Horn Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣯⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠛⠓⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠫⠿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⢀⣀⣀⣀⣠⢤⡶⠶⠿⠿⣦⣀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢀⣴⠋⢀⠀⢀⣒⠝⠛⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢠⠚⠉⠁⣨⠟⠋⠑⠄⡀⠀⠀⠀⠁⠝⣄⣀⡀⠀⡟
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡖⠁⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢼⠀⢀⣾⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣧⠀⢸⠇
⠀⠀⠀⠸⠿⠿⠟⠛⠻⠿⣇⡀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣼⠋⢹⠏⠉⠉⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡀⠀⠈⢿⣿⣇⠀⢠⡇⠀⠀⠀⠀⠀⠀⣾⠀⢿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⢿⣿⡄⠸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠘⡆⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⡸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠿⣿⡄⢧⠀⠀⠀⠀⠀⠀⣷⠀⠀⠙⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⣿⡌⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠀⣤⣤⣀⣤⣤⣤⣤⣶⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠈⢿⣮⡓⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠉⣬⡧⣍⠉⠉⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠙⠿⣦⣄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣩⣾⢟⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠛⢿⣶⣄⣬⣦⣤⣤⣤⣤⣤⣾⠶⠿⣋⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣤⣍⣙⣛⣛⡛⢛⣛⣃⣬⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⢰⡟⠀⠈⢷⣄
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠰⣼⡆⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢰⡏⠀⠶⢼⡇⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⣀⣴⡛⠻⠞⠉⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠚⣛⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠒⠛⠃⠀⠀⠀⠀⢸⡇⠀⠉⣹⠇⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⡄⢰⠀⡟⢠⡇⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣻⡟⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⢦⣾⢁⡟⣸⠇⣸⣃⣼⠟
⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠀⠀⠹⣦⣈⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⣠⡴⠀⢀⡴⠀⣠⡄⠀⣠⡖⠀⣠⡴⠀⢀⡤⠀⣠⠄⠀⣠⠖⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠉⠀⠈⠁⠀⠀⠁⠀⠈⠉⠀⠀⠋⠀⠀⠁⠀⠈⠁⠀⠀⠋

⢸⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⣾⡋⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⢰⡟⠀⠀⠀⣴⡟⠛⢧⡤⠖⠒⣉⣡⡤⠤⠶⠶⠶⠤⠤⣬⣉⡓⠢⢤⣀
⠀⠀⢀⣿⠁⠀⢀⣾⡟⠀⠐⢉⡤⠒⠉⠉⣿⠀⠀⠀⢀⣤⣶⣶⣶⣬⡉⠓⠦⣈⠳⢤⡀
⠀⠀⣸⡏⠀⢀⣾⡿⠀⢀⢴⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡏⠀⢹⣿⣇⠀⠀⠈⠳⣄⠙⢦
⠀⢠⣿⠁⢀⣾⠟⢁⠔⠁⣸⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⠈⠳⣄⠳⡄
⠀⣼⡟⠀⣼⠏⢠⣾⠿⠿⠿⠿⣿⡿⠿⠿⣿⡿⠿⠿⢿⣿⡿⠿⢿⣿⣿⠿⡿⠿⠿⣶⣄⠙⣆⠹⡄
⢠⣿⡇⢰⡏⢠⠏⣿⡀⠀⠀⠀⢸⡇⠀⠀⣿⠇⠀⠀⢸⣿⠃⠀⢸⣿⣿⠀⡇⠀⠀⠈⢻⡆⠘⡆⠹⡀
⣸⣿⠀⢸⠀⣾⠀⢹⡇⠀⠀⠲⣿⣿⣇⣿⣿⣃⡤⣄⣸⣿⣀⡄⢸⣿⣿⠀⡇⠀⠀⠀⢸⡇⠀⢹⡀⢇
⣿⣿⠀⡄⢰⣿⠀⢸⡇⠀⣀⣀⣼⣿⡿⣿⣼⣿⣧⡿⣿⣿⢿⣧⣼⣿⣿⣤⣧⣤⣤⣤⣼⡇⠀⠈⡇⢸
⣿⣿⠀⡇⢸⠹⣷⣼⣷⡾⢛⠛⢻⣿⡇⣿⢹⠁⣿⡇⢹⣿⢸⣿⣾⣿⣟⠈⡏⠉⣉⣽⡿⠁⠀⠀⡇⢸
⢸⣿⡀⠀⢸⣦⣬⣿⡿⠿⠿⠿⣿⣿⡇⢻⣿⠀⣿⡇⢸⣿⠀⣿⣿⣿⣿⣿⣟⣛⠛⠋⠀⠀⠀⢀⡇⢸
⠘⣿⣇⢸⠀⣿⠿⣿⡧⠤⠤⠤⣾⣿⡇⣸⣾⠀⣿⡇⢸⣿⠀⣏⣉⣉⣉⣉⣻⣿⡇⠀⠀⠀⠀⣸⠀⡆
⠀⠸⣿⡄⢇⠸⣤⣽⣧⣀⣀⣤⡿⣿⣧⡿⢸⣶⡗⣷⣾⢿⣴⡏⠉⠉⠉⠉⢹⡿⠀⠀⠀⠀⢠⠃⡸
⠀⠀⠙⣿⣜⣆⠹⣍⠙⠻⢿⣥⣤⣤⣽⣥⣤⣽⣤⣬⣥⣬⣯⣤⣤⣤⣤⣴⡿⠃⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠻⣾⣧⡘⢦⡀⠀⠀⠉⣹⣭⣯⣽⣭⣯⣯⣿⣿⣽⡍⠉⠉⠉⠀⠀⠀⠀⢀⠔⢁⠔⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠐⠒⠲
⠀⠀⠀⠀⠀⠈⠻⣿⣆⡙⢦⡀⠀⢸⣿⣯⣽⣿⣯⣿⣿⣿⣿⣇⣀⡤⠴⠒⠚⠂⠐⠀⠀⣈⡀⠠⠄⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣌⠓⠾⣭⣏⣹⣩⣏⠯⠟⠛⠋⠉⢀⣀⡀⠀⠀⡀⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣌⣉⡁⠀⠀⠐⠒⠒⠛⡉⠁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡿⠚⢡⠤⡀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣠⠿⠃⠀⠀⡴⠊⣇⠀⠀⠀⢀⡀⠀⣇⠀⣸⠛⠁⠀⠀⠙⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣇⠀⠹⣿⠆⠀⠺⡇⠀⠀⠑⠏⠀⢀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽⠇⠀⠀⠀⠀⠻⠽⠂⠀⠀⠀⡏⠑⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡽⠇⠀⡟⠀⢀⣄⣠⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⣤⣞⡆⠀⠀⠸⠽⠃⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣠⣤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⡭⠴⠖⠂⠀⠀⠀⠀⠀⢀⡀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣶⠦⠤⠶⠶⣖⣚⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡷⠶⣤⣤⣀⡙⢯⠉⠉⠉⠉⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣿⡖⠒⠛⠛⡷⠦⣍⠺⣧⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⠞⠁⠀⠉⠓⡦⢰⠇⢰⠋⢳⣼⣧⣀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⡇⢀⣠⠤⠤⣴⣃⣏⡴⣏⣰⢋⡔⢻⡎⣧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡏⢳⢯⠗⡶⢲⢧⣿⣽⣳⢷⢯⣼⣥⢿⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡄⣧⡏⢾⣻⡧⢿⢜⣽⢩⢟⡟⣾⣯⣉⡿⢀⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣈⣇⢹⡙⠲⢤⣝⡋⠯⠿⣞⣾⠴⠛⣦⣼⡧⠎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⠙⢄⡼⡼⣹⢳⠦⠼⠛⠒⣋⡽⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡙⠳⠧⢯⣤⡤⠴⠚⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⠴⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⢀⠔⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠢⡀
⠀⠀⠀⠀⢀⠎⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢱
⠀⠀⠀⠀⡸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢇
⠀⠀⠀⠀⡇⠀⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⣀⡠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⢸
⠀⠀⠀⠀⢡⠨⣿⣎⠢⠄⠀⠠⠤⠀⠒⠒⠏⠘⠃⠀⠀⠀⠛⠀⠀⠀⠝⠋⢹⠒⠒⠀⠤⠄⠀⠠⠔⣱⣿⠃⡜
⠀⠀⠀⠀⠈⠢⣻⣿⣤⣀⢀⣀⣀⣤⣤⣾⣧⠀⠀⠀⠀⠱⣤⠎⠀⠀⠀⠀⣼⣷⣦⣤⣤⣀⣀⣠⣴⣿⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣻⣿⣿⣿⣿⣿⣉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣉⣿⣿⣿⣿⣿⢛⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣄⠤⠶⢶⣤⣤⣩⣽⢻⣶⣅⠀⠀⠀⠐⣤⠀⢀⠀⠀⣨⣾⡝⣯⣍⣤⣴⡶⠶⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡄⠉⠛⠛⢿⡀⠑⢬⣷⡾⠁⠐⠉⠂⠈⢶⣾⡥⠊⢀⡿⠟⡛⠉⠀⠀⡤⠋⢀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡿⣷⣤⠀⠈⣇⠀⠀⠀⠀⠀⣸⠁⠀⣤⣾⢻⡄⠀⠈⠉⠀⢴⣾⡟⠢⠖⢤⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡇⣿⣿⡆⢻⡍⠀⠀⠀⠀⠀⢡⡟⢸⣿⣿⢸⡷⡄⠀⠀⠀⠘⠂⠑⠐⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠇⢸⣿⣿⠦⠿⣦⡀⠀⢀⣴⠿⠴⣿⣿⠇⠸⠀⣿⠂⠤⠄⣀⠐⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⡏⠀⣦⣌⠲⣷⠞⣡⣴⠀⢹⣿⡆⠀⠀⠈⠉⠚⠓⠀⢧⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡀⣄⢸⣿⡷⢦⣽⣿⣿⣿⣿⣿⣧⠴⢾⣿⡇⣀⢀⠄⡆⠀⠀⠈⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣿⣌⠈⣿⣶⣿⡿⠟⠛⠻⣿⣿⣾⣾⠑⣡⣿⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠙⢿⣧⠻⣿⣿⣧⣤⣤⣤⣼⣿⣿⠟⣼⡿⠋⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⠲⣿⡆⠈⣿⣿⣿⣿⣿⣿⣿⠁⣰⣿⠖⠉⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠈⠙⢦⢹⣿⣿⣿⣿⣿⣏⡴⠋⠁⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠦⠈⢳⣿⣿⣿⣿⣿⡞⠁⠴⠛⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢾⣏⣽⠛⠉⠉⢳⡄⠈⠉⠛⢷⣦⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⢰⣿⡟⠛⠛⠲⢾⡷⠦⣤⡀⠀⣿⠻⣍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⢿⣶⣤⡘⢿⣇⣀⡀⠀⢸⡇⠀⠀⣿⣴⠏⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⡏⠀⠙⠛⢿⣾⡿⢿⣯⣯⣿⣟⣻⣿⣿⠁⡄⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣟⣛⣛⠛⣿⠛⠻⣾⣧⠀⣿⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣻⡆⢀⣤⠶⣾⣿⠾⠿⠿⢿⣿⣿⣿⣿⣏⠀⣿⣭⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣿⣷⣾⣵⠛⡟⣿⠛⠒⢶⣾⡧⠴⠶⣿⣟⠀⠁⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣄⡹⣿⣿⣿⣿⣿⡿⠛⣿⠉⢻⡟⠛⢿⣿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣾⣿⣿⣯⣧⣤⡴⠟⠀⢠⡇⠀⠀⣻⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣤⣀⣠⣴⠟⠁⣀⣴⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠏⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢿⡟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣴⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⣲⡟⣱⣶⣶⣶⡵⣤⡀⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⡠⣒⣐⣒⠢⡤⢂⣵⠾⢛⣫⠽⡇⣿⢯⣭⢿⡇⣿⣮⡙⠾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⢈⢸⠁⠀⢠⡇⣿⢟⡥⠚⡿⣓⠴⠆⡟⢶⣝⢺⣿⣿⣹⠿⣦⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠈⡘⢄⡠⣣⣿⡸⠏⠠⠾⠴⠧⠀⠿⠇⣼⣿⣾⣧⣸⣿⡀⠘⢷⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡟⠉⢿⣦⠀⠀⠀⠀⠀⠙⢮⣝⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣟⣷⡠⣿⣿⣷⣽⡎⡿⠋⠷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠋⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠘⢿⣽⣿⣽⢫⠋⣭⣭⡭⢭⣭⡭⡏⢛⠻⣿⣿⣯⣟⢇⠐⠺⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠏⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⢸⢺⣳⡿⠹⠿⠳⠦⠤⠤⠬⠤⠤⣷⣮⣴⣿⣿⣷⣏⡃⢈⣻⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣀⣀⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⡘⣫⡴⢒⣛⣛⣛⣛⣛⣛⣚⣛⣛⣧⣐⣨⣿⡟⠀⢿⡇⠀⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠛⠛⠛⠿⠶⢦⣤⣀⡀⡜⠉⠙⢶⢠⡏⢾⣿⢽⡋⢉⣉⠿⢛⣛⣛⣛⢏⠉⠙⢯⢆⠠⢸⠁⡼⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣮⣿⡳⠦⣄⡘⣇⠻⣌⣛⣛⣩⠵⠒⠋⠉⠻⣿⢻⣧⢡⡀⠜⣼⢂⡿⠊⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠿⠛⠁⠀⠉⠛⠿⠻⣿⣿⣶⣭⣿⣷⣍⡻⢿⣷⡠⠤⠀⡀⠠⢊⣬⠛⣸⣟⣿⡷⠛⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣼⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣷⣶⣭⣝⣓⣒⡒⢛⣡⣴⠟⢉⣁⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣙⢟⣉⡛⣻⣻⣹⡟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠲⠶⠶⠶⠖⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢋⣁⣤⣀⡉⠉⠛⠲⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⣄⣀⠀⢀⣀⡤⢞⣫⡴⠞⠋⠀⠀⠈⠙⠳⢦⣄⡀⠀⠉⠙⠛⠒⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠙⠻⠿⠷⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠒⠛⠉




⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣠⠶⢤⠀⣿⠴⢤⠀⣠⠶⢤⡀⢾⡧⢠⡤⢶⡄⢠⡤⠦⠀⠀⣤⠀⣤⠤⣄⢀⡴⠦⣄⢠⣤⠤⣄
⠀⠀⠀⠀⣙⠓⢶⡄⣿⠀⢸⡇⣯⠀⢀⡗⢸⡇⣰⠖⠒⡇⢸⡇⠀⠀⠀⣿⠸⡇⠀⣀⢸⡀⠀⣿⢀⡇⠀⣿
⠀⠀⠀⠀⠙⠛⠋⠀⠛⠀⠘⠁⠈⠛⠋⠁⠘⠃⠈⠓⠋⠛⠈⠁⠀⠀⠀⠙⠀⠙⠛⠉⠈⠙⠛⠁⠘⠃⠀⠙




⣿⡿⠛⠿⢛⠿⡿⢻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡀
⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⣸⡇⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢸⣇
⠀⠀⠀⣿⠃⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠘⣿
⠀⠀⠀⣿⡆⡄⡈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢁⢀⠠⣿
⠀⠀⠀⣿⡆⢃⠁⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠈⠈⢰⣿
⠀⠀⠀⢻⣇⠸⡀⡌⢿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⢡⢀⠇⣸⡟
⠀⠀⠀⠸⣿⣀⢣⠸⡌⢿⣞⢿⣿⣿⣿⣿⣷⣿⣍⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⣩⣿⣾⣿⣿⣿⣿⣿⣿⡿⢡⠇⡼⢀⣿⠇
⠀⠀⠀⠀⢻⣧⡎⢆⠙⡌⢿⣮⣻⣿⣿⣿⣿⣿⡏⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢹⣿⣿⣿⣿⣿⣟⣽⡿⣡⠏⡴⢱⣼⡟
⠀⠀⠀⠀⠈⢿⣿⡔⢳⠈⢦⡙⢷⣽⡻⣿⣿⡟⠁⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠈⢻⣿⣿⢿⣿⡿⢋⡾⢁⡜⢢⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣟⣆⢳⡱⣝⢦⣹⣿⠟⠋⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠙⠻⣿⣏⡴⣫⢊⡞⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣸⠻⠿⡷⠛⠺⠟⠛⠁⡁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠈⠛⠻⠗⠛⠾⠿⠟⣇
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣦⣶⠶⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠷⠶⣶⣴⠶⠶⠛⠁

⠀⠀⢀⣀⡀
⠀⠀⠙⣮⢃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠹⣧⠀⠀⠀⣀⠴⣺⠭⠉⠀⠀⠀⠉⠉⢓⠦⣄⡀
⠀⠀⠀⠀⢻⡆⠀⠘⡵⠋⠀⣠⠤⣄⠀⠀⠀⠀⠀⣈⣶⣝⠢⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢿⠀⡞⠑⢿⣾⠛⢲⣼⢷⣦⠀⠀⢸⡏⠀⢹⡷⣜⢦⠀⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⠀⣇⢀⣼⠓⢢⣴⢏⡾⠃⠀⢀⡾⢠⡴⠞⠁⠈⣎⣇⠀⠀⠀⢿⠋⣉⡻⣷⡄
⠀⠀⠀⠀⠀⠀⡄⢿⣿⠁⠀⡜⠃⢺⠷⣶⣠⡞⢰⣏⠀⠀⠀⠀⢸⢸⠀⠀⢠⣎⠀⠋⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⡜⣿⣆⢠⣼⠚⣯⣤⣛⡏⠀⣜⡻⣿⣶⣄⣀⡾⡟⠀⣴⣿⣮⡃⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠙⣎⢻⣿⣧⡟⠁⣴⠿⠶⠚⠉⠙⣷⢿⣠⠞⢙⣴⠾⢋⡵⠋⣽⣦⣤⣴⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⡻⢷⡿⠚⠢⠤⠤⠤⢶⣛⣫⡵⠾⠋⣡⣶⣿⠶⠟⠋⠉⠉⠈⠻⢭⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣭⣙⣛⣋⣉⣉⣀⣠⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⡀
⠀⣇⣈⠿⣏⣉⣉⣉⣉⣉⠉⡉⠥⠛⠋⢉⣉⣉⣉⣉⣉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢳⡀
⠀⠀⠀⠀⠀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢧
⠀⠀⠀⠀⠀⢸⠁⣸⠀⠀⠀⠀⠀⣔⣲⢠⣒⣄⣖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠸⡀⢹⠀⠀⠀⠀⠀⢈⡇⠀⣿⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⢇⠘⡆⠀⠀⣀⣀⣿⢸⣸⠉⣇⠃⠀⠀⠀⢀⣀⣀⠀⠀⣼⠃⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠙⣄⡎⡴⠒⣿⢸⢺⠍⠃⠀⠀⠀⢀⠒⠂⠈⠇⡼⢹⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠙⠈⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣸⠼⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⣁⣠⠤⠴⠒⠉⠀⠀⣠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞

⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⢰⡟⠀⠈⢷⣄
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠰⣼⡆⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢰⡏⠀⠶⢼⡇⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⣀⣴⡛⠻⠞⠉⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠚⣛⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠒⠛⠃⠀⠀⠀⠀⢸⡇⠀⠉⣹⠇⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⡄⢰⠀⡇⢠⡇⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢻⡟⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⢦⣾⢁⡟⣸⠃⣸⣃⣼⠟
⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠀⠀⠹⣆⣉⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⣠⡴⠀⢀⡴⠀⣠⠄⠀⣠⠖⠀⣠⡴⠀⢀⡤⠀⣠⠄⠀⣠⠖⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠉⠀⠈⠁⠀⠀⠁⠀⠈⠉⠀⠀⠋⠀⠀⠁⠀⠈⠁⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠀⠀⠀⠀⡀⢠⠀⠀⡄⢀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣠⢐⣷⡏⠀⠀⠹⣾⡂⢄⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⠀⢠⣼⣗⣟⣿⣶⣤⣤⣶⣿⣻⣺⣷⡄⠀⡀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣷⣿⣦⣍⡛⢿⣿⣿⣿⣿⡿⠛⣩⣴⣿⣾⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⠻⣿⣿⣿⣧⠀⠀⠓⠘⠀⠀⢬⣿⣿⣿⠟⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢛⣁⠀⠀⠀⠀⠀⠀⣈⡛⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣶⡶⢶⣶⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢀⠘⢿⣔⠌⢉⡙⣷⣾⠋⡉⠀⣢⡿⠃⡠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⣿⣿⡿⠉⠻⠟⠋⢿⣿⣿⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣼⣿⣶⠦⠤⣶⣿⢹⡿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣍⠙⠋⢩⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠿⣶⣶⠿⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢀⢀⢀⢀⣀⢀⢀⠀⡀⡀⣀⠀⢀⢀⢀⡀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠘⠚⠘⠋⠘⠁⠋⠃⠉⠃⠉⠀⠈⠛⠊⠃⠙⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠽⠉⠉⠉⡉⠉⠉⠉⠉⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⡀⢀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠙⠀⠂⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢡⡄⠀⢀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣨⠁⠈⠁⠆⠁⢀⠒⠀⠀⢀⠀⣄⠀⠚⠈⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠂⠀⠀⢄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠈⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠰⠞⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠢⠂⠀⠠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣂⣦⣾⣿⣿⣿⠀⠀⠀⠀⢀⡴⣶⣶⡶⠞⢻⡟⠛⠷⣦⣄⠀⠀⢸⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣋⣘⣿⣄⣿⣶⣄⠀⢀⡿⢷⣄⣼⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣰⢫⡟⠻⢿⣿⣿⣿⣿⣿⣀⣼⢧⠈⢻⣿⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡟⣽⣁⣀⣺⣿⣿⣿⣿⣿⣿⣇⣾⠀⠈⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⠉⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣷⡔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢈⣙⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣧⣿⣿⣿⡟⢿⣿⠛⣿⡟⡟⠀⠐⢾⣿⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⣦⣾⣿⣼⣿⠿⠃⠀⣠⣿⠃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠐⠂⣀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⣻⡿⠿⠿⠿⢯⣤⣶⣾⠟⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠃⢁⣠⠴⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣾⣿⡿⠿⠛⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠛⠛⠛⠋⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⣤⣀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⢉⣭⣤⣬⣉⣿⠻⢷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⡟⠁⠀⠀⢹⣿⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⣼⣿⢃⣀⣀⣀⣠⣧⣀⣀⣀⣸⣿⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⡇⠘⠒⠒⠒⢲⣷⠒⠒⠚⢻⣿⣾⠠⠤⣤⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢸⣿⡇⠀⠐⢭⢭⣭⣯⣭⣭⠽⢿⣿⣿⠀⠒⠲⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣿⣧⠀⣔⣒⣒⣚⣟⣒⣒⣲⣾⣿⢿⠀⠘⠛⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡘⣿⣧⠀⠉⠉⠉⡏⠉⠉⠁⢸⣿⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⣧⠀⠀⠀⣸⣿⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⠠⠀⠀⠠⠀⠀⠀⠀⠂⠀⠂⠀⠐⠀⠰⠐⠂

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⢀⣄⢤
⠀⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢤⡄⠀⠀⠐⡤⡀⢀⠀⠉⠁⠀⠀⠀⠀⠁⡄⠀⠀⠆⢀⠀⣯⡼⠏
⠀⠀⠀⠀⠀⠀⠀⢐⣦⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡏⠀⠀⠶⠛⠀⠀⢀⡴⢮⣿⣤⠀⠈
⠀⠠⠚⡠⠟⠀⠖⡾⠋⠁⠉⠑⠠⣀⠀⠀⠀⢒⠢⢈⣻⠋⠀⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠙⢏⡺⣄⠸⣯⡀
⠀⠑⠚⠁⠀⢀⠋⠀⠀⠀⣿⣶⣄⠀⠑⠄⠀⠉⠀⠞⠩⣠⠄⠀⠀⠄⢀⣴⡶⢶⠆⠀⠀⠀⠈⠙⢅⠀⠈⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⣿⣿⣤⡀⠀⠀⠐⡀⠀⠀⠚⠁⠀⠀⢀⣲⠋⣞⣲⣶⠂⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠹⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠈⢾⣓⣒⣲⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠙⣿⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠳⠶⠆⠀⠀⠀⠑⠤⢀⡀⢀⣀⠀⠀⠀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠈⢁⠈⠒⠂⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃
⠀⠀⢰⡇⡶⠄⠀⠙⠧⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠰⣄⠀⢀⣤⢄⣀⣀⡀
⠀⠀⠀⠙⢤⣤⡀⠀⠀⠀⠀⠀⠁⣤⠾⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⢹⡄⠀⠻⢆⠰⠖⠚⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢠⠞⠁⢠⡀⠀⠀⠠⠖⣸⠆⡇⠀⠀⢸⠀⠀⠀⠈⠓⠦⣄⡉⠉⠀⠀⢷⡂
⠀⠀⠀⠻⠖⠀⠀⣠⠞⠀⠴⠃⠀⠀⢸⠀⢠⠀⠙⠋⠀⠀⡇⠀⠀⠸⡄⠀⠀⣦⠀⠀⠀⠉⠲⣄⡀⡀
⠀⠀⠀⠀⠀⢀⡴⢃⡀⠀⠀⣷⠀⠀⢸⠀⢘⠀⢠⣤⠀⠀⡇⠀⠀⢨⡅⠀⠀⡧⠀⠀⢀⠀⠄⠈⠉⢫
⠀⠀⠀⠀⡰⠋⠀⠀⡇⠀⠀⣿⠀⠀⣸⠀⢸⠀⠈⠳⠂⠀⡇⢠⠀⢘⡃⡀⠀⡏⠀⠀⠀⡅⠐⠀⠀⠀⢧
⠀⠀⠠⠎⠀⢲⠀⣀⣇⣀⣤⣿⣠⣤⣿⣄⣼⠀⠀⠀⠀⠠⠧⠼⠶⠾⠦⠷⠴⠷⠤⠦⢤⣇⣀⠀⠀⠀⠈
⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠴⣄⣀⠀⠀⠀⠀⡀⠀⡰⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢉⣇⣠⣤⡄⢀⣀⢰⠞⠀⠀⠺⢃⡇⠀⢠⠫⠁⠀⠀⣠⠤⠷
⠀⠀⠀⠀⢀⡱⠖⠁⣀⡟⣹⣖⣢⡾⣿⠃⢨⡿⠉⠻⠀⠀⠀⠛⠃⠀⠈⠈⠉⠀⡴⠕⠀⣀⠐⣄⠀⠀⢀
⠀⢁⢀⣑⣺⠁⠀⣿⣛⠐⠂⣿⠁⠀⠋⣡⡴⠆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠆⠸⠍⠀⣿⠀⠀⠠⣄
⠀⠘⣠⠄⢸⢇⣀⢿⢟⢀⢀⡹⠖⢋⠄⠉⠁⠈⠁⠀⣠⡨⠤⠀⠠⣀⣀⣈⠀⠀⠀⠈⠂⠶⠚⠁⢀⢔⠀⢀⠠
⠀⠈⠀⠀⠰⠺⠿⠞⠟⣄⣀⣉⣼⣀⠀⠄⠂⠀⠠⡎⣸⠻⢌⣪⡝⢤⠛⠳⠄⢀⡀⠂⠀⠀⠐⠊⠝⠄⣶⠀⠁
⠀⠀⠈⠓⢢⣬⣓⠆⢀⣠⠬⠋⠋⠁⠀⠀⠀⠁⠀⢙⡉⣹⢞⡿⠧⠾⠒⠁⠀⠉⠁⠀⠀⠀⠈⡁⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠡⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠛⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠁⠀⠀⠀⠉⠻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠰⠶⢶⣤⣀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠙⠳⣄⠀⣸⠏⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠹⣧⣿⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠈⠀⠉⠉⠈⠉⠉⠈⠁⠁⠉⠀⠈⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠋⠁⠀⠀⠈⠉⠻⠇⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢠⣤⣤⣼⣿⣿⣥⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠈⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁


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

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄
⠀⠀⠷⣅⣡⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠠⣆⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢄⡀⠀⠀⡾⢋⣷⠍
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⡄⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⡀⢡⠆⣴⠇⠀⢀⣠⡄⠀⠀⠀⠀⣈⣦⣤⡀⠉⠀⠆⠀⠙⠋⠁
⠀⠀⠀⣀⠀⣀⠀⠐⠀⣠⣿⡿⢐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢺⣶⠟⠀⠀⠛⠉⠀⠀⢀⡠⠜⠛⠻⠿⣿⣁⠠⣀⠀⢀⡄⠄
⠀⢀⠊⢡⠄⠟⠀⣘⢛⠋⠁⠀⠀⠀⠈⠑⢄⡀⠀⠀⠈⢢⣙⠆⢀⣝⠁⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠈⠫⣘⡛⡆⠈⢷⣼
⠀⠘⠒⠋⠀⠀⠀⠈⠁⠀⠀⠀⢘⢿⣦⣄⠀⠈⠲⡀⠀⠉⠀⢑⠞⢩⣱⡄⠀⠀⡔⠁⣠⡴⣶⣶⡶⠀⠀⠀⠀⠁⠉⠻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣞⣿⣿⣉⠀⠀⠀⠀⢤⡀⠀⠀⠀⠶⠋⠀⠀⠄⣤⡴⢷⠾⡤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⢓⡄⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠐⠓⣇⣼⣁⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣸⣷⢿⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠻⠽⠥⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠏⠀⠀⠀⠀⡘⠀⢀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠑⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠘⠗⠶⡦⠀⠀⠀⠙⠢⢄⠀⠀⠀⢀⡀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠲⠶⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢴⣾⠻⠆
⠀⠀⠀⢠⡤⢀⡀⠀⠀⣶⡆⠀⠀⠀⠀⠀⡀⠀⣸⠇⠀⡆⠀⠀⠀⠀⠀⠃⠀⠘⠓⠂⠀⠀⠀⣀
⠀⠀⠀⠸⡤⠊⠁⠄⠀⠀⠙⠁⠀⠀⠰⡎⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠈⢷⠀⠀⢸⣏⠑⠶⢦⡒⣤⠄
⠀⠀⠀⠀⠈⠠⠵⢦⠀⠀⠀⠀⠀⠀⠀⢀⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠂⢸⡇⠀⠈⠛⢦⣊⡋
⠀⠀⠀⠀⡀⠀⠀⣸⠃⢀⠀⠀⠀⠀⣠⠟⠀⠀⣤⠀⢰⠀⢀⡴⠆⣼⠆⣄⠀⠀⠀⢸⠀⠀⠀⠠⠀⠙⠳⢤⣄⡈⠉⠁⠀⠈⢿⡆
⠀⠀⠀⠀⢯⡴⠀⠁⠀⢀⡶⠂⢀⡼⠃⠀⠀⠀⢸⠀⢸⠀⠘⠞⠋⠁⠀⣿⠀⠀⠀⢸⠄⠀⠀⠀⣄⠀⠀⠀⠈⠙⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⡄⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠻⠀⠀⠀⠁⠄⠀⠈⠻⠶⡶
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠒⣦⠀⠀⠀⡿⠀⠀⢀⡛⠀⢸⠀⠐⣯⠓⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⢰⠄⠁⠆⠀⠀⠘⣆
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠉⠀⠀⣿⠀⡦⠀⢨⡄⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠂⠀⠀⠛⡄
⠀⠀⠀⠰⠋⠀⢐⡆⢀⣀⣧⣀⣴⣼⣯⣠⣤⣼⣧⣤⣼⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠾⠷⠾⠷⠶⠶⠦⠴⢦⣼⣦⡀
⠀⠀⠀⠐⠀⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⡠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⢻⡇⠀⣀⣀⠀⠀⠄⠉⣨⡿⠀⠀⢀⣶⠀⡷⠀⠈⢠⠖⠆⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠤⡀⢀⡤⠀⢠⡤⢀⢀⡀⢀⣸⣿⠀⠸⣷⡷⣿⣾⠃⠀⠚⢀⣿⡄⠀⠀⢀⣡⠔⠁⠀⣰⠋⠊⣀⡀⠀⠑⠛⠉⠉
⠀⣀⠀⡀⠀⢰⣦⠛⠁⣠⣄⡫⠉⠙⣮⡵⠿⢻⣧⡀⢸⠟⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠁⢀⣤⡀⢳⡄⠀⠀⠀⠄
⠀⠨⡈⣅⣙⣹⡇⠀⠈⣿⣋⡉⠉⢙⡿⠀⢀⠚⢀⣽⡗⠒⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠸⠀⠻⠅⠁⣸⡄⠀⠀⠀⣦⠄
⠀⠀⠃⣀⡀⢸⣗⣀⣀⡿⠶⠂⡀⠀⢯⠔⠋⣀⠈⠙⠁⠊⠉⠀⠀⡀⣈⣄⡀⠠⢀⠀⠀⠀⡉⠀⠀⠀⠀⠑⠠⠤⠾⠊⠀⢀⡜⠁⠁⠀⡄
⠀⠀⠒⠑⠀⠀⣤⣽⣇⣻⣮⠄⠉⠀⠂⢀⠊⠁⠀⣀⠠⠄⢀⣤⡾⣹⣯⡒⠠⡞⢟⣉⠹⣾⣄⠀⠀⠀⢠⡒⠄⠀⢀⡤⠒⢷⡈⣤⠀⡙
⠀⠀⡐⢠⣄⠈⠠⣌⣉⠈⠉⠄⠰⠶⣚⠯⠗⠊⠉⠀⢀⢀⠉⠙⣕⠾⢬⣉⣿⣿⠂⢸⣇⣤⠉⠁⡐⠶⠀⠀⠀⠀⢠⠤⠈⢴⠇⠘⠃
⠀⠀⠀⠀⠈⠘⠳⠮⢅⡀⢠⠴⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣴⣭⣽⣛⣛⣋⣉⠀⣀⠀⠀⠀⠀⠀⠀⠀⢘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠐⠒
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣧⣴⣤⣵⣬⣼⣯⣵⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠋⠈⠉⠉⠉⠉⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠀⣀⡀⠤⠀⠐⠂⠒⠒⠒⠂⠤⢄⣀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⡤⠒⠉⠀⠀⢀⡀⠤⠤⠤⠤⠤⠀⣀⠀⠀⠈⠐⠤⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠞⠁⠀⠀⠀⠈⠀⠀⢀⣀⡀⠀⢀⣀⣀⡀⠀⠉⠒⠤⡀⠀⠑⢄⡀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡠⠊⠀⡠⠂⠁⠀⠀⠐⠈⠀⢀⣀⡀⠀⢀⣀⣀⠉⠉⠒⠤⡀⠈⠑⢀⠀⠉⢦⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡔⠁⡠⠊⢀⠤⠊⢀⡠⠔⠊⠉⠁⠀⣀⣀⣀⠀⠀⠉⠑⠂⢄⡀⠑⢄⡀⠑⢄⠀⠱⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡜⠀⡔⠁⡠⠁⣀⠔⠁⢀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⠀⠀⠈⠢⡀⠑⢄⠀⠳⡀⠱⡀⠈⡆
⠀⠀⠀⠀⠀⠀⣀⠼⠤⠴⣀⡜⢀⣎⣀⡔⠁⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠈⠢⠀⠣⠀⠱⡀⢱⠀⢰
⠀⠀⠀⠀⣀⠎⠁⠀⠀⠀⠈⠖⠋⠁⠀⠈⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢱⣀⠷⠤⠷⡀⣦⠚⠁⠒⠒⠒⠐⠢⡀
⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⡠⠒⠠⠔⠚⠁
⠀⠀⠈⠒⠢⠤⠒⠉⠑⠤⠀⣀⣀⣀⡠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠉

⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣀
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢺⡷⣾⣿⣷⢾⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡤⣾⡖
⢀⣴⡆⢸⠀⣶⣤⠀⠀⠀⠀⠀⠘⠇⢻⣿⡟⠸⠃⠀⠀⠀⠀⠀⣿⡏⠉⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⡇
⣾⣿⣿⢼⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣿⡇
⠛⠛⠛⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⡹⠃⠀⣤
⠀⣤⣴⣯⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠆⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⠀⢤⣄⣠⣿⡆
⠐⢷⣿⠟⠳⣤⣀⠀⠀⠀⠀⠀⠸⣿⣿⠳⢤⣀⡀⠀⠀⠀⠀⢠⣾⣿⣤⣤⣀⣘⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⢹⣿⡇
⠀⣼⠛⣧⠸⣿⠿⠇⠀⠀⠀⠀⢸⡿⢿⡄⣿⡿⠷⠀⠀⠀⠀⢘⣿⣷⢾⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠸⠏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠿⢠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇


⠀⢀⣠⡤⣤⣀
⣰⠛⡿⠤⠼⢿⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢀⣀⣠⣆⣰⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⡷⢺⡗⠒⠒⢾⠒⡇⠀⠀⠀⠐⣶⣾⡏⢯⢹⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣴⣿⣿⣿⣿⣶⡆
⠹⣶⣷⠒⢒⣿⡴⠁⠀⠀⠀⠀⢿⠿⠷⣖⣫⡭⠿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠿⠿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁

⠀⠀⠲⣏⢫
⠀⠀⠀⠹⣦⠆⠀⠀⠀⠀⣀⡤⠴⣒⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⢻⣎⡀⠀⡠⢚⡵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠒⠭⡓⢤⡀
⠀⠀⠀⠀⠀⢿⡤⠀⣴⣃⡄⣠⣾⠟⢳⣄⡀⠀⠀⠀⣠⡶⠿⢷⣝⢦⡀⠀⠀⠀⠀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠘⡇⢰⠃⠀⢻⡟⠁⠙⣿⢟⣿⠆⠀⠀⣿⠀⠀⣸⡏⢣⡹⡄⠀⠀⠀⢿⣿⡿⣮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⡟⠉⢻⣾⢃⣾⠃⠀⠀⣼⢏⣴⠟⠋⠀⠀⢳⢻⠀⠀⠀⠘⡏⢀⣤⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠰⡘⣿⣇⠁⠀⢞⠑⣻⠿⢻⣧⡾⠃⣻⣅⡀⠀⠀⠀⢸⢸⠆⠀⢀⡼⣆⠈⠉⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡹⣿⡄⢠⣼⠋⣻⣤⣜⣭⠅⠠⣒⡺⣟⡶⢤⣀⣼⡼⠀⣠⣾⣶⣮⡂⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⢿⣿⣆⣟⠁⣰⡼⠶⠞⠋⠉⠙⣶⢿⣠⡾⠇⣡⡬⠏⣩⠟⢩⣿⣠⣴⣶⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣝⠿⣿⡾⠖⠧⠤⠤⠤⠤⠴⢟⣛⣥⡶⠟⢃⣴⣾⣶⡶⠟⠛⠋⠉⠙⠻⣍⣉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣄⣘⡛⠒⠒⠛⠛⠋⠉⣀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣠⣶⠾⠿⠿⠿⠿⢶⣦⣄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⢀⣀⣠⡏⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠘⣆⠀⠀⢀⣠⣤⣷⣤⣄⡀⠀⢹⣧
⠀⠀⠀⠀⠀⢰⣿⣠⡾⠟⠛⠛⠿⣶⣄⣤⣤⣼⣧⣤⣀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢸⣤⣾⠟⠉⠁⠀⠈⠙⠻⣦⡘⣿
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠈⠛⠉⠁⠰⠶⢭⣙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣠⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⢀⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣤⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠙⠻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣭⣋⣠⣤⣶⣧⡀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠻⠷⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⢀⡄
⠀⠀⠀⢼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⡀⠈⠙⠛⠻⠶⠶⢦⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⣀⣠⣄⣠⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⡿⠟⠟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⢻⠏⣿⠉⠉⠉⠉⠉⠉⠙⠻⣏⠁⠀⠀⠉⠁
⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠐⣇⠀⣀⡀⠀⠀⠀⠀⠤⠄⠀⢸⡇⣸⣦⡿⠤⠄⠀⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠙⠳⢷⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇⢼⡽⣿⠭⠭⠭⠭⠷⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠂⠿




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡠⠄⢤⣀⠀⠀⠀⠀⠴⠖⠒⠀⠀⠀⠀⢀⣴⡾⣆⠀⠀⠀⠀⢠⢲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⣧
⠀⢻⣿⣤⣤⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢺⣿⣿⣧⣽⡶⠀⠀⠐⢡⠏⠀⠀⠀⠀⠀⠲⣿⣿⡿⡀⠀⠀⢰⡄⠀⡀⢰⣆
⠀⠘⠿⠿⠿⠕⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢼⣶⣶⣾⡇⠀⠀⠉⠁⠀⠀⢩⡍
⠀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣏⠀⠀⠂⠀⠅⡆⠀⢠⣶⣶⣶⣶⠄⠀⢰⣿⣿⣿⣿⡇⠁⠀⠀⠀⣱⣀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠄⠀⠀⠰⡥⠤⢽
⠀⣇⠀⠀⠀⠀⣃⡇⠀⢰⢘⢘⣋⡃⡄⠀⢸⣿⣿⣤⣼⡟⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⣹⣿⣿⣿⣿⠂
⠀⠈⠁⠀⠀⠀⠉⠀⠀⠘⠤⠤⠤⠠⠂⠀⠈⠻⠿⠿⠿⠍⠃⠀⠀⠀⠈⠛⠉
⠀⠀⠤⠀⠀⠀⠀⠀⢠⠤⠤⢤⡀⠀⠀⣤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⢠⡤⠤⠤⡄⠀⠀⢀⣀⣀⣀
⠀⠀⠨⣧⠀⠀⠀⠀⠀⠐⠲⡆⡇⠀⠀⡇⣤⠦⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡽⡀⠀⠀⢸⡧⠐⠀⡇⠀⠀⠸⠤⠶⢿
⠀⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠔⢃⠀⠀⢸⡀⠀⠀⡇⠀⠀⢸⠩⠭⢹
⠀⠀⠓⠒⠒⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠈⠉⠀⠉⠁⠀⠀⠈⠉⠉⠉
⠀⡴⠒⢦⣤⣤⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣴⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤⣄
⠀⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣧⣤⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠁⠀⠀⠸⡿⣹⣿
⠀⠛⠛⠛⠛⠛⠃⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠐⠒⠶⠖⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⣠⡶⠒⠶⠀⠀⠀⢠⡖⢲⡄⠀⠀⢀⡤⠲⢤⠀⠀⠀⢀⣀⣀⢀⠀⠀⢀⣤⣄⡀⣄⠀⠀⠀⣴⠛⡆⠀⠀⠀⠀⣠⣄⡀
⠨⣿⡿⠏⠀⠀⠀⠀⠺⡷⠂⠀⠀⠀⢾⢦⣈⢻⡇⠀⠀⢰⡎⣀⣰⡆⠀⢸⣿⣿⣅⣙⠀⠀⠀⣿⣛⣿⠀⠀⠀⠐⣿⣿⠇
⠀⠈⠛⠛⠛⠀⠀⠀⠘⠛⠲⠂⠀⠀⠈⠳⠴⠚⠁⠀⠀⠃⠀⠀⠐⠁⠀⠘⠛⠛⠛⠛⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠘⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣶⣶⡦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠒⠉⡶⠀⠠⣄⠈⠐⠽⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠊⠀⠀⢀⠁⣐⠀⣇⠱⡦⠀⠈⠜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠀⠀⠀⢸⠺⣿⠇⣇⠼⡄⠀⠀⠈⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢦⡄⠀⠀⠀⠀⡇⠀⠀⠀⡤⠚⠂⠉⠀⠀⠀⠘⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣴⠶⠞⠛⠉⠀⠀⠀⠀⠙⣛⢦⡀⠀⢇⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⢹⠃
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠠⠼⠟⣂⡿⠀⠘⢎⢄⠀⠀⠇⠀⠀⢀⡀⠀⠐⠈⠠⣢⠏
⠀⠀⠀⠀⠀⣠⠴⣲⠿⣷⣶⣶⣶⣶⣿⣿⡟⠉⠀⠀⠀⠈⠳⢕⡤⣘⠈⠀⠀⠀⣀⠠⣢⠕⠁
⠀⠀⠀⠀⠊⣠⠞⢁⡾⠁⠀⠀⠀⠀⠨⠭⠽⣦⠀⠀⠀⠀⠀⠀⠉⠓⠒⠿⠾⠓⠚⠉
⠀⠀⠀⠀⠘⠁⢠⡞⠁⠀⢀⡠⠢⠴⡈⢹⣭⣽⣧
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡆⢴⢰⡆⠤⠄⠤⠤⠼⣇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⢰⢠⡄⡝⠄⠩⠭⠭⢽⡄
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠐⡒⠬⠸⠇⠆⡰⠴⠧⠉⣋⣽⣿⣿⠛⠐⠒⠒⠲⠖
⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠈⠒⠤⢆⠮⢥⣀⣀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠸⣇⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⠓⠀⠤⠀⢤⢀
⠀⠀⠀⠀⠀⠈⠙⠓⠾⣦⡀⠀⠀⠀⠀⠈⠉⠒⠛⣿⣿⣤⠄⣀⢀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠒⠛⠀⠀⠀⠀⠿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠉⠐⢄⡀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡘⠀⢀⣀⡀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⣀⣀⠘
⠀⠀⠀⠀⣀⡷⠛⠋⠉⠙⢠⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠀⡠⠊⠉⠙⠛⢷⣀
⠀⡰⠖⠛⠉⠉⠀⠀⢀⣄⡐⠋⠙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⢂⣤⡀⠀⠀⠉⠉⠛⠲⢆
⠀⠁⠀⠀⠀⠀⢠⡞⠉⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠉⣳⠄⠀⠂⠀⠀⠈
⠀⠀⢳⡄⠀⠀⠀⢻⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⢀⣠⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠷⣤⣤⣤⣤⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠋⠳⣤⣤⣤⣤⠾




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠛⠛⠛⠋⠙⠛⠛⠛⠿⢿⣿⣿⣶⣤⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠉⢠⣤⠀⠀⠀⢰⣿⠄⠀⠀⢀⣤⡀⠈⠙⠻⣿⣿⣷⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣠⣾⣿⡿⠋⢁⡀⠀⠀⠈⠋⠀⠀⢀⣀⣀⣀⠀⠀⠈⠛⠁⠀⠀⣀⠈⠙⣿⣿⣧⡀⠀⠀⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠻⠟⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠘⠿⠃⠀⠈⠻⣿⣿⣄⠀⠈⠁
⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⣿⡟⢁⣤⡀⠀⠀⣠⠞⠉⠀⣀⣤⣶⣶⣿⣿⣶⣶⣦⣄⡀⠈⠳⣦⡀⠀⠀⣤⡄⠹⣿⣿⣆⠀⠀⣀⡀
⠀⠀⠀⠀⠈⠁⠀⢠⣿⣿⡏⠀⠀⠛⠁⢠⡞⠁⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀⠈⠳⣄⠀⠛⠁⠀⠸⣿⣿⡄⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⣤⡄⠀⣠⠏⠀⢠⣾⡟⠁⠀⣀⣤⠶⠞⠳⠶⢦⣄⠀⠀⠙⣿⣆⠀⠹⣆⠀⣠⣤⠀⢹⣿⣿⡀⠀⢀⡀
⠀⠀⠀⠻⠀⠀⢸⣿⣿⠁⠀⠙⠁⢠⡏⠀⢠⣿⠏⠀⢀⡼⠋⠀⠀⠀⢀⡀⠀⠙⢷⡄⠀⠘⣿⣇⠀⢹⡄⠘⠋⠀⠈⣿⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⠀⠀⣾⠀⠀⣿⡏⠀⠀⣾⠁⠀⠀⢺⣷⣿⡗⠀⠀⠀⢻⡄⠀⢸⣿⡀⠀⢷⠀⢀⣀⠀⢹⣿⣿
⠀⠀⠰⠖⠀⢀⣿⣿⡇⠘⠛⠃⢠⡇⠀⢸⣿⠁⠀⢸⡇⠀⠀⠀⠈⠿⠟⠁⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⡄⠘⠟⠀⢸⣿⣿⡀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠈⠁⠀⠘⠛⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠃⠀⠈⠁⠀⠀⠀⠀⠛⠛






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

⠀⠀⠀⠀⠀⠀⠀⣿⠖
⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⡾⠀⣠⣶⣿⠿⠛⠋⣉⣉⣯⣉⣉⣛⠻⠿⣶⣤⡀
⠀⠀⠀⣸⣧⣾⠟⠋⠀⣠⡾⠛⠉⠉⡏⠉⠉⠉⠛⠷⣮⡙⠿⣶⣄⡀
⠀⠀⢠⣿⡿⠃⢀⣀⣼⡏⠀⠀⠀⠀⣃⣀⠀⠀⠀⠀⠈⣿⣾⠛⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⢰⣿⡟⠀⠀⣿⠉⣿⡇⠀⠀⣰⡟⡛⠛⠻⣦⣀⣴⣾⣿⠿⢦⡀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀
⣠⣿⣿⣀⠀⢲⣿⡄⣿⣿⣿⣛⣿⠛⠛⠛⣻⣿⣿⡿⠿⠷⢶⣌⣷⣠⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆
⣸⣿⣏⣁⡀⢸⣿⠿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣟⣿⣟⠛⠛⠛⠻⣿⠋⠘⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀
⣹⣿⣍⠉⠁⢸⣿⠓⣿⠿⣷⣼⣯⣤⣭⣭⣽⣯⣿⣿⡷⠄⠀⢈⡷⠀⠴⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢹⣿⡏⠉⠉⢹⣿⠓⣿⣶⡶⢿⡿⠾⠶⠾⣿⢷⣿⣷⣶⣦⣤⣸⡇⠀⢠⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠘⣿⣿⡀⠀⢸⣧⢠⣿⠛⠛⢿⡿⠶⠶⢶⣿⡾⣿⣶⣿⣷⣤⣽⣷⡀⣾⡟⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠘⣿⣷⡀⠈⠷⣿⣿⣧⠀⢸⣧⣾⠛⢛⣿⣧⣿⣩⣿⣿⠀⢻⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠘⢿⣿⣤⡀⢿⣼⣿⣀⣿⣿⢿⣄⡼⣿⣈⣿⣯⣼⠋⢀⣼⣿⠏⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠻⣿⣿⣾⣿⣿⣿⣤⡿⣾⣍⣛⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡴⠤⠭⠭⢭⣭⡥⠴⢶⡴⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎
⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⠀⠀⠀⠀⠉⡁⠈⠁⠀⠒⠀⠈⠀⠀⠀⠁⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠉⠛⠛⠉⠛⠁⢻⡇⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠀⠂⢀⠄⠀⠀⠗⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠚⠓⠛⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡤⠐⠦⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠐⡀⠲⣀⠐⠆⠀⠑⡀⣀⡀⠀⠀⡀⠀⢀⣀⡤⣄⠀⠀⠀⠀⣀⣀⣴⠞⠋⠉⠃
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠻⠛⠉⠉⠉⠉⠉⠉⠉⠉⢻⠏⢿⡍⠻⠏⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠙⠉⠓⠄⠂⠀⠀⠌⠉⠻⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⣀⣠⢸⡰⠼⢓⠦⡇⡀⠀⢠⣄⣀⣀⣀⣜⣠⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⢤⣐⣦⣁⢐⣂⡛⠇⠹⠇⣛⡇⣯⢭⠭⠭⠭⠭⠭⠭⠭⡭⣭⡌⠀⠠⠆⠁⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢀⣽⠃⡭⡇⡍⠉⠉⠉⠀⠉⠉⠀⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠘⠋⠒⠃⠃


⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠘⠄⠀⠃⠀⠂⠠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢀⠔⠂⠈⠀⠀⠀⠈⠀⠢⡈⠠⠆⠀⡠⠠⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠠⡀⡹⠀⠀⠀⠀⠉⠀⢑
⠀⠀⠀⠀⠠⠤⣀⠁⢠⠀⢀⣄⡀⠀⠀⠀⠀⠀⠉⠈⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠈⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⢣⡀⢄⠀⢀⣀⠜⠉
⠀⠀⠀⠀⠐⠤⠤⠀⠰⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠠⠤⣀⠂⠉⠁
⠀⠀⠀⠀⢀⣀⠤⠭⠤⣣⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⢀⠅⠀⠀⠀⠀⡸⠑⢠⠞⠀⠀⡠⠔⠊⠉⠉⠉⠉⠒⠤⡀⠀⠈⢢
⠀⠀⠀⠘⠤⡄⠀⣀⡠⠒⢡⠋⠀⢠⠊⠀⢀⡠⠤⠒⠦⠤⡀⠀⠈⣦⠀⠀⢱⠊⠉⠢⠤⠤⠠⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⠃⠀⠰⠁⠀⡔⢁⣠⠤⠄⠤⣀⠈⠢⡀⠘⣇⠀⠀⡆⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠀⡌⠀⡞⠁⠀⠀⠀⠀⢣⠀⢡⠀⠸⡀⠀⢰⠀⠀⠀⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡨⢄⣀⣇⡰⠀⠀⠀⠀⠀⠀⠀⡀⢨⡀⢀⡀⠀⡼⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠖⠢⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⠈⠀⠀⠘⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣔⣲⡶⠀⠀⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⣾⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣶⠆⠺⡶⠺⡃⠀⣩⠀⢣⡄⠒⠂⢉⣁⡀⠠⣄⣤⣤⡤⠤⠼⣕⢄⡔⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠘⣤⣄⣧⣀⣿⣥⣞⣶⣾⢿⠯⠭⠄⣒⣂⠩⢥⡄⠠⢌⠲⡄⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢒⣂⡉⠭⢴⣿⡾⠿⡿⢿⠛⠉⢹⠹⡄⠀⠈⡆⠈⠉⠀⠀⠀⠀⣧⠀⠀⢡⢸
⠀⠀⠀⠀⣀⣀⣠⠊⡴⠋⣐⣒⣂⠩⠭⢷⠀⠀⢸⠙⡄⠀⠈⡆⢣⠀⠀⢡⠀⠀⣀⣀⡠⠤⠽⡄⢀⠜⡸
⠀⠘⠿⠙⠒⠒⠇⢸⠉⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠇⢣⠀⠀⢡⠘⡄⠀⠘⣯⣥⢤⠶⠶⠛⠓⣚⡡⠔⠁
⠀⠀⠀⠀⠀⠀⠘⢄⠳⢄⡀⠀⣴⡶⠞⣛⣳⠀⠀⠸⣼⡄⠀⠈⣆⣇⠀⠀⢱⠐⠒⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⠭⣜⣭⣥⣤⢴⡆⠀⠀⡏⢣⣀⣀⡵⠘⠤⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠀⠀⠀⠐⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠅⣀⠠⠐⠂⢉⠋⠉⠉⠉⡐⠲⠄⣀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠠⢁⡔⠁⢀⠔⠢⠀⡀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠉⠲⣄⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⠔⠡⠂⢀⡀⢠⠀⠀⠀⠂⠀⠠⠤⠀⠅⡉⣐⠤⢠⠈⢧⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠖⡈⢃⠀⡰⠉⠀⠀⠰⠀⡀⠐⠀⠆⠀⠀⢘⠀⠃⢇⠀⢀⠃⠀⠱⡀⠱⡀
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠠⠁⠠⢈⠃⠀⠤⠀⠀⢈⡓⠓⠀⠈⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⢱⠀⢱
⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢠⠁⡀⠀⠀⡼⠀⢀⠤⡀⠀⣨⢉⢁⠐⠀⠀⠤⢀⠈⡴⠋⠉⢢⢄⠀⠀⠈⡇
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⣼⠀⡇⠀⠀⡇⠀⢧⣀⡜⠀⣌⣉⡀⠀⠀⠀⠀⠀⠠⢱⣄⡠⠚⢒⠇⢀⠀⠑⠠
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢜⢆⠀⠀⢿⠀⡇⠐⢈⠁⠀⠀⠀⠄⠰⢢⠤⠀⠉⠁⠒⠠⠀⢼⠁⠀⢑⡀⣸⠀⠀⠀⡒⠈
⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠸⠆⢀⠞⢈⢀⠀⡀⠀⠒⠦⠶⠀⠄⠀⠀⠀⠀⠌⣈⠳⢔⠊⠈⠏⠀⠀⠀⣉
⡜⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠛⠀⢨⠀⠐⠀⠀⠈⠁⡆⠀⢠⠠⢁⡀⠈⠹⡇⠀⢀⣇⣼⡀⠀⠀⢼⠁⠠
⢇⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠈⠉⠁⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣄⠀⢦⡀⠇⠀⠀⠡⣄⠠⡤⠈⡘⠂⠀⠀⣸⠂⢋⠃⠁⠀⠀⡤⠀⢴⠁
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡂⠲⣄⠢⠄⠀⠤⠛⢀⢼⠀⠀⠒⠂⠄⢠⠃⠀⠀⢀⠜⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠈⠛⢶⣌⡓⠶⠆⠀⠀⠌⡸⠀⠀⣀⠜⢠⠃⠀⣠⠔⠁⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠈⠙⠳⠶⠶⢒⣋⠀⢀⣐⣁⡴⠥⠖⠋⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠸⠀⠀⢸⠀⢀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⠂⢀⡠⠅⠰⠒⠀⠐⠆⠤⣈⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠓⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠈⠁⣀⡔⠊⠑⠢⠤⣀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢆⠈⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡁⠀⠀⠏⠙⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⢱⠒⡛⠤⣀⠴⠒⠚
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠒⠒⠒⠾⢄⡉
⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⡀⣈⢆⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⡖⠉⠉⠀⠀⠀⠀⢫⠳⡀⢀⡜⠁⠀⠀⠀⣀⠤⠖⠒⠒⠒⠒⠤⢄⡀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢰⢊⡄⣨⠟⠁⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠈⢣⠀⡀⠄⣀
⠀⠀⠀⠀⠈⠒⠲⣀⢀⡤⠤⠊⠉⣰⡏⠀⠀⡰⠋⠀⠀⣀⠴⠒⠋⠉⠙⠒⠦⣄⠀⠀⠈⢆⠀⠀⠀⢏⠀⠀⠈⠓⠒⠖⠂⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡸⠁⠀⢀⠞⠁⣀⡠⢤⠤⠤⢄⠀⠈⠳⡀⠀⠹⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠁⠀⢠⠃⠀⡶⠋⠀⠀⠀⠀⠀⠑⡄⠀⠱⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠌⠀⠀⡌⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⠠⠀⠀⡀⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠚⠢⠴⠣⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠧⣀⡰⢇⣀⡠⠛⠒⠊⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⣲⣤⣽⣱⣿⣿⣮⣦⢄⡀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢪⠖⠒⡔⣦⡶⣋⢭⡖⣛⣿⣙⢿⣿⣽⢷⣌⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠡⠀⠔⣷⠻⠎⢠⡤⡎⢡⡇⣸⣷⣿⣩⣟⠈⢷⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠹⣧⠀⠀⠀⠀⠑⢭⣛⣛⣛⣒⣚⣛⣚⣒⣲⣧⣘⣿⣿⣿⣱⠞⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣦⠀⠀⠀⠀⡜⣿⣿⣯⣏⠱⠶⠶⠶⠶⣏⠩⣻⣿⡷⣭⠀⣙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠈⠳⣄⠀⠀⡧⢛⣩⣬⡭⣭⡭⠭⠭⡭⡟⢙⣿⡿⠛⢶⠀⢺⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⠷⠶⣤⣄⡀⢀⠞⠳⣤⢰⢣⣞⣿⠛⢛⠿⢟⣛⣻⠟⠛⢻⠃⠀⡼⢠⠃⣾
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣲⢤⣘⣜⢮⣛⣫⠭⠒⠉⠙⠻⠟⣧⢠⣐⣵⣣⠗⢁⡼⠁
⠀⠀⠀⠀⠀⣰⣿⡿⠟⠛⠋⠉⠀⠀⠈⠉⠙⠻⢿⣿⣿⣷⣭⣛⠧⢄⣀⣀⣈⡴⢃⣾⠿⠛⣁⡴⠋
⠀⠀⠀⠀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣷⣶⣾⣶⣾⣿⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⢠⣶⢆
⠀⠀⠀⡎⠙⠳⢷⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⣶⠶⣶⠄
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣓⣒⣒⣒⣒⣚⣛⣛⣛⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⠒⠤⣤⣤⣤⣴⣖⡆
⠀⠀⠀⣧⣴⣶⢿⡿⠟⠛⠛⠉⣿⡟⠛⣿⠉⠉⠉⢉⣉⡉⠉⢩⣿⣿⡿⣽⡧⣿⡧⣿⡏⡹⠉⠉⠉⠉⠉⠙⢦⠘⡍⠉⠉⠛⠁
⠀⠀⠀⢸⡿⡱⠁⠀⠀⠀⠀⠀⢻⣧⠀⣿⠔⣒⣒⣛⣒⡛⣛⣛⣚⣛⡇⣶⡇⣿⡇⣛⣪⣛⣒⡒⢤⠀⠀⠀⡸⢀⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣄⣛⣛⣛⣛⣛⣛⣛⣛⣛⡇⣛⡇⣻⡇⣋⣉⣛⣋⣋⣀⣁⣀⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢹⣇⣿⣇⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠥⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡏⠀⠀⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢠⣴⣶⠆⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⣧⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠶⠈⢻⠛⣇⡀⢥⠀⣖⣂⣉⠁⠄⢤⣤⠤⠤⢄⠀⠼⢷⣤⡃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢤⣇⣘⣦⣼⣦⣾⣶⣿⠟⠛⣗⣒⠠⠭⠀⠒⢲⠀⠒⠌⠆
⠀⠀⠀⠀⠀⠀⡠⢂⠥⣐⠒⠠⠝⡟⠋⠙⡯⡅⠀⠸⠚⡄⠀⢸⠀⠀⠀⠀⠀⢸⡆⠀⢸⢸
⠀⢠⡄⠶⠀⡞⢰⠋⠒⠂⠀⠉⠉⠙⠀⠀⢡⢠⠀⠀⠇⢡⠀⠀⡆⣐⣂⣈⣭⠭⠵⠔⡡⠂
⠀⠀⠀⠀⠀⠐⡈⢄⡀⠀⢀⣤⠤⠶⣇⠀⠈⣄⡄⠀⠸⠼⡀⠀⢸⠩⠥⠀⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠐⠬⠅⢧⣇⣈⣭⣼⡀⠀⢸⢳⡀⢀⠇⠡⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠜⣙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡹
⠀⠀⠀⠀⠀⠀⠀⢻⡉⣙⣦⣽⢦⣄⠀⠀⢀⣀⢀⣤⣶⢴⣦⣄⣀⣀⠀⠀⢀⡤⢾⣤⣟⡉⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢉⠴⢟⣿⣦⣷⣞⣛⣿⣯⣿⣟⣙⣿⣯⣿⣟⣛⣾⣧⣾⣟⠶⢌⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⡋⣹⡮⢏⣿⣿⡾⢧⣿⢾⣝⢾⣯⠿⣾⣿⣙⠯⣾⡙⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢈⣷⡀⣨⣾⣽⣿⡿⢿⣸⡿⣸⠿⣷⣿⣽⣯⡀⣸⣏⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡝⡢⣍⣗⡍⣈⠉⢹⠖⠺⣷⢼⡷⠒⠾⠁⢉⠈⣹⣋⡵⡚⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⢿⡏⣌⢿⣦⡶⢊⣼⡼⢿⣼⡍⢲⣦⣾⢇⡎⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠙⢸⣇⣿⠢⠌⣙⣲⢏⠇⡈⢏⢷⣚⡉⠤⢺⣇⣿⠘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡏⡞⣛⡻⣷⣄⣄⣽⣿⡺⡿⣺⣿⣅⣄⣵⡿⣛⡛⡌⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣵⣿⡿⣾⠿⣻⣿⡿⣿⣿⠿⣿⣿⡻⢿⣾⣿⣧⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⡛⠙⣲⣟⠛⠳⣿⣿⡶⠛⢛⣷⡚⠙⣋⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣭⣭⣍⣡⣹⣻⣥⣉⣭⣭⣽⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣨⡒⠍⣸⣡⣧⣹⡈⠕⢫⣈⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣝⣿⢽⡂⠀⡇⠀⣿⢽⣟⣴⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢹⣿⢷⣀⣃⣸⣿⣿⢹⡿⠁
⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡻⣷⣾⣽⣶⡿⡻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢛⣷⡈⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⡿⠛⠉⠁⠈⠂⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣴⣾⠏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣪⠴⣊⡅⠀⠀⡙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡟⣹⠇⠔⠵⠋⠡⠈⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠿⢠⣾⢆⠱⣶⡀⢀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⣿⡘⡇⠈⠅⢊⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠦⢀⣴⣶⣾⡜⠷⣆⡀⠀⠀⠀⣠⠶⠟⠙⢿⣿⣿⡿⠃⢷⣘⣶⣾⠈
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠀⠸⠋⠙⠛⡑⢠⣤⠦⠀⣠⠞⠁⠀⠀⠀⠀⢙⣿⣧⡀⣸⣿⡿⡿
⠀⠀⠀⠀⠀⠀⢺⣿⣿⢿⣟⠻⡿⢀⡀⠀⠀⠀⠀⠈⢉⣶⢏⣛⠁⠀⠀⠀⠀⠀⠀⡟⢳⣶⣤⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣌⢹⣦⠀⢸⣿⠆⠀⠀⠀⠰⠎⠻⠛⠙⡆⠀⠀⠀⠀⠀⠀⢸⣿⣯⣄⣽⣺⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣥⣶⣀⠙⠒⠀⠀⠀⠀⣀⣀⣠⣴⠁⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣷⢾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣴⣶⣶⣶⣶⠈⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠛⠿⠿⢿⣿⣿⣿⣿⣿⣾⣿⡟⣿⣿⡏⣺⣿⣿⣿⣿⣿⣿⣇⣘⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣄⡀⠀⠀⣤⣴⣾⣿⠛⠋⠹⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⠇⠀⠀⠀⣻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡟⠛⠛⠻⠟⠛⢋⣉⢭⣅⣀⣠⣔⣼⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡟⠁⢀⣼⡶⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢋⣩⣝⠋⠉⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⣶⣿⣿⣿⣿⣿⣦⡀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⡿⠟⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠙⠿⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠿⠂

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠴⠦⠴⠶⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⢔⡋⣁⣠⠤⠶⠶⠒⠒⠲⠶⠦⣤⣀⡉⠛⠻⣕⡢⣀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠂⠀⠀⠀⠀⠀⠀⢀⡞⢡⠖⠉⣳⣆⠙⣹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⢦⡀⠙⠺⣗⣤⣾⡁⢸
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠂⠀⠀⠀⠀⠀⠀⠀⣼⠀⠇⣠⠎⢁⣽⡂⣥⣤⣤⣈⡉⢃⡐⠒⠲⢄⠀⠀⠀⢸⠆⡏⠳⣄⠈⢏⣸⠉⢹
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢾⠁⠀⣼⠹⡇⢯⡇⠀⠈⡇⢨⠉⠉⢻⡄⠃⠀⠀⣸⠀⠁⠀⢻⣧⡀⠹⡀⢸
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣸⠃⡿⢤⣑⠠⠿⢤⣟⢢⠭⠭⣀⣓⣊⠀⠀⠸⠧⠆⢀⣀⡏⢈⠀⠀⠀⢹⣷⡀⠱⣸
⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢠⡏⡘⠀⠀⠉⢹⠛⣲⠖⠶⣤⠤⣬⣁⣁⣘⣉⠻⠹⠓⣾⢱⡮⣧⢄⠀⠐⠾⠽⢳⠀⢻
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸⠁⠃⠀⠀⠀⣾⠀⣿⠀⣟⢵⢂⣬⢬⠥⠌⣙⣛⣙⠙⣻⣷⣒⣿⣺⡤⠄⠐⠀⠼⡇⠈
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⣾⠀⠀⢀⡠⠔⢓⣚⡃⠠⠬⠑⢲⡶⣮⢥⡤⣜⣃⣛⣹⡱⣷⣹⡷⠯⠭⠭⠿⠶⢶⡏
⣾⠀⠀⠀⠀⡠⠶⢒⡶⠖⠶⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠭⢻⡀⣶⢋⠔⣹⠯⢛⣛⣛⠉⠅⠳⠖⡖⠠⠤⣭⢤⣁⣩⡿⢯⣽⠭⠟⢛⣓⣶⣤⣤⡟
⢿⣄⣀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠈⢢⠈⣧⡇⡈⢰⢣⠎⣽⠁⠍⠉⠉⣿⢒⡒⠒⢶⡭⢥⠤⢿⣜⣇⣿⢙⣶⠒⠲⠤⣀⣈⡇⢰
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠑⢼⣧⢠⠘⣆⠑⠛⠐⠒⣚⣉⣩⣭⠬⢭⣉⠓⢎⠀⠀⡟⢠⠙⠛⠁⠀⠀⠀⢀⡟⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠹⣆⠳⣿⢹⡟⢻⠉⠉⢸⠇⠀⠀⠀⣼⢳⡌⢣⢀⡇⡄⠀⠀⠀⠀⠀⢠⡞⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠈⠳⣌⠚⢷⣈⠒⠀⠋⡰⠃⠀⣰⢃⢜⡇⢸⡼⢁⠁⠀⠀⠀⣀⣴⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠈⠙⢦⣌⡙⠛⠿⠷⣖⣫⣥⠮⠊⣠⠟⣡⠊⣀⡤⢔⡼⠋⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡈⠛⠿⢶⣶⣦⣤⣴⣒⣋⣵⣞⣉⡭⠶⠚⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⣉⠉⠉⠉⠉⢉⣉⣀⡤⠔⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠐⠒⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⢀⡀⠤⠐⠒⠒⠒⠒⠒⠂⠒⠒⠂⠤⢄⣀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠊⠀⠀⠀⠀⢀⡠⠤⠒⠂⠈⠉⠉⠉⠉⠉⠀⠀⠐⠒⠠⠄⡀⠀⠀⠈⠓⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⠔⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠈⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠠⠃⠀⠀⢀⠎⠀⠀⠀⢀⠔⠊⠀⠀⠀⢀⡀⠤⠄⠀⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠑⠢⣀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡰⠁⠀⠀⡐⠁⠀⠀⠀⠘⠁⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢄⡀⠀⠀⠀⠀⣱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠁⠀⠀⡔⠀⠀⠀⠐⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡤⠖⠁
⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠇⠀⠀⠀⠀⠀⣤⣀⣄⣀⡀⠀⠀⡔⠁
⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠢⣀⠔⠂⠁⠀⠀⠀⠀⠀⠈⢓⣬⠀⠄⣀⡀
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡗⠀⠀⠀⠈⠳⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠡⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⠁
⠀⠀⠈⡾⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠢⢀⡀⠀⢀⣂⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⣀⡠⠤⠚⠀⠙⠢⢤⣀⡀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



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

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⢠⡀
⠀⠈⠳⢬⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⣤⡠⠀⠀⠀⠠⣀⠀⢀⡀⠙⠃⠀⠀⠀⠀⠉⠀⠀⠀⢰⠀⡀⠀⣏⣹⠞
⠀⠀⠀⠀⠀⠀⠀⡀⠢⣠⣄⣰⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⣰⣃⡿⠁⢀⣤⠖⠀⠀⠀⢀⣜⣶⣦⡀⠀⠃⠀⠉
⠀⠀⡤⠖⢴⠄⢸⣤⣾⠟⠋⠛⠣⣀⡀⠀⠀⠀⢀⢠⡀⠀⢼⠟⠁⠀⠀⠀⠀⠀⠤⠒⠉⠉⠋⠛⣧⣂⣦⠀⢰⡐⡄
⠀⢰⣠⠎⠁⠀⠸⠔⠁⠀⠀⣤⣄⡀⠙⠢⡀⠀⠠⠗⠂⣴⠿⢀⠀⠀⠀⡐⠉⢀⣀⣀⣀⠀⠀⠀⠀⠲⣧⠃⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢠⣾⣾⠟⠀⠀⠈⠂⡀⠀⠈⢃⣨⠟⠁⠀⠜⣀⣼⣭⣋⣙⣁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⠿⢿⣷⣶⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⢽⣰⡗⠛⡛⠁⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣤⣌⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠈⠻⢭⡭⠼⠃⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠛⢻⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠈⠀⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠁⠀⠀⠛⠲⠆⠀⠀⠀⠑⠢⠀⢀⣀⣤⠀⠠⡀⠠⠔⠊⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛
⠀⠀⠀⡾⢰⠦⡀⠀⠻⣥⠀⠀⠀⡀⠄⠀⡜⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⢉⠀⠀⠀⢠⡄⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠙⢍⣠⣁⠀⠀⠀⠀⠀⠀⠁⢀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⢷⡄⠉⡽⠀⠚
⠀⠀⠀⠀⠀⠉⢩⡆⠀⠀⠀⠠⢂⣰⠏⢀⡀⠀⠄⠀⢠⣤⣶⡄⡆⠀⠀⠀⣿⠀⠀⠄⠘⠧⣄⡀⠛⠶⠃⠀⢠⣤
⠀⠀⠀⢰⣂⡤⠉⠀⢈⣤⢈⢁⡾⠉⡇⠀⡿⠀⡀⠀⢿⡿⠋⠀⡇⠀⡁⠀⣯⠀⠀⢠⡐⠦⡀⠙⠿⢦⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠉⣄⠀⠀⠀⣇⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠐⠀⠀⠙⠶⢶⣤
⠀⠀⠀⠀⠀⢠⠞⠁⠰⡆⠀⠀⣿⠀⠀⢀⡃⠀⡇⠀⠾⡛⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢉⠀⠀⠐⣄⠀⠀⠀⠀⢣⡄
⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⠀⠉⠀⠀⡇⠀⡆⠀⡇⠀⡄⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⢣
⠀⠀⠀⡋⠀⣨⣇⣠⣴⠧⠴⠶⠿⠤⠷⠾⠷⠤⠇⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠚⠛⠛⠛⠒⠓⠶⠷⠤⠄⠀⢀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⢤⡀⠀⠀⠀⠀⠒⡦⢤⡀⠀⡠⠀⠠⡀⠀⠔⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⢹⣄⣤⣆⡄⠀⣀⠀⡞⠁⠀⠀⠟⢠⠇⠈⢠⠫⠁⠀⠀⣀⡤⠼⠇
⠀⠀⠀⠀⠄⣈⠿⠞⠉⣀⠟⠙⣑⣆⣴⢿⡏⠀⣽⣟⠙⠿⠀⠀⠀⠚⠛⠀⠀⠀⠚⡁⠀⡰⠕⠀⠀⠀⢤⠀⠀⠀⢀
⠀⢓⢐⣒⢲⡟⠀⠐⡿⠟⠓⠈⣿⠏⠁⠼⢃⣢⠥⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠊⠀⡄⢰⣏⠁⣸⡇⠀⠀⢨⡀
⠀⠸⠃⠈⢉⣿⠀⢀⣿⡷⠀⠀⢽⣢⠎⣀⠘⠛⠒⠚⠁⠀⢀⢘⣁⠀⠀⠀⣀⢀⣘⠉⠀⠀⠐⠤⣤⠴⠟⠀⢀⡌⠘⠉⢀
⠀⠀⠊⠃⠀⢭⣽⣧⣷⣞⠈⢀⠳⢀⡞⠉⢁⡠⠤⠀⣀⡴⠹⢿⠄⢠⡶⣏⡩⣶⢇⠀⠀⠀⡄⠂⠀⢀⡤⢒⣏⠨⡄
⠀⠀⠐⣤⣈⠀⢬⣍⠈⠍⠊⢵⢖⠿⠗⠊⠁⠀⡄⡈⠙⢷⠶⢮⣹⣷⡟⢈⣧⡤⠈⠀⠰⠺⠀⠀⠀⠰⠤⠈⢾⠄⠋
⠀⠀⠀⠈⠐⠛⠳⠦⠤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠾⣴⣿⣭⠭⠥⠄⠠⠀⠄⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⣛⣛⣛⣛⣳⠶⣤⣀
⠀⠀⠀⠀⠀⢀⣴⢞⣭⢞⣫⡭⠷⠶⠶⠾⣭⣟⡷⣭⡳⣤
⠀⠀⠀⠀⣠⢟⣵⢫⡶⣫⡵⠞⠛⠉⠉⠛⠳⢮⣟⢮⡻⣮⢳⡄
⠀⠀⠀⣰⣏⡾⣱⢏⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡹⣞⢷⢻⡄
⠀⠀⢀⣟⣼⢱⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⡞⣧⣿
⠀⠀⠘⠋⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠀⠙⠃⡀
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⣄
⠀⠂⠄⠠⠤⠀⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠀⠤⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣠⡄⡄⠠⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⣗⢮⣟⢮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⡄⠈⢳⡝⣮⢳⡝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠐⡲⠂⠒⠐⠂⠒⠐⠀⠲⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠛⠛⠁⠀⠀⢿⣸⡏⣿⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣛⣘⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠈⠀⠀⠀⠀⠁⠀⠑⢤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣮⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣄⣀⣤⣀⣠⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡄⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡛⠐⠂⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡀⢰⡀⠀⠀⠁⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⢀⣼⢟⡞⠁⠀⠀⠀⠀⠀⠀⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡿⡿⠃⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣼⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⢰⡟⠀⠀⠀⢀⣀⣠⣴⡾⠟⠛⠀⠋⠉⠉⢹⡶⠷⣆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠺⡟⠉⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⢏⠞⢁⡀⢠⣤⣤⠂⠲⡀⠀⡊⠀⢸⡏⢹⠁⠀⠀⢀⡰⠋⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣋⡟⠀⣸⣤⣿⠋⠉⠗⠀⡦⢰⠯⡧⢸⡇⢸⡀⠒⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⡇⣰⠃⡘⣟⠆⠀⢰⡶⣷⢾⠔⢶⠛⡟⢺⡚⠒⠚⠛⠷⣦⡀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡇⢸⣤⣼⡾⠀⣆⣘⣧⣎⣿⡛⢻⣛⣟⣻⣛⣋⡉⠉⢉⠛⣷⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢸⡌⣽⠖⠇⡞⢻⡟⡛⠛⢙⣻⠛⣟⣹⣏⣉⣘⣿⣈⡙⣶⡐⢺⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢹⡿⣵⣾⣀⣼⠋⠁⣤⡏⠉⣤⡏⠀⠻⠏⠘⠋⠀⢀⡦⢂⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⢳⣻⣯⠀⠀⠰⠶⠲⠶⠶⠶⠶⠶⡶⢖⣪⣈⣹⠟⢁⡼⠡⠀⠀⠀⠀⢀⣀⡠⠐⠩⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⢍⢓⡠⡄⠀⠀⡀⠀⣀⡄⡤⢧⡸⠉⠈⢀⡀⣊⠀⠀⠴⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠙⠓⠭⠄⢁⠀⠀⠀⢀⣈⣧⡤⠖⡫⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠬⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⡀⠛⠛⠃⡅⣀⣌⢀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠀⠒⠀⠀⠀⢢⠈⣯⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣤⢆
⠀⠀⠀⠀⠀⠀⢀⠌⢀⠔⠈⠀⢉⢰⢄⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠃⣸⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠈⠀⡀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣯⠅⣿⡁
⠀⠀⠀⠀⠀⢀⡀⠄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠀⠄⢀⣿⡅
⠀⠀⠀⠀⠀⢈⡔⢰⠈⣆⠀⠀⠀⠀⠘⡀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⢀⡀⢸⡷⠄
⠀⠀⠀⠀⠨⠘⡀⠈⠀⢻⠀⠀⠀⠈⡀⣡⠰⢀⡄⠠⢼⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠣⠄⢸⠷
⠀⠀⠀⠀⠀⠀⡇⠀⣀⠮⠇⠂⠀⠀⠀⠀⠺⡀⢀⡀⠠⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⡔⡄⣿⠋
⠀⠀⠀⠀⠀⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⡴⣣⡁⠆⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⡉⢀⠋⠀⢀⠂⠁⠀⠁⠒⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠂⠉⠀⠤⠐⣊⠱⠞⠐⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⢀⠀⠘⠲⠀⠀⠠⠂⠀⠂⢄⠀⠐⢄
⠀⠀⠀⠀⠀⠈⠈⠹⠻⠆⠒⠈⠁⠀⠀⠀⠀⠈⡐⠤⠤⠒⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡿⢮⠅⡘⠶⠀⠚⡄⠀⠀⠀⠀⠁⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠚⡳⡆⠠⠀⠀⣰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣄⠦⡀⠛⠓⡂⠀⠙⣆⠀⠀⠀⠀⢰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠰⠂⠀⢀⡀⠤⠀⠚⠈⢡⠄⠋⠁⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⡏⡉⢋⢀⣄⡹⢅⡀⠀⠈⠳⢤⠀⠀⠌⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠁⠉⠀⠀⠀⠀⠀⠀⠈⣬⢐⠒⡈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢘⣣⢿⢳⡇⠀⣧⡍⠀⠀⣳⠀⠁⠄⠀⠁⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⠈⠅⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⠿⠈⢙⠉⠰⢆⠀⢀⡟⠀⠀⠈⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢒⠘⠙⠀⠁⠀⠀⠀⠀⠀⠐⠈⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠘⠁⢡⠀⢐⣉⠉⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠂⠃⢀⠀⠀⠀⠀⠠⠀⠀⢁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠻⠶⠂⢷⠎⠀⢹⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⡄⠀⠀⠀⠀⠆⠀⠘⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⢀⠗⠢⠘⠀⠀⠀⢸⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⢀⡀⢄⣸⠀⠂⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢙⠛⠘⣄⡄⠇⢀⣳⣒⠃⠀⢀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⣤⡄⠀⢠⠀⠐⠂⠀⢘⡆⢀⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⢄⠀⠀⠀⡠⠊⠠⠁⡂⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠙⡀⠠⣄⠀⠂⠠⠜⡡⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠠⠂⠁⠀⡅⠀⠂⠉⠀⠀⠀⠍⠐⢀⠀⠀⠈⣇⣀⠀⠈⠂⠐⠈⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⡸⠀⠁⠀⠀⠈⠉⠢⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠤⠀⢀⠀⠠⠐⠈⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠠⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠕⠀⠀⠁⠪⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢠⡥⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⣀⣀⡐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠐⠁⠀⠀⠀⠑⠌⠄⡀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠠⠦⠤⠶⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠁⠀⠀⠀⠀⠀⠘⢏⠘⠁⡃⠀⢀⠀⡁⠀⠉⠀⠀⠐⠒⠊⠉⢉⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠃⠀⠀⢣⡁⠀⠀⠀⣀⣀⣀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⣉⣁⠀⢄⠀⡄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠤⣖⣤⡶⠄⡀⠀⢀⠈⠉⠉⠉⠉⠉⠈⢁⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⠀⠀⠨⢆⠈⢀⠰⣠⠀⠀⠀⠀⡃⠠⠀⢀⣀⣀⣀⡠⠔⡇⠀⠸⠦⢤⠤⢠⣤⣶⣾⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⢀⣀⣤⠇⠀⢀⡦⠀⠀⠀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠈⠓⠒⠲⠅⣀⠀⣇⡀⠉⠉⠛⠛⠛⠙⣸⠀⠀⠀⠁⠀⠀⠀⠤⠰⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢠⠀⠉⠉⠂⠐⠂⠀⡇⠀⠙⠀⠀⠄⠤⠤⢶⠞⢿⠀⠀⠰⠇⠀⠀⠀⣀⣀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠶⠶⠦⠀⡀⠀⠾⠁⠀⠠⣤⣤⣤⣦⡇⠀⢠⡓⠀⠀⠀⠀⠀⢀⡼⠀⠀⢐⠀⠀⠀⠈⠈⠁⠁⢤
⠀⠀⠀⠀⠀⠀⣆⠈⠀⢐⠀⠒⠒⠉⢀⠀⢠⡀⠀⠈⠉⠉⠉⠁⣧⠀⠀⢀⠁⠀⠚⠓⠛⠩⢰⠄⡀⠈⠀⠀⠀⠤⠤⠤⠶⠟⡇
⠀⠀⠀⠀⠀⠀⠑⠠⠄⠠⢤⢤⣤⡴⠊⠁⠘⠢⠤⣀⣤⢤⡤⠶⠁⠔⠈⠀⠅⣀⡤⣤⣤⠔⠋⠀⠀⠲⣌⣀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣀⣤⡶⡚⠋⠉⠁⠀⠀⠈⠉⠙⢓⢶⣤⣀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠦⠠⣾⣿⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣿⣿⣷⠄⠴⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣆⠘⠿⢿⣟⠻⠿⠆⠀⠀⠀⠀⠠⠿⠟⣻⡿⠿⠃⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣡⣤⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⣤⣌⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣷⣿⣾⢢⠇⢠⠀⠀⠀⠀⡄⠸⡔⣷⣿⣾⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡙⠿⠿⠿⡞⠁⢀⡼⠀⠀⠀⠀⢧⡈⠫⠷⠿⠿⠿⢋⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢈⠈⠛⠦⠤⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠴⠛⠁⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⠸⣦⣄⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⣠⣴⠇⢋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⣃⣀⠀⠀⠀⠠⠤⣤⣄⠀⠀⣠⣤⠤⠄⠀⠀⠀⣀⣘⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⣁⣤⣤⣀⡀⢤⣌⣿⡇⢸⣿⣡⡤⢀⣀⣤⣤⣈⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠹⢧⣭⣽⠿⣿⡿⠀⠀⢿⣿⠿⣯⣭⡼⠏⠀⢀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠉⠁⠺⡿⣷⣿⣿⣾⢿⠗⠈⠉⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⢀⣿⡟⢻⣿⡀⠀⠀⠈⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⡇⠀⠀⠀⢸⣿⡇⢸⣿⣷⠀⠀⠀⢸⢸⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣠⢀⠀⠀⠉⠀⠀⠉⠀⠀⡀⣄⣿⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⣌⡟⣾⣤⡄⢀⠀⠀⡀⢠⣤⣷⢻⣩⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠹⣮⡑⣿⢈⡟⢻⠳⠟⡟⢻⡁⣿⢊⣵⠏⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠳⠾⣎⣯⢹⡚⢓⡏⣽⣡⠷⠟⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠁⠘⠋⠛⠛⠙⠃⠈⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠠⢀⣀⣀⣀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠉⢀⣀⡀⠤⠤⠤⠤⠤⢀⣀⡀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣀⠤⠒⠉⠁⣀⣀⣤⠠⠤⠤⣄⣀⣀⠈⠉⠲⢤⡀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠚⠁⣀⠤⠒⠉⠁⣀⣀⠠⡤⠄⣀⣀⠈⠉⠒⠤⣀⠈⠓⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠊⠀⡠⠊⠁⣠⠔⠊⠉⠀⣀⣀⣀⣀⣀⠀⠉⠑⠢⣀⠈⠑⢄⠀⠑⣄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡜⠁⣠⠎⠀⡠⠊⠀⡠⠔⠊⠉⢀⣀⣀⣀⠀⠉⠑⠢⣄⠈⠑⢄⠀⠱⡄⠈⢦⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣜⡀⣰⠃⢀⠞⠀⡠⠊⠀⡠⠖⠋⠁⢀⣀⡀⠈⠙⠲⣄⠀⠑⣄⠈⠣⡀⠘⣄⠀⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠚⠉⠉⠉⠉⠓⢮⡀⡼⠁⢠⠎⠀⡠⠚⠉⠁⠀⠈⠉⠲⢄⠀⠳⡀⠈⢆⠀⡱⠖⠋⠉⠉⠉⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⢀⠇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠱⡀⣈⡞⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢀⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠈⠣⣄⣀⡤⠤⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡴⢤⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠲⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⡏⠀⠀⣀⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡋⠩⣩⣿⣿⣗⠒⠒⢲⡖⢾⡇⠀⠀⠀⣴⣿⣿⣿⣟⠃⢒⣸⣿⣿⡖⠒⠒⢶⠶⣾
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠉⠁⠈⠉⠀⠈
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣾⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⣠⠤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡠⠞⠋⠀⠀⠀⣀⣤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀⠀⠈⠙⠶⣄⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⠴⠋⠀⠀⢀⡤⠖⠋⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠙⠳⢦⡀⠀⠈⠛⢦⡀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢁⡴⠃⠀⠀⣠⠞⠉⠀⢀⣤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠓⠶⣤⣀⠀⠀⠙⠷⣄⠀⠀⠙⢷⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⢠⠞⢀⠀⣰⠋⠀⠀⣠⠞⠁⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠈⢷⣄⠀⠀⠙⣦⡀⠀⠙⣧
⠀⠀⠀⠀⢠⡏⠀⢀⡼⠁⠀⢀⡼⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠙⢦⡀⠀⠈⢻⡄⠀⠘⣧
⠀⠀⠀⠀⡞⠀⢀⡞⠁⠀⢠⠞⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄⠀⠀⠻⣆⠀⠘⣧
⠀⠀⠀⣸⠁⢀⡾⠀⠀⣰⠋⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠹⣆⠀⠀⢹⡆⠀⠹⡆
⠀⠀⢀⡏⠀⣼⠁⠀⡴⠃⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠘⣦⠀⠀⢻⡄⠀⢿⡀
⠀⠀⢸⠃⣰⣇⣀⣼⠁⢠⠏⢀⣤⠖⠒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣷⡀⠘⣧⢀⡬⠷⠒⠺⢧⣄
⠀⠀⣸⠋⠉⠀⠀⠉⠓⢯⡴⠏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠳⢤⡼⠋⠀⠀⠚⠉⠀⠙⣆
⢠⡴⠟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠙⠲⠶⠞⢻⡄⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢰
⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣰⢦⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣀⣴⣤⣀⡀⣀⣀⡴⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶
⣿⣿⣇⣤⣍⣤⣮⣬⣧⣥⣾⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⢠⢊⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⢄
⠀⠀⠀⠀⢠⠃⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⠆
⠀⠀⠀⠀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⡇⢸⣏⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠐⠢⣤⠔⠒⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢘
⠀⠀⠀⠀⢃⠸⣿⡎⠓⠀⠀⠐⠂⠈⠉⣹⠀⠉⠀⠀⠀⠈⢀⠀⠀⠊⠁⢋⡉⠉⠐⠒⠀⠀⠒⢡⣿⡇⡸
⠀⠀⠀⠀⠀⠓⢿⣿⣦⣤⣤⣴⣶⣶⣿⣿⠆⠀⠀⠀⠘⢿⠃⠀⠀⠀⠠⣿⣿⣷⣶⣶⣤⣤⣾⣿⡿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣓⠛⠻⠿⢿⣿⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⡿⠿⠿⠛⣛⠋⠁
⠀⠀⠀⠀⠀⠀⠰⠖⠋⠉⠻⢷⣶⣾⡗⢾⠳⢦⣀⡄⢀⣿⡀⢠⣀⣠⠞⡿⠊⣷⣶⡾⠟⠋⠉⢳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢨⣧⠀⠙⠻⡿⠀⠀⠀⠀⠀⢻⠟⠋⠀⣸⣍⠐⠀⠤⠄⠘⣁⣤⡄⢐⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⡆⢠⡟⠄⠀⠀⠀⠀⠿⡄⢠⣿⣿⣿⡄⠀⠀⠀⠈⣟⠫⡈⣉⠈⠓⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢿⠸⣿⣷⣈⣷⣄⠀⠀⠀⢠⣾⣃⣾⣿⠇⣿⠹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⣿⣿⠋⣉⡹⢧⣤⡴⠏⣉⠉⢿⣿⠀⠁⠀⠳⠤⠭⢌⢢⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢰⣿⣿⣀⡸⣷⣶⣿⣶⣾⡏⣀⣼⣿⡇⡀⢀⠀⠀⠀⠀⢮⡓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⣾⣎⠟⣿⣠⣿⣿⡿⠿⢿⣿⣿⣥⣿⠫⢣⣷⡞⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠹⣿⣦⢹⣿⣿⣏⣀⣀⣀⣹⣿⣿⡿⣴⣿⠟⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠮⣿⣇⠉⢿⣿⣿⣿⣿⣿⣿⠋⢰⣿⠥⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠈⠻⣦⣸⣿⣿⣿⣿⣿⡏⣴⠟⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠳⢦⡈⢳⣿⣿⣿⣿⣿⡞⢁⡤⠞⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣦⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠤⠤⠤⠀⠤⡄⠀⠀⠀⠀⢀⠜⡀⠠⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠸⣧⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠳⣄⠀⢀⡠⠒⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢐⣈⢀⠇⣛⣾⠇
⠀⠀⠀⠀⠯⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠷⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣦⠀⢀⡄
⠀⠀⡀⢀⣀⡠⠎⠀⠀⠈⠉⠉⠙⠙⠉⠉⠀⠀⠀⠀⡀⠀⢀⢰⠀⠀⣮⣤⣦⣴⣶⠀⠀⠀⠀⠙⢀⠀⡀⣻⠇⠉
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠚⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣁⠒⠉⠐⠴⠴⠒⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠤⠀⠀⠀⠀⠀⠀⠀⠛⠓⠦⠆⠀⠀⠐⠀⡁⡀⠖⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠆⠀⠀⠀⠀⣠⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢀
⠀⠀⠀⠀⠀⠠⣤⡦⠶⠟⠉⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠓⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠀⠘⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣛⠭⣭⣹⠚⠁⠀⠀⠀⠀⣀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⡀⠀⠀⠀⠀⢠⠁⡰⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡊⢕⠪⠑⠊⠉⢀⠃⠀⠀⠀⠀⡠⠊⠁⢀⡠⠐⢒⠋⠀⠀⠀⠈⠁⠒⠦⢄⠀⠈⠳⣄⠀⡠⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡡⠊⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⠀⢀⠔⠁⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠢⡀⠈⢲⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⠊⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠎⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⢒⠒⠤⡀⠀⠀⠀⠀⢀⠌⡸⡆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢃⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡣⡁⢄⡁⠢⢑⠢⢀⣀⠎⡘⠁⢱⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡇⠀⢃⠀⠀⠀⢈⠀⡀⠀⠀⠀⠀⠑⠪⢐⠠⢉⠢⢍⢎⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢱⡀⠈⡆⠀⠀⠀⠀⠂⠒⠄⠀⡀⠀⠀⠀⡬⣂⣅⠢⢏⠀⠀⡘⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠻⣷⡀⠈⢆⠀⠀⠀⠀⠀⠀⠁⢆⠀⠊⠐⡈⡴⢥⠫⢂⠙⡶⠁⠀⡞
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⠒⢄⡀⠈⠛⢄⡀⠑⢄⡀⠀⠀⠀⠀⠀⠁⠂⢠⡞⢱⢥⡈⢐⠜⠁⢀⡼⠵⣤
⠀⠀⠀⠀⠀⠀⣿⢃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠉⠲⢄⡈⠑⠢⠤⠤⠤⠴⠞⢓⡤⠉⢀⠕⠁⠀⣠⣎⠛⠶
⠀⠀⠀⠀⠀⠀⢿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠈⠑⠒⠤⠄⠤⠠⠴⠯⠒⠈⠀⢀⡤⠊⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣀⣀⡀⣀⠀⢀⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⢤⢀⣿⣿⣿⣿⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⣤⣄⣀⣀⡀⠐⣶⣞⣹⣀⣀⣤⣤⣤⣤⣾⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⢛⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⠿⣿⣃⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣱⣷⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠀⠀⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢉⡠⠄⠒⠒⠀⣀⡀⠐⠒⠀⠤⢈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⠔⠊⡠⠔⠂⣉⡉⠤⠤⠤⠈⠁⡁⠂⠤⡈⠒⢄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠔⢁⠄⢊⡡⠄⢒⣂⣀⣀⢒⠂⠄⢁⠂⢄⠑⢄⠑⢄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡰⢁⠔⢡⠞⡡⢊⠥⢒⠩⠄⠐⠒⠒⠀⠬⢁⠢⢁⠢⡈⠀⠐⠄⠑⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡐⢀⠊⡰⢁⢎⠔⡡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡑⠌⢀⠁⡀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⣰⣁⡎⣐⢠⣃⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠉⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠁⠀⠀⠀⠉⢂
⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠑⠤⠄⠀⢄⡀⠀⠀⠀⢀⡠⠀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠠⡀⠀⠀⠀⠀⢀⠄⠀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠉

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠋⠉⠙⠛⠒⠦⣄
⠀⠀⠀⠀⢠⠖⠋⠀⣠⠤⠖⠒⠒⢒⠦⣄⡀⠈⠳⣄
⠀⠀⠀⣰⠋⠀⣰⠋⠁⠀⠀⡤⠶⢾⡀⠀⠙⣆⠀⠈⣇
⠀⠀⢰⠇⠀⣸⠁⠀⠀⠀⠘⣧⠀⠀⠙⠀⠀⢸⡆⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢸⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢛⣉⣉⣉⣉⣉⣉⣉⣙⠓⠲⢤⣀
⠀⠀⠸⡆⠀⠸⣆⠀⠈⢳⡤⠴⠞⠀⠀⠀⣰⠋⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡴⢋⣉⡵⠀⢀⣠⠖⠒⢲⣭⠽⠳⡄⠈
⠀⠀⠀⢻⣆⠀⠈⠳⢤⣌⣀⣀⣀⣤⠴⠋⠁⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠲⠤⣽⣛⣋⣁⣀⣀⣨⡥⠴⠚⠁⣠
⠀⠀⠀⠀⠙⣿⣦⣄⣀⡀⠀⠀⢀⣀⣠⣴⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣯⣟⣻⣿⠿⢿⣿⣿⠿⢿⣿⣛⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠶⠖⠲⠶⠶⠤⢤⣀⣸⡿⣷⣯⣟⣻⣿⡿⠿⠿⠿⠿⣿⣿⣻⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⠴⢖⡒⠛⠛⣛⣻⣖⡲⠦⣤⡉⠛⣷⣯⣿⣻⣿⠿⢿⣿⣿⠿⢿⣿⣛⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠻⣔⡒⠻⢤⠤⠞⠁⠀⠀⠟⢉⣠⠟⠀⢸⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣭⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣈⡉⠙⠛⠒⠒⠒⠒⠛⠋⢉⣁⣤⣶⣿⣿⣻⡿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⣿⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡿⢿⣫⣾⣿⣻⣿⣿⡿⠿⠿⣿⣿⡿⢿⣿⣻⣽⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣻⢿⣷⣿⣿⣯⣭⣭⣽⣿⣿⣶⣿⢿⣫⣿⡿⠛⠉⠀⢀⣀⣀⡀⠈⠙⠻⣿⣻⣽⣾
⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⣤⣄⣀⢸⣷⣝⣿⢿⣷⣿⣿⣭⣿⣿⣭⣿⣿⣶⣿⢿⣿⡟⠀⢀⡴⠚⠉⠉⠉⠉⠛⠦⣀⠈⠻⣯⣿
⢀⡴⠋⣉⡤⠴⠶⠒⠒⢒⣒⣶⠶⠤⣌⡙⠻⣿⣿⣿⣷⣿⣿⣿⣭⣭⣿⣿⣿⣷⣿⣿⣿⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⣿
⣏⠀⠸⣁⠶⠾⣥⣤⡴⠋⠁⠉⠿⠛⢁⡽⠀⢈⣿⣷⣿⣯⣽⣿⣿⣿⣿⣯⣽⣿⣾⣿⣇⠀⢸⡇⣤⣾⠁⠀⠀⡴⠚⢶⡄⠈⣇⠀⢹
⣿⣶⣄⣈⠙⠓⠒⠲⠶⠶⠶⠒⠒⠋⢉⣀⣴⣾⣿⣷⣿⣯⣽⣿⣿⣿⣿⣯⣽⣿⣾⣿⣿⠀⠘⣇⠀⠙⢦⣤⠞⠁⠀⣰⠟⠃⣿⠀⢸
⣿⣝⡿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⣟⣽⣿⣷⣿⣿⣭⣿⣿⣿⣿⣯⣿⣿⣾⣿⣿⣧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠉⠀⢠⠏⠀⢸
⣿⣝⡿⣷⣿⣯⣽⣿⣿⣿⣿⣿⣭⣿⣶⡿⣟⣽⡇⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠋⠉⠀⠻⣿⣷⣄⠈⠓⢦⣄⣀⣀⣀⣠⠴⠋⠀⢠⠏
⠘⠿⣿⣷⣾⣯⣭⣿⣿⣿⣿⣯⣽⣿⣶⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠉⠉⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣷⣶⡶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠠⣄⠒⣞⠉⠁⠀⠀⠉⠉⠒⢦⣈⠓⢶⣀⣼
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣸⠀⡰⠂⣼⢠⡤⠤⣌⢀⣀⡀⠀⠀⠈⠊⠑⣄⠉⣹⠈
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣌⠀⢸⢉⡆⣃⠀⠈⠀⠀⠸⠀⠀⢸⠀⠀⠘⢧⡈
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⡖⢰⠂⡀⠠⡅⠁⢁⡒⠒⣞⠭⡧⠀⠐⠊⢣⠘
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠞⠘⠜⣘⣐⣀⠰⠇⠤⢞⠫⣿⠓⠤⣄⣨⡄
⠀⣼⠀⠀⠀⠤⡲⠖⠲⠤⣤⣀⠀⠀⠀⠀⠀⢢⠀⠇⠀⠡⠺⠗⡃⡂⠀⠲⠠⠠⢼⠉⢉⠷⣬⡛⠒⠲⠤⢬⠇
⠀⠙⠦⠴⠒⠉⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠘⡇⠀⡕⢰⠀⠀⠠⠇⠀⠁⠖⡐⠘⠮⠺⠹⠉⠑⠒⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠻⡐⢟⢲⠒⠂⢹⠉⠁⢈⠷⡑⠀⡇⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠢⣙⠦⢀⣩⣂⡠⢞⡀⢁⡜⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⡙⠲⢤⣭⣈⣭⣐⣩⣚⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠭⠉⠉⠉⠀⠐⠊

⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷
⠀⠀⠘⣿⣿⣗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣿⣿⡏
⠀⠀⠀⠈⢿⣿⣷⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣵⣿⣿⠟
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠻⢿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⠟⠛⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⣿⡿⠃⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣱⠟⠋⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢋⠁⢀⣀⣠⠐⠒⠒⠛⠛⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⠧⡄⠀⠀⢀⣠⢖⣯⠞⣋⣡⠄⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢐⢞⡢⡓⢿⣌⢤⢚⡭⠞⢉⣴⡞⠉⣧⡀⠈⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣫⡚⡉⣿⣜⣞⡿⠟⣉⡴⠚⠁⣀⡼⣞⠉⣻⡶⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⣼⡾⢛⣡⠴⠻⣧⡡⡴⠚⢉⣦⠼⠋⣡⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣡⠶⢋⣡⣖⢋⠹⣧⢀⣹⣷⡐⡖⢋⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⠖⢋⠅⠚⣏⣹⣿⣦⢢⠿⣷⡑⣉⣿⣌⠃
⠀⠀⠀⠀⠀⣀⡤⠞⡉⠄⠎⠀⠀⡠⢘⠡⠂⠹⣧⡡⡼⢷⡜⡈⠛⠂
⠀⠀⣶⣦⣼⡯⠐⠈⢁⡀⠢⡀⠀⠀⡥⠀⢁⣠⠿⣷⡄⠈⠉
⠀⠀⠘⠿⠇⠀⠀⠀⠀⠙⠢⠀⠈⠦⠈⠛⠉⠔⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠉⠉⢉⣀⣀⣀⣀⣀⡉⠉⠉⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⢁⣠⣤⠶⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⣤⣄⡀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⣀⡴⠞⠋⠁⣀⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣤⣀⡈⠙⠳⢦⣄⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠞⠉⢀⣤⠶⠛⠉⣀⣠⣤⠴⠶⠶⠶⠛⠶⠶⠶⠶⣤⣄⣀⠉⠙⠳⣦⣀⠈⠻⢦⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣴⠟⠁⣠⡾⠋⢀⣠⡶⠛⠉⢁⣀⣤⣤⠴⠶⠶⠶⠦⣤⣤⣀⡈⠉⠛⠶⣤⡀⠙⠳⣄⡀⠙⢦⡀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⣠⡟⠁⣠⠟⠁⣠⡾⠃⢀⡴⠛⠁⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠈⠙⢶⣄⠈⠻⣄⠀⠻⣆⠀⠹⣆
⠀⠀⠀⠀⠀⣰⠏⠀⣴⠏⠀⣴⠋⢀⣴⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢷⡀⠘⢷⡀⠘⣧⡀⠹⣦
⠀⠀⠀⠀⢠⡟⠀⣸⠇⢀⡾⠁⢠⡾⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢻⣄⠈⢷⡀⠘⣷⠀⠹⣆
⠀⠀⠀⠀⣿⠀⢰⡏⠀⣼⠃⢀⡿⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢻⡄⠈⣷⠀⠸⣇⠀⢻⡄
⠀⠀⠀⢸⡇⠀⣾⠀⢰⡏⠀⣼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⠀⠸⣇⠀⢿⡀⠘⣇
⠀⠀⠀⣼⠁⢠⣯⣤⣼⣁⣀⣟⣀⣼⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣿⣤⣼⡇⠀⣿
⠀⣠⣤⠿⣶⠋⠁⠀⠀⠉⠛⠟⠉⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾⠛⠛⠛⠋⠁⠀⠀⠀⠈⠙⢷⣿
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣄
⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠈⠙⠛⢷⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡷⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠞
⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⣤⠾⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⡶⠶⠋⠛⠲⠶⠶⠞⠛⠁

⣠⢤⣄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣺⣭⣭⣭⠓⠢⣀
⢷⣦⣽⣛⣒⠦⢤⣤⡶⢾⣏⣏⢻⠉⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣾⣿⣿⠟⠉⠀⠀⠀⠈⢢
⠀⠀⠀⠈⠙⢻⡶⢦⣭⣙⠛⠾⢯⡓⣵⣋⣙⠛⠻⠷⢶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢸⢟⣛⣉⡁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢸⠁⣿⣿⢿⠻⡷⣦⣬⣉⠙⠛⠿⣿⣶⣶⣤⣤⣉⣉⠛⠛⠻⠿⠿⠿⣶⣶⣶⣾⣯⠋⢀⣴⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⣄⡿⢿⣸⣟⡷⢿⡉⣿⣿⣷⣦⣤⣀⠉⠹⡿⣿⣿⣻⣷⢶⣦⣤⣄⣀⠀⠀⠀⢀⢆⣾⣿⠟⠋⠀⠈⡇⠀⠀⠀⠀⣠⣤⣿
⠀⠀⠀⠀⠀⠠⣇⠰⣲⢥⡿⠧⣸⠃⣿⣿⣷⣾⣿⣿⢻⣶⡜⢾⣆⠉⠙⠛⠷⣯⣝⡻⢿⣿⡿⡼⣸⣿⣿⡄⠀⡀⠀⣷⣤⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⡏⠀⣾⡶⠶⣸⡆⣿⣬⣌⡈⣹⣿⣾⣿⣿⡘⣿⡄⠀⠀⠀⠀⠉⠺⣕⢭⣃⠃⣿⣿⠟⠘⠜⡰⢱⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠾⣷⣶⣯⣄⠛⢯⣟⡿⣷⣯⣽⣟⢻⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠑⡟⠀⣿⣿⡄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣘⡟⢿⣶⣬⣙⠳⠯⡟⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⣤⡴⠟⠻⢿⢿⣿⣿⣿⣿⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢧⣸⠿⠋⠛⠿⣶⣏⣈⣁⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⢉⣿⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡢⣄⣀⣀⣀⣀⡤⢞⡥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣠⡄⠀⠈⢳⠈⢹⣿⡟⠰⠡⣪⣯⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣾⣾⡟⠄⠂⠀⠀⢺⡆⢸⣿⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣊⠈⠫⣿⢿⣿⣿⣦⠀⠀⠀⠘⣇⢸⡏⠀⡞⠁
⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣦⣄⠙⠪⡻⣿⣧⡀⠀⠀⢻⢸⢇⡜⠀⠀⠀⠀⠀⠀⢀⣠⠶⣢⣴⢦⠂⡐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠳⢦⣈⠪⠻⣿⡄⠀⢰⡾⢸⠇⠀⠀⠀⢀⠄⣊⣥⣿⠤⠐⠼⣼⡇⠒⡪⢔⣬⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠘⣿⡀⠀⢧⢼⡆⠀⠀⡰⢡⡞⡨⠷⡁⠀⠠⢚⠁⠊⠁⠢⡄⠁⠉⠑⢝⢆⠀⠀⠀⠀⠀⠀⡸⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⣧⠀⠸⣸⡇⠀⡘⡐⢍⠜⠀⠀⠡⣰⣷⣫⠐⠄⠀⠀⠈⠆⠀⠸⡆⢈⢣⡀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⣇⢱⡏⢗⣰⢁⣸⠾⣀⡠⣀⣸⣫⣽⡿⣮⣔⠄⢢⠜⠆⠀⠀⠱⠀⠎⣧⠀⠀⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡌⢿⡜⣽⣞⢕⡥⢬⣝⢾⡣⢾⣮⡳⣦⣿⡳⣭⢢⡀⠀⡀⡗⠀⠸⢸⢦⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣷⡘⣧⠹⢹⡸⣆⡁⠊⠑⢌⠳⡝⠺⣜⢿⣿⣎⠿⠴⡪⠊⠀⠀⢠⢸⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡘⣧⠸⣿⠳⠒⠂⠨⠭⡳⣌⡳⣬⣷⣾⣿⣿⢿⣵⣿⡄⢀⣎⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡜⢷⡙⢧⡑⠄⡀⠀⡀⡿⠛⠺⡏⠉⡙⢉⢟⠹⣍⢓⡞⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣍⠙⠓⠲⣈⠔⠁⢀⠴⡃⠃⠙⠛⠻⡣⠙⢛⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⠻⢿⣯⣕⣐⣢⠥⠊⠁⠀⢀⣠⡴⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣉⣙⣛⢛⣛⣛⣛⣉⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉