⢷⣶⠎⠀⠀⠈⣛⡿⡧⣾⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⢀⣴⡿⡿⡖⠊⢠⣿⡄⠀⠀⠀⠛⠄⣀⠈⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠁⣴⠾⠋⠁⠛⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠆⠀⣿⣿⣏⠀⠙⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣿⣶⡧⢔⠈⣿⡇⠀⠰⣿⣿⠶⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⡀⠙⣧⢜⣾⡿⣿⣿⢸⡏⢩⠀⠛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⢟⣨⣿⣿⣿⡋⠀⠈⣿⣿⡋⠀⢀⣠⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢧⣼⣿⡇⠰⠿⠀⠀⠀⠘⠋⠻⡟⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣇⡀⠙⢿⣇⢠⣄⣀⢀⢀⣀⣰⠀⠠⢠⣤⡾⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣶⡄⠀⠀⠉⠻⢦⠔⠋⠁⠀⢀⣼⡏⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣿⣿⣷⡀⠀⠀⢀⣸⣀⠀⠀⢀⣾⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡘⢿⣿⣿⡿⠂⠉⠉⠉⠉⠉⠉⢿⡿⣁⡜⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠈⠙⢿⣿⣿⣧⡀⠀⠀⠀⡀⢀⣠⣾⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠀⠀⠙⠛⢻⣿⣷⡞⠛⢥⠏⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣤⣀⣀⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⣶⡏⣿⣿⣿⣿⠛⠛⣡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣤⣤⣝⣾⣧⣈⠀⠿⠾⠀⢠⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣧⠙⢻⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⢧⡀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡟⠿⣿⣷⣄⠀⠉⠛⠻⠿⠿⣿⢻⣷⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡇⠀⣀⠸⠿⣿⣿⣿⣿⣦⣶⣿⣿⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢰⣿⠿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢄⣨⡄⠴⠛⠁⠈⠃⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣻⣿⡖⣶⣿⣶⠠⢄⣀⠀⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠰⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣦⣴⣿⣿⣿⣿⡿⣦⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⡿⠃⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡿⠿⣿⡿⠿⠻⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⣶⣦⠛⠁⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⣙⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⣤⡴⠂⣤⣤⠀⡀⠀⠤⣤⣘⣻⣄⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣀⡀⠀⠸⠏⢤⠀⢰⣯⣿⣿⣿⣄⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⣀⠉⠉⠉⠛⠛⠛⠛⠓⠀⠀⠘⠛⠚⠛⠛⠿⠿⢿⣦⠳⡄⠀⠀⠀⢀⣥⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠄⠠⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠿⠿⠿⢿⣶⣄⠀⠀⠀⠀⢀⣠⣶⡿⠿⠿⠿⠿⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢹⣷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠻⠿⣿⠿⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡄⠼⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣼⣧⣆⠀⠀⣈⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣦⡄⠀⢰⣿⣿⡆⣻⠛⠋⠀⠰⠿⡏⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠉⢫⡄⠀⠈⣿⣿⡁⠿⢰⠃⢀⣙⠒⠀⠀⣀⡀⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢁⡴⣿⣬⣯⣦⣼⣿⣿⣿⣮⣿⣤⣿⡟⠁⣠⡾⢻⣿⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⢳⣿⠟⠁⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣀⣤⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⡍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠈⠑⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⣶⡀⠀⢀⡀⣠⣴⣶⡞⠷⢾⣿⣧⣼⣿⡷⠮⢳⣶⣦⣀⠀⢀⡀⢀⣶⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⢺⡿⢛⠛⠃⠐⠀⠈⢿⡿⠁⠀⠂⠘⠛⡛⢿⡟⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣯⠩⣤⡄⠋⢀⣶⣤⡤⠇⠄⠄⢸⡇⠠⠀⢸⣤⣤⣶⡀⠙⢰⠶⢉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡘⠁⣾⢘⡛⢉⣡⢀⣈⡀⢸⡇⢀⣉⡀⣌⡉⢛⡇⠚⠈⢃⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⡀⣣⣄⣤⡇⠿⠃⣸⡇⠘⠿⢸⣤⡠⢜⢀⢀⣤⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠏⠙⠂⡈⣇⡞⠑⠋⠙⠪⢳⢸⢁⠘⠃⠸⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡥⣷⡶⣶⠹⢣⠇⠀⠀⢀⠀⠀⠀⠸⡜⠏⣶⣶⣾⢬⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⡿⣿⢰⢣⣶⠀⠀⠉⠁⡀⠀⣶⡜⡆⣿⢿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡌⣬⣀⡀⠻⣿⣿⠟⣠⣤⣧⠥⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⡿⣟⡪⠆⠀⢠⡄⠀⠸⢐⣻⠿⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠹⣶⡖⢾⣷⣾⡷⢲⣶⠏⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡱⠷⠚⣛⣛⠁⠫⢏⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠻⠷⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠋⠉⠉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⣴⣿⠏⠀⠀⣀⣤⣿⣟⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢼⠟⠁⠀⢠⣾⣿⡿⢻⣇⠀⠀⣸⡇⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠙⠻⢶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢠⣶⡶⢶⣄⣀⣿⣿⡿⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠙⠛⢻⣿⢹⣿⣿⠇⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⣼⡟⠀⠀⢿⣿⡀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⢀⣾⡟⠀⠀⠀⠈⠻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣶⡿⠋
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢰⣿⠛⠶⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣦
⠀⠀⠀⢀⣾⣿⡿⠛⢻⣦⠀⠈⠻⣄⣀⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⢷⣄⠀⠀⠀⣠⣾⣿⣿⠿⢷⡄⠀⠀⣰⣷⠀⠀⣰⡿⠋⢠⣿⣿
⠀⠀⠀⣾⣿⡿⠁⠀⢸⡿⠀⠀⠀⠈⠙⣿⡿⢿⣿⣷⠀⠀⣼⣿⣿⠟⠁⠀⢸⣿⡆⠀⣼⣿⣿⠟⠁⠀⣰⣿⠀⢠⣿⡟⠀⣼⡟⠁⢠⣿⣿⠇
⠀⠀⢰⣿⣿⠃⠀⠀⣿⠃⠀⠀⠀⠀⢠⣿⢡⣿⣿⠃⠀⣼⣿⣿⠏⠀⣀⣴⣿⠟⠀⣸⣿⣟⠁⠀⢀⣴⡿⠃⢀⣾⣿⢃⣾⠟⠀⢠⣾⣿⡟
⠀⠀⢸⣿⡿⠀⠀⢠⣦⠀⠀⠀⠀⢠⣿⢣⣿⣿⠏⠀⠀⣿⣿⡟⠛⠛⠛⠉⠀⠀⣠⣿⣿⡟⠲⠾⠟⠋⠀⣠⠿⣿⣿⣾⠏⠀⠀⣾⣿⣿⠁
⠀⠀⠈⣿⡇⠀⣠⣿⣿⡆⠀⢀⣴⡟⠁⣾⣿⡟⠀⠀⣴⢿⣿⡇⠀⠀⠀⢀⣠⡾⠋⢹⣿⣇⠀⠀⣀⣤⠞⠁⠀⠻⠿⠋⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠙⢿⣾⠟⢻⣿⣇⣴⠟⠋⠀⠀⢻⣿⠁⣠⡾⠋⠀⠙⢿⣦⣤⣶⠿⠋⠀⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⣴
⠀⠀⠀⠀⢀⣶⠏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⢀⣾⠏
⠀⠀⠀⣴⡿⠁⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣴⡿⠁
⠀⢀⣾⡟⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⢀⣾⡟⠀⠀⠀⣼⣿⣿
⣼⣿⠁⠀⠀⣰⣿⣿⡇
⣿⣿⠀⠀⣰⣿⣿⡟
⠹⣿⣤⣾⣿⡿⠋
⣤⣬⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠁
⠀⠀⠐⣌⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠲⡀⠙⠢⡀⢣⠢⡀⠀⡄⢁⠠⠗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠘⣆⠀⠈⠲⠧⣀⣀⣉⣹⢚⣛⡦⣌⢹⠲⠄⣠⣀⣦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⠿⡄⠉⠙⠻⠦⢄⡂⠈⠁⠳⣦⡦⣀⡠⠘⡄⢀
⠀⠀⠀⠀⠀⠀⡰⣿⣄⡠⣦⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠈⠳⢄⠀⠉⠘⢻⣦⣀⣵⠎⠰⡄
⠀⠀⠀⢀⡤⠶⠾⠿⠟⠛⠓⠳⠖⠂⠐⢺⣿⣿⡶⢦⡀⠀⠀⠀⣇⠀⠀⠀⠹⡽⡄⠀⠀⡹⡄
⠀⠀⡼⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⡉⠳⡐⢛⣄⣀⣠⡇⠀⠀⠀⠀⡙⣷⠖⠺⣏⣠⣄
⠠⠎⢁⢀⡠⠼⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⠟⠛⠛⠫⡙⠂⠀⢠⡎⣠⣄⡀⠀⠈⠀⠈⠛⠛⡆
⠀⡼⣁⠘⠦⣤⣄⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠈⠧⠈⠻⣷⡀⠀⠀⠀⠀⣰⠃
⠀⡠⠂⢀⡼⠿⠛⠓⠂⠀⠀⠙⢿⣿⣿⣧⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢰⣿⢳
⠐⠉⡇⠙⣶⣶⣄⠀⠀⠀⠀⠀⠈⢿⣼⣿⣿⣿⣶⣞⣯⣧⣦⣸⢳⢄⡀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠘⠃⢐
⠀⠐⣲⣾⣩⣄⣀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣹⠉⠉⠉⡷⠤⣄⣀⠀⠀⠀⣠⠞
⠀⠀⠛⠁⠹⢻⣿⣷⣀⠀⠀⠀⠀⠀⠈⡝⢻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠀⣼⣴⣷⡞⠜⡌⣏⢿⡽⠃
⠀⠀⠀⠀⢀⣿⠿⣶⣤⠀⠀⠀⠀⠀⠀⢷⣔⢩⣿⢷⣄⠀⠘⠻⡿⠛⣿⣼⣿⣿⠗⠃⠀⠉⠀⢨⠃
⠀⠀⠀⠀⡸⡆⠀⠈⠛⣷⣤⣀⡀⠀⠀⠀⠈⠳⡀⠞⣻⣷⡀⠀⠙⠀⢹⣿⣿⣟⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡶⡇⠀⠀⠀⠀⠽⣿⣿⣥⣄⠀⠀⠀⠈⢦⢩⢷⣻⡄⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠁⣷⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣄⡀⠀⠈⢧⠈⡿⣷⠀⠀⠀⢹⣿⢻⡀
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠠⠚⠛⣿⣿⣿⡦⠀⠈⢳⡠⢿⣇⠀⡀⠆⢻⡏⡏⢀⠠⠇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⡀⠔⠚⣟⣿⣿⣿⣦⠀⠀⠙⣧⡿⠊⢀⠀⢠⣿⠛⠁⡔
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⢦⡀⠀⠘⣅⢀⠼⡿⣿⡿⣱⡎
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣧⡀⢧⡀⠀⠈⠛⠲⠧⣥⠷⠟⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠘⠟⢳⡕⢄⠳⣄⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠑⢄⠙⠦⣤⣤⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣧⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠻⣿⣿⣿⣿⠏⠁⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣧⣤⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⠑⠈⠁⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠻⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣻⡿⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢨⣉⠻⣾⠿⠟⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣽⡿⢿⡄⢤⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣏⡑⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⠉⠀⢠⡄⠀⢶⠀⠀⠀⠀⠰⡁⠈⠉⠉⠐⠒⢤⣀⣠⣴⣿⣿⣿⠍⠉⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠘⠃⢀⡼⠳⣤⡀⠀⠀⣿⠀⠀⠠⠀⣀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⡇⡟⠐⠀⢀⣤⣾⣷⣾⣶⣟⡿⣶⣴⣿⣿⣿⣿⡿⠛⠠⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠀⠀⠀⡀⠀⠀⡌⢠⠇⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⠀⢠⠖⠀⠘⠀⠤⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢀⣌⣀⠀⠀⠈⣤⠈⠀⠀⠀⠘⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣟⣿⡿⠀⣿⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠜⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⢿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣴⣤⣀⡇⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡶⠀⠙⢿⣶⣿⠐⢿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⡆⠈⠁⠉
⠀⠀⠀⠀⠸⣄⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⠿⠟⢀⣾⣿⣿⣿⣿⣿⢿⣿⣾⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣛⣵⣿⡟⢹⣇⡇⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣟⢷⣈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡝⠋⠉⠉⠙⢻⡀⢸⡏⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣮⣿⣾⣿⣿⣿⣿⣿⣿⣣⣿⡏⠀⠀⠀⠀⠀⠀⠑⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⡆⠀⠀⠀⠀⠀⠀⢴⣤⣸⡇
⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣿⡇⠀⠀⠀⠀⠀⢸⠀⣀⠼⣿
⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡋⣉⠉⣇⠀⠀⠀⠀⠀⠸⠀⣀⣴⣿⡄
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣏⣉⣉⣀⣀⣀⣀⣀⢀⡟⠀⠀⠀⠀⠀⡄⡾⠋⢹⡇⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢹⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⢀⣠⣤⣿
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⠷⡶⠋⠉⠀⠀⠀⠀⣙⣹⠉⠉⡇⠀⠀⠀⠀⢠⣶⠾⠟⠉⢸⢋
⠀⠀⠉⠢⣄⣀⣀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠙⠐⢶⠊⣿⣿⡇⢻⠀⢀⠇⠀⠀⠀⠀⣈⠁⠀⠀⠀⠈⣸⡇
⠀⠀⠀⠀⠀⠉⠉⠑⠂⠤⢀⠀⠀⠀⠀⠀⠀⠚⠆⣿⠋⢹⣿⡇⢸⠀⣀⡖⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⣀⠀⠀⠀⣼⡃⢀⣿⣿⣀⣿⠳⣿⣧⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⣾⡙⢿⡟⢁⣼⣿⣿⠀⢾⣿⡓⣌⠉⢭⣹⣆⠀⠀⠀⠀⢸⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠀⠀⠀⣿⣾⣿⣶⢛⠈⣿⠀⠘⡏⢹⣾⠏⠲⢌⣍⣻⣆⠀⠀⠈⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠀⠀⠀⠰⠀⣩⡿⢁⡜⠀⠀⠀⠀⣬⣹⡿⠖⠛⢉⣁⣀⣹⠷⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⡾⠋⣠⠞⠁⠀⠀⠀⠀⢹⣿⣿⣾⣿⣿⡿⢿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣨⠟⣿⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠠⡿⢁⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡶⠶⠿⠂⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠴⠶⠛⠛⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢀⣠⠞⠳⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠄⣀⡴⠋⠀⠈⠓⠖⠋⠀⣀⠀⠈⠳⠘⡟⠙⣄
⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡴⠃⢀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠈⠛⠲⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢉⡁⠋⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠀⠀⠀⣠⠀⠙⠋⠉⢇
⠀⠀⠀⠀⠀⠐⠀⡟⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠇⠀⢸⠂⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠢⣄⢀⣤⠴⠲⣤⡀⠀⠈⣠⡤⠢⣤⡀⠋⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠇⢀⣠⠖⠉⠉⠉⠉⠁⠒⠒⠒⠉⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠑⢦⡀⠀⠙⡀
⠀⠀⠀⠀⣠⣞⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠷⢦⡀
⠀⠀⠀⢸⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠄⡇
⠀⠀⢀⡜⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⡆
⠀⠀⡾⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⠀⠙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠌
⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⣠⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⢤⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⣀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠉⠒⠒⠂⠠⠤⠬⠶⠶⠤⠐⠒⠚⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠲⣤⡀⣀⡀⠀⣀⣀⠀⢀⣤⡤⠒⠋⢈⣣⡀⠀⡀⠀⠀⣠⣴⠟⢋⣉⡑⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣤⡈⠉⠉⠉⠉⠉⠉⠀⠀⢀⡴⠞⠋⠙⠛⢫⡙⢦⡴⠟⠓⠂⠉⠉⠉⠉⠉⠑⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⢀⣤⠟⠉⠻⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⢀⣄⠰⠶⣦⣤⣬⣤⣮⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣼⠟⠁⠀⠀⠀⠈⢿⣦⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⡴⢞⡱⠞⠉⣀⠴⠶⠄⠀⠀⠰⠶⠞⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⠁⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡴⢋⡴⠋⣀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠤⣄⣀⠀⠀⢀⡞⠁⠀⡠⠟⡴⠋⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠁⠛⢢⡞⠀⠀⠈⠛⠛⠀⢀⠟
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠈⢻⣷⣤⠀⠀⠀⠀⣠⠖⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡃⣀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢿⣬⣿⣷⣆⠀⠀⠘⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣡⠞⠉⠈⢳⡀⠀⠀⢸⠇⠀⠀⠀⠀⣠⡤⠶⠖⠒⠀⠀⠀⠐⠶⠦⣤⡀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢸⡇⣸⡿⣷⣄⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⡁⠀⠀⠀⢀⣹⡆⢀⡆⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠧⡼⢿⣿⣧⡶⠛⢀⡀⠀⠀⠀⢀⠏⠀⠀⢸⠧⠤⠴⠛⠁⢨⡇⣠⠁⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢯⣠⣴⠟⢡⡤⠀⢠⠏⠀⠀⠀⢸⡂⠀⠀⠀⠀⡜⢀⠟⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⢿⣶⣋⠳⠴⠃⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠇⡸⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⢀⡴⠋⠙⢷⣄⠀⢸⠘⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⡴⠋⠀⠀⠀⠀⠙⠷⣈⠗⠀⠀⢸⣇⣤⡤⠐⠒⠒⠒⠒⠒⠒⠒⠤⣤⣄⡀⠀⢀⡾⠁⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠴⢶⣭⣥⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠛⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣠⠚⠛⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿
⠀⠸⠁⠀⠀⠀⠈⠉⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⢂⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⢀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣥
⣟⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠎⠁⠈⠑⠶⠒⠒⠂⢤⣤⣤⡀⠀⠀⠀⠀⠈⣿
⠈⠳⢌⣵⠶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣤⣠⡇
⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠷⣶⣶⣶⣶⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢴⣤⡿⠟⠁

⠀⠀⠀⠰⠄⠒⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⣀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⠋⠉⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠠⠖⠒⠂⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡀⠠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠘⠂⠤⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⣄⢀⠀⣤
⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠈⠛⠈⠀⠛
⠀⠀⠀⢀⡶⠀⠀⠶
⠀⠀⠀⠀⡤⠤⠀⣤
⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠓⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋⠈
⠀⠀⠀⠀⠤⢀⠀⣶
⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡼
⠀⠀⠀⠀⠀⢹⣷⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⢠⣾⣷⠸⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⢙⣛⣟⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣿⣺⢀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⣀
⠀⠀⠀⠀⠒⠺⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣾⣿⣇⡾⠁⢸⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢘⢷⡉⠡⢉⣁⡃
⠀⠀⠀⠀⠤⢠⡴⣶⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢀⡀⠀⠀⠲⢶⢿⠿⠟⠀⢀⡄⠘⠇⠀⠀⠀⠀⠀⠇⠀⣀⠀⠀⣤⠀⢾⣥⣧⣶⡌⠉⡁
⠀⠀⠀⢀⣀⣀⠀⣀⠀⣁⠀⠀⠀⠀⠀⣰⣟⢿⠁⠓⢜⡿⠋⠡⣄⠙⠛⣀⡘⠀⠀⢂⣿⡆⠀⠀⠀⠀⠀⠀⢠⡆⢸⠟⠀⠀⠀⠀⣿⣈⣉⢀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠘⢏⡸⣄⡀⠀⠠⡿⠟⠐⠂⠀⢬⣧⣴⠀⠸⠖⣀⣀⡇⠀⠀⡌⠈⠀⠀⠀⠀⠀⠀⠲⡿⠁⡌⠀⠀⠀⠘⢶⣟⢫⣉⠘⠡⣠⡍
⠀⠀⠀⠠⠥⢀⠀⠿⠀⣠⡶⠺⢇⡉⠀⠀⠆⠀⢀⠐⡄⠙⠇⠰⠤⢀⣈⠙⡗⠀⢰⠃⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠈⠁⠐⣤⠜⢠⣿⡅
⠀⠀⠀⢀⣀⠠⠠⣤⠀⠛⠦⣴⡞⠻⠀⢠⣤⣲⣿⠄⠀⠀⠀⠀⠀⠀⠍⠘⠇⠀⠀⠀⠀⠀⠀⠺⠈⢳⠀⣠⠄⠀⠀⠀⡈⣯⠁⠀⢱⡶⠋⢀⡟⠉⠇
⠀⠀⠀⠀⠁⢀⠘⣽⠆⠀⣰⢟⠀⠀⠀⠸⢿⣿⣿⣦⠀⠀⢀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢦⣾⠴⢷⣿⣆⡻⣿⠉⠻⣏⠔⡋⠀⢠⡏⠠⢶⡄
⠀⠀⠀⠐⠛⢿⡟⠟⠀⢠⡶⠋⢻⣾⠄⣘⣿⣿⠃⠰⠀⠀⣿⠀⠘⢷⡄⠀⠀⠘⠀⠀⠀⡶⠀⠀⣺⣿⣴⣿⣿⣿⣿⣷⣄⠠⠟⠙⠿⣰⠏⠀⣠⣌⠑
⠀⠀⠀⠠⡶⢺⣶⣶⣰⡿⢁⡟⠀⢙⣾⣿⡿⠳⠀⠠⣄⠈⡏⠀⠀⠀⢿⡄⠀⣂⠀⠀⠀⠂⠀⠘⠋⣹⣿⣿⣿⣏⣿⣿⣿⣦⡄⠀⢠⡿⣶⡼⠻⠃⠎
⠀⠀⠀⢀⣀⢀⣹⣿⠏⠀⡼⠀⠀⠋⠱⡼⠇⠀⢺⠀⣸⣱⡇⠀⠀⣠⣤⣡⢼⣿⣠⣶⠀⠈⠃⢀⣾⣿⣿⣿⣿⡟⠋⣿⣿⣿⣧⣠⣟⣻⠞⠁⠐⠳⢌
⠀⠀⠀⠀⠓⠒⡿⠻⣆⣠⣤⠀⠀⢀⣶⣶⣶⣦⣴⣿⣻⣿⡗⠢⠀⠛⠋⣷⡀⣁⢻⣿⠀⠲⠶⣿⣻⠿⢿⣷⠿⣧⣴⡇⣙⠋⢻⡿⣿⡿⣦⣰⣾⣦⣄
⠀⠀⠀⠰⠶⢶⣶⣾⣯⠀⢰⣿⠏⠀⣴⡯⠈⠙⠳⣿⢿⣿⡇⠀⠀⠀⠰⢿⢇⢃⣼⣿⣦⡄⠀⢸⠛⠆⠚⠋⠀⢹⡿⠉⠀⡴⣿⣷⣾⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠣⠽⠟⠖⢿⠿⠆⠰⠿⠇⠀⠀⠀⠸⠄⢸⠇⠀⠀⠀⠀⠰⠞⠉⠽⠈⠿⠃⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠘⢱⠿⠛⠿⢿⠿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢸⠁⡎⣼⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢡⡴⠒⠋⠀⠀⠀⠘⢄⠱⠜⠂⠘⢦⠀⠀⠀⠀⠀⠀⢀⡶⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠘⠯⣍⣉⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡤⠤⠼⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳



⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠉⣢⠀⠀⢀⠀⠊⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠒⠀⠤⠚⢹⠀⠀⡇⠐⠀⠀⠀⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠦⡤⠀⠤⢤⠊⠠⢄⠈⠒⠄⠤⠀⣀⣠⡮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣰⡆⠀⠀⣸⡇⠁⢞⢀⢸⡄⠀⠀⢀⠿⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⡗⠂⠒⢹⠇⣴⣘⠳⡍⢷⠄⠤⠋⠜⠀⠤⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⡠⠌⢥⡔⠋⡼⠃⢹⡀⠀⠈⢄⠀⣈⠀⠀⠀⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠻⠧⣄⠀⠘⢦⠀⠀⢸⢧⢀⡠⠄⣍⠀⠑⢄⠀⠈⠑⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠠⠓⠈⠓⠠⠼⠋⠀⣸⠀⠏⠀⠐⢸⠣⠀⠈⠑⢌⠳⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣄⡠⣅⢦⠀⢸⢻⡐⠔⠁⠀⡘⠃⡄⠀⠀⠀⣳⠠⠀⠀⣠⠄⠰⢸⡄⠀⠈⢷⠹⡄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢘⣈⠥⠈⡀⡀⠘⡆⠇⠲⣄⡄⠀⠀⠧⠐⠐⠒⢱⠀⠢⠐⠡⠤⣀⢐⡇⢀⡰⣿⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠍⢐⡠⠧⠒⠡⡀⢹⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠘⠲⠄⠴⠂⠑⠚⠓⠉⠀⣿⠀⠈⠇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠒⠉⣅⠀⠀⠀⣧⠑⣾⢸⠀⠀⠀⢠⠀⠈⡆⡎⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡠⠀⠀⠈⢯⠑⠦⠴⣿⠈⢣⠘⡆⠀⠀⢀⠇⠀⣿⡇⢰⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⢁⡀⠈⠒⠈⠑⣦⣠⠇⠀⠈⢆⣧⠀⠀⡸⠀⠀⣿⡇⣸⠀⠀⢰⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠘⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠖⢁⡠⠀⢀⠔⡥⠊⠛⠒⠦⣀⠤⣿⠀⢰⠃⠀⢰⣿⡇⣿⠀⠀⢸⠀⠀⣿⠀⠀⢠⡇⠀⣿⠀⠀⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠎⠀⣠⠞⢸⠅⠀⣀⣀⠠⢬⡂⡈⡄⢸⠀⠀⣸⠸⡇⠘⢦⡀⢾⠀⠀⣿⠀⡇⢸⡇⠀⢻⡀⠀⠀⠙⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡟⠠⡏⠈⢀⡾⢀⡀⠀⢀⠀⠐⣿⠑⣳⠛⠀⠀⡿⠀⢷⠀⠀⠉⢹⡀⠀⣿⠀⠀⠈⣧⠀⠀⠳⣠⠎⠐⠂⠠⠂
⠀⠀⠀⠀⣰⠉⠁⠒⠀⣸⠃⠀⢰⠇⠀⢸⠁⢠⠃⠀⠛⠛⠋⠈⠢⠟⠀⠀⠀⡇⠀⠸⡆⠀⠸⢀⡇⠀⡿⠀⠀⠀⠘⣦⠀⠀⠉⠰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣰⠃⠀⠀⡟⠐⠤⢘⡇⣼⠀⠀⠀⡀⠀⡄⠀⡆⠀⡀⠀⠀⠀⠀⡇⠀⠀⡞⠀⢠⡇⠀⠀⢠⠀⠘⢧⡀⠀⠀⢻⡀
⠀⠀⠀⠀⢿⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⠙⠃⠀⠀⢰⠇⠸⠃⠀⠹⡄⢧⡀⠀⠀⠀⠀⠀⣸⠁⠀⠼⠁⠀⠀⠀⠁⠀⠈⢳⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠷⠀⢀⠞⠁⣀⠤⠀⡴⠂⠀⣰⠏⠀⠀⣠⠞⢉⠁⠀⠳⡄⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢴⡁⠀⠈⠁⠀⢀⡀⠰⠞⣡⠖⠊⠉⠀⠀⢀⡆⠀⠀⠘⠆⠀⠀⠀⠀⠳⠔⠋⠉⠉⠉⠁⠀⠀⠉⠒⠦⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈