⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡝⣆⠀⠀⠀⠀⠈⣻⣿⣿⠟⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⣦⢀⣤⣶⣶⣿⣻⡟⠀⣀⣿⣿⠃⣀⡤⠖⠒⠛⠒⠲⢤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣿⣿⠿⠛⠛⠛⠛⠛⠿⣏⡽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠛⠛⠛⠛⢉⡵⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⢦⠀⣀⣠⠤⠖⠒⠚⠉⠉⣉⣉⣙⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⣧⠀⠀⠒⢠⠏⠀⠀⠀⣀⡴⠚⠉⠉⠉⠉⠉⠙⠚⣿⣍⡀⠀⢀⣠⣴⣾⠿⠛⠓⢋⣉⣙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⠀⣀⣀⠀⢸⡆⠀⢠⠏⠀⠀⢠⠞⠁⠀⠀⠀⢀⣠⣴⣾⡿⠭⠭⠥⢽⣷⡿⠛⢉⣠⠴⣶⣿⣿⣿⣦⡀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡞⠁⠈⣷⢸⡇⠀⡽⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠚⠉⠀⢀⣠⣤⡖⠚⠋⢳⠖⠉⠁⢰⣿⣿⣿⣭⣿⠿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⠀⢠⡟⠀⢷⡀⡇⠀⠀⠀⠀⢀⣴⣺⠵⠋⢀⣠⠴⣺⣟⣿⡿⢿⣶⣤⣼⡤⠤⠤⠼⠛⠋⠉⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠒⠋⠀⠀⠈⣇⠀⠀⠀⠀⠀⠻⠿⠶⠶⣞⣉⡀⠀⣿⣿⡿⠿⢟⣏⡵⠟⠀⣀⠀⠀⢀⣀⣤⠴⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠛⠓⠓⠂⢉⡭⠟⠀⠀⠀⠉⠉⠙⠋⠁⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣾⣿⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡽⠶⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢋⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡾⠋⠀⠀⢹⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠴⠶⢚⣛⣭⣴⠶⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢸⡟⣦⢿⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢋⣉⣉⣉⣉⣉⣉⣥⣤⣤⡴⠶⠶⠞⠛⢋⣉⣡⣬⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠤⣤⣀⠀⣾⠁⡿⠸⡇⠀⠀⠀⠀⠀⠀⣟⣀⠛⢋⣉⣩⣭⣥⣤⣵⣤⡴⠴⠾⠒⠒⠛⠋⠉⢉⡤⠚⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣶⣯⠬⠾⣷⠿⠃⢰⡇⠀⠀⠀⠀⠰⣄⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⢷⡝⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⢿⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⣀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠙⢮⠳⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡽⠃⢸⡇⠀⢀⣤⣦⠀⠀⢷⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇⠀⠀⠀⠀⠀⠈⢷⡝⢦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣱⠃⠀⢈⣧⠀⠸⠿⠋⠀⠀⢸⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡤⠤⠤⣤⠶⠚⠋⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⣟⢳⣖⢺⠃
⠀⠀⠀⠀⠀⢀⣾⢳⠏⠀⠀⠘⢿⣄⣤⣀⣀⣀⣀⣻⣀⣀⡤⠴⠶⠶⢴⣖⠛⠉⣩⠟⢻⣿⣿⠈⠓⣦⢞⣁⣀⣠⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⢢⣿⠉⠀
⣾⡛⠶⡤⠴⢾⣣⡏⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠉⡏⠁⠀⠀⠀⠀⢀⣈⣹⣾⣥⣤⡾⠃⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⢸⡼⣇⠀
⠉⠛⢻⡆⢰⡏⠙⡇⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⡷⢶⣶⣾⣿⣿⣿⣿⠹⣿⣿⣿⣧⣀⣰⡿⠟⠋⠈⠉⠁⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢈⣧⢸⠀
⠀⠀⣾⣀⣌⣃⣸⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣛⣋⣉⣄⣙⣡⣠⣨⣽⣯⣀⣀⣀⣀⣀⣠⡤⠴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠞⠉⠉⠀
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❤凸( •̀_•́ )凸₍ᐢ._.ᐢ₎♡ ༘ʚ♡⃛ɞ(•ᴗ•❁)
⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⡿⣍⣦⠾⠿⠛⢫⠉⢻⣆⣿⣿⣧⠀⠀
⠀⠀⠀⠀⢺⣿⣿⣟⢻⡟⣼⠂⡀⡀⡎⠀⡆⢻⣿⣿⣿⣽⣿⡇⠀
⠀⠀⠀⠀⢸⣷⣿⣿⢼⡸⡟⣻⢯⢻⣾⣿⣿⣾⡿⣿⣿⢿⣿⡇⠀
⠀⠀⠀⠆⣾⣿⣿⣿⣿⡇⣧⣽⢼⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠆⢿⢻⡿⣿⣿⢹⠿⡟⠀⠀⠀⠰⢛⢻⣿⣿⣿⣿⣿⠁⠀
⠀⠀⠀⠀⢾⣧⣷⢿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡏⣿⣿⣿⠀⠀
⠀⣐⣡⠤⠠⣬⡿⣿⣿⡀⠀⠀⢀⡀⢀⠀⠀⢡⣿⣿⣿⣿⣿⠀⠀
⠀⢨⣿⣦⣿⣿⣾⣿⣿⡟⣦⣄⡀⠀⢀⣠⢴⣿⣿⣿⡿⢹⣿⡄⠀
⠀⣸⣿⣿⣿⣿⡏⢧⣿⣿⢸⣏⠟⠂⠁⢇⣸⢿⣿⣿⢃⡆⣿⡇⠀
⢠⣿⣿⣿⣿⣿⣿⡙⣿⡿⣸⣌⣀⣀⣀⣨⣿⣼⣿⠛⣼⠃⣿⡇⠀
⣿⣿⣿⣿⣿⣿⣿⡇⣸⢣⣿⣿⣿⣿⣿⣿⣿⢿⠏⢼⣷⣦⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⢇⠀⠀⠀⠀⢀⠔⢶⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠈⡄⠀⠀⡠⠂⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠘⣀⠜⠁⢀⣴⡿⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⡋⠀⢠⣾⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣴⠟⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣯⣿⡄⢃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡏⢻⠙⣿⣿⡈⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⢸⣇⡠⢧⣿⠟⢹⢄⣱⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⡏⠘⢻⣠⠏⢿⢠⣸⣆⠘⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⣾⡇⠀⢸⢡⠀⠈⡌⣿⣿⡆⠰⡀⠀
⠀⠀⠀⠀⠀⢀⠔⠁⣠⣿⠋⠁⢠⠈⠀⢦⡀⠱⢿⣿⣿⡄⢡⠀
⠀⠀⠀⠀⠠⠊⢀⣴⡟⠁⢠⠀⣼⠀⡆⠘⡇⠀⢙⢿⣿⣷⣀⡣
⠀⠀⠀⡐⠁⣠⣿⠋⠀⠀⠘⢰⢹⣆⢡⢀⣸⣴⣾⢇⠻⠛⠋⠁
⠀⢀⠊⢀⣴⡟⠁⠀⠀⢀⠃⡆⢸⡿⢿⣿⠁⡼⡇⠈⠆⠀⠀⠀
⡠⠃⢠⣾⠋⠀⠀⠀⠀⡜⢠⢠⡟⠁⣾⠃⣰⠀⡇⠀⠆⠀⠀⠀
⠱⠴⠿⠁⠀⠀⠀⠀⡰⠀⢢⠏⠀⣰⠃⠠⡿⠀⠃⠸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠁⡸⠃⠀⢠⠃⢀⠃⢇⠀⠀⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢇⡔⠁⠀⡠⠃⠀⣾⠀⠘⣀⣴⡵⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢴⢲⠀⠚⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀
⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀
⠀⣾⠁⣠⠴⠒⠚⠛⠛⠛⠃⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠀⠀⠀⠘⡆
⢰⡇⠀⢠⣤⣔⡒⢲⣶⣶⡤⠀⠀⠀⠀⠀⣶⠶⠶⠶⠾⠿⠷⠀⠀⠀⣿
⣸⠁⣠⣠⣭⡀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣠⣶⣶⣶⡄⢸
⢹⡴⠿⠟⠿⠏⠀⠀⣀⡤⠤⡤⠤⡤⠴⣶⠦⣄⠀⠀⠀⢿⣻⣋⡞⠀⡿
⠸⣇⠀⠀⠀⠀⠀⠸⢥⡽⠶⡷⠒⢻⠶⠿⠖⠚⠁⠀⠀⠀⠀⠀⠀⣼⠁
⠀⢻⡄⠀⠀⠀⢀⣀⡈⠀⠀⠀⠠⡖⠲⠶⠶⠤⣤⣄⠀⠀⠀⢀⠞⠁⠀
⠀⠀⢻⣄⢠⡾⠋⠉⠙⠻⣶⣤⣤⠿⠶⠶⠒⠀⠀⠈⠷⠴⣶⠋⠀⠀⠀
⠀⠀⠀⢹⡟⠁⠀⢠⠀⠘⣿⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀
⠀⠀⢀⡞⠀⢀⣠⠟⠀⠀⠺⣤⣤⠤⢤⡀⠀⠀⠀⠀⣤⠞⠋⠀⠀⠀⠀
⠀⠀⣼⠀⣰⠋⡅⠀⠀⢀⡾⠃⠀⠀⠀⠙⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⢿⡴⠃⡼⠀⡴⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣀⡼⢃⣼⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀
⠀⠀⢏⣟⣻⢹⣃⣮⡛⣏⢿⢹⠹⢿⣿⡏⡆⢸⢿⣷⠀⠀
⠀⠀⠈⢉⣽⣿⣿⢟⣽⠿⠛⠻⣿⣍⠉⠉⣩⣿⣭⡉⠀⠀
⠀⠀⣰⣿⢿⣿⠃⢿⢡⣶⣶⣦⠀⢻⡗⢸⠛⠛⠻⡿⣄⠀
⠀⣼⡿⠏⣿⡿⣄⠸⣜⠿⣿⣟⡰⠟⠀⠘⣿⣿⣧⢸⡘⡆
⢰⣿⡇⣼⣿⣷⢾⣷⡞⠉⡉⠁⠀⠀⠀⠀⠈⠛⠥⠾⣃⣿
⢸⣿⡿⢿⣿⠟⣿⣿⣿⣶⣬⣝⣓⠶⠤⠠⠄⠀⣸⣿⣷⣿
⠸⣏⠇⠈⠉⢢⡜⢯⡎⠈⡟⠛⢿⠟⢿⡾⢿⣿⡿⠋⣿⡿
⠀⢹⡆⠀⠀⠘⢿⣆⣙⡺⣄⣀⣇⠀⣘⣀⡸⠟⢁⣼⣿⠇
⠀⠀⠱⣄⠀⠀⠀⠉⠙⢷⣶⣶⣬⣭⣥⣴⣿⣾⣿⡿⠋⠀
⠀⠀⠀⠈⠓⢤⣀⣀⣀⣠⣙⣿⣿⣭⣿⣿⣿⡿⠋⠀⠀⠀
⠀⠀⢺⣿⣏⣿⣿⣿⣽⣿⣫⠿⣇⣿⣿⣿⣎⣿⣿⣷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀