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⠀⠀⣿⢦⡀
⠀⢰⣿⡦⡧⠀⢠⣄
⠀⠀⠳⣷⠃⡰⣻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⣶⣦⣨⣤⣟⡯⡼⣠⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢶⣶⣦
⢷⠙⠟⡻⢻⡟⢻⣿⠖⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⣀⡀
⣸⡞⢾⡗⣸⣷⣚⠓⠋⣔⠉⠛⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢆⡟⡀⠃⠖⢻
⡅⢧⣟⡵⡷⣻⡿⠗⢒⣿⠀⠀⠀⣸⠴⠒⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠱⠇⠛⠄⠚
⢿⣾⠧⣽⣱⡇⣇⢶⡇⠘⣶⢰⣠⠏⢀⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⣀⡈⢳⣸⣏⣠⠜⢪⣧⣆⠙⢿⠿⣖⠁⢀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡽⡟⠘⣧⡄
⣗⣿⣛⣿⠀⢹⠖⠉⠉⠉⠳⡄⢻⡿⠷⠬⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠋⠁⠀⠉⠣⣼⠀⠀⠀⠀⠀⣿⣶⣧⣤⠤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⢀⡀⢄⡙⢧⠀⠀⠀⠘⠁⠀⠀⠙⣶⠶⢤⡶⢤⡀⠀⠀⠰⠆⣴⠀⡦⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⣄⠀⣄⡉⠲⡽⢮⣗⡧⣾⡢⠴⠂⠀⠀⢈⠆⣸⡩⠤⢼⡆⠀⠀⠀⣿⠀⠀⢸⡇⠘⣦⠀⠴⠈⣷⠀⢲⡄⠘⡆⠀⣿⡀⠒
⣺⣷⣦⠭⠽⣟⢻⣭⡷⣿⣛⣁⣀⣀⣠⡎⠀⣇⡀⠀⠀⠘⡆⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⣆⡁⣿⡀⢸⡁⠀⣇⠀⣿⠙⣦⡀
⠝⠁⠀⠀⣪⠝⡻⣾⢻⡳⠌⠙⠻⣿⣤⣠⣼⡋⠉⠉⠀⢰⠁⠀⠀⠉⠀⠀⠈⠁⠀⠈⠀⠉⠀⠈⠀⠈⠁⠀⠁⠀⠈
⡆⠀⠰⠊⠁⠚⣹⡇⠁⠙⢦⠀⠀⢸⡆⠙⣯⡙⠻⢶⡒⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠒⠀⠀⡀⢀⠀⡀
⣷⣄⡀⠀⠀⣰⠻⢧⡀⠀⠀⠀⠀⣰⢏⠀⡟⠙⠂⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢦⠀⠙⢸⠀⣷
⣠⡝⠉⢙⣩⣭⣭⣓⣿⠟⠶⠶⣿⣿⡄⠀⢧⣀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⠉⢉⡿⢻⣶⣾⣿⡛⠄⠀⢀⣽⣉⣷⣴⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠈⢧⣴⠚⢡⠃⣯⠳⣍⠉⠻⡟⡿⣿⣉⣉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⣄⠀⣿⠀⠋⠀⣹⠀⠈⠀⠀⡇⢸⢸⣧⣍⣙⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⢦⣆⢀⡴⠆
⣹⢿⠛⣦⣤⢴⡿⡷⣶⡶⠖⠳⣼⡜⡇⢳⣌⡙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣾⠇
⣇⢻⣼⠋⢧⡞⣰⣷⣿⠀⠀⠀⠀⠙⢿⣜⣿⣝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡍⢀⠀⡀
⠞⢫⣟⠀⠸⡟⢹⣟⣿⡆⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡠⣆⠇⡋⣄⣸
⣠⣿⣍⢳⡀⢹⡏⣿⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⠀⡱⠐⠋
⣿⣿⢯⡙⢧⠀⢻⡘⡗⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣴⣴⡶
⣡⣏⢳⡙⡎⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠁
⣷⡟⢧⡽⠁
⠘⢿⣼⠁
⣶⣶⣿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⢀⠀⠄
⠐⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢄⡄
⠊⠀⠑⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣀
⠤⣀⡌⢉⣠⠧⢐⠀⡂⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣀⡈⢢⠐
⠀⠨⠇⠁⠀⡴⢃⠔⠀⠀⠀⠂⢀⡆⠀⢰⢇⠀⣀⠀⠀⣀⢀⣏⡀⠀⠎⠁⢀⡏⠀⢠⡾⢁⡀⠀⠀⠀⠀⠀⠀⢃⡆⠀⠀⠉⠣
⠑⣀⡂⣥⠪⠟⠁⠀⠀⢄⠀⠀⡞⠀⢠⡟⣟⣞⣼⣱⢷⣏⡾⣏⠀⠀⠀⠀⡞⢀⣴⠏⢰⢃⢳⣇⡞⠀⠀⠀⠀⡆⢣⡀
⠴⠯⡥⢐⣉⣀⠠⠤⠂⠁⠀⠸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠑⠉⡞⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣉⠓⢢⠴
⣀⠞⠁⠀⠀⠀⢄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⢀⠉⠂⢄⡀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀
⡘⠠⠀⠆⠀⠡⠉⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠠⣘⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠰⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠰⢔⡆⠉⠉⠱⡖⠈⠀⢠⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠐⠠⣀⣠
⠢⢅⠇⠤⡤⢜⣀⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠂⠑⠒⠦⢤⡍
⢀⣀⠄⠀⠀⠔⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⠙⣶
⡰⠀⠀⠸⡁⠀⠀⠁⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢰⢆⣖⣦⡏⢰⠆⠀⠀⡠⣴⢸⣇⡰⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢾⠻⣤⠏⠷⠿⡄⠀⠰⡧⢿⡎⣯⢤⠯⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣹⠟⠖⠁⠀⠀⠀⠀⠀⠀⠐⠦⠔⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣀⢀⡀⠀⠀⠀⣀⣀⡀⣀⣀⠀⠀⠀⣀⢀⡀⢀⣀⣀⠀⠀⠰⠶⠄⠀⡀⣠⣄⠀⢀⣤⣤⣶⣦
⠀⢀⣾⡟⢿⡏⠘⣿⣶⠟⣿⣿⠿⣿⣇⢀⣾⡟⢿⡇⠘⢛⣻⡿⠃⣾⣶⠀⢹⣿⠟⣿⣇⣾⡿⢩⣿⣿
⠀⢸⣿⣧⢿⣿⡄⢻⣿⠀⣿⣿⠀⣿⣿⢸⣿⣦⢿⣿⡄⣻⣷⣤⣀⢿⣿⣴⢎⣿⡆⣿⣿⠻⣷⣟⣽⣿⣄⡀
⠀⠀⠙⠉⠀⠉⠀⠚⠋⠀⠉⠉⠀⠉⠁⠀⠙⠉⠀⠉⣼⡟⣿⡟⠿⠃⠉⠁⠀⠉⠀⠀⠀⠀⣼⠟⣿⡿⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣄⢹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠂
⣧⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⣀
⠿⠟⠀⠄⢀⣠⣤⣤⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⣄
⣷⡀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣾⡏⠉⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣿⣷⣶⣿⣿⡿⠟⣴⣿⣧⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠈⠀⠀⠀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⣶⣿⣿⣿⣿⣿⣿⡿⠉⠁
⣈⣹⣿⣿⣿⡿⡿⠃
⣿⣿⣿⣿⡜⠁⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣤⣶⣾⡟⠓
⠿⠀⣿⣿⡇⢀⣦⣀⣠⣤⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣦⣠⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⡿⠁
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢇
⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡦⠠⡀
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠹⣧⣼⣷⡄⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⠀⠐⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣟⣁⣀
⠀⠈⠁⠈⠛⠿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⢿⣿⣿⣿⣿⣿⣿⠿⡇
⠩⠿⠀⣄⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⢹⣿⣿⠟⠋⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⢸⣶⢤⣀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⣀⡤⣶⡇⡀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠐⣀⠀⠀⠁⡀⠀⠈⢯⠷⢧⣩⠳⣦⢠⢛⠇⡘⡞⡄⣴⠞⣍⡼⠾⡼⠁⠀⢀⠈⠀⠀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⢀⡀⠀⡀⡙⢶⣉⣱⠿⣿⢺⠎⠹⡗⣿⠿⣎⣉⡴⢋⢀⠀⢀⡀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⡈⠑⠖⠀⢆⢰⠞⢦⡀⠀⣠⡤⠈⠉⠉⢩⢊⣠⣤⡀⡉⢉⡉⢁⣤⣄⠀⢀⡴⠳⡆⡰⠀⠢⣊⢁
⠀⠀⠀⠀⠀⠐⠂⠋⠁⠀⠀⢀⡬⠛⠃⠀⠴⣁⢠⠟⣾⣭⣽⡾⠈⡔⡈⠈⠁⠘⠛⣥⡀⠀⠀⠈⠙⠐⠂
⠀⠀⠀⠀⠀⠠⠃⢖⡁⣤⣾⠟⠁⠀⠀⢠⠊⠀⠀⠢⠌⠿⠋⢀⠇⠈⠈⠀⠄⠀⠀⠈⠻⣷⣤⢈⡲⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠂⡤⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠆⡜⡥⢠⢆⠄⡄⡄⣰⠡⠆⠀⢠⠆⡄⢠⠄⢀⡄⡠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⠤⠋⠘⠁⠃⠃⠚⠚⠐⠑⠃⠃⠀⠀⠈⡸⠁⠃⠊⠘⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠠⡀⣔⠂⠿⣿⣄⠀⠀⠀⠰⡀⠀⠀⡠⠄⣤⡀⠀⡄⠀⠀⠀⠄⠀⠀⠀⣠⣿⠿⠐⣢⢀⠄
⠀⠀⠀⠀⠀⢀⡁⡀⠁⠀⠈⠙⠧⣀⡀⠀⢬⠒⠈⣄⣾⠿⣿⣦⠀⢆⠎⠀⠀⢀⣀⠼⠋⠁⠀⠈⢀⢈⡀
⠀⠀⠀⠀⠂⢐⢀⠉⡕⢰⢄⡴⠊⠉⠳⠦⠀⠁⠀⠹⡘⠻⠿⠊⠊⠐⠊⠐⠷⠞⠉⠑⢦⣠⡆⢪⠩⡀⡂⠐
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠂⠀⠂⢂⡴⠚⢻⣭⣿⣸⡅⢠⣆⣿⣭⡟⠓⢦⡒⠐⠀⠐⠃⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠂⠀⠀⡴⣭⡴⠻⣡⡿⠹⣘⡌⢋⣃⠏⢿⣌⠟⢦⣭⢦⠀⠀⠐⠀⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠀⠀⢸⣷⠵⠓⠉⠁⠀⠀⠱⠁⠈⠎⠀⠀⠈⠉⠚⠮⣾⡇⠂⠀⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⣿⣿⠉⠁⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⣸⣿⡇⠀⠀⠀⣿⣿⡀⠀⡟⠀⠀⣿⣷⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⡟⣿⣇⠀⠀⠀⣿⣿⡇⠀⣧⠀⠀⣿⣿⠀⡜⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣇⣀⣸⡇⠀⠀⢀⣇⣿⣿⠀⠀⠀⣼⢿⣷⠀⢿⠀⠀⣿⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠐⣿⡏⠀⢨⡇⠀⠀⢸⠁⢹⣿⡀⠀⠀⣿⢸⣿⡄⢺⠀⠀⣿⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⣼⠀⢸⣿⡇⠀⠀⣿⠘⣿⡇⣸⠀⠀⣿⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⣇⠀⠀⣿⠀⣿⣿⣾⠀⠀⣿⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⢰⠇⠀⠀⣿⣿⠀⠀⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⢧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⢻⣿⡧⠀⠀⢰⣿⣿⡇⠀⠀⠀⣴⣿⠏⠀⠒⣄⠀⠀⢀⣤⡀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡌⣿⣿⡇⠀⠀⢠⣿⡿⠀⠀⠀⠈⠀⠀⢀⣿⣧⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠰⠀⢻⣿⡇⠀⠀⠸⣿⣧⠀⠀⠀⠂⠀⠀⠸⣿⣿⠀⢀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡠⠃⠀⢸⣿⡇⠀⠀⠀⣻⣿⣄⣀⠌⠀⠀⠀⠀⢿⣇⢀⠂⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⣿⠗⠂⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⣿⣧⠀⢀⠔⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠔⠁⠀⠀⠀⠀⢠⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠘⣆⣀⣀⣀⣀⣀⣴⣁⣀⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣤⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⠄⢠⣔⡇⢀⣴⣂⣴⠆⠀⠀⠀⣀⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡵⠛⠿⢿⣿⣾⣾⣫⣠⣤⠟⠈⡃⣐⠋⣠⡞⢿⡌⠃⢨⡅⠾⠡⣄⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⠉⠀⠀⠀⠈⠁⠍⠙⢻⣿⣿⣶⣿⣇⡋⢁⣃⣤⠀⠀⣀⠀⠀⠀⠀⠀⠐⠞⢉⣉⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠉⢿⡟⣿⣿⣤⣴⣷⡞⠀⢤⡶⠂⢠⣴⡏⠉⠻⡁⢠⠁
⠀⠀⠀⠀⠀⠀⢀⡤⣪⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠟⠓⠘⡟⠛⠿⣷⣾⢳⣼⣏⠥⣤⠄⠀⢀⣤⠀⣴⠅
⠀⠀⠀⠀⣠⢔⡫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣤⣶⠈⣦⢼⣿⢻⠿⢿⣶⣿⣽⣷⠙⠧⢐⡛⢤⡾⠜
⠀⠀⠀⠀⣇⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⣿⠣⣉⠀⠀⠀⠁⢿⠙⠛⠿⡖⡌⠉⠈⠃⠲⡄
⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠀⠀⠀⡇⡇
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢹⡏⢹⡟⢁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠘⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡇⢸⡷⢹⡇⢀⡞⢉⡿⢐⡿⠋⣿⠀⣿⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⠃⢸⠇⠘⣧⡸⣧⠜⣧⣸⡇⠀⢿⣠⡟⠻⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⢀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠸⢀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⣼⣎⠰⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹
⠀⠀⠗⠛⢤⡝⢜⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⢙⣂⣤⢸⡿⠛⠉⠑⢄⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸
⠀⠀⠀⠊⠋⢩⡼⣆⠀⠠⠔⠺⢅⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠕⠋
⠀⠀⠀⠀⠀⣠⣴⠚⠛⣛⣾⠓⣟⡶⢔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡭⠚⠁
⠀⠀⠀⠀⠈⠛⠛⡁⢀⣩⠨⣉⡖⠂⠀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣫⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⣶⣤⢾⠿⠆⠀⠀⠀⠀⠀⣀⠤⠤⣁⢀⣀⣸⠤⠒⢒⣤⡤⠀⠀⢀⣠⣚⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡰⠀⠋⣸⢷⣳⣄⠀⠀⠀⣠⡾⠀⠀⢨⠍⣼⢣⣋⡆⠀⣀⣾⣁⣠⣿⡽⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠋⠲⣟⢾⣷⣤⣤⣝⣿⣄⣀⣤⢾⣹⠛⡤⠟⠛⠿⠿⠿⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠿⢋⣽⠋⠟⢠⣇⡄⠀⢹⠄⠀⢠⣰⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠓⠊⠁⠀⠀⠉⠉⠉


⡿⢿⠿⡿⡿⢟⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣿⣻⣿⣟⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠂⠙⢦⡀⠀⠀⠠⣄⢠⣠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣀⣠⣔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣀⣿⣻⣆⣐⡚⣎⠀⠰⠂
⠀⠀⠀⠀⠀⣠⢤⡞⠋⠁⠀⠀⠉⠉⠉⠙⠒⠒⠢⠤⠤⣤⣄⣈⣳⣄⠈⣛⠟⣹⠄⠄⣠⣄⡤⠄⠄
⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠘⠁⣀⣨⠻⢟⣀⡰⢦⣆⠠⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣠⠔⠑⠠⡄⠽⠇⢉⠘⠚⠋⠀⢠⠞⠁⣠⡉⣦
⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡔⠁⠀⢰⣤⠽⢷⣄⠀⠀⢸⠀⣀⠙⠒⠃
⠀⠀⢀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⢦⡀⢰⡏⢳⣶⡀⠀⠀⢸⣼⠈⠁
⠀⢀⠞⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠙⣦⠁⠀⠙⠃⠀⠀⠈⣟⣶⡏⠀⣀⡖⢹
⢀⡎⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⢠⠠⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⠀⠀⠀⠀⠀⠀⠀⠈⢞⢿⢞⡏⣇
⠈⢷⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢨⠀⡀⡇⢸⠈⡀⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⣀⠀⡈⣿⢸⡄⢸
⠀⠀⠳⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡄⠀⠀⣴⡿⠒⠒⠦⣿⠈⣇⠘⠤
⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⣆⢸⡿⠛⠛⠓⢒⡻⠷⢼⣆
⠀⠀⠀⢸⡟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⠏⠈⠓⠀⢠⠞⠁⠀⠀⠀⠈⠙
⠀⠀⡄⠘⢧⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠏⠀⠀⠀⠀⡟⢀⡤⢤⡀⠀⢠⡖
⠀⠈⢿⣿⠤⠙⢦⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠏⠀⠀⠀⠀⠀⠳⡈⠋⣀⠇
⠀⠰⡎⢉⣉⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢀⣧⡀⠈⠉⣭⠑⡄⢀⣤
⠀⠂⡠⠛⣁⡀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠸⡝⣎⠑⣄⠓⠐⠃⠀⠁
⠀⠀⠀⢀⠉⣄⢶⠤⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡟⠀⠀⠀⠀⠀⠀⢹⠈⢆⠈⢲⠀⣠⡔⢷
⠀⠀⠀⠙⠀⠙⠘⢇⡀⠀⠀⠀⠈⠳⣌⠉⠉⠛⠒⠒⠲⠤⠤⢤⣀⣀⡀⠀⣀⡠⣴⠛⠊⠁⠀⠀⠀⠀⠀⣀⣘⣄⣸⣆⡜⠋⠁⢱⣾
⢄⣀⠀⡦⢄⣠⠒⠙⡇⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠋⠉⠉⠙⠁⠀⠀⠀⠀⣄⣤⠔⠂⢿⠀⢈⠙⡟⢠⢲⠄⣸⠋
⢢⠀⢹⠀⣤⡙⡴⠤⠶⠄⢀⣀⣠⢤⣄⡄⠙⢦⡤⠴⠒⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠉⠓⢼⣄⠹⣈⣿⣿⣫⣴⣥⠒
⢢⡵⠶⢧⣷⣕⡄⣲⣠⡟⣁⠤⠖⡩⠁⠠⢒⡒⠒⢤⡀⢸⢈⠙⡆⠀⢀⣤⣤⢖⠒⠒⢤⡀⠀⠀⠀⣱⠟⢁⡤⢩⣼⡁⣸⢷⠒⣦⠙
⠦⣄⠘⣊⢧⡨⣿⡯⣍⠓⣤⠠⠊⡑⠆⡠⣈⡩⠀⠀⠑⣴⠾⠼⠴⢾⡟⠛⠆⠘⢃⢴⠄⡁⠀⠀⠰⢇⣀⣈⣉⣵⠏⡏⡏⢶⡬⠤⡶
⡒⡚⠋⣉⢠⠅⣷⢬⣁⣀⡨⢆⠀⠉⡊⠀⠀⣀⠀⠀⣸⠵⠒⣉⣉⣭⠽⢦⡀⠀⠀⠑⠊⠀⠀⠀⠀⢠⡞⠉⢀⡼⡇⠉⠁⡜⡇⠀⡗
⣋⣨⣋⣝⣤⣺⣀⣂⣈⣻⣻⣯⣥⣤⣟⣀⣀⣜⣀⣴⣓⣂⣊⣀⣀⣀⣀⣷⣃⣀⣀⣀⣀⣀⣀⣀⣀⣜⣠⣔⣋⣠⣛⣀⣾⣀⣇⣀⣧
⣿⣿⣉⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠒⡂⠭⠍⠉⠂⠈⠁⠫⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣧⠀⠢⡀⠈⢀⣒⡀⠁⠐⠒⠘⠢⠀⠁
⠀⠀⠀⠀⠀⠀⣾⡇⣺⣧⡀⣿⡆⣿⣇⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠠⡙⠲⣌⡐⠒⠀⢀⠐⠄⠞⠀⢠
⠀⠀⠀⠀⠀⠀⣿⡇⣽⡟⢿⣿⡇⣿⡿⣷⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣽⡌⠲⣭⠀⢠⠈⠡⡴⢀⣠⠋
⠀⠀⠀⠀⠀⠀⣙⣃⠙⣃⡀⣙⡁⡙⠃⣈⠛⡀⢛⠁⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣀⡐⠋⠀⠀⢠⣼⡷⢋⡁
⠀⠀⠀⠀⠀⠀⣿⣹⢿⣿⣀⣿⣧⣿⢿⣹⣤⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠬⣭⡭⠤⠀⠊⣈⡵⣶⠏⢀⡴
⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⠐⠀⠀⠀⠀⠁⠠⠚⠁⠐⠋
⠀⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⠀⠀⠀⠀⠀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠟⠋⠁




⠀⠀⠀⠀⠀⠘



⠀⠀⠒⠲⠶⢶⣤⣐⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢬⣟⢦⣄
⠀⠀⣀⠀⠠⠠⣄⡀⢱⣄⡐⡨⠛⢿⣿⣦⣀
⠀⠸⣿⠟⠛⠲⠤⠙⠻⣶⣝⠪⣲⢅⠙⢿⣿⣷⣄
⠶⢄⠙⢷⣄⠀⠀⠀⠀⠠⡙⢷⣮⠳⣕⢤⠻⢮⣿⣷⣄
⣒⡂⠤⢀⠙⠷⣄⠸⣶⣦⣌⠳⣌⠣⠈⠳⡕⡄⠁⠹⣿⡢⡀
⣉⣩⣝⣲⠄⠀⠈⠳⡜⢯⡉⠻⢮⡳⡄⠀⢌⢞⡕⣆⠘⣇⠑⡀
⣁⣠⡤⠤⢤⣀⠀⢠⠈⢎⢷⠀⠀⠻⣌⠆⠈⣎⢞⡼⡆⢹⡆⠐
⣩⠤⠖⠒⠲⠬⡑⣜⡮⠀⠈⠇⠀⡀⠹⡄⠀⢸⡌⣷⢻⠀⣿
⠤⠤⠴⠒⠚⠲⡌⠌⡞⠈⠀⠀⢀⣿⡆⢱⠀⠀⣇⣿⢸⡇⢸⡄
⠒⠒⠒⠚⠛⣦⠈⡆⠀⢠⠀⢀⣼⠿⣿⠀⢀⠀⡇⡿⠀⠀⠀⠇
⠛⠛⠛⢷⠀⡆⡸⠁⠀⠀⢠⣾⠋⠀⡿⠀⣾⡿⠁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠒⣒⡒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⡠⠔⠂⠉⠉⠁⠀⠀⠉⠉⠒⢄⡑⠦⣀⣀
⠀⠀⠀⠀⠀⢠⠞⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⡰⠁⢀⡔⢁⡴⠒⠲⠄⠀⢰⡶⠶⠶⠀⠀⣶⠶⠶⠆⠈⢣⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢠⠃⠀⡞⠀⠈⠓⠒⠲⢦⡀⢸⡷⠶⠶⠀⠀⣿⠶⠶⠆⠀⠀⢇⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⡜⠀⢸⠀⠀⠐⠲⠦⠤⠞⠁⠸⠧⠤⠤⠄⠀⠿⠤⠤⠤⠀⠀⠸⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⡇⠀⡏⠀⠀⢠⠀⠀⠀⣄⠀⣠⠴⠶⢤⡀⢀⡄⠀⠀⣠⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⡇⠀⠀⠘⢧⣀⣠⡿⠀⡇⠀⠀⠀⡇⢸⡇⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⣆⠸⡄⠀⠈⠲⠦⠞⠁⠀⠙⠲⠶⠞⠁⠀⠙⠲⠖⠋⠀⡴⢡⠎⠑⠢⡄⠀⠀⠀⢇
⠀⠀⠀⠀⠈⢦⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠋⠀⠀⠀⢸⡄⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠙⠢⣌⠑⠢⠤⢀⣀⣀⣀⣀⣀⣀⣀⠤⠔⣊⠥⠊⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠈⠙⠒⠒⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⢤⡤⠤⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢠⡿⠀⢤⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣦⡀⢀⣿⢁⣾⠇⠀⢀⠹⣦⡴⠲⣦⡄⠀⢸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢸⣿⡿⣣⡶⢲⣿⠁⣿⣧⠀⣸⣿⢠⣿⡟⢳⣦⠀⠃⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⣿⡿⢿⣿⠰⣿⡇⣸⡇⠀⣿⣿⣴⣿⠏⣸⣿⠃⢸⣿⡇⣰⡿⢰⠋⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠀⠙⠛⠉⠳⠀⣿⣿⠛⠋⢠⣿⣯⣠⣾⡟⢰⣿⠃⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⣾⠟⠀⠸⡿⠀⢠⣤⡀⠀⣸⣿⡏⠀⢠⣿⡟⠀⢩⡷⠒⠈⠻⣿⠏⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣡⠀⠀⠀⠀⠀⣙⠛⠀⠠⣿⡿⠀⢀⣀⠉⠀⣀⠻⣷⣤⣴⡾⠋⣰⣷⠀⠀⢸⠀⠈⣿⣆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠧⣤⡀⠀⢠⣾⣿⣇⣤⡀⠉⠁⢰⣿⡟⢀⣾⡏⡇⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠓⠀⢸⣿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠏⢹⡇⠀⠀⣿⡆⣼⣿⡏⣏⠀⡇⠀⠀⣼⣿⠁⢸⣿⡇⡇⠀⠀⣠⣤⣿⣿⠇⣴⣶⠒⣴⡆⢸⣿⡄⠸⣿
⠀⠀⠀⠀⠀⢠⣾⡏⠀⣿⡇⠀⣰⣿⡏⠸⣿⣇⣘⢿⣶⠿⠀⢿⣿⠉⢸⣿⡷⠀⠀⣼⡟⠀⣿⣿⢸⣿⡇⢠⣿⡇⠸⣿⣧⣤⣿⡇
⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿⡇⠺⠛⠻⢤⡀⠈⠉⢡⣾⠋⠠⣄⠈⠻⣄⣼⣿⡗⠉⣧⢻⣿⣦⣿⣿⣄⠙⠛⠺⢿⣤⠀⠈⠉⠀⣿⡇
⠀⠀⠀⠀⢰⣿⣿⠀⢠⣿⡇⠀⠀⠀⠀⢹⣦⡀⣿⣿⡀⠀⢸⠂⠀⠀⣸⣿⠁⣴⣿⠀⠈⢉⣠⣬⣿⣯⣄⡀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢀⣿⣇⠘⠻⠿⠖⢋⣴⠆⢠⣿⡟⣾⣿⠏⠀⢰⣿⠿⠿⢿⣿⣿⣿⣷⣤⡀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠻⣿⣀⣾⡿⠃⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⣠⣿⡟⠀⠈⠋⠁⠈⠁⠀⠀⢸⡇⠀⠀⠀⠈⠙⠻⣿⣿⣷⣄⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢉⣉⣀⣀⣀⣤⣶⣿⣿⠏⠀⠀⠀⢰⣿⡿⠀⠀⡠⢀⡖⠓⢦⡀⠀⠀⠳⣄⡀⠀⠀⠀⠀⢈⣻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⣫⣤⣤⣄⠀⣠⣾⣿⠟⠉⣸⡇⠈⢧⣀⣨⣧⡴⠂⠀⠀⠉⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢿⣿⣧⠀⢸⣿⠀⠀⢿⣧⣀⢀⣈⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⣿⠀⠀⠙⠷⠤⠌⠛⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⡟⠀⢀⢴⡄⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠀⠀⡀⠀⠙⠳⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣿⠁⠀⡞⢸⡇⣀⠀⣴⡿⠿⣶⣄⣼⣿⠇⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠸⣇⠀⢰⣷⠟⣰⣿⡟⠻⠿⠶⠾⣿⣿⣿⠀⠀⣿⣷⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠙⣻⠏⠁⠀⣿⣿⡇⠀⠀⠀⢠⡿⠘⢿⣷⣤⡿⣿⣷⣄⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⡾⠋⠀⠀⠀⠘⢿⣿⣶⣤⣶⡟⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠔⢊⠍⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠔⢊⣠⣖⠥⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣔⣡⠶⠛⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⢀⡀⠀⡰⠁⡇⣸⠟⢓⣒⡒⠀⣀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⡸⡁⣰⠁⡴⠛⢉⡁⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢰⠀⡇⢃⣜⠴⠊⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⣄⠀⠀⡼⢀⣠⡯⠵⢒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢈⢢⠀⡄⣸⠃⠀⡴⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⢸⠀⠇⣣⠃⡠⠊⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢸⠀⢸⣿⠊⡠⠞⠁⠀⠀⠀⠀⢀⢠⠤⢒⡆⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢆⢸⡧⠊⠀⠀⠀⠀⠰⠊⠉⠀⢸⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠂⠀⢠⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠰⠁⠀⢀⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡌⠀⠀⢀⢒⠀⢀⡤⡀⢀⣃⡀⠀⡀⠀⠀⠀⢇⢀⠴⠁⢀⡧⠀⣤⡀⣀⢀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡖⠹⠔⠛⠈⠢⠾⠁⠑⢺⢿⠡⠊⠀⠀⠀⠀⠀⢁⠔⠊⡉⠀⠈⠚⠁⠑⠺⠔⠁⠀⠀⠀⠀⢀⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⢀⠤⠤⡤⠤⠤⠤⡀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠒⠀⠁⠀⠀⠀⠐⠄⣸⣯⠔⠁⠈⠀⠀⠰⠢⠀⠀⠉⠈⠢⣼⣧⠀⠆⠀⠀⠀⡐⠀⠐⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣽⢀⠀⠀⠀⠀⠶⡷⢦⣿⣅⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣿⣧⢶⡶⠄⠀⠀⠀⡀⢺⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡱⣶⣶⡀⠤⠃⠀⠀⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠀⠀⠐⠤⢀⣶⣶⣇⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠿⡧
⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠸⣿⡴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣿⠏⠠
⠀⠀⠀⡀⢀⠀⠀⣤⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠉⠐⣢
⠀⠂⠄⠀⠈⠀⢀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄⡦⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠁⠈⢀⠐
⠈⠻⣷⣄⣠⣼⣿⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡗⢹⣠⡫⡇⣸⠈⡆⠷⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣦⣀⢠⣴⠶
⢀⣴⡿⠋⠙⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡴⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠟⠻⢷⣄
⠀⠄⠂⠀⢀⠀⠉⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡏⡸⢸⣁⢎⠀⠀⠀⠀⠀⠀⠀⢐⠀⠋⠀⠀⠀⠐⢁
⠀⠀⠀⠁⠀⠀⠀⢷⣢⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⡶⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠄⢰⣿⠳⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠾⣿⡌⠠
⠀⠀⠀⠀⠀⠀⠀⣰⡟⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⢷⣄
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣿⣟⠃⠄⡀⠀⠀⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⠀⠀⢀⠠⠞⣿⣿⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⡏⠈⠀⠀⠀⠠⢶⣶⢶⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⣶⠦⣶⡦⠀⠀⠀⠀⠈⣷⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠤⢀⡀⠀⠀⠀⠠⠈⢻⡗⢄⡀⠀⠀⠀⢄⠄⠀⠀⠀⢀⡴⢺⡏⠡⠀⠀⠀⠀⢀⠀⢀⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠈⠒⠤⠒⠲⠴⠀⠁⠁⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⠼⠅⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠄⠘⠦⡈⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠓⠀⠀⠰⠖⠲⠶⣶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢶⡆⠀⠀⠀⠀⠀⢀⠁⠀⠂⠰⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠀⠀⢁⠀⠀⠀⠀⠀⠀⣠⣾⡥⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣽⣦⡀⠀⠀⠀⠁⠀⠠⠀⠀⠀⣆⡀
⢰⣀⠤⠺⡟⠧⠐⢷⠖⠃⠀⠰⣦⣠⢞⠋⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠎⢙⢦⣠⡶⠀⠀⠙⢻⠋⠐⠛⡛⠐⠠⠴
⠁⠄⢨⠀⠐⠀⠀⠈⠀⠀⠀⣠⣿⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣿⣧⡀⠀⠀⠀⠀⠀⢈⠂⠨⠀⠂
⠀⠀⠀⠀⢘⣀⠀⠀⠀⣠⢞⡁⣾⣏⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⡆⣛⢦⡀⠀⠀⢀⣀⠀⡀
⠀⠀⢤⣠⢺⡇⠐⣠⠞⠱⠯⠟⠛⠩⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠽⠙⠛⠯⠷⠙⢦⡀⠀⣿⢢⣠⠄
⠀⠀⢀⣹⣾⣧⠞⣁⡀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣀⡙⢦⣿⣾⣁
⠀⠀⠉⣹⢿⣏⣿⡿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣏⣿⢿⡉⠁
⠀⢠⠞⠁⢸⡏⠿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⠏⣿⠀⠙⢦
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠊⡀⣠⡤⢀⡄⡀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠈⠀⠀⠁⠈⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡜⠁⡴⢢⠀⡴⢣⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⠀⠀⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⠞⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠀⠀⢀
⠀⠈⠳⣄⢸⣷⣿⣇⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣄⣿⣷⣿⢀⡴⠋
⠀⠀⠶⣾⣿⣇⠿⢿⡟⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⣿⠿⢇⣿⣿⡶⠆
⠀⠀⣠⡜⢹⡍⠳⣄⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠃⠀⢀⡴⠋⣿⢹⣦⡀
⠀⠀⠁⣈⢻⣇⠀⠈⠳⣌⠳⠷⣾⡾⢽⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⠿⣾⠶⠷⢋⡴⠋⠀⠀⣿⢋⡀⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠳⣄⣻⣷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⣁⡴⠋⠀⠀⠀⠀⠩⠀⢀
⡀⠆⢘⠀⠂⠀⠀⡀⠀⠀⠀⠈⣿⣅⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⣽⡏⠀⠀⠀⢀⠀⠀⠀⡀⠨⠀⡃
⠰⠋⠐⢴⣥⡖⠠⡼⠦⡀⠀⠈⠃⠈⠳⣌⢅⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⢅⡴⠋⠀⠙⠀⠀⠴⠚⠖⠘⣾⠾⠂⠉⠙
⠀⠀⠀⠐⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢉⡵⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⢈
⠀⠀⠀⠈⠀⠀⠐⠀⠂⠀⡤⠀⠀⠐⠒⠒⠒⠛⠳⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡶⠛⠒⠒⠒⠒⡀⠀⠐⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢲⡂⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⡽⠉⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣈⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠄⣁⠊⠈⠆⣀⠀⠀⠀⠀⠀⢀⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠠⡂⢀⠘⣸⠃⠆⠀⢀⠀⠁⠙⠉⠈⠙⠙⣿⣥⡀⠀⠒⠀⠄⠀⠀⢀⡀⠔⠛⠀⠀⠠⣀⣀⡀
⠀⠀⠀⠈⢀⡙⣤⡀⠜⠈⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⣿⡇⠘⢦⡂⡒⢐⠒⠆⠲⡀⡶⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠈⠃⠁⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠉⠙⠚⠓⠛⠊⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣻⣄⢀⣠⣿⣿⡟⠀⠀⠀⠒⡀⣀⠀⠀⠀⡇⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠁⠀⠀⠁⠀⠀⠀⡆⠀⢠⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠌⠀⠘⣷⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠙⠈⢀⣤⠃⠀⠀⢇⣀⠀⣀⣀⣦⠆
⠀⠀⠀⠀⡀⠀⠠⡏⠀⠀⠀⠀⢋⡀⠀⠀⠀⠹⣶⣶⣾⡟⣰⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠕⠚⠁⠀⠀⠐⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⡀⠀⣀⣶⠀⠀⡀⣀⢠⡄⢀⠀⠀⠀⠀⣼⠀⠀⣧⡀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠁⢸⠀⠀⠈⠳⠤⡧⡴⠟⠃⠠⠥⠼⡿⠟⠛⠻⣷⡄⠀⠀⣧⠀⠀⠈⠳⢦⣤⣤⡴⠷⠟⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠇⠀⠀⠀⣀⠀⠐⢄⠀⠀⠀⣠⣠⣼⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⣤⡼⠋⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣤⣶⣤⣤⣤⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡞⠀⠀⠀⠀⡸⠿⠛⠏⣗⢀⡀
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠆⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄⣀⣽⡿
⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣯⣄⣀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢦⡿
⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⢩⡟⠁⠉⣹⡟⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⣾⠁⠀⢠⣿⢁⣴⡆⣠⡶⢄⡄⢠⡖⠀⣦⢰⡏⡰⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣾⡇⡎⣿⢱⡏⡀⡾⠁⣿⡄⣰⠇⣾⣗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡠⠋⣾⡟⣸⠇⡀⠉⠀⠣⠀⠛⢠⡟⢀⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡟⠁⣸⢇⠜⡇⡖⢂⢸⣃⡜⡉⢸⠇⠀⠀⠀⠀⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠛⠀⠯⠎⣰⡗⣾⡻⣻⠋⣼⡇⠸⠦⠀⠀⢀⡞⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠈⢂⡤⣀⣔⡉⣸⠑⠷⠃⠙⠊⢸⡁⠀⠀⡀⠀⣼⡠⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢘⣯⣜⣠⡞⣩⣿⠁⢠⡏⣠⡄⣠⠄⣶⠀⠓⠔⢀⡀⢹⡏⣠⡄⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣻⡜⢧⠟⠁⢀⡞⢰⡿⢰⡟⣸⢯⡟⣼⢣⡟⢁⣾⢏⡾⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠋⠀⠘⠦⠴⠋⠀⠀⠀⠀⠀⡿⠘⠋⠛⠉⠓⠁⠈⣼⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠐⠢⠐⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⠘⠀⠀⠛⠢⠀⠀⠈⠁⠀⠀⠀⠉⢀
⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⠀⠘⠘⠀⠈
⠀⠀⠀⠀⠄⠀⣀⣀⠀⠄⠂⠁⠀⠀⠀⠀⠐⠠⢀⡀⢀⠤⠂⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠠⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠠⡔⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠢⡄⠈⠀⠀⠀⠀⠀⠗⢀⠀⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠥⠄⠁⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠁⡀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠰⢆⠠⠄⠠⠔⠀⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠲⠂⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠐⠒⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠦⠀⠀⠀⠀⡀⠀⠀⠀⠈⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠄⠀⠀⠘
⡄⠒⠠⡀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠁⠀⡀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠈⠀⠉⠀⢀⠁⠀⠠⠂⠀⢀⠂⠉⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⠠⠄⠀⠤⠘⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡈⠣⡙⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠠⠍⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠣⠄⠀⠀⠀⠀⠀⠐⠀⣀⣑⡎⠀⠀⠀⠀⠀⢀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠉⠙⠒⠄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣄⣠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣤⠀⠀⢠⣶⢶⣄⠀⣿⣿⠀⢀⣤⢶⡄⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣟⠀⢠⣿⡟⠀⠀⠀⠙⢷⣽⣧⡸⠟⣴⣏⡤⠞⠁⠀⠀⢀⡴⠂⣰⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⢹⣿⠛⠛⣿⡿⢷⣶⣤⣤⣄⣀⣀⣉⠀⠀⠉⠁⣀⣀⣠⣴⠞⠋⠀⢰⣿⠇⠀⢸⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡟⠀⢸⣿⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⢀⡀⠀⣿⡏⠀⡀⠙⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠁⠀⣿⡇⠀⢸⣇⡔⢲⡆⠀⠀⠀⣠⡶⠚⢢⣼⡟⠀⢠⣤⢠⢞⣿⠇⠀⣿⡴⠋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡼⠇⠀⣿⡇⠀⢹⡟⠀⣾⡇⠀⢀⣾⠋⢀⡴⣿⡟⠀⢀⣿⣧⠃⣾⠏⠰⣾⣿⣧⡴⠟⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⢿⣇⠀⠈⠁⠀⣿⠃⢀⡼⠧⠔⠉⠀⢿⣀⡴⠻⣿⠇⠀⣿⣀⠔⠁⠙⢻⣆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⣄⣀⠰⣄⠘⣦⠈⠛⠷⠀⠀⠀⠛⠖⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠁⠀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣮⣿⣮⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⠏⣰⠟⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣍⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⢀⣴⣶⣶⡶⠆⣀⣀⣀⣀⣾⣿⣷⠀⠀⠀⠀⠀⣰⣾⣷⡄⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣼⢿⣿⣿⡇⣰⣿⣿⣿⠋⠀⠸⣿⡏⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⣠⡾⠋⢸⣿⣿⡇⣿⣿⣿⡏⠀⠀⠀⣿⣿⠀⢻⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⡇⠸⣰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⣾⣿⣿⠁⣿⣿⣿⡇⠀⠀⠀⣿⣿⠂⢸⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⣦⣤⣰⣿⣿⡟⠀⢹⣿⣿⣧⠀⠀⢀⣿⡿⠀⠀⢿⣿⣿⣇⠀⢀⣼⠏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡀⠀⠀⣨⣿⣿⣿⣧⠀⠀⠙⠿⢿⣷⣶⡿⠟⠁⠀⠀⠀⠙⠻⠿⠷⠿⠋⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡏⣦⣤
⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣿⣿⣥⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⣀
⠠⠶⣾⠏⣱⠇⠀⣢⡄⢀⣤⣀⢀⡾⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡇⠀⡆
⠀⢸⠏⢠⡟⣾⢳⡏⣿⢸⠏⣿⠸⠷⣆⠀⠀⠀⠀⠀⢸⣯⠄⣼⠃⠛⠏⠽⡏⡛⡿⢰⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⡄⡀⠀⢀⠄⠹⠚⠁
⠀⠘⠀⠘⢃⣝⣠⢣⡞⢄⡷⣰⠆⠀⠉⠀⠀⠀⠀⠀⣿⣿⣸⣿⣸⣠⣼⣤⣡⣇⣇⡌⣻⣿⠀⠀⠀⠀⠀⠘⣿⠃⣼⢣⡄⡴⣆⢰⣦⡄⣸⠣⠄
⠀⠀⠀⠀⠸⢇⡏⢸⠀⠈⠇⠻⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢈⠏⡎⣰⡏⢸⠋⣿⣿⣿⡇⠀⠀⠀⠀⠀⠏⠀⠟⡸⠃⡃⠋⠘⠹⠇⠛⢻⡄
⠀⠀⠀⠀⠀⠨⢔⣶⢲⣓⡏⠿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣞⣼⣧⣿⣷⣾⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡟⡼⣼⠋⢸⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣀⣴⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⣠⠀⠈⠀⠁⠒⠠⠄⠀⡀
⠉⣿⠋⢰⡏⣤⢠⡶⡄⢰⣆⡄⢰⡏⠔⠂⠀⠀⠀⠀⠉⣿⠃⣼⢣⡄⣴⢦⡀⣶⣤⠀⣾⠡⠂⠀⠀⠀⠀⠴⣿⠇⣸⢧⡆⣴⢳⡄⣾⣶⡆⣼⢣
⠸⠏⠀⠿⠀⠿⠻⠐⠇⠻⠙⠇⠸⠛⣧⠀⠀⠀⠀⠀⠸⡇⠀⡟⢸⠇⠿⢸⠰⠏⢿⠀⠟⢿⡄⠀⠀⠀⠀⠀⡟⠀⣿⠸⠇⣻⢘⡇⢟⡸⠇⠿⢻
⠀⠀⠀⠀⣿⢣⠇⣾⠁⢸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⢣⡟⠁⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⢳⢇⡿⠀⣼⠇⡟⠀⠠
⠀⠀⠀⠀⠈⠞⠀⠉⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠟⠘⠃⠀⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠀⠃⠀⠈⠀⠁

⣴⣶⣶⣶⣴⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠣⠀⠀⠀⠀⡠⠒⠉⠁⠀⠈⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠃⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠂⠁⠀⠁⠀⠁⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠙⢦⣀⠀⣀⣀⡴⠂⠀⠀⠀⠀⢁
⠐⡀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠢⢀⠀⠀⠉⠢⠤⢀⣀⠀⠀⠔⡣⠒⠒⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠄⢀⡠⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠰⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠀⠐⠠⠄⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢄⡀⠀⡠⠂
⠀⠈⠀⠀⢀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠁⠀⠀⠈⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠠⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠦⢠⡴⠶⣦⡀⣠⡶⠶⣤⠀⣶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢿⡀⠀⢸⡇⣿⠀⠀⣸⠇⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠛⠛⠋⠀⠈⠛⠛⠋⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣾⡄⢠⠞⠓⠀⣦⠆⢠⠞⣓⡂⣾⣳⢠⠞⠓⡆⢶⢰⠆⡷⣄⡆⣾⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠘⠉⠙⠈⠓⠚⠀⠋⠓⠈⠓⠚⠁⠉⠑⠈⠓⠚⠁⠙⠚⠀⠃⠘⠃⠛⠚⠁

⠀⢀⣀⣀⡀
⠑⡈⠶⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠖
⠄⢰⠸⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁⠀⠀⢄
⠀⠀⡠⠀⠐⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠐⠢⠄⣀⠉⠂⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⢀⠀⠀⠀⣀
⠒⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⣀⠠⢔⡡⠒⠀⠀⠁⠒⠢⢄⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠂⠀⠀⣠⠖⠛⠛⠛⠶⣄⠀⠀⡴⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢰⡆⠐⠀⠀⠀⡅⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠘⡀⠈⠂⠄⣀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠙⠲⢤⡟⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣠⠟⡧⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⣠⠴⢾⣁⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⣀⣀⣤⠖⠛⠛⣆
⠀⠀⠀⢀⣶⡈⠉⢻⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠁⣠⠖⠺⣅⣛⢳⡿⣆⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠉⣉⣀⡀⠀⠀⢿
⠀⠀⣤⣏⣤⠟⠉⠙⢷⣄⠀⠀⠀⠉⠓⠶⠤⣤⢶⠀⠀⠀⢠⣧⣯⡞⡿⠛⠙⢯⣹⠏⠀⢹⡀⠀⠀⠀⠀⠀⠈⠳⣦⠀⢠⠞⠉⠈⠉⢳⡄⠈⠳⢤⡀
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⡤⠿⣼⠀⠀⠀⠘⡇⠉⠙⢛⠒⠒⣿⠃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣠⠞⠀⢺⡀⠀⠀⠀⢸⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠉⠙⠛⢦⠶⡆⠀⠀⠀⠀⠹⡄⠀⠉⠀⡼⠁⠈⠀⠀⠀⠸⡆⠀⠀⠀⠀⡞⠁⠀⠀⠀⠓⠦⠤⠶⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣤⣤⣤⠴⢿⣸⠇⠀⠀⠀⠀⠀⠙⣆⢀⡾⠁⠀⠀⠘⠃⠀⠀⢻⡀⠀⠀⠘⣇⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠈⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡄⠀⠈⠁⠀⠘⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢳⣄⠀⠀⠄⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣩⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡇⠀⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣄⣴⠓⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡄⠀⠀⠀⠀⣴⠂⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⣄⣽⢾⣄⣽⣴⠻⣦⢶⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣾⣼⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⡶⠧⠤⠽⢶⡄⠀⠸⡇⢺⠁⠀⠀⠀⣠⣾⠟⠀⠀⢠⡶⠤⣽⣃⣠⠟⡇⣸⣆⠈⢷⡃
⠀⠀⠀⠼⣧⡤⠤⠤⠤⠤⠴⠶⠶⠶⢶⡾⠇⠀⠀⠀⠀⠀⣸⠶⠶⠶⠶⠾⣇⡀⠀⣿⠘⢡⡟⠙⠻⣿⠁⠀⠀⠀⠘⠓⠒⣦⡉⠻⣼⡇⣿⠙⣆⠈⢳⣀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢳⠀⣩⡤⢬⣷⣄⣈⡿⠀⠀⠀⠀⠀⠰⣶⠿⢷⣤⣬⣷⣧⡼⠿⠖⢛⡏
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠀⢯⡀⠀⢘⡏⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡟⠛⡻⣿⠳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⡴⠶⣴⣷⣴⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⡀⠁⠀⠙⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⣲⣾⠷⠮⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⣰⡯⠶⠓⠶⢭⣉⢿⣩⣽⣇⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠶⠖⠒⣖⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⢠⡄⠀⠈⠙⠶⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠁⢀⠀⠈⠳⣄⢿⣍⢳⣈⠳⣄⠀⠀⠀⠀⢀⡾⢋⡐⠶⠀⣦⠀⠁⠶⠈⠻⣆⠀⠀⠀⠀⠀⠀⢸⡷⣄⡉⠙⠛⠦⣄⣤⠀⠀⠈⠳⣄
⠀⠀⠀⢸⡇⢀⡀⠀⠀⠈⠀⠠⠄⠈⢧⡈⣿⣩⡗⢹⠀⠀⠀⠀⠸⣇⠛⠀⠀⠀⠠⠴⠀⠀⠚⠃⠸⡇⠀⠀⠀⠀⠀⢸⣧⠉⡷⠀⢘⣂⣀⣙⠳⣴⠞⠀⠘⣧
⠀⠀⠀⠀⠙⢦⡀⠀⠃⠀⠀⠀⠀⠘⠂⢷⠀⣻⣿⢸⠃⠀⠀⠀⠀⣿⣉⣛⣛⡛⠛⢛⣻⣟⣛⢛⣿⣇⠀⠀⠀⠀⠀⢸⠷⠚⢁⡾⣫⣤⢲⢯⣳⣬⠷⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠃⠀⢀⠀⠘⣧⣿⣼⡟⠀⠀⠀⠀⠘⣯⣉⣁⣨⣭⣿⣭⣥⣤⣭⣭⣥⡾⠀⠀⠀⠀⠀⢸⣤⡀⠘⣏⣳⡼⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣈⡀⠀⢻⢡⡞⠀⠀⠀⠀⠀⠀⠘⢯⣉⠀⠀⠀⠉⠛⢶⣄⢀⣴⠃⠀⠀⠀⠀⠀⣿⣠⡽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠖⠛⠉⠋⠁⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣦⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠙⣿⣿⡟⠉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⡧⠀⠀⠀⠀⣿⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⡀⠀⢀⡶⣾⣿⠀⢠⣿⡟⢀⣦⠀⢸⣿⣏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⣿⡇⣼⣿⠀⣸⢃⣿⣿⠀⣾⣿⣧⡞⣿⠀⣼⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⣿⣿⡇⣿⢀⣿⡾⢻⣿⣼⠁⣿⣿⠁⠹⡼⢻⣿⡇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⢿⣿⠁⢹⣼⠃⠀⠀⠙⠁⠀⣿⡟⠀⠀⠀⢸⣿⡇⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣤⣬⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣷⠀⢀⡀⠀⢰⣷⡄⠀⣠⣤⠀⢀⣾⠋⣉⢻⠟⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⣿⠀⣾⣿⡞⢻⣿⡇⠀⣿⣿⠀⣼⣿⡀⠘⠋⠀⣶⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⡏⣿⣿⡶⣿⡏⢳⢸⣿⡇⢰⣿⣿⠀⢿⣿⠟⠀⢸⠀⢿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣿⣿⡇⢿⡇⣸⠘⣿⣇⡾⢹⣿⡆⠘⣿⡘⢃⣼⣧⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⡇⠈⠻⠟⠀⠘⠿⠃⢸⣿⡇⠀⠈⠙⠻⠿⢛⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⣿⠃⠀⠀⠀⢀⣀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⡟⢠⣶⠀⢠⣿⣿⣷⠀⠀⠀⢠⣤⠀⢠⣶⣆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡋⢠⡟⣿⢀⣿⠃⣿⣿⡇⣾⠀⣿⣿⠀⣾⡿⠇⢸⣿⣇⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣸⠇⣿⣸⡏⢠⣿⣿⡇⣿⢠⣿⣿⢀⣿⡇⢀⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⣿⣿⠁⢸⣿⣿⠀⢿⡞⢻⣿⡼⢿⣇⡞⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⢿⡟⠀⢸⣿⡿⠀⠀⠀⠀⠉⠁⠀⠉⠀⠘⠿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠘⢃⣀⣠⣤⣤⣤⣶⣶⣶⠶⠶⠶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠈⠁⠒⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘
⠀⠠⣀⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠖⠂⠀⠀⠀⠀⠀⠀⠠⢠⠄⠀⠠⠆
⠀⠀⠈⠀⠀⣠⠴⠉⠀⠀⠀⢀⡀⡈⠀⠠⠤⠒⢀⣰⡾⣍⣉⠜⠀⢂⣻⣄⠄⠀⢀⠀⠀⠀⡀⠘⠁⡠⠀⠀⠀⠀⢀⠀⡀⣣⠋
⠄⠀⠀⠀⠀⠈⢖⢒⣛⣥⡽⠁⠤⠖⠛⠓⠂⠀⣀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠩⠭⠤⠔⠂⠉⠀⠀⠰⡁⠀⢀⡀⡄⠄⠈⣽⠏⠓⡲⠖⠚⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠉⠉⠤⠤⠤⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠐⠀⠀⠨⣟⠓⠂⠈⠀⠀⠀⠆⠀⠨
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⡼
⠓⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠀⣠⠞⠁⠠⠄⠀⠀⠁⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣽⡇⠀⠀⠀⠀⠀⢈⠀⠀⠉
⠀⠀⠀⠀⠒⠛⠖⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠸
⠀⠘⡿⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠃
⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡊
⢀⣠⠌⢉⢻⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠄⠀⠀⠀⠀⠀⠘⠰⢒
⠀⠀⠀⠈⡟⢶⡤⠄⠒⠊⠁⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⠀⠀⢂⣤⠖⢃⠀⠀⠁⡨⢶⢀⡄
⠀⠁⠀⠀⠘⣟⠛⠁⠠⠀⠀⠀⠐⠊⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⡈⠀⠂⠀⠂⠀⠂⢀⠠⣾⠃⠀⡇⠈⠃
⠀⠁⡆⢀⠀⠘⣟⣀⣐⣃⡀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠄⠤⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⡀⠒⠘⢭⣁⡀⢀⣊⠀⠁⠂⠤⣖⡁⠀⣿⡧⠘⠁
⠂⠐⠁⠀⠀⠀⠘⡝⠋⠀⠀⠀⠰⠀⠀⢀⠇⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣶⡟⠠⢤⠤⣞⡀⠀⢠⡬⠗⢻⠁⠀⣿⡧⠐⠁⠀⠀⢀⣀
⠤⢄⠀⠐⠂⠰⠀⠙⢄⣀⣀⣀⣈⠀⠀⠁⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢻⡀⢠⣤⡀⢸⣯⠉⣷⡀⢸⡻⠐⠁⡀
⠀⢸⠂⢀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠿⣶⢆⠄⢀⡀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠈⠉⠁⠒⠻⠧⠽⠗⠈⠀⡀⠀⠀⠀⡀⠀⢀⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣧⣤⣾⡦⠦⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⢸⠃⠀⣿⢀⣠⣤⠀⢀⡀⣾⠀⣈⠀⣠⣤⠀⣸⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿⡞⢡⠏⣼⡟⢹⡏⢰⣿⡾⢁⡞⢰⣿⣤⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠿⠀⢿⣸⠃⠛⠛⣧⡟⠘⠃⢸⣡⡟⣿⢸⣀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠦⠄⢀⡴⠦⣄⠀⢷⣈⣁⣴⡇⣤⣬⢸⡇⣼⠀⠉⠀⣀⡀⠉⠀⠒⢋⣁
⠀⠀⠀⠀⠀⠀⠤⠄⠀⣿⠀⠀⣿⠀⠆⣨⠤⢾⢧⣝⠋⠸⣦⠏⠀⠂⣰⠋⠙⣦⠀⢈⣉⡉
⠀⠀⠀⠀⠀⠀⠠⠶⠀⠹⡄⠐⠋⠠⠀⠻⠴⠋⢀⡈⢃⡴⣦⠀⠉⠀⣿⠀⠀⡿⠀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣤⣿⣳⡆⢰⣦⡏⣿⡟⢠⡟⣴⢀⣶⣿⠀⢠⣄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⣇⠹⣭⠟⠀⠿⠀⠿⢡⡏⣸⠻⡾⠻⡼⣦⣼⢻⡉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠈⠷⠏⠀⠀⠀⠀⠈⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⡣⠤⠞⠃⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠨⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⡀
⠀⠀⡠⢀⠀⠃⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢀⢤⠴⠇⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠈⠃⠒⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠃⠈⠲⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⢀⠀⠄⠂⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⢀⠀⢀⣀⡤⠀⠠⠖⠂⠈⢁⠀⠤⠀⠒⠒⠚⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠈⠈⠃⠉⠉⠉⠉⠙
⠀⡀⠀⠀⠑⠀⠠⠄⠰⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⢩⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠒⠈⡉⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠔⠉⢀⡂⠀⢸⠀⠀⢨⠁⠀⢀⠀⣄⡠⠄⡄⠤⠈⠡
⠀⠀⠣⣀⡁⠐⠒⠈⠅⠀⠀⠠⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠁⢂⠘⠈⠀⠐⢸⠀⠀⠸⠀⠀⢈⣉⠀⠈⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠜⠀⠈⠁⠀⢰⠈⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠉⠁⢀⠀⢀⠀⠐⠀⠀⡆⠀⣥⠤⠀⠀⠀⠀⠀⢀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⡀⢤⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⡍⠍⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠁⠠⠤⠃⠣⠄⠸⠯
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠄⢀⣀
⠀⠀⢀⡁⡀⣀⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡁
⠀⢀⠊⠈⠁⠢⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠄⠀⠐⡂⠀⠀⠀⠀⠀⠚⠀⠀⠰⡄⣈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠇⠊⠀⠀⠀⠠⠀⠄⠀⠒
⠀⠀⠀⣀⠀⠨⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄
⠀⢰⡄⠀⡶⢢⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⠾⢁⣈⠷⠆⠀⣀⠁⠀⠀⠀⠀⠀⠰⠒⠠⠒⠃⠀⠐⡀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠘⠠⠀⠀⠠⠜⠀⠀⠈⠀⠀⠀⡄⠉⡁⠘⠒⣒⡓⣊⠉⠹⡄⠀⠀⠀⠀⠀⠀⢐⠠⠤⣃⡀⠀⠀⠂⠀⠀⠀⠁⡠⠒⠉⠀⠤⠠⠈⠉⠒⣄
⠀⠀⠀⠐⢤⡂⠐⠀⠀⠀⠀⠀⠀⠀⢱⡜⠥⠲⠷⠈⠇⠭⠆⢠⠇⠀⠀⠀⠀⠀⠁⢃⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡴⠡⠠⠤⠤⠀⠤⠰⠔⠆⢸
⠀⢠⠀⠀⣈⠑⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣄⠀⠐⠒⠒⠀⢀⡠⠋
⠀⠘⢀⠀⠀⠀⠀⠈⠰⡀⠂⠀⠀⠀⠀⠀⠀⢄⡀⠀⡀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠉⠙⠛⠛⠋⠁⣀⡤
⠀⠀⠈⠁⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠑⠒⣾⠇⢸⣿⠉⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣾⣇⠃⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠠⠤⠀⠀⠌⠃⠀⠠⠜⠁⠀⠀⠀⢠⣿⠀⢸⣧⢸⡇⢠⣿⣿⡇⢰⣦⠞⣿⠇⣿⣧⡿⠀⢀⠀⠀⠀⠀⠓⠤⡀⠐⢂⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⣿⢀⠎⠛⢸⣧⠛⠟⢿⣷⠻⠏⢸⣿⡔⠹⡿⣧⠀⠎⠉⠙⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣤⡀⠈⢁⡄⢀⡤⢤⡀⠀⢀⣉⠀⠀⠀⡈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠎⢠⡿⠀⠀⢻⡆⢸⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⢸⡷⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⣠⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠟⠠⠒⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣷⠀⠀⣼⠇⠘⣿⡀⠀⢰⠁⠀⠀⠀⠀⠒⠄⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⠉⠑⣚⢿⠀⠀⠈⠛⠚⠁⠀⠀⠈⢹⢞⡓⠉⢹⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⢰⠀⠀⠀⠀⠪⢁⡀⢸⠁⠉⠐⠒⠂⠀⠀⠀⠀⠒⠒⠈⠁⢹⠀⣀⠩⠂⠀⠀⢰⡄⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠤⠼⢤⣰⡃⢹⠤⠄⠀⣴⢒⣇⠀⠀⠀⠀⠀⠈⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢰⡄⠀⠀⠀⠀⠈⠁⠒⠠⠤⠤⠒⠁⠑⠪⠤⠤⢼⣿⠊⠀⠀⠀⠀⢠⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠄⠰⣠⣠⣿⣦⣤⠀⣤⠀⣤⠀⢠⣤⠄⢼⣇⢰⡆⠇⠠⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣿⡇⣿⢡⡿⠐⣿⣼⡿⢠⢿⣧⠴⠻⠟⣿⢡⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠸⠟⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀
⠀⠀⢠⢴⡧⠖⠚⡛⣷⠀⠀⠀⢀⠌⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢣⠀⢀⠔⣄⠀⢠⣄⣀⡀⠀⠀⣤⢤⡤
⠀⠀⢸⡄⠁⢁⡀⠤⠸⡆⠀⠘⠀⠀⠀⠀⠀⠀⠱⠑⡀⠀⢀⠔⠲⡒⡀⠀⠀⡆⢹⠀⢇⠀⢸⠀⢘⠿⢿⠂⠀⠀⡆⠀⡆
⠀⠀⠀⡧⠠⠀⢒⣂⠁⣳⠀⠰⣄⠀⠀⠀⠀⠀⢀⡴⠁⡠⠃⢄⡄⠈⢶⡀⠀⢁⠸⡆⠀⡕⡅⠀⢀⢳⠁⠀⠀⢀⠇⠀⢇
⠀⠀⠀⢳⡈⠭⢄⡐⠦⠝⠀⠀⠀⠍⠹⠹⠉⠍⡇⠀⣰⢁⣀⡈⠀⠀⣀⢳⠀⠈⢸⠃⠀⠀⡇⠀⠈⢠⠀⠀⢠⡋⠀⠀⠘⢷
⠀⠀⠀⠈⠊⠀⠒⠂⠁⠁⠀⠀⠀⠺⠈⠙⠛⠋⠇⠀⠈⠉⠉⠉⠉⠉⣉⣁⣀⣀⣀⣋⣀⣉⣁⣀⣈⣁⠀⠀⢠⠀⣉⡀⠈⠈
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠒⢲⣲⡒⠒⠒⢒⣖⡖⠒⠒⠒⠁⠀⢸⡀⣊⣈⣀⢀
⠀⠀⠀⡔⣡⠔⠂⠁⠐⢌⠢⡀⠀⠀⠀⢀⡔⢈⠉⠈⠉⠒⢄⠀⠀⠀⠸⠘⠉⣠⠖⠒⢦⡉⠙⣮⠀⠀⠀⠀⠨⣀⡀⠀⢀⡰
⠀⠀⠈⠘⠔⠒⡄⣀⢀⣀⢡⠡⠀⠀⠐⠑⠁⡎⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣿⣶⢠⣴⡇⠀⣿⠀⢀⠄⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⢀⢸⡀⢀⠇⡣⠿⢝⡜⠠⠀⡤⣦⣤⣤⡄⠐⠒⠒⠒⠒⠚⠢⡄⢰⢀⠀⠈⡷⠒⡏⠁⠀⠿⠀⠸⠼⡆⣷⠲⢷⢸⢈⢼⢃
⠀⠀⠀⠢⠋⠅⣀⣈⡒⢛⠔⢁⣀⡈⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠁⠀⠈⠈⠉⢀⠀⠀⠀⠈⠉⠈⠀⠐⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⢸⡸⠁⠐⠀⠀⠀⠠⣟⣒⡨⡆⠀⢀⠔⡒⡒⠒⠈⢕⠌⣇⠀⠀⡏⠀⠤⠀⠐⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠌⠃⠹⠁⠀⣤⠰⠀⠃⠆⠃⣀⠴⡍⠀⠀⢳⠛⠉⣽⣟⡇⠀⠀⢹⠓⣛⡛⣛⠛⡛⢛⢛⠻⣻
⠀⠀⠀⢀⣀⣀⢀⡀⠀⠀⠙⠦⠦⠄⠀⠀⠘⠵⣲⣖⡆⠽⠚⠉⠀⠀⠀⢸⡴⣶⠏⢹⠇⠀⠀⢧⠘⣅⠗⠟⠹⡇⠟⠾⢀⢿
⠀⠀⢰⠧⠤⠄⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠛⠀⠀⠀⠂⠁⠀⠈⠈⠀⠀⠀⠀⠁⠘
⠀⠀⢈⡺⠡⠀⠘⡂⠀⠀⢰⢑⡠⠤⠤⠤⠴⠥⠯⠮⠬⠤⠤⠤⠤⡄⡏⡇⠀⢀⠤⠐⣑⣒⣑⡒⠠⠤⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠘⡔⢄⣀⠰⡱⠀⠀⢸⢸⣈⣉⣉⠉⠉⠉⠋⢹⠉⠉⠉⣉⣉⣁⡇⡇⠀⢨⠢⠉⡇⠆⠀⡂⢉⠢⣤⠛⡂⢠⠇⠀⠘⡆
⠀⠀⠀⠈⠑⣲⠉⠀⠀⠀⢸⢰⠒⠒⡆⡇⠀⠀⡀⢸⠀⠀⢸⢰⠀⠀⡆⡇⠀⢸⡤⣀⡃⠂⠀⠃⡈⠴⠙⢄⡀⠀⢙⠉⠉⡅
⠀⠀⠀⠀⠰⠾⠄⠀⠀⠀⠸⡸⠀⠀⡇⠇⠀⠀⣅⡸⠀⠀⠸⡸⠀⠀⠇⠇⠀⠀⠘⠚⠩⠉⠍⠃⠊⠁⠀⠀⠀⠈⢤⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⣀⣀⡀⣀⡠⢜⠵⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢄⡢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠁⠑⠋⠭⠝⠋⠁⠀⠀⠀⢀⠔⡩⠢⠀⠰⠉⠀⠠⢆⠀⠀⠀⠀⠀⠀⠹⣿⣧⠎⠉⠀⠀⠀⠀⠀⠀⡔⢮⣹⠖⢤⡀
⠀⠸⡡⡁⠁⠁⠀⠀⠀⡇⠀⠀⠀⠈⠰⡁⢣⡄⠈⠂⠒⡡⡸⠀⠀⠀⠀⣠⡤⢦⣽⣦⢦⣤⡀⠀⠀⠀⠀⢸⡠⠤⠒⣊⡩⠇
⠀⠀⡱⠕⣂⣀⢀⡠⢮⡀⠀⠀⢄⠘⠂⠘⠒⠃⠀⠀⠁⠚⢁⠄⢠⡤⠼⠿⠿⠿⠿⠿⠿⠿⠷⠤⢤⠀⠀⡗⠐⣋⡉⠥⠔⠃
⠀⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣴⣶⣶⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⡵⠚⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣤⠤⣴⠧⠤⠶⠚⠀⠀⠀⠀⠀⠀⢀⢠⠚⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠉⠛⠛⠉⠉⠑⣷
⠀⢠⢠⡀⠈⠛⠶⢾⠴⠀⣠⢄⡀⢀⣤⣄⣠⡀⢈⠹⣠⠀⡀⠀⣀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢠⡏⣽⠅⠀⠀⠀⠈⡆⠸⣅⣨⣇⢸⡅⢸⡀⣿⣸⣀⣿⣀⣿⣀⣿⣠⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠁⠀⠀⠀⠳⠤⠞⠀⠀⠈⠁⠈⠈⠀⠈⠀⠈⠀⠀⠀⠁⠀⡡⠛⡇⠀⠀⢳⡄⣀⡔⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠢⠀⠀⠒⠶⣄⠀⠀⠀⠀⠀⠣⠤⠇⠀⠀⠘⢿⣷⣶⠟⠀⠀⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠶⠋⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣁⣤⡶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣀⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⢠⡾⠋⠀⠀⠀⣠⣴⣶⣾⣗⠦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢠⠏⠀⠀⠀⠀⠜⠋⠉⠉⠉⠙⢷⡔⠀⠀⠀⠀⠀⢀⠞⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⢠⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢸⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢠⠋⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠳⣽⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⡌⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠛⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡿⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠾⠛⠻⢦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠾⠿⠶⢶⣤⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣨⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⢰⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣤⣤⣤⣴⠾⢃⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛⠒⠶⠶⠛⠀⠀⠈⢷⣭⣭⣤⠤⠶⠖⠒⠒⠒⠒⠶⠶⢶⣾⣿⣯⣭⣭⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⢀⣄⣠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⠀⣀⡰⢦
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠗⠲⠔⠂⠳⠀⠀⠀⠀⠀⠀⠀⠀⡐⣇⢠⠉⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠖⠀⣔⠉⢂⣸⡭⡤⠖⢈⠅⣠⠋
⠀⢠⣆⣀⣀⡀⢀⣀⣠⣀⡖⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠬⠕⠶⠀⠀⠀⠀⠀⠀⠜⢭⡀⢆⣐⠶⠶⠒⠉⠅⡀⢀⣠⠕⢋⣼⠑
⢤⠃⠊⣿⣓⣀⣐⣛⣓⡿⣅⣀⣄⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠃⠐⠒⠀⠘⠀⠀⠀⠀⠀⠘⣆⠘⢷⣺⡔⣺⠃⣠⢒⣓⡩⢿⠓⢚⡭⠊⢠
⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡉⠙⠶⠓⠯⠥⠖⠉⠉⣁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣴⡿⢦⣠⡞⠁⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⢀⡜⢩⡀
⠀⠀⠀⠈⣾⠛⠀⡀⢉⠀⠘⠈⢦⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠤⣞⣉⡉⣱⠞⢫⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡟⠝⠓⠈⠋⠢
⠀⠀⠀⢰⣟⡲⠦⠤⠬⠴⠾⠤⣺⣶⠀⠀⠀⠸⡁⠩⣍⡖⣲⠋⠀⠀⢈⢷⣓⡾⠧⣮⠏⡀⠀⠀⠀⠀⠀⢸⣤⣤⣅⣠⣤⣾⡇
⠀⠀⠀⠀⡏⠈⠉⠉⢹⠉⠉⠉⢹⠁⠀⠀⠀⢠⢹⣫⠍⡾⣻⠦⣤⣐⣫⡾⠤⠓⢔⣩⠿⡇⠀⠀⠀⠀⠀⠀⢛⠛⠛⠛⠛⡋
⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⡷⣝⠚⠚⠤⢞⡚⠋⠉⠗⣲⡶⠊⣩⣾⠃⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠄⡇
⠀⠀⠀⠀⠘⣦⣀⠀⢘⠀⢀⣠⠞⠀⠀⠀⠀⠀⠑⣌⠉⠓⠒⠶⢷⡀⢀⡼⠷⠚⢋⡡⠋⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠈⠫⠭⠭⠭⠭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⢦⣽⣯⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⣸
⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⣠⠀⢠⣀⣀⡀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⣠⠚⢱⠀⠑⣀⣹⣫⣏⣠⣮⢞⣥⡷⠀⠀⠀⠀⠀⠀⠀⣙⡦⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢰⠃⢁⣌⣵⣷⠛⠁⣤⢸⡍⠁⢺⡀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠉⠙⠒⢄
⠀⣿⡤⢛⡻⣭⡇⠐⢾⣀⣀⡆⠀⣸⣷⠀⠀⠀⠀⠀⠀⢰⢻⠟⠲⠖⠒⠲⡗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⢈⡀
⠀⣿⢮⣷⣾⣧⣵⣦⣤⣬⣥⣶⣿⡿⣿⠀⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⠀⢠⠇⡇⠀⠀⠀⠀⢀⣰⠂⠀⠀⠲⠁⠉⠉⠉⠈⠡⠃⠀⠐⣆⡀
⠀⠹⣷⣌⡉⠉⠛⠛⠛⠛⠉⠉⠁⣠⠇⠀⠀⠀⠀⠀⠀⠘⢦⣙⠲⣤⠖⣋⡠⠃⠀⠀⠀⠘⠃⢭⢀⣷⣆⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⡬
⠀⠀⠈⠳⣍⣙⠒⠂⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣉⣯⡁⠀⠀⠀⠀⠀⠀⠀⠸⣌⣛⣙⣛⣁⣀⣀⣀⣀⣈⣋⣀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠲⢤⣀⡀⠀⢀⣇⣠⣀⠊⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⡤⠛⠉⠀⠀⠀⠀⠑⢄⢀⡴⠋⠉⠁⠈⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⣰⠁⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⠋⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⡇⠀⡀⠀⢴⣹⠗⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣅⣀⠀⠇⠀⡿⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠈⠁⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⡠⠚⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠎⠀⠀⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠶⣦⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⠇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣤⠾⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⣤⣤⣀⡀
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠐⢦⡀
⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣀⠴⣫⠗⠂⠒⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣤⠚⠁⣼⠏⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃⣠⠴⣄⠀⣼⡟⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠃⠀⢸⣿⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠉⣩⡄⡼⠁⢰⣿⣸⣿⠁⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⣀⡠⠔⠊⢹⣿⡏⠀⠀⣾⡿⠀⢀⡴⣋⣀⠀⣴⡶⠲⣤⡄⢠⣿⣿⠁⢀⣾⡟⣿⡟⠓⠒⣩⠿⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣾⣿⠁⠀⠀⣿⣷⠀⢉⡞⢹⣿⢼⣿⠃⣰⣿⠁⠘⣿⡇⠀⣼⣿⠁⣿⡇⢠⡞⠁⣀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣿⣿⠀⡞⢠⣿⡟⠘⡿⠖⢻⣿⠀⢰⠃⠀⢰⣿⠇⠀⣿⣿⣯⣴⠞⠋⠁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢀⡟⣿⣿⠁⣾⣿⠁⢰⠇⠀⠘⠿⠴⠋⠀⠀⢸⡿⠀⣰⠛⠟⠀⠈⢿⣦⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠞⠀⠈⠉⣼⣿⠃⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠁⢀⡀⠀⣀⡘⣿⣧⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣰⠃⣰⣿⠀⣾⣧⠀⢰⣥⣾⠓⣦⢰⣿⡇⢰⣿⡇⠹⣿⡄⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⠏⠀⣿⡇⢠⣿⣿⠀⣼⣿⡇⠀⢸⡾⣿⡇⣸⣿⣷⠀⠈⠙⠋⠉⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠸⣯⡴⠋⠀⠀⠙⠳⢋⣿⡿⠋⠈⠿⣷⡴⠟⠀⠛⠷⠋⠈⠙⠓⠀⠀⠀⢀⣾⠏⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⢹⡆⠀⠀⠀⢀⡤⠀⠀⠀⣀⠤⠐⠈⢹⣿⠃⢀⡤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠘⢀⣴⣿⢻⡉⠀⠀⢠⠞⠁⠀⠀⢠⣿⠏⣴⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⢀⣤⡶⠳⣆⣼⣿⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡉⣿⣿⣸⠇⠀⠀⢻⡀⠀⣀⡴⠟⢁⡾⠁⢀⣿⡿⠀⣰⡿⠀⣴⡇⠀⣾⣿⠁⠒⠃⣿⣿⢈⡴⢻⣿⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢻⣷⠈⠛⠉⠀⠀⢠⣤⠈⢉⣥⣄⠀⡞⠁⢀⣾⡿⠁⠀⣿⡇⣸⣿⡇⢀⡿⣿⡀⢀⡴⣿⣿⠎⢠⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣤⠿⠃⠀⠀⠀⠀⠀⣿⣿⢠⡟⢹⣿⡾⠁⢀⣾⣿⠃⠀⠀⡿⠛⠁⠉⠛⠉⠀⠈⠉⠉⠀⠈⠉⠀⣾⡟⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⢸⣿⠇⠀⣼⣿⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢠⣿⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢹⡌⢿⡀⠀⠀⠀⢀⡔⠂⠁⠀⠀⠀⠀⠀⠀⢠⡾⣣⠶⣤⡀⠀⠀⠀⠀⠈⠉
⠀⠀⣀⣠⣄⠁⣤⢀⡀⡄⠀⠀⠂⣤⠤⠄⠐⠀⠀⠀⠀⢸⡀⢸⡧⠈⠣⡄⠀⠠⠉⠶⠀⠀⠀⠂⠀⠀⢀⡰⢿⣜⡱⢶⡌⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤
⠀⠀⠉⠉⠉⠀⠸⠼⠃⠇⠀⠀⠺⠶⠒⠺⠛⠁⠀⠀⠀⠘⡗⠈⠡⠀⣰⠇⠀⠀⠤⠼⠒⠈⠤⠀⠀⠀⠚⢵⢏⣺⡷⠋⠀⠀⠀⠀⠀⠂⠀⢺⣗⣤⠟
⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠛⠁⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠰⣦⠙⠃⢰⠀⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠀⠃⣀⠈⠓⢄⣤⠀⠀⠀⢠⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢞⠁⣽⠆⠉⠀⠀⠀⠀⠛⠇⠀⢀⠴⠋⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢠⣴⠏⢠⣤⠀⠙⠃⠀⠀⠀⠀⢠⠙⠃
⠀⠀⠀⠀⡀⠻⠀⠀⠀⠀⠀⠧⠀⢠⣠⡤⠀⠁⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠙⣁⣠⣤⣤⣤⡀
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⣦⠤⡤⠀⠀⠀⠀⠀⠙⠇⢠⣄⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢘⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡉⠀⠀⠈⠀⣀⠀⠐⢦⡀⠀⠛⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡷
⠀⠀⠀⠀⠀⢾⡧⣴⠄⠀⠀⠧⠀⠁⠀⠈⠀⠀⠛⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡆⠀⠀⠀⠒⠃⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠘⢷⠜⠃⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⢀⣴⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠉⠟⡄⠀⠀⠀⠀⢰⠄⠸⣀⡴⠆⠀⡀⠀⠀⢄⠀⢰⠀⠀⠀⢠⣴⣶⣶⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⡇⠀⠠⣠⡁⣠⡤⠘⠋⠀⠀⢈⡁⢀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠉⢀⡰⠶⠃
⠀⠀⠆⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠀⠀⢸⠁⠀⠀⠀⠀⠈⠁⠈⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⣩⡁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⡗⠀⠀⠰⠟⠆⠀⠀⢀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢤⠄⢨⠀⡀⠚⠁⠀⠀⢀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠈⠀⢀⠟⠀⠀⠀⠂⢀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣟⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠐⢣⠀⠈⠒⠂⠀⠀⠀⠀⠠⢠⡄⢠⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠟⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⠜⠃
⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣴⠶⠿⣷⣶⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣦⠀⠀⢠⣶⡄⠀⠀⠀⢦⠀⠘⢇⠀⠀⠀⠀⠀⢀⣤⣤⣠⡄⠠⠶
⠀⠱⣤⣤⣄⣄⣰⣤⠆⠀⠀⠙⢿⣤⣿⣦⡼⠇⠀⠀⡀⠀⠋⠉⠃⠀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠰⠀⠀⠸⠃⡰⣿⡟⢶⡏⠓⣤⠀⢿⠯⣍⡼⠇
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠴⠶⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠈⢱⣏⣠⠼⠷⠞⠁⠀⠀⠀⠈⠁

⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡤⣤⡤⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣄⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠀⠀⠀⠀⠆⣠⠆⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠺⠀⠀⠠⠀⠀⠀⠈⠻⠶⠶⠶⠟⠁
⠀⠀⠀⠀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠂⠠⠀⠀⢑⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠀⠀⠀⢠⣬⡑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠄⠀⠄
⠀⠀⠀⠼⠃⣀⠴⢟⠀⠀⠠⠖⠐⠒⠀⠈⠆⠀⠀⠀⢴⣬⣿⣷⣤⡈⠀⠀⠀⠀⠀⣰⡀⠊⣱⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠒⠶⠐⠿
⠀⠀⠀⠲⠀⠙⢄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠋⣹⡄⢣⡷⠀⠀⠀⠀⠂⠙⠁⡟⠁⠀⠀⠀⠘⠛⢃⣀⡄⢶⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⢠⣤⣤⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠐⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢱⡄⠀⠀⠀⠀⣀⣤⣴⠾⠟⠂⢀⠀⠀⡤⠈⠀⡀⠺⠆⠀⠀⠀⠀⠘⠛⠛⠂⠀⡄⠀⠀⠀⢈⠛⡀⠀⠰⣺⠉⣧
⠀⠀⠀⢶⣿⠿⠃⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⣶⠟⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠦⣿⡶⠀⠀⠀⢄⣠⡀⣀⣸⣀⡲⠐⠀⠀⠀⠀⢀⡀⠀⠀⢑⠊⠁
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠐⠋⠀⠀⠀⠀⣼⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠺⢀⢈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠙⢻⡋⠉⡟⠛⢻⡙⠏⣙⡿⢛⡋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣾⠃⠀⠜⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣸⡏⢠⠊⢀⡾⠋⠀⠀⠀⠀⠀⠀⣸⡏⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⣿⠀⠀⠀⣿⠇⠁⠀⣾⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⣿⠀⡀⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣸⡏⠠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠀⢸⡟⠀⠀⢸⣿⢀⡀⠀⣀⡀⠀⢠⣄⢠⡄⠀⣿⡇⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⢾⡇⢰⡇⣾⠇⠀⡿⢸⡇⢠⣿⣣⠟⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢺⣿⢸⣧⠜⠓⢿⡄⠘⠃⣼⡇⢾⣿⣿⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠜⠈⠋⠀⠉⠀⠀⠀⠀⠀⠀⠹⠟⢸⡿⠘⢧⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢶⡄⠀⣄⢰⠁⣠⡔⡄⠀⣴⡄⢠⡄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⠈⣿⡀⢱⣿⠀⠖⢠⣿⠀⣿⠇⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣿⣰⢻⡧⠈⣿⢀⠃⠸⣿⡰⣿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠼⣿⠀⠈⠁⠀⠀⠈⠀⢻⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠇⣿⣿⣿⢇⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣋⣀⣈⠋⢱⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⠈⠛⣛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣠⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⢀⣠⣀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⢠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠈⠛⠛⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠸⣿⣿⠏⠸⣿⡿⣯⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣆⠘⠛⣛⣁⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣦⣄⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡷⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡆⢀⣭⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⠀⠙⠛⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠺⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠛⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⣶⣆⢠⣾⣿⣧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⣿⣿⡏⢺⣿⣿⢟⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣉⣡⣤⣤⡉⠁⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⢈⣩⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠁⢺⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁

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

⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠛⠉⠉⠉⢿⣿⡏⠁⣸⣿⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡴⢡⣿⡇⣰⣿⡆⠀⣼⡿⢛⣶⣿⠃⢸⣿⡇⣰⣿⠆⢸⣿⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠞⠀⢸⣿⣴⢹⣿⠀⢰⣿⠃⢘⣿⡏⢠⢿⣿⣰⢻⣿⠀⣿⣧⠎⣠⠟
⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠀⠀⣼⣿⠇⠈⣿⣤⠿⣿⣀⠾⣿⣤⠋⢸⣿⡇⢸⣿⢸⣿⡟⢻⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠈⠛⠀⠀⠉⠁⠀⠉⠉⠀⠉⠁⠀⠈⠉⠁⠀⠻⣌⠙⠁⢀⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢠⣤⠀⣴⠞⠉⠛⣦⠀⢀⣀⠀⠀⣀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠉⢻⣷⠀⣾⣿⠙⣿⣿⡇⠀⢸⣆⣼⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⡼⣿⡏⠀⣿⣿⠶⠖⡾⠁⣿⡇⢠⣿⠃⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢡⣿⠁⠀⠻⢿⠦⠞⠁⠀⠿⠷⠋⠿⣄⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣾⠷⠶⠶⠶⢶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣷⡀




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⡄⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠹⣿⣿⣿⡿⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⣿⠈⢿⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣛⣳⣶⣶⣦⣤⡀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠘⠀⢠⣿⡟⠀⠀⠀⢀⣀⣤⣤⣤⠶⠶⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢻⣿⣧⠀⠀⠀⣰⣿⡟⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⡇⠀⢰⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠔⠀⠀⢄⠀⣀⡴⠥⠒⠒⠠⣄⠀⠀⠄⠀⠀⢰⠀⠀⠀⠀⣀⠤⠄⠒⠂⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠤⢤⣤⣤⣘⣀⣄⠈⠂⢀⣴⠞⠁⡇⠀⠀⠀⠀⢸⡆⠀⠈⠂⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡴⠋⠁⠁⢦⠀⠀⠀⠀⠀⠀⠀⠀⢯⡟⠉⣱⠟⠓⠂⠀⠙⢄⣀⣀⣠⠞⠁⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠢⣀⣀⣀⡤⠋
⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⢾⡿⢁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠘⠢⢄⣀⠈⠒⠠⠐⠊⠀⠀⢀⣀⠤⠚⣡⡟⢁⣾⠧⣄⠀⠀⢀⡤⢤⣀⡀⢀⣄⡠⢤⠀⠀⣰⡿⠤⣤⠀⠀⠀⢀⡀⠤⢀⣀
⠀⠀⠀⠈⠉⣩⠽⠿⢭⡉⠉⠁⠀⠀⣰⠟⢀⡾⠁⢀⡿⠀⢠⠏⠀⣸⡟⠀⣼⠋⠀⣾⠀⢰⣟⡠⠞⠉⠀⢠⠊⠁⠀⠀⠀⠀⠉⠢⡄
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠘⣆⠀⣠⡾⠁⠀⢸⠇⠀⢸⠃⡠⢿⣀⣰⡟⢀⣾⠃⠀⢸⠃⡠⣿⠻⣄⡀⢀⡴⠁⠀⢀⡄⠀⠀⠉⠲⣄⠈
⠀⠀⠀⠀⠳⣄⠀⠀⢀⣀⡼⡞⠁⠀⢀⣠⠤⠤⠤⠤⢉⣀⣀⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠠⢌⣇⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢃⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠱⠈⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡨⠀⠀⢳⣄⠀⠀⣀⡤⣠⡾⠋⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠉⠉⠁⣼⠏⠀⠀⠀⢀⣼⡟⠀⢀⣀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⠀⢀⠀⠠⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣼⠃⠀⠀⠀⡰⣻⠏⢀⡴⠙⢿⡷⢲⡟⠀⣸⠇⠀⡴⠋⠀⠀⢀⡇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢰⡏⠀⠀⢀⠞⣱⠏⠀⣾⠁⢀⡞⢁⡟⠀⣰⠏⠀⠊⠀⠀⠀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡧⠠⠔⠁⣰⠏⠀⠀⠈⠓⠉⠀⠀⠙⠈⠈⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠢⡀⠀⠀⣠⠟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢈⣢⠞⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⡬⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⡐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢪⢧⡀⡇⠀⠀⠆⠡⠀⠁⢀⠀⠀⠀⠀⠄⠈⠀⠀⢀⠀⠀⠀⠀⠀⢀⠰⣬⠶⣂⡀⠀⠀⠀⠉⠁⡒⣷⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀
⠀⠠⠂⡠⢁⠆⢵⠃⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠅⢀⢀⠀⠀⠀⠐⠀⠀⠀⠨⠑⢆⡃⣩⠶⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⢀⡀⢊⠔⠀⢠⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠃⠘⡣⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⡤⠁⡤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠊⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣿⢻⡟⠄⠀⠀⠀⠀⠀⠀⠀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢰⣦⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠷⠇⠆⠀⠀⠀⢀⠅⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠀⠆⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⢀⣀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠄⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠺⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⠅⠈⠀⠀⠀⠀⠀⠀⢀⢂⡄⠀⠀⠀⢀⠒⠑⢄
⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠀⠀⢦⠆⠀⠀⠀⠐⠛⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠔⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢀⠠⠀⠈⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⢤
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠈
⠀⠀⠀⠀⡄⠀⠰⢄⠀⠀⠀⠀⠀⠥⠀⠁⠀⠀⠀⢀⡀⠸⡧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠈⠒⠂⠀⠀⠀⣀⠄⣬⠇
⠀⠀⠀⠰⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠿⣿⣻⣶⡝⢪⢁⢄⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠘⠛⣋⣽⢤⣾⣩⣌⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⡣
⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠄⠀⠀⠀⠀⠙⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⡀⢀⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠏⠉⣱⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⠃⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣰⡏⢠⡀⢀⣤⣀⡀⣠⣄⢔⡦⣸⣧⠞⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⣿⠀⣼⢳⠏⢠⡿⢁⣿⠁⣾⢁⣿⠰⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡀⠀⠁⠸⠃⠉⠈⠘⠇⠂⠀⠀⠃⠀⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⣿⣠⣿⢀⢆⡾⠁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠔⣹⡏⠻⠠⠋⠘⠇⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁⠀⢸⡁⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠛⠀⠐⠀⠀⣹⠘⠀⠁⠀⠀⠀⣠⠞⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⢀⡀⠀⠀⠀⠠⠼⠃⠀⠀⠀⠀⠀⣼⠃⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢰⡇⢸⠇⠀⡶⠀⣴⠀⢠⡶⠂⣸⢇⠎⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠿⢠⡟⠀⣸⠃⠈⡏⠀⣾⠁⠀⡿⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣾⢀⠜⠀⠀⠀⠓⠀⠈⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣷⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⣃⣠⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⣫⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄⡀⢸⣿⣿⣿⡿⢣⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠙⢿⣿⣿⣿⣿⣿⣏⣿⣿⡿⢱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣜⢿⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡹⢿⣿⣿⣿⣿⣿⣧⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⡏⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣻⣿⣿⡟⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⡿⢈⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⢟⣣⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠛⣁⣴⠿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⢀⠀⠀⡠⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠾⡫⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣒⡀⠀⠐⠛⠠⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣦⣶⣦⠦⡀
⠀⠀⠀⠀⠀⠀⢀⠏⠉⠾⠉⠻⠝⡏⣩⣗⣜⢩⠗⠀⠀⠀⠀⠀⠀⠀⠙⡛⠉⠸⠛⠈⠙⠏
⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⢡⠎⠀⠀⠁⠊⠀⠀⠀⠀⠀⢀⡶⠞⠃⣠⣾⠿⠋⣰⡾⠓⠐
⠀⠀⠀⠀⠀⠁⠤⢀⠀⠀⠐⠒⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠨⣤⡶⠟⠋⣀⣴⠾⠋⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠠⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠶⠟⠋⣠⠴⠿⠫⠔⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣪⣭⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐
⠀⠀⠀⠀⠀⠀⠠⠞⣁⠀⡉⠉⣀⠬⠉⠃⠀⠀⠀⠀⠀⠀⡠⢀⠔⠀⠀⠀⠀⠀⠐⢄⠀⠀⠙⢷⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⢁⣀⠀⡆⠀⠀⠀⠀⠀⠀⢘⠠⠩⡀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⢐⡏⠃⢀⠀⠈⠠⣿⡇⠀⠀⠀⠀⠀⠈⠎⣓⠊⠒⣤⢄⡀⡀⢤⣊⡔⠁⣧⢤⡀⣀⡀⣀⡆
⠀⠀⠀⠀⠀⠀⢸⣬⡋⠅⣸⡿⠗⣀⢴⠈⠂⠤⠄⠒⠀⠀⡁⠌⠓⢒⠉⠁⠄⣀⡤⢋⣁⠀⠀⠉⣁⣙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢬⡓⣶⢒⠭⠒⠋⠀⡄⠀⠀⢁⠀⠐⢘⠀⡀⠀⠠⠶⠘⠷⠢⠤⠊⠀⠀⠊⡀⢀⡑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠁⠀⠁⠀⠐⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠀⡀⠀⡠⠀⠀⠀⠄⡀⠄⠒⠀⠉⠀⠉⠉⠀⠂⠀⠢⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⡀⠒⠰⠉⠘⢀⠉⣂⠈⠄⠀⠀⡈⠑⡄⠀⠀⠀⠰⠀⢠⠄⠀⠀⠐⠀⠀⡜⠛⡂⢎
⠀⠀⠀⠀⠀⠀⢀⡈⠣⠄⠸⢡⣾⡆⠈⠡⠢⠀⡄⠸⠀⠀⠠⠀⠅⣀⡀⠤⠖⡀⠘⠀⠀⠀⠔⠀⠄⢰⠀⡠⠈
⠀⠀⠀⠀⠀⠦⠤⠥⣤⣤⣤⠴⠶⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠁⠄⠘⠃⠂⠀⠈⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠁⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣇
⠀⠀⡠⠄⠀⡀⠀⠀⠲⢄⠐⣿⠉⢳⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢫⢧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢐⡂
⠀⠀⠬⠁⢐⣦⡀⢦⣤⠀⠨⢿⡆⠈⢷⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣌⢇⡀⠀⠀⢀⣠⡤⠔⡦⣶⠂⠉
⠀⠀⠈⡁⠀⠀⠽⣤⡉⠳⣶⣤⣤⠈⠚⢧⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢹⡈⡆⠀⣰⢿⡏⣠⡏⢥⠇⠀⠀⢀
⠀⠀⠀⠒⣀⠀⠠⢙⣿⣮⣤⡼⡏⣷⡄⠀⠑⢬⠇⠀⠀⠄⡀⠀⠀⠀⠀⠸⣄⠻⠄⠟⣠⣾⢃⡟⢵⣒⡾⠃
⠀⠀⠀⠀⠐⢂⠈⣐⢠⠼⠫⡪⣫⠈⣧⣶⡀⠀⠀⠀⢀⠊⠀⠀⠀⢠⠀⠀⠈⠃⣤⡾⠋⣳⢏⣞⣭⠾⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢞⢷⣶⠎⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⣷⣽⠿⡿⠓⠂
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢴⢶⠃⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠯⣀⣤⠤⠾⣿⣽⡷⠚⠫⠏⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⠼⢿⡶⠊⠟⠆⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠐⢤⣄⡀⡄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⣠⣴⢾⡟⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣰⠂
⠀⣀⣠⣴⠶⠛⠉⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣶⣄⢶⣷⢀⣿⡟⠁⢀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⢸⣷⠘⣿⠀⣠⣶⡄⢰⡿⢹⣿⡏⢸⣿⡏⢸⣟⡸⣿⠙⠲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣸⢡⣿⠁⣿⡇⡼⢿⡤⠚⠟⠀⠈⠉⠀⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢻⡇⠀⠻⠖⠉⠉⠀⠀⠀⠀⢀⣴⡾⠗⠀⣸⣷⢼⠖⠁⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢀⣤⡀⢰⣦⠀⢸⣿⠀⠀⠸⡥⠒⣳⡞⠹⣿⡜⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⡄⢠⣿⠖⡗⢻⡿⢀⠿⣏⡰⠀⠀⠀⠐⠷⠃⠀⠀⢀⣤⣤⠀⣿⢃⣴⣶
⠀⠀⠀⠀⠀⢸⣿⡆⢀⣿⣿⠛⣿⣠⠇⠈⢓⠋⠀⠈⠀⢀⡀⠀⣤⡄⢠⣿⡆⣼⡿⠀⢠⣿⡼⠸⣿⢠⠆
⠀⠀⠀⠀⠀⠀⣿⠁⡼⢹⣿⠀⠀⠀⠀⠀⣿⡇⣴⣶⢠⢿⣿⢠⣿⡇⡾⢿⡠⠋⠻⠤⠃⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠀⣼⣿⠀⠀⠀⠀⠀⣿⣷⠿⣿⡜⠘⣿⠜⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠸⠿⠀⠉⠁
⣶⡶⠶⠶⠶⢶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠓⠚⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⣠⡰⠊⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠈
⠀⠀⠀⠀⠀⠀⡀⢀⡠⠠⠄⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠄
⠀⡀⠀⠀⠈⠉⠠⠤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠀⢁⠀⡀⠀⠀⡀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⠀⠈⠀⢠⡀⠤⠄⠄
⠀⠈⠀⠂⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⡀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⢀⢀⠀⠀⠀⢠⠠⠀⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠈⠁⠉
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡁
⠀⠀⠁⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠈⣠⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⢒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠴
⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⢁⠐⠒⣒⢂⠉⢱⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠠⠁⠐⢀
⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠱⠐⠐⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⢂⡐⠐⠊⢠
⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠑⠠⠤⠤⠤⠒⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠶⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡾
⠀⠀⠀⠀⠀⣠⢖⣚⡭⠭⡄⣰⠒⡶⠶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⡿⠀⡇⢸⠀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⡇⡇⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡗⡇⠀⡇⡏⡰⠋⠉⢆⢸⡕⢮⢢⡀⣾⢧⠙⠶⠉⠈⠱⡄⡇⡇⣸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⡇⡿⢁⢤⡀⢸⢠⠞⣙⠃⢳⠀⠈⡆⢠⠖⡄⠀⣿⡇⢟⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡇⠀⡇⠇⡞⠀⡇⢸⡞⠸⡼⠀⠀⢧⠀⡇⡞⠀⢸⠀⣟⡇⣬⠙⠢⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣇⠀⡇⢸⠁⢀⡇⡟⡇⠀⣠⢻⠀⠘⣆⡇⡇⠀⡎⢸⢹⠀⣯⠱⡄⠙⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠛⠀⠓⠚⠀⠘⠚⠁⢙⣊⠁⠈⡓⢒⠛⠓⠃⠐⢓⢃⡘⠒⠛⠀⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠙⢉⡉⠳⣄⡙⠛⣠⠎⢉⡙⠋⢉⣽⡉⢉⠓⠲⡏⠙⢹⠃⠀⠘⡍⠏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡑⢦⡀⠀⠃⠠⠉⠩⠄⠚⠀⢀⡴⠛⡤⠶⢮⡰⠄⠘⡇⢸⠀⠀⢠⠃⢶⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠈⠓⠬⡑⠦⢄⡀⠀⡠⠴⢾⠁⣉⢸⠘⠾⣲⠃⡃⠀⡧⠈⠣⠤⠋⠒⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣐⣀⠀⠀⣃⢸⠃⠀⠊⢦⠀⠂⠩⠍⠡⠈⢁⡼⠁⠀⠁⠀⢁⣀⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠉⣀⣀⠸⠞⠀⠻⠛⠀⢙⣲⣤⣤⠴⠒⢩⡏⠓⢲⠒⠺⣯⡌⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠁⠉⠁⠈⠀⠀⠙⠁⢀⣂⣑⣀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠒⠀⣮⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⡀⢴⣶⢠⠶⡆⠐⣶⣤⡆⡤⢤⣠⢤⣤⣤⡄⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣁⡀⢀⣀⡀⣙⣛⢉⣛⣁⣘⠛⠛⠃⠓⠛⠻⠶⠸⠹⠇⠈⠁⠈⠃⠈⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠤⠄⠀⢀⡀⠀⠀⠀⠈⠡⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⡀⠄⢀⠆⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⢀⡠⠔⡠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠯
⠀⢀⢢⢨⣰⠁⢐⣤⡆⠀⠀⠘⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⢰⠶⠂⠀⠀⡰⠉⠴⡀⢁⠸⠆⠀⠀⠀⠀⣤⠤⠤⠤⠾⠿⢦⡄
⠀⠸⠈⠉⠉⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⢰⡆⠀⠀⠀⠀⠐⠿⣾⢉⡀⠒⢰⠀⠀⠀⠀⠣⠀⠀⢀⡀⠤⠀⡁
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣤⠁⠀⢀⡄⠄⠀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠉⡄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⢡⡖⠀⠀⠀⠋⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⢣⠀⠀⠀⠁⠀⠀⣀⢀⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⢠⠖⠑⠊⢤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⠋⠁⠀⠈⠀⠄⠀⠀⠀⠀⠀⣰⠈⢷⠀⢠⣠⠀⠀⠀⠀⡄⠲⡀⢀⡀⠀⡀⠀⠄⢀⠸⠙⠀⠀⠀⠀⠠⠈⡃
⠀⠀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⣘⢦⠀⠀⠀⠈⠢⡑⠄⢠⠀⠀⠀⢂⠀⠁⠂⠀⠀⡀⢀⠀⠀⢁
⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠈⡿⠔⠀⠀⠀⠀⠀⠐⠀⠄⠂⠀⠠⠔⠁⠀⠀⠸⡀⢀⠀⠀⠏
⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⢀⠶⠁⠈⣀⣀⣀⠀⠉⠀⠀⠀⠈⣲⣷⡀⠘⣶⢧⠖⠀⠀⠀⠀⠀⠀⢈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⡀
⠠⠀⠤⠘⠉⡽⢏⠉⠐⠀⠀⠀⠀⢹⠀⠁⢸⡙⠳⠀⠀⠀⠀⠀⡠⠉⠉⠀⠁⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⡠⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠉⢰⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠑⠚⢻⠟⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠈⡘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠖⡄⠀⠀⠀⠀⠀⢠⡴⡆⠴⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⢌⠄⡝⠁⠉⠀⠀⠀⠀⢠⡇⠂⠀⠀⠀⠀⠹⡀⠀⠀⠀⡠⠂⠉⠀⠀⠀⢀⠀⠆⠀⠀⡠⠚⠀⠀⠀⡌⠃
⠀⣀⠀⠐⠒⠛⠛⠛⠓⢲⠃⠀⠀⢀⢈⠂⢜⠖⢀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⠀⠀⠀⠀⠀⠨⠀⡁⠀⠀⢁⠄⠒⠀⠘⠢⣄⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢃⡠⠌⡠⠂⠀⠀⠀⠀⠀⠘⡆⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈


⢀⠉⠃⣶⣴⣶⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣄⣀⡀
⠈⠀⠀⠹⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⢰⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡟⠛⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⣀
⠀⠀⢾⠀⠀⡇⠀⢸⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣉⣷
⠀⠀⠘⠀⠀⡇⠀⠸⠀⠀⣠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠄⠀⠀⠀⢁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⠀⠀⠀⠀⢀⠐⠤⠀⠠⠌⠁
⠀⠀⠀⠂⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠐⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡄⢥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠠⠄
⠈⠒⠂⠀⠀⠀⠈⢉⠶⠒⠒⠒⣿⠓⢺⣷⠈⣷⢀⡤⢤⣤⣆⢴⡜⣿⣴⠃⠐⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣿⠀⠈⣿⢸⡇⣾⢁⣿⢸⡇⢸⠁⣿⠳⢄⡀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⢀⢿⡐⠀⠀⠻⠊⠀⠀⠘⠃⠙⠉⠈⠘⠃⠈⠁⠉⠀⠀⠀⠘⣆⡕⠁
⠀⠀⠀⠀⠀⠀⢀⣈⠑⠆⣾⠀⠀⠀⠘⡄⣸⠏⢰⠁⢹⡆⠀⠀⢸⡇⠀⢠⡀⢤⢹⠐⠒
⠀⠀⠀⠀⠀⠀⠀⣩⣥⠐⢇⢰⠇⠴⠂⠘⡏⠀⢸⠀⢸⡇⡆⠀⢸⠃⢲⣦⡃⢘⡇⣄⣂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠄⢪⡜⠀⠀⠠⠧⠀⠘⠀⠼⠃⠣⠤⠞⠀⠀⠉⢳⢪⠴⣮⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢸⡾⠁⠉⠂⢀⡠⠤⠶⠒⠒⠒⠒⠒⠦⠤⣄⡀⠐⠁⠘⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠔⠊⢧⣐⣂⣤⠬⠤⠤⣬⣀⡀⣠⠛⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⣠⠤⠛⠃⠀⠀⠀⠀⠀⠐⢿⡥⠄⣹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠰⣦⢲⠀⡶⢠⡆⣠⠆⣾⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠷⠻⠸⣇⠸⠧⢏⠠⡿⢰⠃