⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀
⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠹⣧⡀⠀⠀⠀⡏⠉⠙⠋⢹⡆
⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠉⠛⠿⣦⡀⠀⠸⣷⡀⠀⠀⠹⣄⠀⠠⡹⠃
⠀⢀⣿⠃⠀⠀⠀⠀⣠⣶⠾⠟⠛⠉⠉⠛⠿⣶⣄⠀⠀⠀⠀⣰⠃⢀⣠⣤⣄⠀⠈⢻⣦⠀⢻⣧⠀⠀⠀⠈⠓⠋⠀⠀
⠀⢸⣿⠀⠀⠀⣠⣾⠋⠁⠀⠀⢀⣠⣤⣤⣤⣈⠙⣷⣄⠀⢠⣏⣶⣿⣿⣿⣿⣿⣦⠀⢻⣧⠘⣿⢀⢸⡛⠛⢲⠀⠀⠀
⠀⣾⡇⠀⠀⢠⡿⠁⠀⠀⣠⣾⡿⢿⣿⣿⣿⣿⣷⣌⢿⣆⢸⣿⡋⠉⢉⣿⣿⣿⣿⣧⠈⣿⠀⣿⣿⠀⠹⠶⠋⠀⠀⠀
⠈⡟⣿⠀⠀⢸⠇⠀⢀⣼⣄⠈⠀⣸⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⣿⣤⣿⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⢷⢹⡆⠀⢸⡄⠀⣾⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣇⣿⢹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠏⣸⢧⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠈⣇⢳⡀⠈⢧⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢎⠃⠀⠈⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠁⠀⠀⠈⠻⣿⣿⣿⣿⡿⠟⠁⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⡻⠿⢿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⣀⡀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⡤⠬⣷⣄⡲⣤⣈⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢛⡵⠚⠁⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⣇⠀⠀⠀⢻⣿⠒⠭⣟⣷⢶⣤⣤⣤⣤⣤⣤⣤⣤⣶⢾⣟⠯⠗⠊⠁⠀⠀⠀⣾⠟⠁⠀⢘⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣧⠀⠀⠈⣿⡄⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠋⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣿⠇⠀⠀⠙⠛⠛⠒⠒⠛⠛⠛⠻⠿⢷⡢⣤⣔⡶⠿⠟⠛⠛⠛⠻⠿⠽⠟⠁⠀⠀⢺⡿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⠒⠲⠛⠙⠲⠦⢼⣿⡿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀
⠀⠀⠹⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⢀⡾⡿⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⢯⠞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⣖⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⠮⣷⣶⡤⡤⠴⠾⠟⠋⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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:: :: :: ::
:: :: ˙˙:::::::::::::˙˙
:: :: ::
˙˙::::: ::::: ::
::.....:: ::....::
˙˙˙˙˙˙˙ ˙˙˙˙˙˙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠿⠯⣭⣉⡉⠁⠀⠐⠒⠛⠛⠒⠒⠒⠒⠒⣲⠂
⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀
⠀⢀⣴⡯⠴⠒⠚⠛⠛⠉⠉⠙⠛⠓⠲⠤⣄⠈⢳⡄⠀⠀⠀⢴⡋⠀⠀⠀
⢀⡾⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢈⠑⢆⣳⠀⠀⠀⠀⠙⢄⠀⠀
⣼⠁⠀⡈⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⡤⠖⢻⠀⠀⠹⠀⠀⠀⠀⠀⠈⢳⡀
⣿⠀⢰⠁⠀⠀⢠⢻⠁⠀⠀⣠⠖⣉⣀⡄⠘⡇⠀⠀⣠⡀⠀⠀⢹⠛⠒⠻
⣿⠀⡏⠀⠀⣰⠃⡇⠀⡠⠚⠋⠁⠀⠀⠁⠀⢷⠀⢸⠀⢻⠀⠀⠀⢳⡀⠀
⢸⡄⡧⣤⣴⡧⢼⢠⠞⠀⢠⠤⢖⢒⠛⡻⠛⢹⡀⠀⡇⠀⡇⢀⡀⠀⢧⠀
⠀⢿⡇⢰⡓⠨⢹⠃⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⡇⠀⡧⠞⠁⠀⡏⠳⢼⠀
⠀⠈⣇⡟⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣰⡇⠀⠁⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠿⡇⠈⠲⣄⠀⢀⣀⡀⠀⠀⠀⣀⡤⡾⠁⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀
⠀⠀⠀⣧⢰⠀⠀⢳⣌⠀⣈⡤⠖⠋⠁⠀⡇⠀⣇⠀⡄⠀⠀⣼⠁⠀⠀⠀
⠀⠀⠀⢸⣸⠀⠀⢸⣘⡯⢽⠇⠀⠀⠀⠘⡗⢤⡏⢸⡇⠀⣰⠇⠀⠀⠀⠀
⠀⠀⠀⠘⣿⡆⠀⡟⠉⠲⡞⠀⠀⠀⠀⣠⠽⠒⣧⢿⠇⣰⠏⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢿⢳⢠⠗⢄⠀⠹⡄⠀⠀⣰⠃⠀⢰⠟⡾⣰⠛⢦⠀⠀⠀⠀⠀
⠀⠀⣰⠃⠀⠊⢿⠀⠀⠑⢄⠙⣄⣰⠃⢀⠔⠁⠠⠟⠁⠀⠈⣇⠀⠀⠀⠀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠉⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣒⣒⣲⡄⠀⠀⢀⡴⢦⠀⠀⠀⣴⢶⠄⠀⠀⣀⠤⣖⣒⣒⣲⣦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢯⡀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⣿⢸⠀⠀⠸⣇⢺⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠬⢭⣓⠲⢄⠀⠀⢸⡇⣞⠀⠀⠀⢸⢸⡇⠀⠀⠈⠒⠤⠭⣁⡒⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡷⢸⠆⠀⠘⢇⠸⣄⣀⣠⠏⡼⠁⠀⠀⢀⣀⣀⣀⣀⡹⠆⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠞⠛⠉⠉⠉⠛⠦⡀⠀⠈⠑⠒⠒⠒⠋⠁⠀⠀⣠⠿⠛⠉⠉⠉⠓⠫⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠉⢦⠀⠀⠀⠀⡠⠞⠃⠐⠒⠉⠉⠑⢆⠀⠀⠀⠀⢳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⣆⡀⠀⠀
⠀⠀⢠⡆⠀⡴⠋⣸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣄⠀⣀⣀⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠙⣆⠀
⠀⢰⠋⡇⢰⠁⠀⡏⠀⠀⠀⠀⠀⠙⢄⡀⠀⠤⠤⠤⠤⣾⠁⠀⣤⣀⡜⠉⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣷⠀⠘⡄
⠀⢸⠀⡇⣾⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠙⢦⣽⣷⡒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡇
⢰⣿⢶⢷⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢸⠛⠾⣧⣤⢄⡽⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠸⣿⢯⣠⣿⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇
⠀⠙⠿⠛⠻⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠷⠤⠤⠾⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠧⠤⠴⠇