Robot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠲⡋⠖⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡀⠀⢩⠀⢀⡀⠀⠀⠧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⡀⠀⠉⠁⢀⠶⡀⠉⠀⠀⠠⠂⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠐⠀⠒⠁⠀⠀⠐⠐⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡒⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠲⢬⠀⠀⡋⠀⠀⡏⠀⠀⢹⠀⠀⡇⠊⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⠷⠤⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢏⠀⣠⡇⢸⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠈⠀⠀⠀⠀⠀⢸⠀⣇⠀⢈⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣄⣠⠞⠉⠀⠙⠺⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⡼⠒⠁⠈⠛⢄⠀⡼⢆
⠀⠀⠀⠀⠀⢠⡁⠀⡜⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠹⡄⠀⣠
⠀⠀⠀⠀⢀⠆⠈⡛⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠸⡋⠀⠆
⠀⠀⢠⠊⠉⢉⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡶⠶⠶⠦⠤⠤⠤⠤⠴⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠦⢴⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠂⠉⠙⢦
⠀⠠⠚⠈⠁⡈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠉⠀⠉⠙
⠀⠀⠀⠀⠀⠉⠀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠀⠀⠀⠀⠀⠀⢸⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠈⠁⠁
⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⢺⠀⠀⠀⠀⠀⠀⢸⠐⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠤⠴⠀⠀⠀⠀⠀⠀⠸⠠⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⡇⠀⠀⠀⠀⠀⡶⠤⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠀⠀⠉⠓⡄⠀⠀⠀⡴⠋⠁⠈⠁⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠈⠀⠀⠩⠆⠀⠸⠁⠁⠈⠉⠉⠉⠁⠈⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⡠⠤⠤⠐⠒⠒⠠⢤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⣰⣶⡤⠒⣿⣲⡀⠂⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠁⡘⠀⠀⠀⠁⠀⠀⠀⠁⢢⠋⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡇⢸⡉⠉⠉⠃⠒⡶⢰⢸⠀⠀⢠⡇
⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣑⡢⠤⠤⠔⠃⢸⠈⠓⢺⠋
⠀⢎⠳⡱⡘⣀⡀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠍⠉⠉⠙⠛⢿⡛⠚⢻⡉
⢎⠢⣳⡵⠋⠉⣗⣲⢦⢲⠒⠒⢲⠃⠀⠀⠅⠀⠀⠀⠀⠀⠱⡀⡤⠷⠤⠤⣄⡀
⠀⠱⡏⠀⠀⠀⡏⠀⢸⢸⠤⢤⡏⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⡇⠧⠤⣤⢤⡇
⠀⠀⠈⠒⠤⠤⠣⡀⠈⢸⠀⢰⡇⠀⠀⠀⠀⠀⢄⡘⣸⠀⠀⡇⠀⠀⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠑⢕⣤⡈⠀⢸⠀⠀⠀⠀⠀⠠⠀⠠⠠⡀⠀⡇⠀⠀⢸⢀⡇⡇⠀⠀⠀⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠀⢸⠀⠀⠀⠐⠀⠉⠁⠈⠒⠁⠀⡇⠀⠀⣿⢸⣇⡇⠀⠀⢀⡨⠖⠛⢻⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡏⠘⠶⠃⠀⠀⠻⣄⡀⠀⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢄⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢇⠀⠀⠀⠈⠉⠉⠙⠛⠖⡚⠒⠞⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠙⢦⡀⠀⠀⠀⠀⠀⡰⠐⣄⣀⣀⣰⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢧⠀⠀⠀⠉⠒⠲⠦⠤⠖⠓⠒⠉⠁⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠤⢆⡰⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢤⠔⠂⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢘⣀
⠀⠀⠀⠀⠀⣔⡉⠒⠤⣀⢰⣀⠤⠂⢉⡶⠀⠀⠀⠀⠀⢰⠚⠭⣄⣐⠖⢀⡩⠴⠆
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣙⣅⡤⠖⠁⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣉⠑⠂⢾⠐⠈⣁⡤⠗
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣾⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡴⠀⠀⠀⣎⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⠀⠀⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣻⣀⣀⣀⣀⣸⣛⣛⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠉⠉⠉⠉⠉⠏⠀⠀⢀⣠⠔⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠒⡒⠒⠒⠒⠒⠶⠶⠶⠶⢶⠚⠉⠀⣀⣀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠉⡇⠶⣿⣷⣲⠀⠀⢰⣶⣾⢷⡆⢸⠀⠀⡼⠀⡼⢙⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣈⡇⠀⠛⠛⠛⠀⠀⠸⠻⠿⠿⠀⢸⠀⠀⢹⡀⣳⣘⠤⠃
⠀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⣇⣀⣀⡀⠀⠐⣶⠂⠀⢸⡀⠀⠀⢀⣠⠞
⢠⡉⢙⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢒⡶⠶⠦⢭⣭⣭⣍⣉⣀⣀⣸⣇⡤⠞⠉⠳⣄
⠣⣈⣩⢿⣉⡷⣶⡀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠈⣦⣇⠙⠲⠤⢤⣤⠤⢴⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⠚⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠒⠤⢤⣤⣤⠤⡾⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠹⠶⠶⢶⣤⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣀⡂⢀⢐⠸⡇⠀⢀⡠⢼⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢈⣉⣉⡉⠉⠛⠒⠒⠒⠒⠲⠦⠬⠤⠴⡷⢊⡡⠖⠉⣧⡼⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⡏⠁⠀⠀⠀⣇⡷⢦⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣤⢫⠟⢿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠶⡶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⢸⣇⣠⠖⠋⢷⣻⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⠁⡟⠉⢩⡟⠉⠉⠀⠀⠀⠈⠁⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠶⢾⣷⡀⠀⠀⠀⠀⣯⡀⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⣉⣀⢀⡴⠋⢀⡇⢀⣠⠔⠋⠀⠉⢻⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠬⠭⠽⠶⠚⠉⢰⣛⡒⠦⠤⢤⣤⠏⣠⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣷⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣼⣄⢀⡀⡀⡀⢀⡀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠒⠉⡉⠐⢄⠀⠀⠀⠀⡠⠂⢉⠉⠒⡄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⢰⠡⠌⠣⠀⡆⠀⠀⢰⠀⠜⢡⠌⡆⢸⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⠀⠀⠀⠀⠣⡈⠑⠒⢁⡴⠁⠀⠀⠈⢦⡈⠒⠊⢁⠌⠀⠀⠀⠀⡏⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⢀⡀⣀⡀⢀⣀⡀⠀⣀⡀⣀⣀⣀⡀⠠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢤⠤⢠⡄⠤⠤⠤⠤⠤⠤⠀⠇⠀⠀⠸⠀⠤⠤⠤⠤⠤⠤⢤⡤⠤⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⡀⡀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⡆⠀⠀⢸⠦⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠴⠇⠀⠄⢰⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠛⠐⠃⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠘⠀⠛⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠒⠒⠒⠀⢴⡃⠀⠀⢘⡦⠀⠐⠒⠒⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⢄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠋⢰⡟⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠈⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠻⠦⠤⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⢶⣀⡇⠀⠀⣼⠀⠀⣀⣀⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢿⡀⠀⠀⣸⢿⡇⠀⠀⡿⢠⡟⠁⠀⠈⢷⠀⢿⠀⠀⠀⠀⠀⠀⠘⢳⠦⣤⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⡈⠛⠲⠚⠉⢸⠀⠀⠀⡇⠈⢧⣄⣀⣠⠟⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠶⠦⢬⣭⣥⣤⣄⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠶⠴⠤⠤⠄⠀⣀⣤⣀⣀⡀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⡀⢀⣀⣀⣀⣧⣠⣤⣤⣤⣤⣄⣄⣠⣄⣠⣄⣉⣉⣉⣽⣇⣈⣁⣀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢸⣧⡅⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢸⠤⢴⠇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⡏⠀⡆⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⣸⡇⢸⠀⡾⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠓⠒⠃⠈⣿⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣿⠃⢸⣺⡃⠀⣀⡴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠇⠀⢹⣷⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⡏⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾⠀⠀⠀⡀⠀⠀⠹⣷⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣼⠟⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⢰⠇⠀⠀⠀⡇⠀⠀⠀⠙⢿⣦⡀⠈⠳⣄⣀⠀⠀⠀⣀⣤⠞⢁⣠⡿⠋⠀⠀⠀⢸
⠀⢀⣤⣾⣁⡀⠀⠀⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣉⣛⣛⣋⣉⣤⡶⠟⠉⠀⠀⠀⠀⠀⢸
⡴⠋⠀⠀⠈⠉⠙⠛⢳⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡇⠀⣿⠙⠒⣶⠄⢰⠏⠀⠀⠀⠓⠂⡄⠐⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢶⡶⠢⠞
⠀⢻⣀⣻⡄⣰⠋⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠉⠉⠉⠛⠲⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣦⣀⣀⣿⢦⣄⣠⣤⣿⣀⣀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠤⣤⡤⣬⡉⠙⡏⠀⠀⠀⠀⡏⠉⣽⠷⠶⠶⢦⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠾⠃⠀⠀⠀⠈⠷⢶⡇⠀⠀⠀⢀⣷⠞⠃⠀⠀⠀⠈⠓⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠛⠉⠉⠙⠛⠶⣄⣿⠀⠀⠀⢸⣠⠶⠛⠋⠉⠙⠛⠶⣤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⣁⡢⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣇⠀⠀⠀⠀⠀⠀⠀⡵⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠷⠀⠀⠀⠼⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⡀⠀⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠀⠀⡀⠻⠀⡈⠀⠀⡓⠐⠴⠆⠙⠀⠸⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡈⡂
⠀⠀⠐⢰⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠋⣾
⠀⠀⠈⠲⡞⡉⠛⠑⠶⣄⢄⡀⠀⠀⠀⠐⠂⠤⠄⠀⣀⣀⣂⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⢀⡤⠊⠀⢔⠏
⠀⠀⠀⠀⠘⢎⠂⠀⠀⠀⠉⠚⠶⣤⡤⠀⠀⠀⠀⠀⠘⠻⠛⠻⠄⠄⠠⠤⠠⣀⣀⡤⠔⠁⠀⢀⢀⡜
⠀⠀⠀⠀⠀⠀⠑⢕⠀⠀⠀⠀⠀⡞⠀⠩⠂⢸⠩⠅⠀⠀⠀⠀⠂⠀⣀⡀⣠⡌⠺⠁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢄⠀⠀⠃⠀⠭⠂⢸⠀⠀⠀⢀⠀⠂⠀⠀⠶⠬⠰⠂⠀⣀⡠⣔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣀⠀⠬⠄⠸⣀⣐⣒⠓⡦⠒⣀⠀⣉⠥⢜⡁⠀⡇⠱⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⢽⠀⢆⠄⠀⢠⣤⣤⣤⣄⠀⠀⠀⠴⢝⡡⠄⠸⠆⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⢸⠀⠀⠀⠀⠀⠁⠹⠀⡇⠀⠀⠀⠙⠁⠘⠁⠠⢌⣑⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠭⢱⣐⠀⠀⠀⠀⠘⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠟⠚⠮⠥⠭⠭⠽⢏⣽⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠀⡉⠲⡀⠀⠀⠀⠠⠒⠈⠉⠓⠢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠒⠼⣀⣀⣂⣁⣧⡵⠃⠀⠀⠀⠳⢬⣐⡀⠠⠀⠈⡷⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠁⠈⠉⠉⠁⠀⠈⠀⠀⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⡄⠠⠤⠀⠄⠠⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠁⠀⠀⠐⠒⠚⠛⠁⠉⠉⠙⠒⡠⠤⢄⣤⡭⠖⠦⠀⠀⠀⠀⠀⠀⠀⠠⠈⢁⠞
⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠀⠛⠠⡀⠀⠐⠀⠀⠀⠈⠂⠀⢀⣈⠀⠀⠒⠤⠤⡀⡄⠂⠀⠀⡠⠃
⠀⠀⠀⠀⠤⢀⣀⣈⣡⣴⣢⠀⠂⠄⠀⠁⠀⠀⠄⠒⠐⠒⣀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠴⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠨⠄⠤⡄⠠⠀⠀⠐⠖⢈⡀⠀⠠⠐⠤⣄⣂⡱⠖⠃⠀⠘⠻⣋⣀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠒⢚⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡀⠀⢉⡉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠐⠛⠉⠀⠀⠘⠛⠶⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⡀⠀⠀⠸⠾⠇⠀⠸⠾⠇⠀⢠⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣦⣄⣀⡀⠀⠀⢀⣀⣠⣴⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⠿⣿⡿⠛⠛⠛⠛⠻⣿⣿⡿⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠉⠛⠲⠶⠤⢤⣤⣀⣀⡀⠠⠤⠄⠐⠋⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡂⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡴⠂⠀⠀⠐⠁⠀⠀⠀⠀⢸⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣏⠘⣧⠞⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⢈⡆⢰⡿⠁⢈⠀⠀⠀⠐⠆⢸⠀⢀⡞⢦⣼⠉⣷
⠀⠀⠀⠀⠀⠀⢠⡇⢨⡟⠁⠀⠈⠙⢶⠀⠀⠠⠄⠀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠙⣞⠁⣇
⠀⠀⠀⠀⢀⡠⠼⢬⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠊⠀⠠⠀⠀⠄⠀⠠⠀⠲⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⢸⣆⣹⣀
⠀⠀⠀⣀⡎⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⡞⠁⠀⠈⢳⡄
⣠⠖⢋⡩⢧⡀⠀⠀⣰⢧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡼⣇⠀⠀⠀⣰⡗⠲⢄
⠙⢦⡏⠀⡜⠉⢻⠉⠳⡄⣙⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⢰⡋⢀⡜⠓⡖⠚⣇⠙⢦⣀
⠀⠀⠉⠺⣧⣀⣼⠀⠀⠉⠁⠀⠀⠀⠘⠢⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠁⠀⠀⠀⠈⠉⠀⠀⣇⠀⣸⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠈⡟⠋⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⣅⡐⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡙⠻⠾⠿⠯⠒⠉⠐⢄⠀⡤⠒⡛⣶⠭⡤⠭⠒⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠉⠀⠀⠘⢦⠀⠀⠀⠀⠀⢹⡔⠉⠀⠀⠳⡄⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⡀⠀⠀⠱⣀⣀⡀⠠⢤⠂⠀⠀⠀⠀⠈⢆⣀⣀⣀⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡇⠈⠀⠀⠀⠁⠀⠀⠀⢸⠀⣰⠁⠑⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⣇⡀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠹⡀⡀⠀⠀⢀⠁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⡆⠀⠀⠈⠀⠀⠀⡜⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⢀⣀⠜⠁⠀⠀⠐⠁⠈⢦⡀⠀⢀⣠⠎⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠋⠉⠉⠉⠉⠉⠙⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠐⠀⠀⠀⠘⢢⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠀⠉⠁⠀⠴⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠉⠢⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⣀⠴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠉⠒⢢⠤⠤⠤⠤⡄⠒⠘⠁⠀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠔⠁⠀⠀⢸⠁⠀⠀⠀⠀⢠⡄⠒⡄⠀⢰⡏⠉⢲⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠃⠉⠁⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⢷⡀⣹⡤⢾⣀⣠⠎⠈⣳⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠈⣧⠴⠚⠁⣳⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⡤⠛⡿⠿⠛⠛⢿⡟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⢀⡟⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠏⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⠁⠀⡝⠉⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⡜⠀⠀⠀⠀⠙⠦⢄⣀⣀⣄⡠⠴⠚⠁⠀⠀⠀⠹⡋⠛⢤⠀⠀⣀⡤⠚⠅⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠉⢸⠁⠀⢠⠃⠀⣠⣴⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣯⠉⠀⢁⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⢀⡷⠚⢧⢀⡏⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⣽⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠋⠀⠀⠀⢹⠁⠀⠀⠘⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠃
⠀⠀⠀⠀⠀⣀⡤⠯⣼⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡸⠁⠀⠀⠈⢳⠀⠀⠀⢠⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⠴⠚⣉⣧⡀⠀⠀⠀⣼⣄⠀⠀⠸⢇⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢸⡇
⠹⣤⠞⠁⢠⠛⢲⠶⠻⣄⠘⢦⡀⠀⠀⠘⡋⠲⠦⢤⣄⣀⣀⣀⠘⠂⠀⠀⠀⣀⣀⣀⣠⣤⡴⠖⠊⠁
⠀⠈⠳⢴⡇⠀⢸⠀⠀⠈⠚⠋⠀⠀⠀⠀⣱⠖⠒⠶⢄⠈⠉⠉⠉⠉⠉⢉⡽⠿⠿⢤⡀⢀⠃
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⡶⠒⠠⡀⠳⡀⠀⠀⠀⣴⠃⡠⠄⠲⣾⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⣿⠴⠁⠀⠀⠀⢹⣴⣇⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠓⠛⠓⢾⡁⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠛⠛⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⣠⠗⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠛⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠚⠛⠒⠒⠤⢄⣧⠀⠀⠀⠀⠀⠀⢠⣇⠤⠄⠒⠚⢓⠶⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢠⠈⠁⡠⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⢤⠤⠤⡤⠤⠤⡶⠴⠤⠤⠤⠤⢾⠀⠀⣾⢠⡤⠤⡤⠤⢤⡤⠤⢤⡤⠤⣤⠄⢾
⣀⣀⣀⣀⣀⣀⣀⣀⣽⣶⣾⣶⣶⣷⣴⣴⣷⣦⣶⣶⣴⣶⣾⣄⣀⣿⣴⣦⣤⣷⣤⣴⣦⣤⣴⣦⣤⣶⣤⣾⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⠙⣧⠀⠀⠀⠀⢹⠋⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣧⠀⠀⠀⠀⢺⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡎⠙⡾⠃⠀⠀⠀⠀⠙⣷⠀⢶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⠀⠀⠀⠀⠀⡿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠾⠷⠶⠷⠶⠶⠶⠶⠶⠶⠿⠶⠾⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇⠀⠀⡟⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⡗⠀⠀⢸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⠀⢸⡇⠀⢀⣤⣤⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠛⠶⠖⠲⠚⠃⠀⢸⣤⣬⠀⠀⠘⠳⠖⠶⠖⠃⠀⠀⢸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣧⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢼⠁⠀⠀⠀⠀⠀⠀⣻⠀⠀⢠⡄⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⣤⠤⡤⣤⡀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢀⠀⡀⠀⡀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢻⠁⠉⠉⢻⠀⠀⢸⠋⠉⠉⢹⡇⠀⢸⠉⠉⠉⢹⠁⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⣇
⡶⠛⠓⠒⠒⠒⠒⠒⠒⢿⠀⠀⢸⠀⠀⠺⠤⠤⠤⠾⠀⠀⠸⠧⠤⠤⠼⠃⠀⠸⠤⢤⠄⠼⠀⠀⢸⡇⠀⠀⡿⠉⠉⠉⠉⠉⠉⠉⠉
⣷⢤⡤⢤⡤⠤⢤⣤⠤⣿⠀⠀⢸⡀⢀⡀⠀⢀⣀⣀⡀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⡀⢀⡀⠀⠀⣸⡇⠀⠀⣿⠠⠦⠤⠤⠤⠤⠴⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆⠀⠀⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢉⠀⢀⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⠀⢀⡀⠀⡀⠀⢸⡇⠀⠀⡿
⠿⠀⠶⠠⠶⠐⠲⠖⠶⣿⠀⠀⠘⠀⢸⡏⠀⠉⠀⠀⡏⠁⠉⢻⠀⠀⡏⠈⢸⡇⠀⢸⠁⠈⠙⠀⢸⡇⠀⠀⣿⠲⠒⠶⠶⠶⠶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠘⠗⠒⠚⠂⠀⠛⠒⠒⠛⠀⠀⠛⠒⠚⠃⠀⠚⠒⠒⠛⠀⢸⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠺⡟⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠹⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠚⡇⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠤⠆⡀⠂⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⣦⣎⣄⠀⠀⠀⠀⠘⠋⠄⣀⢄⣀⣀⣠⠀⠀⣀⡄⣉⡁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢻⡿⠀⠀⢠⠂⣦⡅⠀⡇⠊⣠⠍⢯⣴⢋⠉⢋⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢇⠀⠧⢀⣠⠄⠀⢃⡄⢀⠆⢾⣿⣦⣤⣿⠀⣇⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⣰⠗⠻⣥⣶⠋⣼⣁⠞⣄⠔⣯⣀⣸⣿⠀⢹⡆⣧⡀⢤⡀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢻⣉⠓⢤⣹⣿⣿⠋⠀⠀⠠⡹⣿⣿⠇⠀⠀⠙⢿⡷⠸⢧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⢿⢸⠄⣙⠢⣿⣏⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⣾⡓⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⡚⠈⠇⢨⣿⣿⣿⡀⠀⢾⡇⠀⠀⠸⠀⠀⠀⠀⣰⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⡴⠖⢻⣧⡿⠦⡄⢸⠏⠛⢿⡷⢴⡿⠿⠿⠿⠛⣷⣶⣤⣶⣿⠇⠸
⠀⠀⠀⠀⠀⠀⠀⡰⢦⠚⣿⠅⠀⠁⠀⣀⡴⢺⠷⣠⠄⠉⢈⢉⣭⣿⣿⣿⡟⢯⡑⠔
⠀⠀⠀⠀⠀⠀⠀⢹⢚⡄⠘⠄⠂⡀⠀⡿⠲⠥⣤⠀⢀⠀⠀⠀⡄⠈⣧⠬⣿⠆⡉⠳⢴⡂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠠⠀⠀⠀⣟⠀⠀⠀⠀⣾⡀⠀⠀⣷⡄⠆⠀⢻⡇⠀⠀⢸⣯⣾⣤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠀⣏⠀⠀⠀⣷⣾⣷⠀⢸⣿⡟⠀⠀⠈⣷⡄⢀⣾⣼⣡⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡼⢀⣴⣿⣿⡇⠀⠀⠀⣾⣿⠉⠈⢿⣿⡗⠀⠀⠀⢹⣿⣿⡿⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⡇⣰⢏⣼⣿⣿⡿⢻⡄⠀⠂⠀⣿⠇⠀⠀⠀⢻⣿⣄⢀⠀⠀⠹⡟⠫⠆⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⠋⢹⡏⠀⠸⣇⠀⠀⠀⣿⣄⠀⠀⠀⣴⣧⣿⣀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣼⡇⠀⠀⣏⣙⢻⣿⢻⣿⠀⠀⡞⣿⠋⡉⢿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢿⢉⣿⣿⣾⡃⠰⣞⣷⣡⡞⠠⡮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣴⣾⡈⠋⣹⣿⣿⣶⣧⠬⢽⣤⣾⣿⣛⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣴⣶⣿⣿⣿⣿⢧⣴⠧⠿⣿⣿⣿⡄⠀⠈⠁⠹⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⢠⣶⣹⣿⣿⣿⣿⣿⣿⣿⠏⠘⠁⠀⠀⠹⢹⣿⣇⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠦
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣤⣀⣀⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣾⣿⣿⣿⣿⠍⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄⢀⣤⣖⣒⣒⣒⡒⣒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠠⠤⡀⠀⣈⡉⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢒⡆⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠂⠔⠁⠸⠄⠝⡇⠀⠘⡇⠀⠀⠀⠀⠀⣎⠐⣅⡠⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠤⣄⣀⡠⡤⡀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣼⠲⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠒⠧⠦⠓⠃⢸⢸⣠⠟⠀⠀⠀⠀⡼⠣⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⡟⢻⠛⠓⠛⠋⠀⠀⠀⠀⢀⡞⢤⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠛⠧⠾⠛⠓⠒⠒⢲⣤⣄⡴⠫⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡅⠈⠉⠉⠉⠉⠓⠒⠒⡖⠉⣯⢺⠹⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢤⡇⠀⠒⡂⡒⠒⠒⢲⠀⡇⠀⣙⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢤⣷⠀⠟⠲⠃⠀⠀⢸⡗⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣸⢹⠀⢰⣞⡳⠀⠀⢸⣷⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⡼⢸⡄⠘⠒⠒⠒⠒⠚⠁⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣤⡇⠈⡇⠀⠀⣤⣤⣤⣤⣤⣿⠔⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣤⠗⡄⠓⣒⢲⡖⠒⠒⠒⠲⡿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⢇⣸⠀⣍⣹⠀⠀⠀⠀⠀⡷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠘⠉⠀⠆⢾⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡟⠀⠀⠀⠀⠀⢹⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣇⠀⠀⠀⠀⢠⢾⠚⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠘⠒⠃⡇⠀⠀⠀⢻⣈⡉⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠒⣒⣒⡭⠃⠀⠀⠀⠈⠒⠬⠭⡩⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠛⠛⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡇⢎⣏⠇⢎⣏⠆⣷⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠷⠶⠶⠶⠶⠶⠶⠿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠀⡔⠶⢲⢲⢲⠲⠖⡄⢰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣤⡺⡱⣱⠀⠉⠉⠉⡭⡍⠉⠉⠁⢰⡱⡹⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠆⢹⠊⠁⠀⣀⢄⣀⡨⣀⣀⢄⡀⠀⠈⢺⠁⠐⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠌⠀⠀⠀⠑⠚⠀⠂⠂⠸⠒⠁⠀⠀⠈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠤⢤⠤⢤⠤⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣻⠀⢸⣃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠒⢾⠀⢸⠖⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⣽⠀⢸⡍⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⡁⣇⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⣸⣿⣿⣿⣷⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡵⠟⣫⣭⣭⡛⠭⣉⠉⠙⠻⢿⣿⣿⡄⢀⡾⠁⢰⣶⣶⣤⣤⣤⣶⠶⠾⣛⠿⢟⣛⡻⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⣿⣿⣿⣾⣿⣮⡳⡑⡦⠄⣙⣿⣿⣿⡇⠀⣿⣿⡿⠿⠛⠛⠁⡠⠊⠁⣿⣩⣭⣝⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⢀⡿⣴⣲⣶⣶⡟⢿⡿⣿⣿⣿⣇⢱⢱⣤⣤⣿⣿⣿⠏⠀⠻⣆⣠⠒⠒⡲⣰⠁⠀⣀⣻⠛⠛⠙⠽⣿⠿⣦⢿⡄
⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⠗⣀⠸⣿⣿⣿⣾⢈⢶⣭⣽⣿⣿⣷⠀⠀⣸⡏⠛⣿⣿⡃⡇⣾⢩⣿⣍⢸⡉⡱⠸⣿⡿⣸⢸⣷
⠀⠀⠀⠀⢀⣿⣿⡸⣿⣷⣾⣿⣿⣷⣿⣿⢗⣋⣘⡺⢟⢿⣿⣿⣿⠆⠀⠘⣧⠀⠳⠞⠋⣣⠹⣌⣿⣧⣦⢨⣄⣸⣿⣿⡿⣼⣿⣇
⠀⠀⠀⠀⣸⣿⣿⣿⣜⡻⢿⣿⣿⣯⠟⣵⠟⠁⠀⢨⣷⡹⣹⣿⡏⠀⠀⠀⢸⣧⠀⡔⣫⡶⠒⠚⣭⡻⣿⡻⢿⣛⢿⢟⣽⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⡇⠈⠛⠿⣷⣶⣶⣶⡄⣇⠀⣀⣴⡿⢋⡇⣿⣿⣿⠀⠀⠀⠻⣏⢸⢼⠋⠀⠀⣼⡿⣣⢨⣭⣭⣥⣶⠿⢿⣿⣿⣿⢿⡀
⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣧⡹⢿⣿⣷⣶⢟⣴⣿⣿⡟⠀⠀⠀⠀⢹⣷⡸⣦⣠⣾⣟⣵⡟⣼⣿⣯⠉⠀⠀⠈⢿⣿⣿⡞⣧
⠀⠀⢠⡟⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⢿⣟⠲⠦⣴⣶⣿⡿⠁⠈⠀⠀⠀⠀⠀⠸⢿⣷⣬⣛⣛⣛⣩⣾⣿⣿⣿⣇⠀⠀⠀⠈⢿⣿⣿⠹⡆
⠀⠀⢸⠧⣿⠁⠀⠀⠀⣾⣿⢟⡽⠃⠀⠙⠛⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣉⣹⣿⠿⠋⠘⣯⡻⣄⠀⠀⠀⠈⢿⣿⡇⣿
⠀⠀⠸⣧⡙⠀⠀⠀⠴⣟⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠚⠋⣠⠟
⠀⠀⠀⠈⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⠀⢀⠔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠢⢄
⠰⠁
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣤⣀⣼⠿⢿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣿⣿⣇⣀⣨⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣹⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛
⡇
⡇
⡇
⠰⡀
⠀⠈⠢⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⠈⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⢀⠀⠀⣄⠀⠈⠈
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠂
⠀⠀⠘⠀⠀⢀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠂
⠀⠀⠠⡀⠀⠈⠿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠁⠒⠂⠐⠀⠀⠀⠠⠤⠤⠄⠀⠀⡀⢀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠓⠦⠀⠀⠀⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⢀⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠂⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠔⠐⠐⠇⠒⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢅⣀⣴⣀⣀⣀⣐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⣀⣽⣤⣭⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣄⣽⣧⣈⠃⣿⣗⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣿⣿⣿⣿⣛⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣶⣾⣿⣿⣿⣿⣿⣷⡶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢸⣿⣷⣿⠀⣨⣟⠛⠛⠛⠋⡀⢸⣼⡟⢹⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⠛⣿⣄⣧⣿⡀⢰⣿⡆⠀⣼⢻⣇⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⢿⣿⣿⣿⣃⣀⣉⣀⣠⡟⠀⢻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠆⠀⠸⣿⣯⣴⣶⡶⣾⣭⣽⠇⠀⢺⣿⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⣀⠛⣛⡷⣷⡛⢛⡀⠀⠀⠀⠹⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣿⡿⠿⠷⠾⠿⣿⣽⡄⠀⠀⢀⣜⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠛⠆⠀⣿⡿⣿⠀⠀⠀⢸⣏⠉⠁⠀⠠⠟⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠁⠀⢿⣿⣿⡀⠀⠀⢸⣿⣷⠏⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢿⡿⠟⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡏⣸⠖⠀⠀⠻⣿⣩⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠴⠶⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢄⣀⡈⠙⠛⠛⠛⠋⠉⣁⣀⣀⡴⠶⠛⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠗⠀⢸⠀⠀⠉⠁⠀⠛⠚⠛⠛⠫⣤⣍⢡⡀⠀⠀⢸⡇⠀⡀⢨⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢀⡀⠶⢾⡀⢶⣿⣟⡄⢱⠄⠀⡎⣿⣶⡞⢀⠙⠀⣰⠚⠓⠐⡇⢸⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣤⠛⢛⠀⢶⣽⣿⣧⠴⠂⠀⢷⣈⣿⣣⣼⠶⠀⠙⢳⡶⠺⣷⠸⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⢸⠀⠀⠁⢀⡤⠀⠠⣄⠀⠉⠀⠉⠀⡀⠀⠀⢸⡇⠀⠘⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⣤⠶⣍⠀⠀⢀⣼⠂⠀⠀⣠⡾⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⣄⠀⢀⣷⠴⠛⠁⣀⣠⠾⠋⠀⡃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠉⠛⠿⣥⠤⠚⠟⠋⠉⠀⠀⣿⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠿⢧⡀⠀⠀⠀⠈⠢⣄⠀⡀⠀⢂⣴⠏⠀⠀⣠⡾⠉⠉⠀⠀⠉⣩⡟⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠋⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠙⠛⠚⠛⠁⣴⣿⠶⠛⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⡴⠷⢦⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢀⡴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣴⠶⠒⠶⢾⣧
⠀⠀⠀⠀⢰⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⣧
⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠘⠷⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠂⠛⡈⢳⠀⠀⣿⠀⠀⣿⡀⠀⠀⠀⣀⣀⣤⣴⡶⠿
⠀⠀⠀⠀⠀⠀⡏⠀⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠦⣤⠀⣄⠀⣀⠁⢸⠀⠀⣿⠀⠀⠉⠛⠛⣿⡟⠛⠉⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⠛⢿⡷⠶⠶⢦⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠉⠀⠀⣿⡀⠀⠀⠀⢀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠳⠶⠶⠶⣶⣤⣤⣤⣀⣀⣽⣇⣠⡶⠞⠋⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿⠀⠀⠙⠛⠶⢦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠉⠁⢀⣠⣾⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⣾⣦⣤⣤⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠿⠿⠶⢶⣶⣦⣤⣤⣤⣄⣀⣀⠀⠀⣘⣠⣴⠿⠋⠀⣿⠀⠀⠀⣾⡇
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠄⣀⣀⣀⣀⣀⣉⣉⡿⠟⠛⣿⡟⠛⠋⠀⠀⠀⢀⣿⣄⣀⣀⣽⣇
⠀⠀⠀⠘⣧⣀⣀⣀⠀⢀⣾⠀⢀⣀⣠⣤⡤⣾⡇⠀⠀⠀⠤⠀⢉⡉⠉⠉⣉⣉⣤⡴⠛⣿⣇⠀⠀⠀⠀⠀⡏⠁⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠈⣿⠋⠉⣿⠋⠁⠀⣾⢿⡄⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢁⣤⡴⠚⣿⠋⠉⠛⣻⣶⣶⢃⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⡟⠀⠻⣦⠀⠀⠻⢶⣤⣀⣀⠉⠁⠀⠀⠙⠋⣉⣀⣴⠾⢋⣠⡶⠟⠁⠀⡟⠘⣷⠶⠶⠶⣶⠿⠃
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣷⡀⠀⠘⢳⣄⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⣉⣡⡶⠛⠁⠀⠀⠀⣀⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡯⠀⠀⣟⠀⠀⠀⠙⣷⡄⠀⠀⠙⣷⠶⠶⠿⠟⠛⠛⠚⠛⠛⠛⢿⡏⠁⠀⠀⠀⢀⣠⣶⠟⠁⠀⣿⠀⠀⢨⣿
⠀⠀⠀⠀⣰⡷⠶⢶⣏⠀⠀⠀⠀⠈⠻⣦⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⢀⣠⣶⣿⡏⠀⠀⠀⠀⣿⠀⠀⢸⡟
⠀⠀⢀⡾⠁⠀⠀⠀⠙⡿⣆⡀⠀⠀⠀⣿⠻⣾⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣧⣾⠟⠋⠀⣿⡇⠀⠀⠀⠐⣿⣤⣤⡾⢷⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⣰⡟⠀⠀⠀⠀⠀⠀⣇⢸⣧⠀⠀⠀⣿⠀⠈⠉⢹⣟⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⠀⠀⠀⠀⣿⠁⠀⠀⠀⢰⠋⠳⢦⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⣩⡿⣦
⣰⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣻⡀⢀⣿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⢦⣀⠀⠀⠙⠛⠶⣄⣀⡰⠶⠛⠋⠀⣠⣿⣀⣀
⡿⠒⠦⢤⣤⣤⣀⣸⡀⣿⠿⠾⣿⠇⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡽⠳⣤⣀⠀⠀⣼⠋⣀⣀⣤⠞⠻⣯⣤⣩⡽
⠘⢧⡄⠀⠀⠀⠀⠉⣻⣿⣤⠾⠁⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢰⣯⡀⢠⡞⠛⠳⢾⢿⡖⠉⠉⢳⣤⣤⢾⠇⣏
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⣀⣼⣏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡇⢹⡇⠀⠀⠀⠀⠀⠙⣶⡶⢾⡏⠀⢸⠄⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠿⢷⣶⣿⣧⠀⠀⠀⠀⠀⠀⢰⡿⠿⣶⣶⣶⣶⡿⢿⣷⠀⠀⠀⠸⣧⣄⣳⠀⠀⠀⠀⠀⢰⠏⠀⢸⡧⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠿⣤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⢷⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠿⠷⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⡁
⠀⠀⠀⠀⠀⣀⣠⠶⠞⠛⠁⠀⠀⠀⠀⢀⣴⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⠉⠉⠉⠛⠶⢦⣀⡀
⠀⢀⣤⣾⣛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⡀⠀⠀⠀⠉⠙⠛⢦⣄⣀
⠀⣼⠋⠉⠉⠙⠛⠶⠶⠶⣤⣤⡶⠋⠁⢀⣴⡟⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣄⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠈⠉⣷⣦⡀
⠀⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠇⢀⣴⠟⠉⣥⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⡷⣦⡀⠀⠉⢻⡶⠶⠚⠛⠛⠛⠁⢹⡃
⠀⠀⣹⠉⠹⠛⡿⢳⡶⣤⣤⣴⡿⢻⠅⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣶⣾⠃⠈⠻⣦⣄⣸⡇⠀⣀⣀⣀⣀⣠⣼⠇
⠀⠀⠹⣄⣼⡄⠁⣼⡀⠠⡄⠀⢀⣼⠷⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠃⣴⠏⠛⠛⠿⠙⡟⢛⠙⡏⢹
⠀⠀⠀⠀⠀⠙⠶⠜⠛⠶⠼⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠹⣆⠀⢀⣸⣠⣷⣀⣶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣀⠀⣤⣥⣾⣦⣴⠢⢠⣄⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉⠀⢹⡉⣷⣴⣷⣿⣿⡿⢷⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢸⣱⡿⠟⠛⢿⡳⣬⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠈⠻⠞⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠖⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠃⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⡿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢰⠀⣌⠀⠛⠋⢁⣠⣤⣾⣷⣦⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢐⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡐⢀⠁⠀⠀⠀⠀⣨⠑⠒⣂⠐⢀⡚⠒⢉⡀⠄⠀⠀⠀⠀⠛⠛⠀⠀⠀⠄⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⢄⠠⡀
⠀⠀⢀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⢁⣂⣉⠀⢈⣁⡢⠓⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡔⠇⠐⠂⠂⠀⢄⠠⡄
⠀⠀⠘⠐⠁⠦⠤⠤⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⢤⡤⠀⢀⢠⠁⠀⠀⠀⠀⠀⠀⠤⠁⠀⠈⠠
⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠤⠄⠀⢠⠀⠄⠅⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠴⠰⠤⠤⠤⠤⠤⡴⠴⠄⡀⠀⠀⠀⠈⠀⠈⠁⠁⠈⠀⠃⠀⠀⢠⠈⠁⡄⠀⢀⠀⠤⠐⢈⡁
⠀⠀⢘⠁⠀⠅⠀⠀⠀⠀⠀⠀⡇⡀⠘⠄⠀⠀⠈⠀⠔⠢⠈⠉⠁⠉⠉⣀⠒⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠒⠈
⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢀⠈⢘⠀⠀⠄⠑⠄⠐⠀⢁⠀⢀⠠⠀⢒⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠠⠂⠂⢀⢀⠀⢀⠀⠀⠁⠺⠗⢈⠀⠀⠈⠀⠀⠒⠀⠉⠁⠉⠁⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⢤
⠀⠀⠄⠀⠐⠀⠉⠈⠁⠈⠈⠁⠐⠀⠀⠄⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⢀⠀⠀⠀⠠⢊⠠⠀⣤⠀⣤⠀⢴⢵⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⢄⠐⠀⡀⡀⠘⡁⠂⠂⡁⠀⠀⠀⢀⠤⡀⠀⠀⠄⠀⠆⠀⢀⠀⠒⠀⢀⣤⣤⣤⡀⠈⠊⠀⡀
⠀⠐⠃⠃⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠳⠀⠀⣂⠐⠀⠐⢡⠉⠁⠁⢄⠁⢄⠄⢁⠀⠀⠀⠐⠤⠈⠀⠉⠄⠁
⠀⠀⠖⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠢⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⢙⠀⠀⠁⠀⠀⠀⣠⣄⠘⠀⠃⣀⣄⠀⠀⠐⠀⢀⠀⠀⠁⠀⠈⠀⠀⢀⠀⠁
⠀⠀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠄⣀⡀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡁⢈⠂⠀⠀⠀⠀⠖⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠀⠀⠉⠠⠂
⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠄⠉⠀⠀⣄⡀⣠⢠⠀⠜⠃⠈⠀⠀⠸⠿⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⠒⢢⠄⠀⠀⠀⠀⠀⠀⢀⡔⢢⡄⠀⠀⠀⠀⠈⠀⡼⢀⢀⠀⢨⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠉⠁⠀⠤⠀⠄
⠀⠀⠀⢑⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⡁⡡⠀⣉⢘⠁⠀⠀⠀⠀⠂⠔⠀⠒⠀⠒⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠃⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡏⣠⡝⣿⣿⣿⠟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡿⢿⣶⡆⠀⣿⡟⢿⣿⣿⣿⣿⣽⣡⣿⣿⣧⣿⣻⣿⠀⠀⠀⣀⡀⠰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣤⣿⡇⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⣿⣁⡀⡈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣈⡛⠛⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣛⣿⣿⠀⠀⢢⣽⣭⣾⣥⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣿⠿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠸⡿⠀⣿⡇⣤⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣷⡐⡘⣿⣿⣿⣷⣿⡟⢿⣽⣿⡿⠛⠻⣾⣗⡄⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣶⣶⣦⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠀⣿⣿⣇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣶⣦⣤⣭⣿⣿⣿⢻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣼⣏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣷⣶⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣾⡟⢿⣿⠟⣡⣶⣼⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣉⣙⠛⠉⠀⠈⠻⢸⣿⣿⣇⣽⣿⣇⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣿⠿⣿⡇⠀⡈⢺⣟⣲⣍⢹⣿⡏⠃⢸⣷⣮⠉⣿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⠿⣶⣿⡇⠈⡏⠀⣿⣯⣾⣾⣿⡷⠃⢸⣿⣵⣷⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣈⠉⠀⠀⣍⠀⣿⣿⣿⣿⣿⢧⠀⢸⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠉⠉⠻⣷⡀⠀⠶⠀⣿⣿⣿⣿⣿⣿⡗⣾⣿⣿⣿⣿⣿⣟⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⣼⡇⠀⡷⠘⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠁⠘⠛⠀⠟⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠟⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⣉⣉⡉⠉⠉⠉⠉⢳⡞⠉⠉⠉⠉⠉⣉⣉⡙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⢈⣉⣉⠁⠀⣾⣷⡀⣾⣷⢀⣾⣷⠀⠈⢉⣉⡁⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣤⡞⣭⡭⠉⢹⡏⠉⠉⠉⢳⣤⣄⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠻⢿⡇⡅⣴⠲⢋⡙⠒⣦⠀⢸⡿⠿⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣾⣿⡇⢰⡋⢰⣿⣿⡆⢙⡆⢸⣿⣷⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣨⣽⡇⠀⢿⣈⡙⢋⣁⡿⠀⢸⣯⣅⠀⠀⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠻⠿⣇⠀⠀⠁⢹⡏⠀⠀⠀⣸⠿⠟⠀⠀⢸⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠐⠒⠒⠀⠈⣿⣿⠋⣿⣿⠙⣿⣿⠁⠀⠒⠒⠂⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠘⠛⠛⠃⠀⠈⠁⠀⢸⡇⠀⠈⠁⠀⠘⠛⠛⠃⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⣿⣷⣶⣶⣶⣶⣿⣷⣾⣿⣿⣿⣿⡬⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⣉⣀⠀⠀⣈⣉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡟⡄⢎⢸⣉⠆⢸⠰⣉⡇⡜⡕⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠵⢧⡬⠭⠭⠤⠤⠭⠭⣤⠵⠵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠋⠙⠛⠛⠛⠛⠛⠛⠉⠋⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡇⠀⡖⢲⠓⡖⢲⠒⡖⢲⡖⢲⠀⣿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡙⣌⢲⡇⠀⠓⠛⠒⠓⣚⡒⠛⠚⠓⠋⠀⣷⢊⡼⣱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣉⣉⡉⠙⣦⣑⡬⠛⡇⠀⠀⠀⠀⠘⠧⠟⠀⠀⠀⠀⠀⡟⠫⣔⣡⡞⠋⣉⣉⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡆⢸⠋⠀⠀⡇⢠⡴⠧⣄⣠⠜⢤⡀⡤⠷⣤⠀⡇⠀⠈⢻⠀⡎⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡜⠀⠀⠀⡇⠀⠷⠾⠅⠸⠶⠾⠀⠹⠶⠇⠀⡇⠀⠀⠘⢤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡏⠭⢭⡏⠉⠉⣯⠭⠭⣽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣗⣒⣒⡇⠀⠀⣗⣒⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡧⠤⠴⡇⠀⠀⡷⠤⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡯⠭⢭⡇⠀⠀⣯⠭⠭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣓⣒⣒⣇⡀⣀⣗⣒⣒⣻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠓⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⢦⡏⠉⠉⠳⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣾⢿⣇⣒⢐⠂⣿⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢺⠓⠛⠛⠛⠋⠉⣉⢉⡉⠉⠙⢛⣛⣙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢰⣇⡰⠀⠀⠀⠋⠘⠉⢷⣦⣶⢡⠎⠹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠚⠋⠉⠑⣠⠆⠀⣦⡙⣖⡽⠋⠻⣄⣓⣺⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⢀⠀⣀⠟⠀⠀⠈⠛⠋⠀⠘⠁⠉⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⢟⠆⠀⠀⢠⡶⢶⠒⣶⣒⣖⢒⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢋⡜⠀⠀⠀⢸⣏⣻⣉⣻⣉⣯⣽⡏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠧⠤⢧⣤⣤⣦⣤⣥⣤⣤⣤⣤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣦⣴⡶⠶⠶⠖⠺⣿⣿⣿⣿⣿⣿⣭⣤⣞⣉⣉⣁⠐⠶⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠙⠉⠉⠑⢆⠀⠙⡇⠉⠁⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢱⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⠙⣦⠈⡀⠀⠈⠀⠀⡇⠀⠀⠷⠀⠃⠀⠀⠀⡀⠀⠀⢀⣀⣉⡋⠀⠀⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣆⠸⠤⠂⢀⣁⣤⣼⠀⠀⡇⠀⠀⡰⣤⣤⣷⣞⣻⣿⡶⠶⠾⠛⠻⣜⠀⠀⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠻⣦⣄⣤⣾⣿⣿⡇⣸⠇⡇⠀⠀⣿⠀⣠⢦⠟⡙⠁⠉⠘⠛⢷⠄⣿⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⣿⣿⣿⣿⣿⢃⣿⢀⡇⠀⠀⡏⠀⢣⣀⡠⠞⠀⠒⠀⠤⠛⠀⣿⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢦⡀⠀⠀⣼⣿⣻⣿⣿⣿⣸⣯⠸⡇⠀⠀⡟⢀⡀⠁⠀⠀⠀⠀⠀⠐⢞⠀⣿⠀⠀⣧⡀⠠⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡤⣄⠙⣦⣼⣿⣳⣿⣿⣿⡇⣿⡿⠀⡇⠀⢴⡇⠧⡀⠤⠠⠀⠘⠶⠒⠐⢊⡁⣿⠀⠀⣿⣷⠖⠀⠉⠒⠂⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡀⣀⠀⡟⠛⠛⣿⣿⣿⣿⢻⣿⡇⢀⡁⠀⠀⡿⠀⠤⠰⠆⠰⠀⠋⠈⢁⣈⣀⡻⠀⠀⣿⣿⣀⣀⣀⡀⠐⠀⣤⣽⡗⠒⠲⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣠⠧⠀⢺⣹⣿⣿⡟⣿⣿⢠⡜⠄⠀⠐⣧⣠⣴⢶⣶⣶⣿⠿⠿⠏⠉⠉⠀⠀⠀⣿⣽⡿⠉⠉⠁⢀⡛⡠⢄⠙⣤⣀⡀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠿⠛⠃⠀⠘⣿⣿⣿⣿⣿⠏⣸⠇⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⡇⠀⠈⠉⠐⠒⢠⣤⡑⠊⣠⣿⠿⣅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣆⠀⠀⠀⠀⢻⣿⣿⣿⡿⢦⣿⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⣉⠉⠁⠀⠀⣌⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⣇⠀⠀⠀⢸⣿⣿⣿⣇⣼⡏⠀⡇⠀⠀⠀⠀⠀⣇⣀⣀⡀⠠⠴⠎⣀⣀⠠⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠹⣆⣄⠀⣘⡛⠛⠻⢿⣿⣇⣠⠛⣄⢀⣀⣠⣤⣤⣶⣶⣶⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢢⠘⡿⣍⠉⠲⡀⢸⣿⣿⣿⣿⡿⠟⠛⠛⠉⣿⣿⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠛⣠⡇⠘⠀⠀⠃⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡏⠈⣷⠀⠀⣠⡄⠀⠟⢻⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢠⡤⠿⠖⠚⠋⠁⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠶⢀⣇⣠⣤⠤⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠓⠲⢤⣀⠀⠀⣀⡠⣦⠈⡏⢀⣀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠴⢄⠈⢿⠋⠁⠀⠀⢨⠀⣾⠾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢈⠆⠀⣶⣦⡽⠿⢷⡶⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⠈⠀⣴⠉⠀⠀⠀⢸⡁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢻⡷⣿⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⣘⠀⣾⠁⡿⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⢀⣀⣤⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⢠⡟⢸⠇⠀⠀⠀⠀⠀⠘⢀⣂⣠⣤⠿⠐⠒⠤⠤⢄⡀
⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠃⡏⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⠀⠀⠀⠀⣀⣠⣤⣗
⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣙⠳⢨⣍⢦⣄⣀⣀⣤⣴⠾⠿⠿⢿⣿⣦⠞⠛⣭⠵⠚⡛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣍⣙⠉⠁⠀⠀⠀⠀⣀⣠⣸⣋⠡⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⢿⣿⣿⣷⣯⣟⡻⠿⣿⣿⣿⣿⣿⣷⣄⡴⠖⠛⣉⠠⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣷⣾⣭⣛⠻⢿⡿⠒⢈⣁⠴⠖⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣵⣷⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞⠉⠙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢉⣭⣭⣿⡀⠀⠀⢀⡷⠀⠀⠀⠀⢀⠘⡁⢋⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⢀⡴⠋⠀⣠⡾⢻⡶⢶⠋⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⣠⡾⠋⠀⣸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠾⠋⠀⣠⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⢠⡶⠀⠀⣀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡀⠀⡀⠀⠈⣥⢠⠁⠀⠀⠀⠀⠈⠀⠀⠘⠁⠀⠀⠁⠀⠀⠀⢀⡴⠶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢙⡦
⠀⠀⡐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠠⠄⣄⢀⠈⢹⡆⠀⠀⠀⠀⠀⠚⠛⣩⢷⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠒⠒⠒⠒⠒⣟⠁⢠⠘⠀⠀⠀⠈⡄⠀⣷⠀⠀⠀⠀⠀⣠⠞⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠰⠆⠀⠀⠀⠀⠴⠀⢸⡆⠀⠀⠀⠘⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⣇⠀⠛⠰⠂⠤⠉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠛⣶⠦⣤⣄⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠁⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡆⠀⠀⢀⠀⠀⠀⡾⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠰⠀⢠⠚⠻⣷⡄⠀⣇⠀⠀⢸⠀⠀⠀⠃⠀⠀⠸⠁⠀⠀⣰⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⠀⢠⠀⠀⢀⠀⠀⠀⠰⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣄⠀⠀⢸⡄⠀⠰⣿⠀⠀⣀⣀⡘⠀⠀⡿⠀⢀⡟⢠⠖⠳⣦⠀⢀⡴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⡀⣼⠉⠈⠉⣷⡴⠶⢦⣄⠀⡏⠀⢀⣿⠀⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣴⠾⠛⠛⠛⠋⠉⢉⣭⣭⣭⣭⣭⣭⡇⠀⠀⠀⣿⠀⠈⠁⣿⣭⣭⣭⣭⣍⠉⠉⠙⠛⠛⠻⢶⡄⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠖⢂⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢀⡀⢸⣿⠀⡀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣠⣶⡒⢦⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡾⠰⢡⣄⡸⡟⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣏⣀⣿⣿⣿⣿⣿⣿⣿⢠⣤⡝⠛⣡⣤⡱⠸⡆⠴⠒
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡇⠀⠘⠛⣩⣥⠛⠛⣸⣿⣟⣩⠴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⣣⣦⡙⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⡇⠀⠀⠀⠙⠛⣰⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣙⠛⠁⠀⠀⠀⡇⠐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢹⡀⠀⠀⠀⠉⣿⠀⠀⠀⠛⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠀⠈⠉⠁⠀⠀⠀⣸⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⢧⣀⡀⢀⠀⢹⡀⠀⠀⠀⢸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⣀⣀⡀
⠀⠀⠀⠐⠒⠛⠋⠉⠉⠀⠀⠀⠀⢉⡛⠛⠛⠛⣧⠀⠀⠀⠈⠉⠉⠙⠛⠿⠛⢿⠿⡿⠿⠿⠿⠿⣿⣿⠛⠛⠛⠛⠛⠛⢛⡋⠁⠀⠀⠀⠈⠉⠉⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠖⠛⠉⠁⠀⣀⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠈⠙⠦⣄⡀⠀⠉⠀⠀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⣀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⠀⠑⠂⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠠⠚⠁⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠁⠀⠀⠀⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣠⠆⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠶⣿⡟⠁⠘⠆⠀⠀⢦⡀⠀⠈⠳⠀⠀⠀⠀⠀⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣁⣀⣀⢀⣞⣁⣀⣠⣆⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣧⣀⣀⣀⣦⣀⣀⣙⣆⣀⣀⣀⣙⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⠾⠉⣉⣉⣩⣉⣉⠉⠿⠿⠉⣉⣉⣩⣉⣍⠉⠿⠉⣉⣍⠙⠟⢉⣉⣍⠉⢉⣉⣉⣉⣍⣉⣩⡉⠉⢉⣉⣉⣉⣉⣍⡉⠹⣶⣀
⠀⢰⡶⠶⠶⠶⠾⠂⣶⣿⠉⠉⠉⠻⠿⠀⠀⣾⣿⠉⠈⠉⣿⣶⠀⠠⣿⣿⣧⣴⣾⣿⡿⠀⢸⣿⡏⠉⠉⠀⠉⠁⠀⢸⣿⡏⠉⠁⢹⣿⡆⠈⠻⠶⠶⠶⣶⣀
⠀⢸⡇⠸⠿⠿⠂⠀⣿⣿⠀⠀⠺⣿⣿⠀⠀⣿⣿⣤⣤⣤⣿⣿⠀⠈⣿⣿⠘⠟⢸⣿⡇⠀⢸⣿⡿⠾⠿⠿⠂⠀⠀⢸⣿⡿⠿⠷⢿⣯⡀⠀⠺⠿⠿⠀⠈⣿
⠀⠘⢻⣦⣤⣤⣤⡀⠛⣿⣤⣤⣤⠟⣿⠀⠀⣿⣿⠁⠀⠈⣿⣿⠀⢀⣿⣿⠀⠀⢸⣿⣇⠀⢸⣿⣧⣤⣤⣤⣤⡄⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣠⡄⢠⣿⣿⣿⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⡄⣴⣿⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣇⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⣿⣿⡟⠈⢻⠇⠀⠘⠋⢻⣿⣿⣿⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⣀⣠⡏⣙⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣤⣾⡸⣆⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢡⠱⢻⡀⣼⣿⣿⡇⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⡇⣿⣿⣧⣀⡟⠎⡌⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢫⡳⣄⠰⢦⠘⣀⣄⣿⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣿⣿⢰⣀⠃⡴⠂⣠⢞⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠃⠀⠱⣭⣾⣿⣿⣇⣝⠷⢹⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⣻⣿⣿⣷⣭⠎⠀⠘⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣳⣌⢻⣿⣿⢿⣹⣷⣌⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡿⣿⣿⣟⣡⣞⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠗⠛⣛⣛⠛⠛⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣿⣿⡛⠺⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡿⠞⠁⣀⣀⣀⠀⢀⡀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠶⠏⠀⢀⣀⡀⠀⣀⠀⠈⠳⣿⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⢀⡯⠟⢹⡀⠈⠉⠀⠀⠀⠀⢀⣀⣈⡀⠀⠀⠀⠠⠋⠉⠀⣞⠉⣹⠀⠀⠈⠻⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⡀⢸⣏⡉⠳⣤⠛⢉⣽⡇⠀⢀⣀⣀⠻⠭⠭⠟⣀⣀⡀⠀⢠⡞⠓⡎⠛⢡⠞⠳⡄⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢰⠇⠘⠓⠳⡞⠈⢳⠞⠚⠃⣴⢫⠀⠈⢳⡄⢀⡞⡁⠀⠙⣦⠈⠓⠞⣡⠶⢬⠳⠞⠁⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⠸⠀⠀⠀⠀⠷⠷⠾⠀⠀⠀⠻⣜⠳⢂⣼⣿⣿⣧⡛⠦⣠⠟⠀⠀⠀⠻⢤⠞⠀⠀⠀⠀⣆⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⢱⠀⠀⠀⠀⠀⠀⠠⠄⣠⣴⠆⣈⣭⣭⣤⣭⣭⣤⣭⣭⣁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢾⡿⣼⠟⢡⢞⣁⣀⢀⣀⡀⣀⠀⣀⢀⣈⠳⡌⠻⣦⣄⠀⠀⠀⠀⠀⠀⠐⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣷⠀⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⢸⡇⣤⣸⣭⡗⣿⣾⣿⢸⡧⠀⠙⢦⡘⢿⣷⠀⠀⠀⠀⠀⠀⣠⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣿⠀⠀⠀⠀⠀⠰⠏⣠⠞⠀⠀⠘⠳⠟⠘⠀⠛⠛⠋⠛⠘⠳⠂⠀⠀⠳⣄⠹⠀⠀⠀⡀⠀⠀⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣷⣦⣀⣀⠀⢀⣾⠁⠀⠀⠀⠀⢉⡿⢸⡟⣿⣿⣶⡇⣿⣉⠀⠀⠀⠀⠈⣳⡀⠀⣘⣿⣩⣾⣿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡙⠛⠟⠛⣉⣴⣻⣦⢳⡀⠀⠀⠀⢼⣧⢸⣧⡿⢿⠹⡇⣿⡤⠀⠀⠀⢀⡞⣵⣟⣦⣈⠛⠻⠛⢋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⢿⣿⣿⣧⣙⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣋⣼⣿⣿⡿⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠦⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡔⠓⢦⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⢧⣤⡞⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣧⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣎⡆⢾⡆⠀⠀⠀⠀⠀⠀⠀⢸⡁⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣧⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⢿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⡇⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣸⡆⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣛⣛⣛⣻⣏⣉⣻⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⡾⠁⠀⠀⠀⡤⢤⡀⠀⠉⠉⣷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡹⠊⢩⡇⠀⠀⠀⠀⡹⢯⠀⠀⠀⠀⢸⠓⢞⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠯⣈⡿⠁⠀⠘⡇⠀⠀⠀⠀⢙⣚⠀⠀⠀⠀⣸⠀⠀⢳⣏⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠄⢀⠈⡇⠀⠀⠀⢻⣄⣀⣀⣀⣙⣴⣁⣀⣀⣠⠇⠀⠀⠀⡟⠀⡠⣄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢾⣠⡇⠀⠀⠀⠀⣼⣅⣽⠈⠉⠉⠉⢹⣅⣽⡄⠀⠀⠀⣄⣸⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣄⣼⡀⠀⠀⠀⣸⣄⡤⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠺⠧⣤⡼⠧⡄⢀⠴⠿⣤⡤⠿⠒⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠚⠛⠛⠛⠛⠛⠓⠶⠶⠶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡤⠤⠤⠤⠤⢤⣄⣀⣀⠀⠀⠀⠉⠉⠙⠛⠓⠶⠦⣤⡀
⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠒⠤⠤⣀⣀⠀⠀⠙⢷⣤⡀
⣠⠶⢾⠋⢹⠁⣸⣤⠞⢻⡄⠀⠀⠀⠀⠀⠀⣼⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⢸⠈⠛⢶⣄
⠙⢦⡈⢣⣀⣦⡿⠁⣰⠟⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⠃⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡜⠀⠀⠀⠙⢦⡀
⠀⠈⢳⣬⠿⠼⠷⢾⡁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡎⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠀⢀⠇⢰⠁⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡧⠖⠒⠚⢻⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠃⠀⠀⠀⠀⠘⠉⠉⠓⠒⠲⠦⠤⡆⠀⠀⠀⠀⠀⠀⡸⠀⡆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠇⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠁⠀⠀⠀⠀⠀⣾
⠀⠀⢸⣄⣀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⣿⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡜⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠘⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢰⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⣧⣀⣀⠤⠀⠀⢹⡄⠀⠀⠀⠈⣧⠀⠀⠉⠓⠒⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠈⠛⠶⣤⣤⣀⣀⠀⠀⠀⠉⠉⠉⠛⠒⠶⠤⣤⣀⣀⣀⣰⠏⠀⡇⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⢻⡀⣀⠔⠉⠀⠀⠙⢦⣄⠀⠀⢀⣰⠋⠀⠀⠉⠙⠓⠒⠶⠤⣤⣄⣀⣀⠀⠀⠀⠉⠉⠀⢀⣼⣠⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠻⡅⠀⠀⠀⡀⠀⠀⠈⠛⠲⢿⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠋⠁⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠙⢦⣠⠞⠁⠀⠀⡴⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢀⣠⣷⡤⢶⠶⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡴⠁⠀⢠⠏⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡞⠉⢹⠀⠀⠀⢸⠀⠀⠀⢠⠋⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠺⣯⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠀⠀⠀⠘⠀⠀⠠⠃⠀⠀⠀⢀⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠳⣤⠶⠖⢶⠖⠶⠶⢶⣄⡀⠀⠀⠰⠃⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢸⡇⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡖⠒⠒⠒⠒⠒⠶⠦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠖⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠃⠀⠀⠀⡇⠀⠀⠀⠀⣀⣠⡤⠾⡶⠤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⣉⣠⡴⣾⠳⡶⡎⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠉⠁⢠⠇⢸⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠿⣤⠾⠶⠷⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢼⣷⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼
⢴⡦⣤⡀⠀⢀⣀⣠⡶⠲⣟⠀⠀⠸⡀⠉⠛⠒⠲⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⠤⠴⠶⢶⡶⠞⠃
⠘⣧⠀⠉⠻⣏⠉⢻⠀⠀⠈⢆⠀⠀⠱⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠈⢷⠶⣦⡴⣤
⠀⠈⢧⠀⠀⠈⢆⠀⢳⡀⠀⠈⠀⠀⢀⣤⣄⣤⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠄⠂⠀⠀⣠⠀⠀⡀⠀⢘⣷⣤
⠀⠀⠈⢧⠀⠀⢸⡀⠀⠁⣠⡾⠳⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡞⠉⠀⠀⠀⡰⠁⠀⢀⠃⢠⠊⠀⢸
⠀⠀⠀⠘⢷⣤⠟⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⣰⠁⠀⢀⡎⢀⡏⠀⢀⡾
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠓⠖⠛⠛⢻⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⢿⣿⠟⠉⠉⠀⠀⠀⢀⣴⠟⠁⠀⠉⠉⠙⠛⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠉⠀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⡏⠀⠀⠉⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋⣽⠂⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⢀⣿⠀⠀⠀⠀⣿⠀⣴⠟⢶⡟⠁⠀⠀⠀⢠⡶⠻⣾⡋⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠈⢿⣤⣼⠇⠀⠀⠀⠀⠘⢷⣤⡾⠃⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢠⣾⡿⠋⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⣰⣿⡟⠁⠀⣰⡟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⣰⣿⡟⠀⠀⠀⣿⠀⠀⢀⣤⣤⣴⠟⠁⠀⠀⠀⣠⣶⣿⡿⢿⡿⠋⠉⠉⢉⣽⠿⠟⠻⢿⣿⣶⣄⠀⠀⠈⠻⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣆
⠀⢠⣿⡿⠀⠀⠀⠀⣿⠀⠀⢿⣅⢈⣿⠀⠀⠀⣴⣿⡿⠛⠁⢀⡿⠀⠀⠀⢰⡟⠁⠀⠀⠀⢀⣈⣻⣿⣿⣦⠀⠀⣿⣁⣨⡿⠀⠀⠀⠀⠀⣠⡾⠋⢿⣿⡄
⠀⣾⣿⠃⠀⠀⠀⠀⣿⠀⠀⠈⠙⠋⠁⠀⢠⣾⣿⢿⡇⠀⠀⠸⣧⠀⠀⠀⣿⠀⠀⠀⣠⡾⠋⠉⠉⠉⠙⣿⣷⡄⠈⠙⠋⠁⠀⢀⡀⣠⡾⠋⠀⠀⠘⣿⣷
⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠸⣇⠀⠀⠀⠻⣦⣀⣀⣿⣦⣀⣸⡏⠀⠀⠀⢀⣀⡀⠈⢿⣿⡄⠀⠀⠀⣾⠛⠻⣏⠀⠀⠀⠀⠀⢹⣿
⣼⣿⠃⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⠀⠙⢷⣄⣀⣠⣾⣿⠿⠟⠻⠿⣿⣷⣄⣴⠞⠛⠉⠙⠛⢾⣿⣷⠀⠀⠀⠻⠶⠾⠋⠀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⢀⣴⠶⣶⠟⠁⠀⠀⠀⠀⠀⢸⣿⡿⣧⡀⠀⠀⠀⠈⢹⣿⡟⠁⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿
⣿⣿⠀⠸⣧⣀⣿⠀⠀⠀⣀⣀⠀⠀⢸⣿⡇⠈⠻⠶⣤⣤⣤⣿⣿⡁⠀⠀⠀⠀⠀⠀⢈⣿⣿⠛⠛⠻⠶⣦⡀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣿⠉⢹⡆⠀⠀⣿
⣿⣿⠀⠀⠈⠉⠁⠀⠀⣼⡋⠙⣷⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⢀⣼⣿⣇⡀⠀⠀⠀⠈⢻⣾⣿⡇⠀⠀⠀⠀⢀⣴⠿⠷⠟⠁⠀⠀⣿
⢻⣿⡄⠀⠀⠀⠀⠀⣠⡾⠳⠾⠋⠀⠀⢿⣿⡷⣦⣄⣀⣤⡴⠟⠙⢿⣿⣶⣦⣴⣶⣿⡿⠋⠉⠙⢷⣄⠀⠀⢈⣿⡿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⣿
⠸⣿⣇⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠈⠉⠀⠀⠀⠀⣸⠏⠉⠛⣿⠉⠉⠻⣆⠀⠀⠀⢹⡆⢀⣾⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢿⣿⡄⣠⡾⠋⠀⠀⠀⠀⠀⣰⠶⠷⣦⠘⢿⣿⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿⠀⠀⠀⢻⡆⠀⠀⢸⣷⣿⡿⠃⢀⣠⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣿⣦⣴⠟⠀⠀⠻⣿⣿⣯⡉⠁⠀⠀⠀⢀⣼⠇⠀⠀⠀⣿⠁⢀⣤⣾⣿⠟⠀⠀⣿⡁⢉⣷⠀⠀⣿⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠹⣿⣧⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣴⣶⣟⣁⣀⣀⣠⣾⣷⣾⣿⠿⠋⠀⠀⢀⣴⠟⠛⠛⠁⠀⠀⣿⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠹⣿⣧⡀⢀⣴⠟⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠏⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠘⢿⣷⣟⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠈⣿⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣷⡀⣿⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠻⢦⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⣴⠟⠁⣿⠀⠀⠀⠀⣿⠁⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢀⣠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠀⠀⣀⣤⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣤⣄⣀⣀⢀⣴⠟⠁⠀⠀⠀⠀⣀⣴⣿⣥⣶⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠀⠀⠀⠀⢄
⠀⠈⠒⠤⢤⠤⢄⣀⣁⡀⠀⠀⠀⠠⠤⠄⣀⢣⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⡀⣑⠀⠀⠀⠘⠆⢧⡄⠀⠀⡱⠊⢍⡉⠉⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠠⡀⡀⡀⠀⣄⢀⠀⠠⣀⠀⡀⠀⠃⠔⢱⢸⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⡄⠐⠒⠁⠇⠀⠠⡀⢠⠀⣀⠀⠠⠂⠉⠻⣄⠀⠀⡸⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⠚⠀⣬⡀⠀⢀⣱⢀⢀⣀⠞⣧⣤⣤⣴⠏⡷⠊⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⢾⠈⠙⠿⠍⠉⣼⠄⢠⡀⠀⠈⠉⢀⣼⡅⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣠⣤⣶⠗⠛⠻⣌⣐⣓⣫⠽⠷⣄⠠⡀⠀⠀⠀⠀⠀⠈⣇⠄
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠉⠉⠀⠄⠀⠀⠀⠘⠛⠒⠒⠒⠾⢷⣾⣦⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢥⠀⠀⠀⠀⢻⠿
⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⢎⠓⠤⡄⠤⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⢉⣹⠧⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢧
⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢾⢾
⠀⠀⠀⠀⠀⠘⡆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⣟⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣤⣵⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢦⡀⠸⡅⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⣬⢽⡀⠀⠀⣠⣤⡴⠦⡿⣅⠈⢸⡇⢠⡇⠀⡇⢀⣴⣶⣾⡿⠿⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⠃⠀⠰⣩⣯⡉⠰⣇⣼⣇⠀⢳⣸⣿⣦⡷⠋⢩⣴⣶⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣤⣷⣤⣀⣿⣿⣷⠀⠻⢿⣿⣦⣤⠽⣯⣷⣶⣶⡿⣋⣵⢖⠛⣻⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣭⠾⠟⣛⣭⣴⣶⣮⣭⣿⣦⡶⠞⣿⣷⣤⠀⢿⣿⣿⣿⣵⣿⢿⣼⣾⣿⢟⣛⣻⣿⣦
⠀⠀⠀⠀⠀⠀⡞⣀⣰⣿⣻⣿⣟⣛⠿⣿⣿⣿⣷⣦⠈⠻⣿⣷⣦⣿⣿⣿⣿⣴⣼⣾⡿⠿⣿⣶⣿⡟⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣯⣭⢻⣿⣿⣮⢿⣿⣿⡿⠿⠾⠿⠋⢻⡉⠻⠿⢿⣾⣿⣿⣿⣿⣼⣿⡿⣱⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⣙⠿⣵⣦⡌⣿⣿⣿⣿⣿⣶⣤⣶⣤⡿⠤⢤⣤⣄⣀⡏⠀⠟⢻⣿⣿⣵⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣗⢿⣿⣷⣿⣿⣷⣿⣿⣿⡿⢛⣛⢻⣿⣟⢴⣿⣷⣿⣿⠟⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡙⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣇⣿⣻⣷⣿⣿⣾⣿⣿⠟⠁⢀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣭⣭⣭⣭⣾⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⡯⠀⢀⡎⠀⠀⠀⠙⠮⣽⣿⡿⠿⠿⣿⠟⡼
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⠻⣿⣿⠛⠋⠀⣽⣿⢤⣤⣂⡼⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⡿⢣⢻⢦⣀⣹⣧⣀⠀⢜⣻⠁⣀⣽⠇
⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣶⡇⠀⢠⠃⠈⡷⠤⢤⣾⣟⣛⠛⠛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠙⢬⣛⠿⠿⣁⡴⠋⠀⠀⠉⠋⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣻⣭⣿⣯⣽⣿⣭⣿⣯⣽⡛⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡁⣿⣿⡿⢿⢟⣟⢻⠿⣿⣿⣿⢨⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣛⠣⣿⣿⢸⣷⢟⣛⢷⣿⢸⣿⣿⢘⣛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⢇⣿⡰⢿⣇⢿⣿⣧⣿⠷⢸⣿⠸⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆⣿⣿⡰⡿⢶⣾⠿⠿⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⡞⣿⣿⣿⣾⣮⣭⣾⣷⣿⣿⡿⣰⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣮⡍⣭⣭⢩⣽⡍⣭⣯⣭⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣿⣿⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡆⠀⠢⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠤⠼⠿⠓⠒⠸⠷⠤⠼⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣥⠀⠀⡾⡉⢳⣶⢙⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⠀⢀⡆⠀⠳⠾⠋⠏⠳⠟⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⠁⠀⢰⣖⣶⣶⣦⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠯⢸⣀⣄⣘⣓⣓⣓⠁⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⡄⠀⣜⠿⠿⠿⠿⠿⠞⠀⠒⠒⠒⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠄⠄⠈⠀⢸⠀⠠⡱⠀⠀⠀⠀⠀⠘⡒⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠒⢀⣤⣴⢠⢸⠀⣰⠶⠶⣿⡿⡟⣛⡛⡟⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠙⣳⣿⣿⡟⣾⡠⠀⣿⠐⡋⡈⠄⠀⠀⠏⣇⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⢠⠜⠢⡄⣠⣿⣿⣿⣷⣿⠃⠀⣿⡠⠈⠀⢀⡀⠐⠤⡿⠀⣷⣀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⡇⡎⡃⠸⠿⣿⣿⣿⣿⡇⠀⠀⣷⠉⠈⠄⠀⠂⠂⣀⡯⠀⣿⣏⣀⡀⠀⢁⣶⠦⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠰⣄⣠⡟⠀⣽⣿⣿⣿⢰⠃⠀⠛⠖⢺⠿⠛⠛⠩⠀⠀⠀⠿⠻⢍⡁⢈⡁⡂⣳⣤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⢻⣿⣿⡏⣾⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠱⡂⠛⠉⣑⠆
⠀⠀⠀⠀⠀⠀⠀⠘⡿⡄⠀⠸⣿⣿⣽⡇⢸⠀⠀⠀⠘⣠⣄⣰⣮⣤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣉⠳⢞⠃⠄⢿⣿⣷⠿⠞⠛⢻⣿⠟⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠬⢾⠈⠀⢸⢸⣿⡟⠀⠀⠀⠸⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠼⠧⠛⠃⠘⣸⠃⠀⠀⠀⠀⡦⣄⡠⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠲⡤⠄⠐⠇⢸⣤⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⣈⠆⣳⡶⠶⣧⠊⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣤⣤⠇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⢀⡀⠀⠀⣀⣤⣄⣀⣀⠀⠀⣀⡇⡏⡏⠀⠀⠀⡿⠀⠀⢀
⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃⠀⠀⠀⣧⣴⠖⠋⠉⠁⣐⣦
⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣓⠈⠤⣄⣠⠤⠞⠻⠿⢦⡖⠯⢉⡋
⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣠⡤⠖⣲⠗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣽⣛⣿⢂⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠇⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠒⣛⠒⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⢀⡀⠀⠘⠀⢀⡄⠀⠀⠱⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠐⠀⠀⠈⠁⠀⠜⠀⠈⠁⠀⢀⠆⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠲⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠑⢼⡀⠀⢹⠀⠀⢸⠀⠀⢸⠅⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠠⠼⠄⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⢹⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢣⠀⣀⡄⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⡇⠀⠀⠀⠀⢀⡇⢸⣀⠀⡝⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠚⢄⣠⠞⠉⠀⠘⠺⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⡇⠀⠀⠀⠀⢸⠗⠃⠀⠉⠢⣄⡠⠓⡀
⠀⠀⠀⠀⠀⢠⣁⠀⡜⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⢧⠀⣀⡄
⠀⠀⠀⠀⢀⠃⠀⡝⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠐⡀
⠀⠀⢰⠊⡉⢉⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⣉⠉⠑⡄
⠀⠠⠊⠈⠁⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣤⠀⢠⠄⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠁⠀⠈⠉⠓⢄
⠀⠀⠀⠂⠁⠉⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡇⠀⠀⠀⠀⠀⢸⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠉⠈
⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠚⡇⠀⠀⠀⠀⠀⢸⠓⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠲⠲⡇⠀⠀⠀⠀⠀⢸⠖⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⢒⠲⡇⠀⠀⠀⠀⠀⢸⠖⢒⣺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠀⠀⠀⠈⠙⢆⠀⠀⠀⡰⠋⠉⠀⠀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣦⣰⣦⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣤⣤⣽⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣷⠟⠑⠂⠀⠀⠀⠀⠀⣤⣀⡀⢀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣤⣎⣁⣠⣤⣤⠠⢬⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⣿⣩⢹⡿⣿⡟⢡⣾⣿⣫⣿⣥⣼⢿⣿⣿⣧⣀⢹⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⣿⣿⣿⡟⢻⣇⣿⣦⡿⣼⣿⣿⣾⣿⣿⣿⡿⣿⣛⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢻⣿⠉⣿⠀⢸⣿⣎⢿⡇⣿⣿⣿⣷⣤⣼⣿⣿⠉⢵⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⣿⣧⣿⣄⢠⣿⣿⣷⡇⣿⣿⣏⡻⠉⣿⣿⣿⡇⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣯⣿⣿⠛⢻⣿⣿⣃⣴⣶⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⢿⣿⣿⡉⢸⣿⣿⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣾⢿⣿⣤⣬⣏⣖⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⣿⠛⠛⠉⣿⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣿⣷⣿⣶⣶⣿⣿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⣿⡿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⢰⣦⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⣼⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠈⠢⠀⠀⠀⠀⠀⡀⡀⡠⠂⠀⠘⠁⠀⢀⡴⠤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠱⡀⠀⠀⠀⠀⢀⠈⠀⠘⡀⠀⠀⢸⠋⠹⠇⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⢢⣼⣄⡀⠀⠰⠆⠀⢸⡀⠀⡆⠀⠀⠈⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠗⢾⠁⠀⠈⢧⣠⣴⡦⠤⠾⠀⠀⣹⡄⢀⠔⢀⠀⠦⠸⣦⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟⠟⢿⠐⠀⠙⠢⣀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢰⠀⠀⢘⣦⣤⣴⣾⣿⣿⣿⣿⣿⣄⠴⣬⣧⡀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡼⠀⠂⠀⣸⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣄⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠒⠊⠀⣿⣇⣸⣦⣿⣿⣿⣿⣽⠿⠛⠛⠉⠀⠀⠀⢀⣾⣿⣷⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡅⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠙⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⡀⢀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⠌⢂⠀⢸⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣏⣍⣹⣏⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⡿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣧⢠⢱⢷⢦⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣷⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠒⠒⠓⠒⠒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡖⢲⡇⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⢻⠒⢦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠸⠀⠀⡄⢤⠀⠀⠀⠀⢀⠄⢄⠀⠘⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠳⠼⡇⠘⡀⠀⠀⠀⠒⠤⠤⠄⠂⠀⠀⠀⡰⠀⣼⠤⠞⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠤⠚⢁⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠓⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠢⠀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉
⣿⣿⣦⣀⠀⠀⠀⠙⠁
⣿⣿⣿⣿⣷⣄⢰⣿⣶⣾⣿⣷⡄⠀⢰⣦⠀⠀⠀⠀⠲⠖⠀⠀⣄⠀⠀⠀⢰⡆⣤⣐⢾⣷⣶⣶⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣠⣶⣿⣿⠗⠀⣾⠇⠀⠀⠀⠀⠙⠇⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⡄
⣿⣿⡿⠿⠿⣿⣿⣟⣿⣿⣿⣧⣼⣿⣿⣿⠀⠀⠀⠀⣤⢠⡆⠀⠀⠀⠀⠀⣰⡟⣿⣿⣿⣿⣿⡾⠁
⣿⣿⠀⠀⠀⠈⠙⢿⣿⣿⣿⠟⢸⡿⢿⠀⠀⠀⠀⢠⡿⠿⠀⣴⡶⠃⠀⠀⠛⣿⢠⣿⣿⣿⡇⢠⡆
⣿⣿⣷⣶⣶⡆⠀⢸⣿⣿⣿⡀⢸⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣾⣷⡀⠀⠀⣴⣿⢻⣿⣿⣿⡗⣼⡇
⣿⣿⣿⠟⠛⠓⠶⠛⠿⠀⠀⡧⠀⠁⠀⣀⠀⠀⢰⡿⠀⠁⢠⣾⣿⠁⠀⠀⢻⣟⣶⡟⢻⣿⣷⣾⡧
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡆⠀⠛⡿⠒⠛⠳⠀⠀⠀⢻⣿⣤⣤⣤⣼⡿⣿⡿⠿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣿⣿⣿⣷⠶⡶⢦⣤⡔⠂⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣿⠃
⡀⢈⣿⣿⣿⡿⠟⠻⣧⣀⣨⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⢙⣿⣤⣤⣤⣤⣠⣤⣄⣠⣀⣀⣀⣀⣀⣀⣀⠀⣀
⣀⡀⠛⣿⡿⠛⠛⠻⣿⣯⣍⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡼⠛⣿⠁
⡈⠁⢠⣿⠀⠀⠀⠀⣿⡏⠉⠁⠀⠀⠀⠉⢿⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⣿⣿⣿⣿⣷⠄⠠⠞⢯⣀⡀⠀⠀⠰⢄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⠉⠀⠈⢻⣿⠠⣾⠉⠁
⣿⣿⣿⣿⣿⠀⠀⠀⢈⣿⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣤⣿⣧⣤⣀⣤⣼⣿⣦⣤⣶⣤⡀
⣿⣿⣿⡿⠛⠀⣤⡤⠈⠁⠀⠀⠀⣤⠈⣦⠰⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠉⠘⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣉⣽⣦⣾⣿⣿⣿⣿⣟⠉
⣿⣿⣿⣷⣤⣤⣄⢀⣰⡿⠿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠿⢿⣿⣿⣿⣿⣿⣍⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣀⡀⠰⣿⣿⡟⠿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣅⣽⣿⣿⣿⣿⡿⠛⠃
⣿⣿⣿⡟⠉⠀⠀⠀⢀⣿⣿⠆⠀⠙⠛⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣿⡟⠋⠀⠀⠁
⣿⣿⣿⣇⠀⣰⣶⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⠄⠀⣀⣠⢀⣤⡀
⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⣤⠀⠀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣶⣼⣿⣶⣿⣿⠁
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢘⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠠⠉⢻⣿⣿⣿⡟
⢿⣿⣷⣄⠀⠀⠀⠀⢲⣄⣠⠄⠀⠀⠀⠀⠀⠈⠡⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣏⣄⣈⣂⠀⡇⠾⢿⢹⣿⡇
⠀⠉⠉⣿⠆⠀⠀⠘⠋⠉⠻⠀⠀⠋⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢒⣀⢸⣿⠉
⠄⠀⠀⠋⠀⠀⠀⢰⣦⣤⣾⡀⠀⠀⠾⠀⠀⢸⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣟⣻⣯⣸⡷⠀⠀⠤
⠀⠀⠀⠀⠀⠐⠒⠛⠛⣿⡟⠁⠀⠀⠀⠀⠀⠸⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣸⠇⡇⣿⣿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠆⠀⠀⠀⠀⠙⠀⠀⢠⣦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⠋⠁
⠀⢀⡄⠀⠀⠀⣘⣸⣿⣻⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠀⠈⡥⠈
⣾⠟⠀⠠⠀⠀⠀⠈⠉⠈⠉⠀⠀⠁⢀⣠⣶⡾⢿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠐⠂
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠛⠛⢶⣷⣿⣿⣬⣦⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣴⡃⠀⠐⠂⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⢋⣿⡷⠀⢰
⠭⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⠛⠋⠉⠛⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⠀⣤⠤⣴⡟⠉⢙⣶⠾⠛⠲⣄⠀⣾⠟⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣁⣀⣀⡀⠈⠓⢾⣇⣀⣿⠁⠀⢸⡇⠀⠀⠀⠘⡄⣿⣀⣤⠴⠗⠒⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠚⠉⣥⣶⡍⠙⢷⣄⠀⠉⠉⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⢻⠉⢀⠴⠞⢛⣛⡛⠶⣌⣆
⠀⠀⠀⠀⠀⠀⢀⡟⢀⣤⡀⠙⠿⠃⣀⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢿⣿⠇⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⣸⠁⠻⣿⠏⠀⠀⠐⣿⣿⠀⡇⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡇⠀⠀⢠⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢦⡀⠀⣾⣿⡆⠀⠀⠘⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠁⣴⣶⡀⠙⠋⠰⢻
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⠲⠾⠭⠤⣠⠖⣂⣀⡀⠀⣷⠀⠀⠠⠈⠠⡀⠀⢀⠤⢒⡂⠙⠦⣄⣻⠟⢁⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⡾⠉⠉⢻⡄⣿⡆⠀⠀⠀⠀⠘⣆⠀⡼⠋⠙⢷⠀⡄⠀⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⡾⢡⣏⣿⡄⠀⠀⠀⠔⠉⠳⣷⣀⣀⡾⢡⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⠲⠬⠤⠶⠋⠀⠈⠻⣤⡀⠀⠀⠀⠀⠙⢏⣩⣴⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠈⣧⡈⣿⣿⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢀⡜⠀⢿⣇⠀⠀⠀⠀⠀⠀⣰⣿⢿⡇⠀⠀⠀⠀⠚⠁⣿⠏⢿⣷⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡤⠴⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⠃⠀⢻⡀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣄⡀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡄⠀⠢⠠⠂⡱⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⣶⣶⣶⡾⠇⠉⠁⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊⣯⢉⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣤⡐⠄⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⢸⡀⣅⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠘⡀⢾⣯⠇⡀⠀⠀⠀⢸⡼⠟⠀⠂⠀⠀⢸⣷⡏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣷⠀⠀⠈⠒⠐⠊⠀⠠⠤⠄⠀⠀⠈⠀⠀⠀⠀⢸⡟⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⢄⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠠⠀⢸⣇⠌⡋
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠑⠠⢄⣀⠀⠀⠀⠀⠀⢀⡀⠔⠁⠀⢸⡕⢩⡏
⠀⠀⠀⠀⠀⠀⠀⣀⡆⡀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠤⠀⠀⠀⣀⣉⣉⣉⣉⣁⠀⠠⠤⠤⠼⠷⠒⠁
⠀⠀⠀⠀⠀⠀⢩⠀⠤⣧⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣙⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠾⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠒⠚⠒⠒⠛⠛⠓⡶⢤⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢶⡾⠧⣮⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠁⣄⣓⡶⠒⠂⠢⠶⠦⠶⢿⣶⠗⣦⣤⠜⠒⠒⠂⠠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣆⠀⠀⠀⠀⣀⠔⠈⡀⢠⠙⠂⣿⠘⠀⠀⠀⡐⡑⠡⢀⢺⣿⠒⢫⢷⣄⡀⠀⠀⠠⠀⠀⠀⠁⠁⠂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠐⠈⣆⠀⡠⠊⠀⠄⣈⠖⢻⠀⠀⣿⠀⠀⠀⠀⡜⡔⠀⢢⢻⣿⠀⢸⡢⠉⢽⠑⠓⠳⠦⠦⣤⣁⣈⣶⣧⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡼⢏⢀⢀⡼⠊⠁⠀⢸⠠⠄⣿⡀⠀⣀⣚⣀⣈⢀⡡⣸⣿⣀⢸⡪⠂⣽⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⡤⠖⠋⠀⠀⠀⠀⠈⡀⠀⠓⠒⢀⡛⠛⠛⠛⠛⠻⠻⠏⠀⣸⡈⣂⡍⠀⠀⠀⠀⠀⠀⣰⠃⢀⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘⠃⠀⠀⠀⣿⡿⣿⠇⠀⠀⠀⠀⠀⢰⡟⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣂⢁⠆⠀⠀⠀⠀⠀⢀⡟⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⣂⠤⠄⠐⠛⢚⠛⡋⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠞⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢾⣶⣶⠀⠀⢸⣶⣷⣷⣶⠀⠀⠀⠀⠀⢀⣠⣴⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⡯⠀⠀⠈⠀⠈⡉⡇⠀⠀⠀⠀⢠⠋⡠⣄⠙⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢁⣿⠀⠀⠈⠀⠨⢪⡇⠀⠀⠀⠀⠀⠉⠀⢀⢠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃⣿⠀⠀⢠⠀⠨⢪⡇⠀⠀⠀⠀⠀⠀⡠⢊⡼⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣽⠀⠀⢸⡀⠸⡘⡇⠀⠀⠀⠀⠀⠀⠑⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠈⠉⡆⢀⡎⠉⠉⠓⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠃⠈⠓⠶⠒⠒⣧⡜⠓⠠⠤⠤⡴⡾⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠠⠐⠠⠐⠦⠂⠔⠈⠐⠉⠐⠡⠛⠖⠶⢶⠧⠶⢶⠏⡧⢤⣼⣬⣤⣥⣽⡫⠐⠬⠊⠅⡊⠑⠪⠐⠁⠊⠕⠊⠒⠘⠂⠑⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⢰⣟⠉⢹⣷⣤⣄⠀⠀⠀⠀⠀⣤⣤⣾⠋⠙⣿⡄⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⣀⠈⠻⠶⠿⠃⠈⣿⠀⠀⠀⠀⢸⣿⠀⠹⠷⠾⠟⠀⣀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⡟⠛⣷⣤⡀⠀⠀⢀⣿⣰⣷⣀⣿⣸⣿⠀⠀⠀⣀⣴⡟⠛⢿⡆⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠻⣷⣴⡿⠉⣿⣤⣼⡟⠛⠛⠛⠛⠛⠛⠛⢿⣦⣼⡏⠹⢷⣴⡿⠃⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⡇⢰⣶⠶⠶⠷⣶⠀⢸⣯⣁⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢈⣉⣿⡇⢸⣿⠀⠀⠀⣿⠀⢸⣏⣉⡁⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢈⣉⣿⡇⠘⠿⠶⠶⠾⠿⠀⢸⣟⣉⡁⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣴⡟⠻⣷⣤⣿⠋⠻⣧⣤⣤⣤⣤⣤⣤⣤⣾⠟⠛⣿⣆⣠⣾⠛⢿⣆⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠻⣧⣴⡿⠋⠁⠀⠀⠈⣿⠈⣿⠈⣿⢸⣿⠀⠀⠀⠈⠉⠙⣿⣤⣾⠏⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣾⠟⠻⣷⣄⣠⣿⠀⠀⠀⠀⠸⣿⣀⣀⣀⣾⠟⠻⣷⠀⠀⠀⠀⣰⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠿⣦⣴⡿⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣦⣴⡿⠀⠀⠀⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠋⡉⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡧⢧⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡞⠉⣉⣉⠉⣉⣉⣀⣀⣉⣉⣉⠉⠉⣉⣓⢢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠤⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠤⣤⠀⠀⠀⠀⠀⡀⠤⣄⠀⠀⠀⠀⠀⠀⡤⠂⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁⢰⡢⠀⠀⠀⠀⠀⠘⠠⡤⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠠⠴⠊⠀⠀⠀⠐⠀⠠⠎⠎⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠨⡀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢲⠀⠠⠀⠠⡄⠀⡖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡼⠿⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⢀⢠⡀⣸⠀⢀⡇⢀⣇⣀⣀⠀⠀⠀⠀⢀⡀⠠⠐⠈⠈⡆⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⣀⠴⠋⠀⠑⠂
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢀⣀⠠⠴⠚⠁
⠀⢠⠃⠀⠀⠀⢀⣠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⣜⠀⠀⠀⡰⠊⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣠⠌⠫⠼⢿⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢧⣤⠊⣦⣀⠇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠋⠤⠤⠤⠄⠀⢀⣀⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠂⠀⠀⠀⠂⡶⠶⠶⠤⠤⠤⢤⡄⠠⠤⠞⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⣸⠀⠀⠀⠀⠀⢀⡎⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠀⠐⠚⠑⡄⠀⠀⡠⠚⠓⠠⠼⢻⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⢸⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣠⣶⡾⠛⠛⢻⣿⣶⣦⣄⣀⣀⣀⣀⣀⣰⣿⣿⣁⣀⣀⣻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣠⡾⠿⣛⣛⣛⠻⢭⣍⡉⠉⠙⠛⠿⣿⣿⣿⣆⠀⣼⣿⣿⣷⣿⣿⡿⠛⠉⢉⣩⠭⢟⣛⣛⡻⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⣾⣭⣽⣭⣻⣿⣮⡙⢮⡲⣄⣀⣈⣛⣿⣿⣿⠋⠉⢁⣀⣀⣀⠀⢀⠔⠉⠀⢰⡏⣩⣭⡙⣾⣷⣽⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⢳⣽⣀⣀⣀⣿⣟⣿⠀⣾⣇⣀⣀⡼⢠⠏⠀⠀⠀⢸⡟⢿⣿⢟⣿⣿⣿⣷⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡇⡞⣻⣿⣿⣿⡙⠿⠛⣹⣿⣿⣿⢳⡌⣇⡿⠿⣿⣿⣿⣿⠀⠈⡹⠿⢿⡁⡏⣼⠛⣻⣿⡛⢃⡤⢤⡘⢛⣿⣟⠛⡆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡄⡇⣿⣿⣿⣿⠇⣀⡘⣿⣿⣿⣿⢸⡇⡟⣳⣶⣾⣻⣿⣿⡀⠀⠳⠶⠾⠃⡇⣿⠸⣿⣿⠇⠘⢄⡠⠇⠺⣿⡿⠀⡇⣼⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣧⢹⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⠿⣋⣀⣙⣛⠿⣿⣿⣿⣿⣷⡄⠀⡠⢖⣚⣁⣈⠻⢿⣿⣽⡇⣤⣤⣰⢿⣿⣿⣿⢡⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⡻⣿⣿⣿⢿⣿⣿⢿⡿⣱⡾⠛⠁⠀⢨⣷⣌⢧⣹⣿⡿⢣⢞⣴⡿⠋⠀⠀⢩⣶⡹⣿⡟⢿⡿⢿⣿⣿⡿⣫⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣦⣍⣙⠓⠒⠒⣋⠀⡏⠀⠀⠀⣰⣿⡟⣻⢸⡿⣿⠀⣾⢸⡏⠀⠀⢀⣴⣿⠟⡇⢈⡛⠒⠒⢚⣉⣭⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠈⠙⠻⠿⣿⣿⣿⣿⡆⢿⣦⣠⣾⡟⣋⣴⡿⣸⣿⣿⣿⣿⡸⣷⣄⣴⣾⣟⣫⣼⠇⣾⣿⣿⣿⠿⠿⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⢻⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⡙⠿⠿⠿⠿⢋⣴⣿⡟⠛⠛⣿⣷⣌⠻⠿⠿⠿⠟⣩⣾⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⡿⣾⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⡟⠛⠶⣶⣿⣿⣿⠏⠀⠀⠀⠈⢿⣿⠛⠳⢶⣾⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⢹⣿⣿⣿⠸⡇
⠀⠀⠀⠀⢸⡇⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣻⠏⠀⠈⠛⠳⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠾⠿⠛⠉⠀⠈⢷⡹⣧⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿
⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⣸⣿⡟⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠃⠀⠀⠀⠀⠘⣿⣿⡇⣿
⠀⠀⠀⠀⠘⢧⡈⠓⠀⠀⠀⠚⣛⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠚⠛⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠈⠀⠀⢀
⠀⠀⠀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⣀⡀
⠀⢸⣾⣇⠀⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⡿⡇
⠀⠀⠙⠻⣧⣾⣣⣺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣼⣿⣿⡿⣡⣇
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣟⠦⡀⠀⣀⠀⠀⠀⠀⠰⣾⡛⣿⣿⡟⠱⣿⣉⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠛⢻⣶⣿⣶⣿⣷⣦⣴⣣⣤⣬⣿⣿⣿⣷⣄⠣⠙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠙⢷⣴⣟⣿⣿⣿⣿⣿⣾⣿⣻⣿⣿⣟⢻⡀⠀⢙⣿⣿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣽⣿⡟⠉⠉⠛⠻⢿⣿⣿⣻⣦⣈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣿⣿⣿⣿⣿⠋⠀⠀⢀⣴⣾⣿⣄⣀⣀⠀⠀⠀⠀⣴⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⣦⣄⠈⠀⠀⠑⣰⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⡹⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣦⣤⣀⠝⠻⡻⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣨⣿⡆⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡻⠶⣌⣩⣾⣋⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⠃⠀⠠⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⢲⣾⣿⣟⣣⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠁⠀⠈⠉⠉⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠀⠘⠿⣦⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠉⠀⣠⣴⣶⣿⣿⣷⢦⣄⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣶⣾⡟⣻⣶⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠬⠥⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡿⣻⣷⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⣤⣄⣀⣀⣼⣷⡄⠀⠀⠀⠀⠀⠈⠙⠻⢿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⢩⣥⣬⣿⣦⡀⠀⠀⠀⠀⠀⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠀⠈⠛⠁⠈⠃⠐⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣩⠆⠀⠀⢀⣀⣤⣤⡤⠤⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡁⣤⣄⡀⢀⢠⣤⣄⡀⠈⡆⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⢌⠁⠛⠛⣺⠋⠈⠿⠷⢿⡄⢼⠀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⠁⠀⡏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⡄⣼⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠀⠀⡇⢠⡖⠒⠲⠲⠦⠤⠤⢤⡄⠀⡇⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡇⠈⢧⡀⠀⠀⠀⠀⠀⣰⠇⢸⠃⢿⣀⣀⣠⡾⠁
⠀⠀⠀⠀⣠⠴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣉⡛⠲⠶⠶⠚⠋⠀⢸⠀⠈⠉⠉⡏
⠀⢠⠚⢣⡘⣄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢋⠉⠉⠙⠛⠛⠛⢶⡶⠿⠖⠲⣟⠛⠁
⣠⢦⠳⣄⠱⣘⡦⠗⢳⣀⣠⡄⢠⣄⣀⣀⣀⣀⣴⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠈⢧
⠱⣄⠳⣬⠖⠉⠀⠀⢸⡧⠤⠃⡟⢸⠀⠀⠀⣸⠃⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢠⡟⠉⠉⠋⠉⢳⡤⡄
⠀⠈⢻⠃⠀⠀⠀⠀⢸⡇⠀⠀⡇⢸⣦⣤⣤⡟⠀⠀⠀⠀⠈⠀⠀⠠⠐⠦⢦⡀⠀⠀⢿⠘⣧⣤⣤⣠⣀⣼⠀⡇
⠀⠀⠀⠙⠦⣀⡀⠀⣸⡀⠀⠀⡇⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠙⢧⡑⢄⠀⡇⡿⠀⠀⣸⠀⠀⠀⠀⠀⠒⠀⠀⠘⠶⠧⠼⠁⠀⠀⢸⠀⠀⠀⠈⡇⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢳⣇⡇⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠂⢐⠲⠰⢠⠀⠀⣾⠀⠀⠀⠀⡇⠀⡏⢠⡇⠀⠀⠀⠀⠀⡖⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠗⠀⠀⡟⠀⠀⠀⠀⠀⠄⠈⠉⠑⠒⠛⠒⠘⠀⠀⣿⠀⠀⠀⠀⡇⠀⡇⢸⠇⠀⠀⠀⠀⠀⣹⡬⠗⠚⣷⠒⢂⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⠄⢠⠀⠀⠀⠀⡀⡏⠀⠀⠀⢸⡇⢰⡏⢻⡒⠒⠒⢒⣲⠋⠁⠀⠀⠀⣿⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠀⠀⠀⢸⡇⠀⠙⠉⠉⠉⠉⠉⠉⠳⢤⣀⠀⠀⣿⣉⣙⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠗⠂⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡰⠛⠒⣏⠀⠈⠉⠉⠉⠛⠛⠓⠲⠶⠶⢶⠤⠼⠧⠤⠴⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⣤⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠋⠉⠉⠉⠉⣷⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⣤⣀⣀⣀⣀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣿⠀⠀⠀⠀⠈⠙⠲⠤⠤⣤⣤⣤⠴⠥⠴⠖⠚⠉⠀⠀⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣇⢸⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠳⠤⣀⣉⠉⢀⣠⠴⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠈⠙⡏⠀⠀⠀⣸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠲⡇⣿⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⢮⣁⠀⠉⠒⠤⣄⣀⣇⣠⠔⠋⠀⣀⡽⠃⠀⠀⠀⠀⠀⠀⠀⡾⠉⠒⠦⣤⣀⡓⢋⣀⣤⠤⠞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡀⠉⢯⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⢙⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠙⠒⠤⣄⣀⠀⡇⢀⣀⠤⠒⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⣀⠈⠙⣿⠉⠀⣀⣤⠶⠋⠁

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡻⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⣾⣿⡀⠙⢿⣿⣿⣿⣿⡽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣇⢀⣿⣿⣯⣀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⡿⣆⠀⢿⣿⠀⣠⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⣽⣿⣿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⣾⢹⣿⣿⣮⡻⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⡏⣧
⠀⠀⠀⠀⢠⡏⣾⣿⣿⣿⣿⣶⣽⣿⣿⠟⣛⣟⡻⣿⣿⣿⣿⣿⣿⢟⣻⣛⠿⣿⣿⣫⣶⣿⣿⣿⣿⣷⢻⡄
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⢹⣿⣿⣿⣿⡏⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣮⣛⣛⣵⡿⠋⠉⠉⠙⢿⣮⣛⣛⣵⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷
⠀⠀⠀⠀⢻⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠻⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⢸⣇⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⣸⡏
⠀⠀⠀⠀⠀⠻⣦⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠛⣿⠀⠟⢨⡷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠏⠸⡏⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣧⣀⣀⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡤⠶⢶⣄⢠⡇⠀⠀⢸⠁⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⡏⠀⠀⠀⢹⣿⡇⠀⠀⣿⠀⣴⠛⠛⠛⢦⠀⢻⡄⠀⠀⠀⠀⠀⠀⢿⣄⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠻⢦⣤⡤⠟⣹⠁⠀⠀⡿⠘⣇⠀⠀⠀⣸⠇⠈⣷⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠛⢳⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠒⠶⠶⠤⠤⠤⣤⣿⣤⣤⣤⣇⣀⣈⠓⠒⠚⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠓⠿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠉⠉⠁⠀⠛⠛⠓⠒⠒⠀⠤⠶⠦⠤⡤⠀⠾⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⣶⠠⠀⣶⡶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠴⢶⣶⠄⡄⠀⢸⠇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢰⡟⢲⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⡏⢉⡿⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣾⠀⣰⠀⢿⣧⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⢸⣿⠀⣇⣸⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠏⠉⠩⠀⠸⣿⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⣾⠇⠀⡏⢿⣤⣤⠴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡟⠀⠀⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣸⠃⠀⠀⢠⠀⠀⠀⠻⣷⡄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⣼⠟⠀⠀⠀⡇
⠀⠀⠀⢰⡇⠀⠀⠀⢠⡟⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⣦⣀⠀⠛⢦⣄⣀⣀⣀⣠⡴⠛⢀⣴⠟⠁⠀⠀⠀⠀⡇
⠀⣠⠶⠾⣤⣀⣀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣉⣉⣉⣤⡴⠾⠛⠁⠀⠀⠀⠀⠀⠀⡇
⣾⠁⠀⠀⠀⠀⠉⠉⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⣀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⣧⠀⣿⠉⠛⢳⡆⢀⡾⠀⠀⠀⠘⠒⢰⠆⠒⢶⡒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⣶⠖⠒⠒⠒⠒⠒⣶⠒⠴⠃
⠀⢻⣀⣹⡆⣠⠏⠀⡾⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠉⠉⠉⠛⠲⠾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⣄⣀⢀⣿⢦⣄⣀⣤⣾⡀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⣤⣤⣤⣤⣌⠉⠙⡏⠀⠀⠀⠀⣾⠉⣹⠧⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠾⠁⠀⠀⠀⠘⠧⣴⡃⠀⠀⠀⠀⣽⡴⠋⠀⠀⠀⠀⠻⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣤⠴⠖⠲⠶⢤⣀⠀⣿⠀⠀⠀⢸⡇⣀⣤⠴⠖⠶⠦⢤⣀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠠⠄⡄⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠘⢄⢨⠀⠀⠀⠆⠀⡄⠐⠊⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⠀⠐⢕⢀⢀⠜⠀⠀⠀⠀⠀⠀⣚⢒⢔⢑⢝⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠲⠀⠰⠁⢑⢀⠐⠠⡂⠠⡃⢠⢔⠅⠑⢵⠀⠀⡠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠦⠆⠆⠆⠖⠠⡀⠰⡈⠰⠄⡇⠄⢈⠒⠁⠀⠰⡊⡀⡊⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠑⠁⢄⠀⠀⠈⠪⢀⠪⢬⠅⡆⠀⢽⠀⡦⠠⠌⠈⠈⠁⢀⠔⠅
⠀⠀⠀⠀⠀⠀⠀⠘⢪⠯⡧⡄⡄⠄⠀⠀⠀⠈⠈⢁⠀⡄⠀⠀⠑⢤⢀⠀⠈⢠⠀⡅⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣐⣂⣀⣀⣀⡀⠄⢐⢂⢀⢠⠀⠀⠂⡂⠀⣖⢄⠠⢀⣀⢀⣠⡀⠁⠣⡂
⠀⠀⠀⠀⠀⠀⠀⠐⣒⠒⠂⠀⠀⠂⠠⡁⢐⢕⠕⠀⠀⡇⠈⠂⠒⠀⠑⠀⠀⠀⠀⡀⣊⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢆⠆⠆⡈⠢⡀⠨⡀⠰⠀⠐⠀⠀⣆⠆⢀⢴⠰⠠⠔⠄⢆⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠑⠔⠡⠀⠬⠢⢈⠀⠀⠀⠀⠀⠡⠀⠭⠨⡠⠪⡆⠑⢕⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣔⠑⠤⠀⠤⢑⠀⠀⠀⠀⠇⠀⠀⢽⠄⠘⠈⢢⠨⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡀⣚⢐⢀⠘⡀⠲⡃⠠⡐⠀⠑⢅⢄⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⢠⢐⡂⠨⣢⡈⠒⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣾⡷⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣷⣶⠤⠄⠞⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠄⠀⣀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠛⠻⣿⣿⠀⣾⡟⠻⣷⣼⡷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣼⣿⣿⠀⢿⣧⣴⡿⢻⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⡻⢟⡛⢠⠤⠠⠤⢶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣾⡇⠤⣤⣔⣲⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⣴⣾⣿⣶⣿⣿⣿⣷⣦⣽⣿⣿⣷⣿⠓⠲⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣀⣀⢰⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠏⠡⠂⠀⡆⣀⣀⣿⡜⡇⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⢿⡛⠛⠉⠀⠀⠀⠀⣀⣿⣿⣿⣿⣧⢸⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠙⣿⣿⣿⣿⣿⢘⡄⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣴⣶⣶⣶⣶⣶⣶⠎⠇⣠⡀⠀⠘⣿⣿⣿⡟⢸⡇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⠈⠓⠃⠠⢀⡟⢻⣿⣧⣼⣧⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⢀⢀⣽⣇⠹⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⣿⣿⡃⢸⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣄⠀⠀⢀⣾⣿⣿⡧⠘⣿⡿⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⠿⢿⡻⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠙⠦⠀⣾⡟⣿⡿⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠈⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠉⠀⠀⠉⣹⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣭⣭⣿⣿⣿⡏⠀⣹⣾⡁⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡁⠀⣹⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢷⣄⣀⣴⠟⠛⠛⠛⠛⠻⣦⣀⣠⡾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⣀⠀⣀⣀⡀⣀⡀⢀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⢀⡀⠀⡀⠀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣿⢸⡏⢹⡇⣿⠽⣇⢹⡇⣿⠀⠀⣿⠭⠁⠀⣾⠉⣥⢀⣟⣷⠀⡿⣦⣿⡇⣿⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠙⠈⠓⠛⠃⠛⠛⠋⠛⠃⠛⠛⠃⠛⠛⠃⠀⠙⠓⠛⠘⠉⠙⠃⠃⠀⠘⠃⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠛⠳⢦⡄
⠀⠀⠀⢀⣠⡶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⡆
⣼⠃⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠛⠛⠋⠉⠉⠀⠸⣇
⣿⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣤⠀⠀⠀⢸⡆⢸⡆
⢸⡇⠀⠀⠀⣀⣤⣤⠿⠀⠀⠀⠀⠹⠂⠀⠀⠈⠁⠘⣧
⠀⢷⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿⡀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠶⠶⣤⡀
⠀⠘⢷⡀⠈⠷⣤⣄⠀⠀⠀⠀⢀⣤⡴⠞⠛⢉⣿⠀⢸⡇⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠙⠢⣤⣤⣿⡀⠀⠀⠀⠀⠹⢦⣤⣤⠞⠃⠀⣿⠀⠀⣼⠃⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠈⢿⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡟⠀⠀⠀⣾⠁⠀⠀⠀⢻⡆⠀⠀⢸⡇
⠀⠀⣠⠀⠀⠀⠀⠀⣠⠿⠳⢦⣤⣤⣤⣤⡴⠾⢯⡀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⣀⡾⠁⠀⠀⢸⡇
⠀⣼⠋⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⣦⡀⠀⠹⣆⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣠⡟⠀⠀⠀⣸
⢰⡏⠀⠀⠀⢠⡟⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠷⣤⡀⠘⣧⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠉
⣼⠁⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡀⠀⠀⠀⠈⢹⡟⠛⠛⠛⢻⡍
⣿⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣨⣿⣄⣀⡀⠀⢸⡇⠀⠀⠀⠸⣇
⠉⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⡀⠀⠀⠀⢀⣴⠏⠀⠀⣴⠏⠁⠈⠉⠙⢷⣼⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⢦⣄⣠⡾⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣰⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠙⢷⣶⣤⣀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇⠈⠉⠛⠳⠶⣼⣧⣀⣤⠾⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠶⠶⠶⢾⡃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣫⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣭⣽⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⡿⠋⢀⠀⣀⠀⠉⢻⣿⣿⣿⣿⣿⡿⠋⠀⠀⣤⡀⠈⢻⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⡇⣿⣿⡇⠀⣧⣴⣿⣿⡆⠀⣿⣿⣿⣿⣿⡇⠀⣾⣤⣾⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⡇⣿⣿⣇⠀⠻⢿⣿⠿⠃⢠⣿⣿⣿⣿⣿⣇⠀⠙⢿⣿⠿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⡇⣿⣿⣿⣦⣄⡀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣼⡇⢸⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣤⣾⣿⣿⡻⣦
⠀⠀⠀⠀⢸⣿⣿⣁⣽⣿⣿⡇⣾⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣯⣨⣷⣼⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⢿⡇⣿⣿⡇⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⢸⣿⣿⣿⠙⢿⣿⣿⣿⣟
⠀⠀⠀⠀⢰⣟⣿⣿⡟⠀⢸⡇⣿⣿⡇⠀⢀⡴⣤⡀⠀⣤⣤⠀⢠⠶⣦⡀⣴⢶⣄⢠⡶⢦⡀⢸⣿⣿⣿⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⡾⣼⣿⡿⠀⠀⢸⡇⣿⣿⡇⠀⠀⠻⠟⠁⠀⠛⠛⠀⠘⠿⠟⠁⠻⠿⠋⠘⠿⠟⠁⢸⣿⣿⣿⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⢸⣷⣿⣿⠃⠀⠀⢸⡇⢻⣿⣿⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣇
⠀⠀⢀⡿⢿⣿⡏⠀⠀⠀⢸⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⠿⢿⡄
⢀⣴⣿⡷⢶⣿⣦⡀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⢀⣴⣿⡶⢾⣷⣦
⣾⣿⣿⣤⣤⢿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣤⣴⢿⣿
⠻⣿⡇⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⢸⣿
⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣤⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⡾⣻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣾⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣷⣶⣤⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣿⣯⣿⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠭⠍⠭⠉⠭⠉⠩⠭⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡀⢰⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠡⣀⣠⣬⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡿⠿⡿⠟⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠁⠒⠒⠈⠃




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

⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣿⣿⡇
⠀⠘⢿⣿⣿⡟⠻⣏⡉⠉⠉⣉⠉⠉⢉⣩⡟⠹⣿⣿⣿⠏⠁⠈⢻⣿⣿⣿⠙⣯⣉⠉⠉⣉⡉⠉⠉⣩⡿⠉⣿⣿⣿⡏⠁
⠀⠀⠈⠛⠛⠁⠀⠙⢿⣦⣸⣿⣆⣴⡿⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠈⢻⣷⣤⣿⣧⣤⣾⠋⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠃⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣀⡤⠚⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠚⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠑⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠲⠆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢀⡀⠄⣀⡀⠀⠀⠀⢀⡠⠄⣀⡀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠀⢀⠋⢠⣤⠉⢿⡆⠀⠰⠋⣠⣄⠙⣿⡄⠀⢠⡶⠙⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠰⠄⠈⢦⡈⠋⣠⣾⠇⠀⠸⣆⡙⠋⣠⡿⠁⠀⠉⡇⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣤⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠙⣧⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣯⢸⣿⡅⣿⣧⣸⡇⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠉⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠟⠛⠻⣿⠛⠛⠛⠛⠻⣿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⢢⣤⡐⠒⢺⣇⠀⣘⣿⠗⢺⣆⠀⣘⣿⠒⠒⠒⠒⠲⡶⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢠⣶⡟⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠙⡿⠀⣰⣶⡄⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣀⠘⠿⡇⢸⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠂⣷⡀⠛⠟⠃⣠⡿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡌⢶⠉⢶⠶⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠁⢶⢶⡖⠉⣤⢡⡄
⠀⠀⠀⠀⠀⠀⠀⢇⡸⠀⢠⡎⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣯⠀⠸⠀⢱⡄⠀⠆⠱
⠀⠀⠀⠀⠀⢀⠻⡜⠃⢀⠏⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣏⠀⢰⠀⠀⠱⠀⠘⢃⠾⡀
⠀⠀⠀⠀⢠⡌⣶⠁⢠⠟⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢸⡆⣿⠀⢸⠀⠀⠀⠻⡀⠈⣤⢃⡄
⠀⠀⠀⠠⣆⡸⠀⢠⡎⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⣿⠀⢸⠀⠀⠀⠀⢡⡄⠈⢎⣱⡀
⠀⢀⡰⠛⠉⠓⢲⡆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⣿⠀⢸⠀⠀⠀⠀⠀⢱⡖⠊⠉⠛⢦⡀
⠀⠘⠁⣴⣿⣦⠀⠃⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⡇⣿⠀⢸⠀⠀⠀⠀⠀⠈⠀⣶⣿⡆⠈⠃
⠀⠘⣆⡈⠛⠋⡠⠇⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⠀⠀⠀⠀⠀⠰⣄⠙⠛⢁⣴⠃
⠀⠀⢹⠉⠀⢻⠁⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠙⠏⠀⠉⡇
⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠘⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⣏⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⡿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣸⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣇
⠀⢘⣋⣉⣉⣉⣉⣱⠀⠀⠀⠀⠀⠀⢀⡴⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣎⣉⣉⣉⣉⣹⠇
⢀⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⠁⢠⣿⣆⡀⠀⠀⠀⠀⠀⠀⡇⠀⣾⣿⣷⠀⢈⡆⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧
⣾⣿⠿⠋⠉⠉⠙⠻⣿⣧⠀⠀⠀⣰⠿⣆⡈⠛⠛⠉⠶⠆⠀⠀⠰⠾⢿⡀⠙⠛⠋⢀⣼⢷⡄⠀⠀⠀⣾⣿⠟⠋⠉⠉⠙⠻⣿
⣿⣿⡄⠀⠀⠀⠀⠀⢻⡿⠀⠀⣠⡿⠟⠛⣻⣶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⠃⢶⣶⣟⠛⠻⣿⡀⠀⠀⢿⡏⠀⠀⠀⠀⠀⢠⣿
⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣡⡶⢋⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠻⣦⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⢉⡴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⠾⢋⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠻⢦⣜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠳⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⣴⣶⡄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣤⣶⣆⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘⠿⠿⠃⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠻⠿⠟⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣤⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⢯⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣭⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠉⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡍⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠋⣸⡇
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠚⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠚⠛⠛⠛⠻⠂
⠀⠀⠀⠀⠀⣴⣿⣶⣶⣶⣶⣶⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣶⣾⣶⣶⣿⣷⡆
⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠗⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡟⠀⢢⠀⠀⡀⠒⢉⠄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣨⠁⠀⠚⠋⡁⠤⠤⣈⡹⠛⠀⠈⣅⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⡇⠀⠀⢸⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⢤⣤⣀⢀⡤⣤⣤⣤⣤⣤⣀⡀⣤⣤⠖⢁⠔⠁⠀⠀⡀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⠢⠤⡌⢹⡇⠀⢀⣀⠀⠀⣿⢩⠤⠄⠂⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠉⠁⠠⠤⠔⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⣀⣀⣀⣀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠤⠤⠀⠉
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠃⠠⠋⠉⡆⠀⠒⠚⠒⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠤⠤⠶⠶⠤⠤⠤⠤⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠑⢤⡀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⡀
⠀⠀⠀⠀⡄⠉⠀⢀⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠴⠦⡚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠜⠀⠀⠀⠀⠱⣌⠄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⣀⣬⡁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⢧⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⠉⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠞⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣾⣶⣾⣶⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠤⢀⢀⣀⣀⣀⣀⣀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣴⠚⣻⣿⣿⣭⢏⠍⡩⠋⢊⢞⢅⠝⠉⠉⡩⢒⢅⢷⡵⠟⡛⡿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⢻⠋⢻⡟⠉⣿⠛⠙⣶⠛⠛⠓⢄⢼⠉⢉⣹⣗⡯⠚⠛⠲⣖⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣾⡆⠈⠃⢠⣿⠀⠀⣿⠀⣺⣦⠘⣿⠀⠛⢻⡞⠀⣴⢲⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⢱⢳⠀⠀⣼⣩⠀⠀⣿⡀⠚⠋⣰⢿⠀⠿⢽⣷⡄⠉⠋⠀⡸⣻⣧
⠀⠀⠀⠀⠀⠀⣿⢸⢼⠖⡹⣋⠁⣓⠖⣡⠳⡶⠫⡵⠚⠖⢖⡟⠃⠹⠓⠲⢏⡀⢺⣿
⠀⠀⠀⠀⠀⡃⣿⡋⢡⠞⢋⣉⡈⢳⣊⣸⠋⠉⢷⢞⡏⠱⣬⡟⠉⢿⡏⣉⣹⡆⢹⣿
⠀⠀⠀⠀⠀⠀⣿⡑⡞⠀⣿⡞⠛⢾⣷⡏⠸⠆⠸⡵⡇⢤⠹⢠⠄⢸⡇⣉⣉⡆⢸⣟
⠀⠀⠀⠀⠀⢠⢸⠊⡼⢄⣈⠛⢁⣈⣿⣀⡶⢦⠀⢿⡇⡼⡆⣸⡀⣸⡇⠙⠛⣗⢘⣿
⠀⠀⠀⠀⠀⠈⠘⣌⣁⣀⣝⣴⢃⣀⡨⡡⢋⡉⣏⣩⣞⢅⡀⡄⣟⡕⣟⣑⠉⣱⣩⡟
⠀⠀⠀⠀⠀⠀⠑⠒⠒⠐⠀⠀⠀⠘⠛⢻⣿⣿⡿⣻⣿⣟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠓⣦⣤⣤⣤⡤⣤⣀⡀⠀⢀⡤⢤⣴⣏⡀⣀⣀⡴⠃
⠀⠀⠀⠀⠐⠂⠒⠒⠒⠒⠀⠓⠖⠶⠖⠲⠖⠖⠖⠴⠰⠲⠂⠖⠖⠖⠒⠒⠋⠑⠒⠐⠀⠀⠀⠙⣷⠿⠖⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠉⠉⠉⠙⠒⢠⣄⣶⠛⠉⠛⣌⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⡢⠄⡤⢿⡄⠒⢲⢆⠀⠀⠀⠀⢸⣏⡓⠒⠚⣃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢎⠕⢀⣭⠨⡅⠒⢨⠒⡱⡀⠀⠀⠀⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢶⠂⠀⠀⠀⢀⠁⠲⡄⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠢⠛⠟⠐⠈⠂⠊⠐⠑⠂⠙⠺⠻⠓⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢤⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄
⠀⠀⠀⣴⣮⣿⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⣯⣶⡄
⠀⠀⢸⠉⡇⢸⠀⣿⣿⣿⣿⣷⡀⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠠⠤⠤⠄⣰⣿⣿⣿⣿⠇⢸⠀⡏⢹
⠀⠀⣸⠀⡇⢸⠀⢸⣿⡟⣧⣴⣶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠷⢶⣶⣤⡟⣿⣿⠀⢸⠀⡇⢸⡀
⠀⠀⢳⡆⢡⠀⡆⠈⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⡏⠀⡆⢠⠁⣾⠃
⠀⠀⠀⠱⡈⣆⣰⣠⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣰⣀⡎⡰⠁
⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁