Weiss Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣴⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠠⠤⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠤⠤⠤⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣆⠀⢀⡀⣸⣀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⡀
⠀⠀⠀⣀⡀⡠⠀⢤⠀⠀⠀⢀⠔⠑⠒⠊⠑⠂⠐⠊⠑⠄⠠⠛⠢⠤⠜⠢⠔⠓⠠⠚⠑⠚⢢⡀
⠀⡠⠼⠀⠈⠁⠀⠐⠓⢄⣠⠃⢀⡀⠀⠀⢀⡀⠀⠀⡄⠀⠀⣀⠀⠀⢠⡀⠀⠀⣠⡀⢀⡤⡀⢑⣄
⠀⣇⠀⠀⢠⠒⠀⠀⣠⠋⠈⠁⠉⠈⠐⠒⠀⠈⠉⠁⠈⠁⠉⠀⠉⠈⠁⠈⠁⠀⠀⠈⠁⠀⠈⠁⠀⠳⡀
⠺⡁⠀⡠⠅⠀⠀⠐⣅⠀⡠⣄⠀⡰⠦⠀⠴⠆⣀⡠⠲⣀⣀⠴⢄⣀⡀⠢⢀⣀⠴⢧⣀⣀⡴⡀⠀⢀⠏
⡔⠁⠀⠀⠀⣆⡀⠀⠀⠉⢱⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⣀⠀⠀⠀⢀⠀⠈⠉
⠑⡦⠀⠀⠀⠀⠙⠀⠀⢀⠇⠀⠀⢀⠒⠐⡖⠀⡖⠈⠐⠈⠉⠆⠀⢠⣏⡦⠶⠩⠆⠀⠀⠸
⠀⠑⢤⡄⢀⣤⡤⢤⡤⠋⠀⠤⠀⢸⠀⠀⡦⠄⡗⠒⠒⠀⢰⠀⠀⢠⠔⠒⠒⠒⢦
⠀⠀⠐⠈⠇⠀⡀⠇⠀⠀⠀⠶⠀⠀⣄⣀⣇⣀⣇⣀⣰⣀⡸⠀⠀⢸⢸⠉⠈⡇⠘⠀⠀⢸
⠀⠀⢠⠀⡀⠀⡇⢀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢸⠰⠤⠴⠃⡄⠀⠀⢠⡀
⠀⠀⠯⠭⠭⢗⠃⠸⡀⠀⢀⠾⠒⠊⠉⠈⠉⠉⠁⠁⠈⠀⠁⢹⢆⢸⠀⠀⠒⠂⡇⣔⠚⠉⠹⢲
⠀⠤⢀⡴⡦⠾⠤⠤⠵⠤⠬⢆⠀⣀⢀⡀⢀⡀⢀⡀⢀⠀⢀⢐⣎⣸⣀⣀⣀⣀⣅⣹⣤⣀⠀⣌⡡⠤⠤⢴⠛⡤⠤
⠀⠀⠀⡄⣇⠀⠀⣠⠀⠀⠀⠀⠁⠀⠀⠈⠁⠈⠁⠉⠀⠉⠉⠁⠀⠸⡀⠀⠀⠀⠙⠢⣀⠀⠉⠀⢀⢦⡠⢞
⠀⠀⠀⣇⡦⣙⢺⠀⡇⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠉⠒⠒⠾⢸⡕⠻⠤
⠀⠀⠀⠈⠁⠀⢹⠀⣍⡒⠎⢸⢄⣀⡔⢱⠀⠀⢀⡴⡄⠀⠀⢠⢆⠀⠀⠈⡞⣀⠀⠀⠀⠀⠀⠀⢰⣸
⠀⠀⠀⠀⠀⠀⠨⠤⠂⠈⠃⢺⠠⢤⡅⢸⠒⠤⢼⠀⡧⠤⠤⢼⢸⡤⠄⣰⡆⡋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⢣⣸⠉⠉⠙⠆⠓⠂⠈⢹⢈⡉⠁⠀⣇⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠁⠈⠉⣳
⠀⠀⠀⠀⠀⠀⠀⡞⡟⠉⠉⠉⢲⠀⢸⢯⣛⣒⣚⣻⡟
⠀⠀⠀⠀⠀⠀⡴⣏⡓⠒⠒⢺⡿⠀⢸⡓⠲⠿⠟⣻⡇⠀⢀⣀⡤⢖⣋⡳⢄⡀
⠀⠀⠀⠀⠀⠀⣟⠦⠭⠭⠽⣿⠁⠀⠀⠈⢉⣩⣭⣷⣖⠚⠉⠁⠀⠉⠀⠈⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⣿⠥⠖⠒⠛⠉⠉⢠⡝⢿⣿⣷⣄⡙⢦⡀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡞⠳⣄⠀⠀⠀⠀⠈⢷⣄⠙⢿⣿⣿⣦⡃⠀⠀⣠⠴⣚⡥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣍⠢⡈⠑⢤⡢⣄⠀⠀⠉⠛⠒⠉⣻⣿⣿⣾⣿⣶⢿⣓⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡳⣕⢌⠳⣄⠙⠦⠭⢭⣶⣶⣿⣿⣿⣿⣿⡯⣗⣺⠽⠒⣫⡵⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣝⢮⡓⣒⣫⠭⣗⣫⢭⣖⣿⡿⠟⠋⠁⠀⠀⢸⡯⣥⣀⣀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣭⢵⣒⡯⠟⠚⠉⠁⠀⢀⣀⣀⠀⠀⠀⣯⣛⡲⢶⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⢠⡾⠟⠉⠉⠉⣳⠀⠳⠬⣭⡭⠽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⣓⠒⢒⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣀⠀⢠⣤⠀⠀⠀⣀⠀⠀⠀⣠⠰⠀⣠⠀⢀⡠⠆⣠⠄
⠀⠀⠎⢲⡚⠯⡤⢤⣍⣤⣰⣧⣡⣾⣥⣄⣩⣭⣤⣭⡤⠜⠧⣔⠻⠶⠂⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠦⠄
⠀⠀⢸⡈⠷⠀⢼⡹⠇⣠⣤⠈⠶⠈⡶⡂⠀⠐⡷⡆⠼⠛⣴⠌⠃
⠀⡀⠸⣇⣐⢲⡌⢣⡙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓
⠘⠿⠆⣿⠋⣘⠎
⠀⢀⣤⣿⠐⠟⠃
⠀⠔⣁⣿⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠉⣿⢺⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡐⡂⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⢀⡅⣟⢺⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠉⢀⡟⣢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡼⠛⣧⡩⠟⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠰⢾⡅⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣧⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⠝⢾⣁⡴
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⢟⣸⠉⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣿⠐⠋
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡿⢸⡑
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⣽
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⣿⠛⡠
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡉⣿⠾⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⢀⣿⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢆⠡⣇⣛⢻⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡾⣁⡤⢰⣴⡀⠀⢐⣴⠀⣤⠀⠿⠗⢩⡼⡧⠀⣤⠘⡇
⠀⠀⢨⣠⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣤⣌⡳⢄⡭⠵⠶⠾⠶⠒⠶⠶⣷⠖⢷⡖⠾⠾⠯⠵⢤⣘⣆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⠜⠃⠀⠟⠀⡍⠟⠀⠈⠁⠚⠈⠀⠀⠿⠆⠀⠛⠁⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠛⠻⠿⠿⠿⠟⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⠀⠀⠀⡀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠉⠀⡀⠀⡀⠀⠁⠐⠀⠀⣀⠀⠀⠀⠐⡇⠀⠸⡇⠀⢰⣶⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠉⠁⠀⠀⡠⣄⢀⠠⡀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠁⠀⡀⡤⡀⠸⡟⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠭⠁⠀⠈⠁⠀⠀⠄⠀⠀⡀⡀⠊⠀⡄⠀⠀⠀⠀⠀⢣⠈⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⠁⠀⠀⠔⠀⠀⠉⠀⢰⡗⠀⠀⠀⠀⠀⠻⡆⠀⠉⠀⡀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠠⠃⠀⠄⠀⢀⠄⠀⠀⠀⠀⢀⡨⠀⠃⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠠⢀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠘⢷⡄⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⡀⠀⡀⠀⠀⠀⠀⠀⢀⢀⠶⠶⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠘⢀⡄⠀⠀⠀⠀⠀⠀⠸⣀⠃⠀⠀⠀⣀⠀⠴⠴⠠⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠋⠀⢀⣠⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠈⣃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠂⢀⠌⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢆⠈⣠⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⢈⢁⡾⠀⠀⠀⣤⣩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡈⢢⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⢀⠸⠁⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⡇⠀⠀⢀⣀⠀⠀⠀⠤⠐⠁⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠀⠈⠀⠀⠀⠀⠂⠤⠀⠀⠸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡁⠀⠈⣆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⢹⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⢀⠠⠤⠄⠀⠤⡀⢀⠀⠁⠉⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠀⠀⠁⠂⠤⢀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠠⠂⠀⠀⠀⠀⠉⢶⡀⠀⠀⢀⠌⠀⠀⠀⠀⠱⡄⠀⠀⣠⠟⢁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠬⠶⠶⢧⣄⣀⣀⣀⡀⠰⠽⠶⠟⠂⠀⠀⡀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠈⠑⠂
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⣿⣿⠁
⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠚⠶⡄⠀⠀⠀⠲⠒⡀⠀⠀⡀⠪⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⣆⠀⠈⢢⡀⠀⠀⠄⠀⠳⠃⠀⠈⢋⡉⠓⢰⡖⠤⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣰⡾⢏⠱⣌⢢⢴⣿⡇⠀⠀⠈⠰⡆⢲⡀⠉⢈⠁⠚⠀⡄⠓⠨⢆⡀
⠀⠀⢀⡴⡏⠸⡄⢹⡀⢣⡈⠣⡈⢹⠀⠀⠀⠀⠀⠀⠀⡆⢤⡀⡣⠬⠤⠉⢰⠄⠸⢠⡓⠈⠁⠿⠛
⠀⠠⢾⣇⡙⣄⢱⡀⢳⡀⠳⣄⡍⢺⠀⠀⠀⠀⠀⠀⠀⣦⠀⣥⠀⠀⠀⠀⠀⣧⠆⣦
⠀⠀⢸⣬⠻⣼⠁⠀⠀⠀⢧⡌⢘⢺⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⢤⣤⣤⣤⡤⣤⢤⡠⠒⠲⢄⣀⣀⡀
⠀⠀⠈⠚⠐⠚⠀⠀⠀⠀⠰⠆⠘⠛⠀⡀⠴⠖⠛⠊⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⢤⣘
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠑⠢⡁⠒⠒⠒⠒⠠⠤⡀⢠⡀⡠⠤⠄⠀⠀⠤⡀⣤⢀⠔⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⡎⢰⡄⠀⢀⠀⡀⡤⠘⡀⠀⠀⠀⠀⠀⠐⠃⠙⠁⡠⢄⣀⣀⡀⢙⠀⠛
⠀⠀⣠⠴⠒⠒⠚⡅⠀⡃⠁⢲⠂⠁⡅⢀⠇⠀⡠⠗⢺⡀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠙⣝⣀⣀
⢠⢾⠁⠀⠀⠀⠀⠱⢄⠈⠒⠀⠀⠊⣀⠜⠀⠮⠥⡀⡀⢿⡆⢠⠪⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢤
⠚⢸⠀⣄⢀⣀⣀⣀⣀⡉⠁⡒⢲⠉⠁⠀⠀⠀⠀⢹⠁⣼⣯⠈⢆⡀⠘⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣸⢸⠸⠀⠀⠀⠀⢸⠰⢀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠈⣿⡇⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠁⢈⡁⠀⠀⠀⠀⠈⣁⣈⠁⠀⠀⠀⠀⠀⠀⠀⢸⢃⣹⣳⠀⠀⠈⠀⢰⠆⠀⠋⠉⠉⡆⢀⡆⠀⡇
⠀⢠⠊⠁⡜⠋⠉⠉⠑⡎⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢸⡈⠙⣖⣇⣀⣀⠁⣁⣐⣚⣀⠀⠀⠈⠉⠉⠊
⠀⢸⠀⢰⠄⠀⠀⠤⠀⢸⠀⠀⠀⡥⣀⡀⠀⠀⠀⠀⠀⣧⡀⠘⠛⠀⢶⡦⠀⣀⠀⠛⢋⡉⢢⡀
⠀⠀⠑⠒⢆⠀⠀⠀⡀⠀⠁⠒⠊⠀⠀⠈⠓⢄⠀⠀⠀⢯⠀⠐⠿⠂⢠⣦⠀⠛⠠⣤⠘⠋⣠⡇
⠀⠈⠯⡓⠋⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⢸⠸⡆⡟⠓⠒⠓⠒⠒⠊⢻⠰⡇⠇⣳
⠀⠀⠀⠀⠉⢹⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡏⠀⠀⢸⡄⢸⢠⠀⠀⠀⠀⠀⠀⢀⢀⠃
⠀⠀⠀⠀⠀⢸⠀⢸⡆⠀⠇⠀⠀⠈⢣⠀⡇⠀⡇⠀⠀⢸⡇⠘⣸⠀⠀⠀⠀⠀⠀⢸⣸⡆
⠀⠀⠀⠀⠀⠈⢄⠌⢣⡠⠀⠀⠀⠀⠰⡠⠣⡠⠃⠀⠀⠸⠇⠀⢿⠇⠀⠀⠀⠀⠀⠰⠏⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢈⢣⣼⡄⢒⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣼⠧⠃⠂⠀⠑⠘⠜⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠈⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⢀⡀⡀⢀⠆
⠀⠀⠀⠀⠀⡀⡀⠀⠄⣀⣠⠄⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠈⠉⠀⠂⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠂⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⢀⡄⠐⠂⠠⡄⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠤⠠⠐⠒⠂⠀⠰⡀⠀⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠁⢸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠒⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋
⠀⠀⠁⠠⠼⠂⠈⠀⡆⠇⠀⠀⠐⠊⣡⡄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡄⢰⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⣒⠒⠒⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠆⠀⠡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠄⠠⡀
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡬⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣌⠈⠉⠁⠀⠀⠀⠀⢠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⢤⠂⠑⠀⠀⠀⡀⠀⠠⠟⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠈⣅⠀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠆⠉⠂⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢰⡋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠁⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡛⠲⠶⠶⣤⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡐⠂⣂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠉⢀⡱⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⠃⠂⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢱⠮⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣤⠀⠈⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⢀⣠⣔⠺⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢸⠆⣄⡈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠧⠨⠯⠉⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣹⠆⣁⠉⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠠⡐⠃⢈⠄⠰⡄⠸⠦⠻⠗⠨⠇⣛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣫⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠲⡀⣰⢙⠃⣀⣄⡁⠐⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⡽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠉⠉⠉⠒⠶⠶⣤⣤⣤⣾⠇⢀
⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠂⠐⢶⣴⣷⣄⠘⠳⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢿⣿⣷⣶⣾⣷⠀⠀⠀⢀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣄⠈⠀⢂⣶⣾⣿⣿⣦⠀⠀⠀⢹⣿⠿⠺⡏⠋⠂
⠀⠀⣠⡖⠈⢰⣿⠙⢿⣿⣷⣦⣄⣹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢳⣀⣀
⠀⠐⡟⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠚⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠉⢛⠁
⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠂
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⡀
⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⣀⡇
⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⡶⠺⣿⣯⡛⠀⠀⠀⠀⠀⠀⠀⠄
⠺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄⠀⠁⠀⢸⣿⣧⠀⠀⠲
⠀⠙⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠚⠢⡤⡤⠐
⠀⠀⠈⠻⢷⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠽⠿⣿⣶⣦⣴⣀⠀⣠⣤⣴⣆⣠⣀⣠⡶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢿⣿⠟⢿⣟⣋⢉⣽⢿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠿⠌⠢⡈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢆⠈⢢⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⠈⠣⡐⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠾⠿⠟⠛⠻⠿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀⢀⣠⣤⣶⠶⠶⣶⣾⣿⣆⣠⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⣠⠄⠒⠈⠉⠉⠉⠀⠒⠤⡀⠀⠀⠙⢿⣟⠉⠀⢠⡶⠀⢀⡀⠈⠙⠻⣽⣷⡄⠀⠀⢀⣴⣿⣹⣷⡄⠀⠁
⠀⠀⠀⢠⡿⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠓⠆⠈⠀⣴⣿⢄⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠻⠿⢿⣗
⠀⠀⠀⣾⠇⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠨⡆⠀⠀⠀⠀⠀⠀⣿⣿⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡂⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠈⣿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⢿⡆⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠘⣿⡀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠊⠉⠒⠠⠤⠤⢤⡄⢲⢻⡿
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣸⣾⠇
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣿⠃
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠃
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⠃
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡞⠹⣷⣄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠙⠿⣦⡀
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠼⣿
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠈⢷⣦⣌⣲⣿
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⠻⢶⣤⡀⠀⠀⢿⡟⠛⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠈⣿⠀⠀⣾⡇
⠀⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠾⠿⣧⣄⠀⠀⠀⠀⠀⢿⣾⡶⠟⠁
⢰⡿⠁⠀⢠⣤⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣯⣥⣤⣤⣦⣬⣽⡇
⣿⠃⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠷⢶⣤⡀
⢿⡆⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⢉⣠⣤⣶⠶⠶⠶⠶⢿⣿⠄
⠘⢿⣄⠀⠀⠙⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠛⠉⣀⣴⡾⠟⠋⠁
⠀⠀⠙⢷⣦⣀⠀⠀⠉⠛⠛⠿⠶⠶⠶⠶⠶⠶⠾⠟⠛⠉⢁⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⡶⠾⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⢂⠀⠀⠀⠀⡀⠀⠀⠂⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡱⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⠀⠈⢠⣔⠃⣀
⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢞⠁⠀⠐⠀⠀⠈⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣥⠖⠀⠄⡃⠂⣡⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⢅⡒⠁⠀⠰⠋⢄⣰⠄⠒⣁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡾⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢤⠪⡄⠜⢶⡁⣠⠄⠀⠀⢉⣩⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠂⣈⠓⠀⢁⡄⠀⠀⠀⠒⡄⠘⠿⢿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⠉⡓⠐⠂⡳⠀⠀⠷⢀⡹⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠤⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣄
⠀⠀⠀⠀⢸⣿⠀⣼⡟⠉⠉⠉⢩⣭⣭⣭⣭⣭⣭⣭⡉⠉⢉⠉⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⡉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⣿⡇⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⢸⣿⣀⣿⡇⠘⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠙⠛⣿⡇⠀⠀⠀⠻⠤⠤⠤⠤⠤⠤⠤⠄⠀⠻⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠼⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣴⣶⣿⠛⣭⡉⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⠛⠉⢁⣴⠲⣶⣤⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⡇⠘⠷⣶⡄⢸⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢸⣧⣈⡉⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⢷⢻⣿⣿⣿⣿⣦⣈⣙⣤⡿⠿⠛⠛⠛⠛⠀⠀⠀⢸⣿⣿⣿⡘⡄⠀⠀⠀⠈⠻⠷⠶⠛⢀⣀⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠈⣧⠻⠿⠿⠿⠿⠛⣋⣡⣴⣶⣿⠿⠿⠿⠀⠀⠀⢸⣿⣿⣿⣷⡹⣄⣀⡀⣀⣠⣤⡶⠿⠛⠉⣁⣀⣀⣈⣿⣿⣿
⠀⠀⠀⠀⠘⣷⣶⣶⣶⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠐⠀⠉⠉⣁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⣠⣤⣶⣾⠿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⣁⣤⣴⣶⠿⠟⠛⠉⠁⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⣁⣠⣤⣶⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠂⡉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⣀⣠⣤⣶⣾⡿⢛⠩⣀⠀⠀⠐⢨⠙⠿⠖⠐⠀⣀⣀⣠⣤⣴⣶⣶⠿⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⣀⣤⣶⣾⠿⠛⠛⢉⡼⠿⠴⠀⠀⠉⣀⣀⣤⣤⣴⣶⣶⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠉⠀⢴⣶⣿⠿⠟⠉⠁⠀⢀⣈⣀⣤⣤⣶⣶⣾⣿⠿⠛⠉⠉⠉⠉⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣶⣶⣾⡿⠟⠛⠛⠉⠉⢀⣰⡟⣁⠊⢀⣨⣧⣤⣄⠀⠈⢂⣻⣷⡀⠀⣀⢊⣠⣤⡈⣀⢸⣿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⡏⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡿⠟⠋⠩⢉⠭⠙⠛⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠀⠁⢀⣤⣿⣤⡀⠀⠐⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠄⠀⠀⣿⠇⠀⠀⠀⣾⡁⣿⠸⠿⠀⠀⠈⠈⣿⡄⠀⢀⠐⣀⡀⠢⠀⢸⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣿⣿⢰⠀⣿⣿⡧⢈⠀⢸⣿⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠆⢸⣿⣿⠆⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⠀⠀⠈⠉⠠⠂⠀⠀⣿⡄⠀⠀⠀⣾⡆⣿⠈⣿⠀⠀⢀⢀⣿⠃⠀⠈⠀⠉⠁⠔⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠀⡀⠈⠛⣿⠛⠁⠀⠠⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣷⣦⣄⣒⣈⣐⣀⣤⣾⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡀⡠⠀⢠⠀⠀⣤⣤⣤⣤⡔⣶⡴⣲⣦⣠⣤⣴⣶⣶⣦⢲⣪⡹⠧⢼⢰⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠴⠛⠃⠁⠀⠉⠁⠈⡉⢉⣵⠃⢇⣣⡎⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⢠⠸⣷⢿⡇
⠀⠀⠀⠀⠀⠀⢔⡕⠁⠀⠀⣔⠦⡢⠀⠀⢰⣵⣿⠏⠚⣸⣿⡃⠀⠇⠀⠀⣶⣶⣶⡷⠀⠀⠀⠀⡌⢢⣧⡿⡇
⠀⠀⠀⠀⢀⢪⠊⠀⠀⠀⠀⠈⠀⡀⠀⠀⢘⣿⡿⣡⡏⣯⣿⠃⠔⠀⢠⡀⠀⢁⡀⢠⣠⣤⡄⡜⣌⢞⣿⡿⠁
⠀⠀⠀⡠⠱⠿⣶⣶⣃⣂⡐⠼⠭⠡⢶⣶⣾⡿⡱⠱⡇⣿⣶⣷⣶⣦⣤⣤⣤⣤⡠⠖⢞⣂⣚⣼⣿⣼⡿⠁
⠀⠀⠀⢹⣿⣿⠟⣽⣯⣽⣿⣿⣿⣿⣶⣶⣶⣷⣷⣶⣷⣿⣾⣟⣛⣛⣿⡶⠒⠛⠻⠭⣍⣉⣙⣵⣿⣿⠁
⠀⠀⠀⢸⣼⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⣿⠿⠿⠷⠾⠯⠽⠿⠷⢶⣿⣿⣿⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠰⡶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⢲⡆
⠀⠀⠀⠀⢠⡴⠶⣷⠲⠖⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠗⠲⢶⡄
⠀⠀⠀⠀⠈⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢸⠃
⠀⠀⠀⠀⠀⢻⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⢶⡿⡶⠶⠟
⠀⠀⠀⠀⢀⣠⡼⢃⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠋⠠⡛⠶⣄⡀
⠀⠀⣠⡶⠋⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡛⠶⣄⡈⠙⣦⡀
⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⠞⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣌⠻⣆⠈⢷
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢉⣉⣉⣛⡛⠛⠚⠓⣶⢬⣙⠓⠶⢦⡄⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠘
⣿⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠏⣀⠤⣌⡉⠉⡗⢰⠇⠀⣨⠟⣠⡞⠓⠲⠶⠖⣶⠆⠀⠀⠀⣞⣳⡄
⣿⠘⠀⠀⠀⣠⣄⠀⠀⠀⣼⢣⠏⡞⡅⣠⢦⢘⣶⢣⡟⢀⠞⢁⡴⠋⠉⠉⣹⠂⢠⠏⠀⠀⠀⣠⣌⠉
⣿⠀⠀⠀⠸⣧⣈⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⣶⠞⢉⣼⠇
⣿⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡇
⣿⠀⠀⣴⢤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⢠⡏⢠⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣿⠀⢸⠇⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⣼⠀⠧⢼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣯⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⣿⠀⠛⠲⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄
⣿⠀⢀⡼⢋⡼⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣝⢷⣼
⣿⠐⠿⣤⣿⣀⣴⠿⣶⣶⣶⠶⣤⠤⠤⠤⣤⢤⡤⢤⣤⡤⢤⡤⠤⠤⠴⠶⠶⠶⣶⡶⢦⡾⣅⢈⡿⣣⢿
⣿⠀⠀⠀⠈⠉⠙⣳⣶⣬⣁⣰⠋⣀⣀⣀⡿⢒⣋⣭⣥⣶⣾⢡⣶⣖⣟⣻⢿⣆⣀⣹⣦⡙⠻⢋⡴⠋⢸
⢸⠀⠀⠀⠀⠀⣼⡏⢀⣶⣋⣿⣿⢟⡋⠛⣿⡛⠛⣿⣿⣿⣷⣾⣿⣾⣶⣳⣾⣭⣿⣀⣹⣿⠶⠋⠀⠀⣾
⢸⡄⠀⠀⠀⠀⢿⣿⣾⣿⡩⣝⡳⡌⠷⢀⡿⣭⣿⣭⣿⣯⣯⣽⣿⣯⡷⠿⢭⣀⣼⡿⠿⠋⠀⠀⠀⠀⣿
⠈⣧⠀⠀⠀⡀⠀⠉⠛⠿⣿⣶⣷⡟⠚⠋⠀⠀⢰⣿⣟⣿⣿⣿⠿⠿⢷⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⡟
⠀⠈⠳⣄⡀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠛⠛⠉⠉⣀⣠⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡄⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣿⠋⠀⠀⣷⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠀⠘⠀⠀⠀⢸⡄⠀⠂⠀⠀⠀⢻⠘⠳⣤⣶⠶⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠀⠀⠀⣀⡈⢢⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣼⠇⠀⠲⠒⠠⣤⣀⢀⣀⡤⠴⠦⠴⠼⢏⡁⡀⠘⣿⣿⡄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⡿⠄⢠⡤⠖⠋⠙⠉⠉⠁⠀⠀⠀⠐⠀⠀⠈⠱⢶⡙⣿⠃⠖⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⢿⡿⢾⣇⡰⠊⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠈⢻⣄⠠⡃⡀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠴⠁⠀⠀⡾⢻⡏⠻⣦⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢻⣿⣯⡐⡀⠀⠄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡀⠸⣇⣸⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠿⢿⡁⠒⢀⠅
⠀⠀⢨⠀⠀⠀⠁⠉⠀⠀⣈⡩⠁⠀⠙⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣾⠟⠂⠉
⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠂⠀⠄⢀⡀⢹⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠠⠘⠋⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠐⢀⣐⠀⠀⠀⠀⠀⠀⠀⡳⣤⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢀⡆⢀⣴⡏⡁⠒⠍⣿⣷⠆
⠀⠀⠀⠀⠰⠀⣒⡎⠄⣀⡀⠀⠀⠀⢳⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣠⣾⣿⣿⣿⣕⡬⣭⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠯⢀⠒⠒⠀⣨⣄⣈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠢⣿⡿⠋⣨⡿⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⡇⠀⠀⠀⠀⠀⠞⢁⣾⡯⢇⡀⠀⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⢰⣿⡏⢄⣛⣮⢶⣿⡟⠁⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣭⣵⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠈⢁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠔⠉⠐⠂⠀⠂⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠆⡠⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡈⠢⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠊⣢⠤⠒⠒⠒⠢⢤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⡀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⢠⠎⠁⣤⣀⣤⡀⠀⠀⠈⢦⠢⠀⠀⠀⠀⠀⠀⠀⢀⠐⡠⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⡡⢋⠔⠀⠀⠂⡏⠀⠀⣼⡟⠙⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢚⡠⢉⠐⠁⠀⠀⠀⠘⡄⠀⠀⠻⢷⣶⡿⢿⣦⠀⠀⡸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠔⠀⠀⠀⠀⠀⠀⢀⢣⡀⠀⠀⠀⠀⣤⣾⡟⠀⢠⢃⠀⡠⢂⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⢤⣀⠀⠀⠈⢀⣠⠔⠁⠠⢊⠔⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⢂⠁⠀⠀⣀⠔⡁⠁⠀⠀⠀⠀⡂⠁⠀⠂⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⠚⠓⠉⠁⠀⠀⣂⠁⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠱⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠊⠀⢀⡀⠄⢀⡰⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠇⢳⡈⠢⡀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠨⠥⠒⠢⠤⠔⢒⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡢⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠒⠢⢤⠬⠦⡔⠂⢀⠤⡄⠂⢎⣄⢉⢁⣀⣀⣀⣀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠄⠈⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠈⠐⠻⠦⡈⠒⠁⠔⠁⡙⣄⠀⠹⠿⠿⠋⠀⠀⢀⠜⡉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠆⠁⠀⠀⠈⡊⠑⠢⠤⣤⡤⠤⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠅⠀⠀⠄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠊⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⡀⠀⠈⡽⢿⣝⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡓⠶⠾⠋⠀⢉⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣼⣤⣀⡀⠀⠙⠳⠶⠾⠿⠛⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠶⠶⠟⠉⢁⣠⣿⡄⠲⠤⠶⠶⡶⠶⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠛⠛⠛⠛⠛⣛⠛⣿⡷⠠⢤⣤⣤⣤⠆⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢉⣙⣛⣋⡉⠀⣿⡇⠠⠤⠤⠤⠴⠖⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠠⣤⣭⣭⠄⠀⣿⡃⠰⠦⠤⣤⡴⠶⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⠤⠤⠴⠆⠈⣿⡇⠠⣤⣤⣤⣤⣤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠐⠶⠶⠶⠒⠀⣿⡇⠀⠀⣀⣀⣤⠤⠼⠿⠒⠛⠁⠀⠲⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢐⣲⣶⡾⠿⠶⠛⠛⠋⠙⠛⠦⣤⣀⣀⠤⠐⠺⢥⣄⣴⠒⠈⠙⠛⠷⢶⣤⣄⣀
⠀⠀⢀⣀⣠⠤⠴⠖⡒⠋⢉⠭⠙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣄⡀⠀⠀⠉⠐⣞⣚⣓⡄⠉⠙⠛⠛⢆
⢠⠀⢀⡀⠀⠠⣔⠦⣭⣳⡖⠉⠉⠁⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢦⣔⣊⣩⡭⠭⠶⣒⣋⠭⠤⣾
⢼⡄⠀⠉⠢⣄⠈⠓⠛⠿⣭⣱⠦⠤⣤⡀⢀⣀⡤⠍⠓⠢⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⠋⣿⠁⠐⢒⣋⡭⠥⠒⠊⣡⣿
⠘⡏⠳⢄⡀⠀⠙⠲⣄⡀⠀⠩⡘⢲⣦⣹⣏⡠⠖⠉⣉⣠⣤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⠤⠶⠒⠋⢉⣹⣿
⠀⡗⠢⣀⠙⠲⣄⠀⠀⠉⠲⢤⣈⠙⠛⣉⣥⡶⠾⢛⠋⠉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣀⣤⡤⠶⠖⢛⣉⣽⣿
⠀⡟⠦⣈⠑⢦⡈⠙⠦⣀⠀⠀⠀⣴⣻⠭⠖⣚⣩⡤⠴⢒⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣀⣤⠴⣖⠛⠋⣉⣹⣿⣶
⠀⠀⢦⣈⠓⠄⠈⠒⠄⠈⠓⠦⡀⣹⡖⢛⣉⣭⠶⠒⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣒⣮⣽⣶⣾⡿⠿⠛⠋
⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠁⠀⠀⢀⣀⣤⠄⢸⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⢿⣇⡤⠤⣶⠿⠛⠉⢁⣀⣼⣶⣾⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⣿⣇⣠⡤⢶⣪⣵⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣫⠵⠞⠋⠉⠁
⣿⣷⠿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡱⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠈⢂⠤⠶⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣷⣶⣮⣷⣦⣄⣀⠀⠘⠋⠙⠈⠀⠀⢀⣠⣴⡞⠉⠀⠀⠀⠀⠒⠒⣂⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⡏⠀⠉⠐⠒⣤⣴⣬⣬⣉⣉⣁⣀⡀⣀⠄⠐⠚⠫⠤⣖⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⠴⣟⡛⠙⠛⠧⢤⣀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠦⠤⣉⣙⣻⣿⣟⣿⡇
⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠉⠉⠁⠀⠀⠀⠠⣿⣷⣦⢄⡀⠀⡉⠛⠛⢿⣿⣿⠿⠟⠛⠿⠷⡖⠦⢬⣝⣿⣛⣽⣷⠄
⠀⠀⠀⠀⠀⠀⣇⡉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣬⣕⢺⡟⠉⠁⠀⡤⣉⣙⣲⣾⡿⣉⠓⠶⠶⣿⣿⣿⡟⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢻⡉⠑⠂⠀⠉⠑⠲⢶⣶⣿⣿⣿⣿⣿⣿⣿⣯⢸⡁⠀⣀⣀⣵⠤⢬⣝⣛⣋⣭⣉⠓⠶⣿⠿⣿⣇⠒⠶⣶⣶⣿⣏
⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⠿⠷⣼⣋⡽⠟⠋⢛⠒⠶⢶⣿⡿⠦⣀⣉⣙⣿⣿⣿⣏⣈⣛⡿⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣉⠒⠢⠤⢌⣛⣭⡿⠛⠋⣁⣤⣶⣾⣿⣿⣀⣠⣴⣭⡙⠒⠶⣾⡷⠖⠢⠤⣭⣿⣯⣽⣷⠦⣭⣽⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣴⣎⣉⠑⠒⠦⢄⣀⠀⠀⠀⠈⠉⣭⣷⣿⣿⣿⣿⡟⠉⣀⠥⡧⢌⣉⣛⣻⣿⡛⠡⠤⣤⣬⣭⣽⣛⠐⠶⣾⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠓⠲⠤⣄⣀⡈⠐⠀⠴⢞⣫⣵⣿⣿⣿⠿⠊⠁⠀⠁⠀⠈⠉⠛⠛⣩⣉⠒⠲⣿⣿⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠧⣄⡀⠀⠀⠀⠀⠈⠉⣙⠒⣚⣭⢶⣻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠢⠤⠤⣀⣉⣩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁







⣿⢿⠿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣿⣛⣻⣿

⠀⠀⠀⠃⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⣣⣼⣦⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⡿⢿⠟⠿⠆
⠀⠀⠀⠀⠀⢠⣶⣿⠞⢹⢻⣿⡦⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠
⡀⠀⠀⠀⢀⣟⠛⠀⠀⢰⠀⠋⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢂
⢿⠀⠀⠀⠾⣿⡆⠂⣄⣸⠀⠈⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓
⣿⡆⠀⠀⢸⣿⣃⠀⠙⠛⠀⠀⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣡⣴⠀⠀⠀⠀⠀⣼⣾⣷⣆⢠
⢿⠃⠀⠀⠀⠼⢿⣶⣶⣖⣦⣼⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⠟
⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣶⣶⢰⣶⣶⠎
⠀⠀⣀⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃⠀⢻⡟⠀⠀⠀⠀⠀⠀⠠⣄⠠⣶⠖⣀⠂
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠈⢣⣶⣦
⠀⠀⠀⠀⠠⠶⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠠⣽⢹⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⡮⣭
⠀⠀⠀⠀⢀⠀⡀⠀⠄⠇⠀⠀⠀⠀⣀⣀⠔⠒⣒⠒⠠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢏⢀⠀⢐⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⣿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣷⣾⡧
⠀⠀⠀⠶⢤⣤⡤⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠛⠛⠛⣿⣷⣿⠆
⠀⠀⠀⠀⠸⢿⡿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⣰⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⡏⠀⠀⠀⣿⣿⣧⠄
⠀⠀⠀⡖⢢⣤⣴⠐⡇⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉⠖⠚⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⣶⠄⢿⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠠⣄⠀⠀⠈⠢⣦⡀⠀⠈⠢⠱⡄⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⡠⢤⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢁⠀⠀⢀⠤
⠀⠀⠀⠀⠈⠓⢄⠀⠀⠈⠈⠳⡄⠀⠀⠈⠀⠀⠀⡇⠁⠀⠀⡀⠀⠀⢐⡏⠀⡀⢤⠌⣰⢟⣷⠂⠀⠀⠀⣀⣤⠾⠦⠀⡀⠀⠀⠊⠀⠘⠁⠠⠾⠃⠁⠚
⠀⠀⠁⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠤⠀⠀⠐⡇⣎⠀⠀⠠⠀⢀⡿⠁⠴⠆⠀⣩⡅⠀⠀⠀⠀⠱⡖⠀⣀
⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠀⠀⡇⠈⠙⠢⢄⣀⠀⠀⠀⠠⠜⠃⠉⠀⠀⠀⠀⣠⡤⠄⠀⢨⠀⠄⠀⡀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠀⢰⡄⠀⠀⠐⠀⠀⢠⡇⠀⠀⠀⠀⠈⠉⢉⡀⠠⠞⠁⠀⣠⠀⠓⢋⣩⡤⠀⠠⢀⠀⣀⠄⠈⢀
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣽⡁⠀⠀⠀⠀⠀⢨⣄⣀⠀⠀⣄⠀⠀⠀⠈⠢⡊⠠⠼⠅⠤⠀⠀⠀⠀⣀⡴⠞⣫⡁⡀⠁⡜⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⠀⠐⠒⣠⠀⢨⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠘⢦⡀⠀⠀⠀⠀⠀⡀⠠⢤⠴⠞⣀⠁⠈⠀⡰⢋⡰⠂⡰⠁⠈
⠀⠀⠒⠄⠀⠀⡆⠀⠀⠀⣸⠀⢐⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⣶⡶⠿⠵⠾⠷⠗⠓⢉⣴⣿⣅
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠰⠀⠀⠀⠀⠀⠀⣿⠠⠤⠀⠀⠀⠀⣼⣿⣿⠀⢠⡄⠀⠀⠐⠀⠠⢉⠓⠂⠤⡉⠲⣤⣤⣀⠀⢀⣴⣿⡋⠻⠿⠷⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠀⡤⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠈⠁⠀⠐⠂⣼⣯⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠍⠀⠀⠠⣤⣴⡟
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠯⣀⠀⠀⠀⠀⠀⠼⠟⠛⠋⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠩⣽⣷⣄⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢥⠀⠋⠀⣀⣀⠀⣀⠀⠓⠲⠆⠈⢉⠁⠀⠀⠀⢲⣬⡂⠀⠀⠀⠀⠀⢸⡟⠀⠓⠀⢀⡀⠀⣤⠀⠀⠀⠀⠃⠀⠀⡂⠀⣿⡿⣿⡿⠃
⠀⣀⢀⠀⠀⠰⠦⠀⠐⠛⠉⠀⢀⣤⡤⢤⣄⠉⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠰⠦⣼⣦⠤⠒⡒⢻⡅⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢐⣿⣯⣥⣀⡀
⠀⠉⠁⢂⠙⠓⠢⠤⢀⡀⠀⠀⠘⠒⠞⠛⢛⣛⣙⣡⡤⠀⡀⠀⠀⠀⠠⣶⣤⠄⠆⠀⠀⠀⠨⡭⣿⡇⠀⣤⣐⠿⠖⢋⠀⠀⠀⠐⠻⠅⣀⠀⠠⠶⠿⠿⠿⠛⠃
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠐⠂⠤⠄⢀⠀⢉⣀⡠⢘⡠⠍⡀⠚⠛⠋⠉⢐⠀⢀⠀⠀⠀⠀⠉⣿⣧⣀⣭⠭⠶⣿⠛⢣⣾⡀⠚⠫⣤⠀⠀⠀⠀⢀⡀⠠⣤⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣭⣄⣬⡥⠎⠵⠒⢛⣾⡿⣻⠀⡒⠈⠁⠀⠠⢭⣿⣯⡤⠀⠀⠒⠛⠓⠂⠼⣿⠆⠀⠙⠀⠀⢀⠀⠊⣁⣲⣿⠷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣩⡍⠉⣩⣶⠮⢒⣺⣿⣿⣶⡿⠿⠂⠀⠀⠀⠀⢽⣿⣿⣿⣄⡀⠀⠐⠀⢂⡼⠓⠠⣶⣤⣶⣾⠿⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⣀⢒⡊⠭⠴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢈⡀⠀⠀⠀⢀⠠⣾⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠛⠿⠿⢛⠿⠟⠛⠿⠟⠿⡿⠟⣿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣷⣾⣿⣿⣾⣿⣶⣷⣷⣷⣿⣿⣾⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠠⢄⣀⠀⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣓⠄⠰⡱⠙⠦⠠⢀⣩⠤⢀⠝⡦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠁⠀⠀⠉⠛⠒⠀⠒⠛⠋⠉⠁
⠀⠀⠀⣀⠤⣆⡒⠺⡺⢻⡅⡀⠲⣦⠴⠀⠀⠁⢐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⢤⣀⡀⠀⢀
⠀⠠⠎⠘⠒⠛⠒⠬⠠⣈⠽⠛⠚⠁⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⠀⠀⢀⠖⠀⠀⠠⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⠔⢠⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠿⣥⠭⡅⠂⠀⢀⢄⠞⠁⠀⠀⢀⠀⡂⠤⣀⠠⡀⢀⣼⠇⠒⠒⠀⣲⣤⣤⢀⣀⣠⡷⢶⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠲⡀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠐⠃⢈⡠⠤⢍⡿⢀⠾⢇⡀⠀⠀⠈⢚⣈⣡⠁⡠⢄⡀⠹⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⣩⠔⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠢⠀⠀⠀⠉⠀⠀⡀⠞⠩⠝⡟⡳⠁⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢆⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠸⠀⠀⠀⠀⠀⠊⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠢⢧⣀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠥⠠⠄⠀⠀⠀⠐⠒⠦⠒⠂⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠕⣒⠀⠀⢰⠽⠒⠰⡦⠤⣔⢃⢷⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠒⠒⠒⢀⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⠚⠉⠁⠊⠀⠑⠫⣢⡰⠂
⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⣀⡠⣄⠀⠺⠄⢲⣆⣀⠨⣟⠐⠯⠕⠶⠤⠤⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢥⣀⠐⣀⠀⠀⠈⠑
⠀⠀⠀⠠⡊⠁⠀⠏⠃⠁⢠⣄⡜⢢⠀⠈⠀⠀⠀⠈⠁⠛⠬⠤⠤⣥⡤⠤⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠉⠀⠉⠐
⠀⠀⠀⢠⡁⠂⢅⠱⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡖⠚⠒⠲⠄⡀⠄
⠀⠀⠀⠈⢏⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠯⠷⡷⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠉⠊⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆⢁⣠⠬⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠉⠙⠋⠀⠀⠀⠀⠠⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⠃⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠳⣄⡄⠀⢤⠀⠀⠘⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠘⣿⣦⣸⣇⣴⣢⠀⢀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⡿⠿⣦⣈⣷⣤⣄⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢛⣽⢻⠟⡏⡏⠹⠀⠀⠈⡍⠉⠉⠉⠉⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣸⣧⡟⡼⠁⣠⠀⠉⠠⢀⣡⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢠⠄⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠠⠀⠀⣿⣤⣤⠦⠿⡛⠛⠛⠛⠛⠃
⠀⠀⠀⣠⣴⣴⣾⣶⣷⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠓⠀⠀⢀⡌⠉⠀⠀⢀⣧⠀⠀⠀⠀⣀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣲⠟⠀⠀⠀⣠⣾⣿⣦⣭⣥⠾⠟⠛⠛⠛⠛⠫⡀
⠀⠀⠸⣿⠏⠹⠛⠟⠹⡇⣧⠀⢹⠖⠒⠀⠀⠀⣠⣴⣿⠏⠀⠀⠀⢸⣿⢻⡏⣝⠉⠀⠊⣿⠀⠀⢀⠀⣠⣷
⠀⠀⢰⠀⠠⣀⢠⣸⣼⣷⢸⠀⠈⣷⣶⣶⠤⠿⣿⣿⠏⠀⠀⠀⠀⠘⣿⣾⣇⢸⣷⣦⡾⠿⠿⠿⠿⡿⠿⢧
⠀⠀⣼⣷⣶⣿⣾⣿⣿⣿⡘⡆⠀⢹⣷⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠈⣿⠛⠉⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⢿⣷⡀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡸⡄⠀⠹⡄⠀⠀⣿⡀⠀⠀⠀⠀⠀⢹⣿⣄⣿⠀⣿⣸⣿⡇⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⢳⡆⠀⠀⠀⠀⢳⣿⣦⣀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠻⠻⢿⣿⣿⣿⣿⠿⠿⠿⠇⠀⠀⠀⠰⢼⣿⣿⣿⣿⣖⣀⠀⠀⠀⠙⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢲⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢄⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⠀⡻⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠤⠤⣾⣿⣟⣶⣃⠀⠊⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠑⣤⡟⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⢿⣿⠁⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣤⡏⠀⣼⠇⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⢸⣦⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⡄⠘
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠋⠀⣴⣿⠃⠀⠀⠀⠁⠀⠀⠀⠘⡿⢽⡢⢄⡀⠀⣀⡆⠸⣿⣷⣷⣼⠋⠀⠀⠀⠀⡄⠀⠀⢣⠀⠸⠇⠤⣿
⠀⠀⠀⠀⠀⠀⠀⡻⣦⡀⠘⣶⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠘
⠀⠀⠀⠀⠀⣎⢰⡇⢸⣿⡆⢹⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⢸⣯⠀⡘⠉⣷⡀⢹⡄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡀⠄
⠀⠀⠀⠀⠀⠈⠁⠘⠁⠈⠙⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣰⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣷⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠶⠶⠶⠶⠾⠑⠛⠁⠀⠀⠀⠠⠒⠛⠹⠉⠃⠁⠙⠁⠀⠀⢢⠄⠋⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⣄⠀⠼⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠀⠀⠀⠀⡀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢸⡦⣤⠀⢠⣤⠿⠛⢿⠀⢹⠀⡁⠀⠀⠀⢀⡌⣠⣦⠀⠂⢸⠤⠀⡀⣴⠆⢀⡴⠀⠀⠀⠀⢠⣿⠀⣰⠟⠦⠀⠀⠔⠃
⠀⠀⠀⢠⠊⢀⡾⠀⠈⠀⠀⠀⠀⠀⠈⣆⢘⣴⠃⠀⠀⢀⣼⣴⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠎⠇⠀⠀⠀⠀⠘⠁⣼⠁⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣼⣠⡏⣇⠀⡂⠀⠀⠀⠀⠀⠀⢻⣸⡟⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠸⣁
⠀⠀⠀⢽⣿⡄⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢋⣉⡛⠛⣋⣤⡌⠛⣉⣴⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⣰⣿⣿⣿⣿⠟⣋⣴⣾⡿⠛⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⣡⣾⣿⣿⠟⢋⣤⣾⡿⠟⣡⣴⣿⣶⣤⣭⣥⣦⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢋⣴⣿⠿⢋⣥⣾⡿⠟⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⢋⣡⣶⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠛⣡⣶⣿⣿⣿⡿⠋⠉⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠟⣡⣴⣿⠿⢋⣿⣿⠋⠀⣴⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⣉⣴⣿⡿⢋⣡⣶⣿⣿⡏⠀⣸⡿⠋⢁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣉⣴⣾⡿⠛⣡⣶⣿⣿⣿⣿⣿⡇⠀⠀⣠⣶⣿⠿⠋⢉⣿⣿⢿⣿⣿⣿⣿⡿⠛⣡⣶⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⣴⡿⢋⣤⣾⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣇⣠⣄⠀⠻⠛⢁⣠⣶⣿⡿⠃⠀⣿⡿⠟⣡⣴⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⣠⣾⣿⡇⣾⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⠀⠉⠙⠋⠉⢀⣠⣾⣿⣶⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣧⡘⢿⣿⣿⣿⡏⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣷⡌⢻⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣽⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⠹⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡈⢿⣿⣿⣿⣌⣩⣥⣬⣉⠻⢋⣡⣾⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠻⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠭⣓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢕⡢⠭⣐⠈⠙⠓⢦⣄⠀⠀⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⢴⣪⠥⣂⠅⠀⠈⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠠⠬⠿⠶⣥⣀⡀⠀⠀⠀⠀⣼⠀⠀⢀⣀⣠⡤⠶⠶⠿⠯⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠿⠗⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⣶⢶⣶⠶⠶⠶⠶⠖⠒⢒⣒⣒⣒⣶⣶⠶⠶⠓⠛⠛⠛⠛⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠁⢳⣸⡟⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⠐⠊⠙⢦
⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃⣀⣠⣤⣿⣷⣶⣶⣶⣖⣒⡚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⢀⡴⠖⠛⠉⠁⠀⣿⠟⠉⠀⠀⠀⠀⠐⠋⠭⠭⠭⠭⠭⠍⢉⣛⣛⢛⣛⡒⠒⠒⠒⠲⢶⡾⠓
⣰⠋⠀⠀⠤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⠉⠡⠭⠭⠭⠤⠄⠀⠀⠀⣼⠉⠓⠦⣄⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⣀⣀⣈⣉⣉⡀⠐⠒⠒⠒⠠⠭⠍⠉⠉⣀⣀⣀⠀⠀⣇⠀⣀⣀⣈⣹⣷
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⣀⣭⣽⣯⣤⣈⣿⠋⠉⢹⡇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⣠⠴⠖⠒⠛⠉⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡘⢷⡠⠀⠀⠀⠀⠤⠤⠶⢒⣒⣛⣫⡭⠭⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠈⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⠀⠀⠀⣀⡀⠙⣿⣷⡿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠙⠛⠋⣩⣿⡿⠛⠁⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⠟⠀⠀⣀⣴⡿⠟⠁⣀⡴⣄⡀⢀⣀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢠⡞⠁⣀⣴⡾⠟⠉⣀⡴⠞⠁⠀⠈⠉⠉⠉⢷⡀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢀⣤⣾⠟⠋⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⣴⣾⠟⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣠⣾⠿⠋⢁⣠⠆⠀⢈⣷⡾⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⣿⠆
⠀⠀⠀⠀⠀⠀⣰⣿⣫⣶⡿⠋⠁⣠⠶⠋⠁⠀⠀⢸⣿⣀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠀⣠⣶⡿⠋
⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⢻⡇⠀⠀⠀⢀⡴⠞⠉⣠⣴⡿⠛⠁
⠀⠀⠀⣼⡿⢹⣿⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣾⠃⠀⠀⠞⠉⣀⣴⡿⠟⠁
⠀⠀⠘⣿⣄⠀⠹⣿⣄⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠇⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠈⠻⣷⣄⡘⢿⣦⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠃⢀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣻⣷⡄⠘⢧⣀⣀⣀⡀⠀⣠⠴⠋⢁⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣆⠀⠉⠀⠈⠙⠛⢁⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⣠⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠠⠔⠚⠈⠉⠀⠀⠀⠀⠀⠈⠉⠃⠶⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠓⠢⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠄⣄⠠⠐⠖⠨⠛⠒⠑⠒⠀⢄⣠⠄⠀⠀⠐⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠶⡡⠥⠄⠒⠄⠀⠉⠀⠀⠤⠆⠼⠬⠐⠦⠀⠈⠉⠀⠀⠒⠀⠽⠄⠒⠄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢆⠀⠆⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠒⠴⠶⠖⠂⠠⠀⠀⠀⠀⠰⠶⠀⠀⠒⠀⠀⠀⠀⠠⠤⠴⠲⠤⠦⠒⠦⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠐⣈⣐⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣤⣦⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⢀⣤⣤⣔⣦⡤⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠈⠄⠀⠀⠐⠁⠀⠀⠀⠂⠆⠌⢁⠐⠆⠄⠀⠀⠈⠂⠀⠀⠩⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠈⠚⠛⠊⠙⠃⠀⢀⡀⢠⡀⣀⢀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢀⣂⠀⠀⠀⠀⢠⡄⢀⣀⡀⢀⡙⠈⣩⣿⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣿⣦⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⠦⠦⠤⠆⠠⠤⠶⠤⠴⠀⠀⣾⢻⣿⠿⢿⣽⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣺⣻⣿⣿⠀⠀⠀⠃⠀⠰⠂⢠⠤⠀⠄⠀⣆⠀⠀⠀⠀⠀⠀⠠⠀⠤⠄⠸⠄⠀⠀⠈⠀⡄⠀⠀⢸⢿⣋⣾⣟⡙
⠀⠀⠀⠀⠀⠓⡄⠘⢻⢿⠾⠛⠉⠀⠐⡀⠀⢀⠀⠀⠄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠡⠄⠀⠀⢀⠀⡹⠅⢀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⢀⣼⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠀⣟⠒⠒⠀⠀⡀⠀⢀⠀⡀⠀⢰⢤⣖⡂⠀⣀⣀⣉⣀⡈⠈⠀⠀⢀⡄⠹⠀⠘
⠀⠀⠀⠀⠀⠀⠋⠀⠚⠻⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠀⢭⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⡷⠼
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣗⠀⡀⠀⠀⠀⠀⠈⠉⠉⢙⠃⠈⡐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠤⢡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⢊⠀⠀⢐⠠⠂⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⢠⠀⠐⠀⠂⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⡇⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⣿⢸⡇⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⢸⡇⣿⢰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⢹⡇⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⢸⡏⢿⣄
⠀⠀⠀⠀⢀⣀⣼⡏⡇⢸⠇⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠸⡇⢸⢹⣦⣀⡀
⠀⠀⠀⠀⣿⣿⡿⣇⡇⢸⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⡇⢸⣸⢿⣿⣿
⠀⠀⠀⠀⣿⡿⣧⢸⠇⢸⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡇⡇⠸⡏⣼⢻⣿
⠀⠀⠀⢰⣿⡇⢻⢸⠇⣸⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⣇⠘⡇⡟⢸⣿⡄
⠀⠀⠀⢸⣿⠁⠸⢸⠀⡟⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⢻⠀⡇⠇⠘⣿⡇
⠀⠀⠀⢸⣿⢸⠀⠸⠀⡇⣸⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣇⢸⠀⠃⠀⡇⣿⡇
⠀⠀⠀⣿⡟⠈⠀⠀⠀⡇⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⢀⠀⠀⢹⣿
⠀⠀⠀⣿⡇⠀⠀⠆⢀⠇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⡀⠸⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⢸⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⡇⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠃⠀⠀⠀⢸⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠀⡇⠀⠀⠀⠘⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠸⠀⠂⠤⣀⡒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⣀⠤⠐⠀⠃⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢄⣀⣀⠈⠙⠲⢭⣓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⡭⠖⠋⠁⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⣿⡟⠀⠀⡠⠀⠠⠀⠤⠤⠈⣉⣙⡒⠦⢬⣙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⣋⡥⠴⢒⣋⣉⠁⠤⠤⠤⠄⠀⢄⠀⠀⢻⣿
⠀⠀⣿⡇⠀⠐⠀⠀⠀⠀⠠⠄⣀⣀⠒⠩⠭⣓⡚⠿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⢓⣚⠭⠍⠒⣀⣀⠠⠤⠀⠀⠀⠀⠢⠀⢸⣿
⠀⢠⣿⡇⢄⣀⠀⠤⢬⣍⣐⠒⠦⢬⣍⡓⠲⠤⣍⡙⠲⢬⣙⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⣋⡥⠖⣋⣩⠤⠖⣊⣩⡥⠴⠒⣂⣩⡥⠤⠀⣀⡠⢸⣿⡆
⠀⠰⣿⣷⣦⣬⣉⣙⠒⠦⢬⣍⣓⠒⠤⣍⣙⠒⠤⣍⡑⠲⢌⡙⠢⣍⠻⣷⡄⠀⠀⢠⣾⠟⣩⠔⢋⡡⠔⢊⣩⠤⠒⢋⣩⠤⠒⣊⣩⡥⠴⠒⣋⣉⣥⣴⣾⣿⠆
⠀⠀⠈⠉⠛⠛⠿⠿⣿⣷⣶⣤⣌⣉⡑⠒⠬⠉⡀⠂⠈⠀⠀⠀⠀⠀⠁⠀⠙⣦⣴⠋⠀⠈⠀⠀⠁⠀⠈⠁⠐⢂⠉⠅⠒⢚⣉⣡⣤⣶⣾⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠉⠁⠒⠒⠦⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣶⡖⠛⠋⠉⠉⠉⠓⠦⢄⡲⠶⢦⡄⠀⠀⠀⠀⠀⠀⣠⠴⢋⣉⣭⣷⠶
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣩⣿⣶⢤⣀⠀⠀⠀⡀⠀⠉⠐⢒⠠⠤⣄⣤⣶⣿⣯⣽⣿⣿⣿⣶⣾⡇
⠀⠀⠀⠀⠀⣿⠲⠤⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣀⣉⣲⣦⣤⡽⠒⠒⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⢸⡷⢦⣄⠉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡄⢸⣿⠿⠿⠿⠿⠛⠉⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⢉⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣤⠧⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢏⡓⠦⢍⣐⠒⠤⠤⢙⣫⡭⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠒⠈⠉⠀⠈⠑⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠬⣍⣙⠒⠒⠶⠊⣡⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠄⠒⠚⠻⣥⣀⠀⠈⠁⠚⠳⠦⠀⠀⠀⣀⣀⣤⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠋⠀⠀⠀⠀⠀⠀⠀⢚⠃⠤⣀⡀⠀⠀⠀⠀⠈⠙⠛⣉⣀⡄⣶⠔⠚⠀⢸⣿⣿⣿⣿⠿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠂⠀⠉⠓⠢⠴⢶⣾⣿⣿⣿⣿⡇⣿⠀⣀⣠⣼⣿⣿⣭⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡒⠾⣿⡿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠤⣀⠀⠀⢻⣿⣿⣿⡿⠿⠛⣓⣽⡝⠛⠉⣠⣿⣿⣯⠄
⠀⠀⠀⠀⠀⢐⡁⠠⢤⣤⡽⠢⠬⣭⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠇⣀⠈⠁⠂⠚⠋⠡⣠⣶⣿⣿⣿⠷⠒⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣋⣉⠒⠶⣿⡿⣿⡁⠤⢤⣤⣿⣇⠀⠀⠀⠀⠘⡏⠓⠒⠤⢍⣁⠀⠀⠀⠨⢿⣿⣿⣿⣿⠖⠂
⠀⠀⠀⠀⠀⢀⡓⠲⠶⣿⡿⠦⢍⣙⣿⣿⣿⠅⣙⣻⣿⣛⣿⡄⠀⠀⠀⠀⠻⠄⣀⠀⠀⠀⠉⠁⣒⣺⢽⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠌⣙⣛⣿⣟⡓⠒⠶⢿⣯⣿⡗⠰⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠦⠗⠋
⠀⠀⠀⠀⠀⠀⠒⠢⢬⣿⣯⣥⣐⣒⣶⣶⣿⣧⠙⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠢⢬⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⣻⣿⢿⡿⠷⠖⠒⠐⠶⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⣎⠻⡀⠀⢀⣴⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠻⣿⣿⡄⣷⠀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣍⣭⣿⣿⣿⠿⠿⠆⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢡⣾⣿⣿⣿⢿⠋⢿⣧⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠃⠞⣀⢈⡉⠃⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢀⣤⣿⣾⣧⣄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡇⠀⣾⣿⣻⢸⡟⠛⠃⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠙⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠿⠄⠀⢀⢸⡟⣿⣿⣆⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠐⣷⣶⣶⣦⡤⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣇⣧⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠛⠿⣿⣷⣴⣭⣿⣿⣿⣿⣷⣆⣠⠤⢲⣿⡏⢻⡿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⣸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡽⢛⣫⢤⣤⣼⣿⣷⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆
⠀⠀⠀⠀⢈⣿⣷⣶⣶⡦⣤⣤⣉⡙⠃⠀⠀⣀⠤⠊⣹⣿⣦⠀⣸⣦⣤⡄⠀⢀⣀⣤⠖⠋
⠀⠀⠀⢠⣤⣀⣉⢉⠉⠛⠛⠻⠿⢿⣿⣟⣉⡤⠔⣲⣿⣿⣿⣿⣶⣺⣾⡆⣲⠿⠉
⠀⠀⠀⠘⠿⠿⢯⣿⣿⣶⣶⣤⣤⣀⡤⠟⠓⢊⡉⠭⠃⠀⠈⡉⠉⠉⢹⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠒⠉⠁⠀⠀⠀⠀⠀⣓⡿⠀⣚⣅⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣯⡀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⡴⠦⠌⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣄
⠀⢿⢿⣦⡇⠀⠀⠀⠀⢠⣶⣾⣷⢄⣀⠀⠀⠀⠄⠀⢀⡀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣏⠀⠀⠀⠠⠦⡐⢿⣾⠋⠀⠈⠙⠻⡀⠀⠀⠀⠀⠐⠖⠚⠻⠟⠛⠇⠀⠀⠀⠈⠀⠀⠀⠰⠶⠶⠶⠦⠑⠺⠾⣷⣮⡂⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⢸⣧⣶⣦⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣗
⠀⠀⢦⣈⡤⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⣄⠀⠀⡆⠘⣿
⠀⠀⠙⠿⠿⠶⠿⣿⣿⣿⡞⢹⣿⣿⣿⣿⡏⠀⠀⠀⠐⣦⣀⣄⡀⠀⠀⠄⢀⡀⠀⢠⠀⠀⠀⠀⠀⣼⣿⡾⡣⠊⢿⣷⣿⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⣀⠙⣿⣿⣿⢆⣴⣾⡆⠀⠈⢿⣿⣷⣤⣸⣾⣼⣧⣤⣿⣇⣾⣤⢠⣾⣿⣿⣿⣜⢄⡀⠊⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠛⢿⣻⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣦⣂⣤⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠠⢸⣿⣿⣿⣧⡈⣻⠟⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣀⣈⣛⢿⣿⣿⣿⣿⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣄⣄⢿⣿⣸⣿⣿⣏⣀⣜⣿⣿⣿⣿⣿⣿⣿⣿⡟⡉⢘⡩⠙⠉⢿⣭⣏⣹⡾⣿⡟⠛⢿⣮⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢭⣶⣿⣿⣿⢧⠉⠁⠀⠈⠉⠉⠛⠛⠻⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⢳⣤⣿⡿⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣮⠀⠀⠹⣿⣇⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⠀⠀⠀⠀⢹⣿⡷⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠁⠀⠀⠀⠀⠀⣟⠣⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣯⡝⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡡⠀⠀⠀⠀⠀⠀⠀⠓⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠰⡟⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠂⠉⠉⠩⠐⠒⠒⠢⠤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⠤⠐⠒⠒⠈⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠉⠉⠛⠒⠒⠶⠦⠤⣤⣀⣀⣀
⠀⠀⠀⠀⡎⡀⠀⠉⠉⠒⠒⠲⠤⢤⣄⣀⣀⠀⠀⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠬⢿⡟⠁
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠁⠀⠀⣼
⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⢤⡴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣠⣄
⠀⠀⠀⠴⣴⣖⡛⠛⠉⠓⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣀⣀⣀⠀⠀⠀⠀⢀⣤⣶⠾⠟⣋⣁⡠⠭⠽⠗⠲⡀
⠀⠀⠀⠀⢹⠉⠉⠛⠲⠦⣄⣀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠈⢻⣷⣤⡶⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⣀⣀⣉⣩⡿⠷⠶⠶⣦⣤⣿⣤⡶⠿⠛⡿⠇
⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡞⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⣀⣠⣽⠿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠖⣲⣿⣓⡶
⣰⡿⠛⠻⠾⢷⣤⣍⣻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⠒⣋⣉⣭⣥⡴⠶⠶⠛⠛⠛⢹⡏
⣿⠃⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣶⣤⣀⡀⢠⠏⣌⠀⠀⢀⣀⣀⣤⣴⠶⠶⠒⢛⣋⣉⣭⣥⡴⠶⠖⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣾⣟⣋⣩⣭⣤⣶⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀
⠀⠛⠿⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡶⠶⠞⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣤⣄⣀⠀⣀⣆⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⠾⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣯⣵⡶⠶⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢩⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⣶⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⢀⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠈⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⡿⡿⣿⣿⣶⣶⣦⣄⣀⣄⡀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠑⣿⣯⣳⣶⡈⠉⠀⠀⠀⢀⠍⢉⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⣧⠈⣳⡀⠀⠀⣴⡧⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣯⣿⣇⢻⣧⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⣻⣾⣿⣿⣿⣿⣿⡤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠟⢛⣿⣿⣿⣯⣭⣭⣡⣤⡞⢔⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣀⣴⣿⣿⣿⣿⣿⢃⡏⠀⠻⣿⣆⠑⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⠃⠼⠃⡀⢀⡙⠈⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⢀⣤⣿⣼⣧⣄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢰⣿⣿⣿⢻⣿⣿⣷⡄⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣿⣿⣿⠀⠀⢿⣿⣿⣾⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣟⣶⠀⠈⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠻⠿⣿⣿⣿⣥⣄⡀⠀⠀⣿⠟⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⢠⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠔⣷⣿⡇⣿⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠠⠼⢿⣿⣿⣿⣿⡉⠛⠿⣶⣶⣦⣤⣤⣄⣀⠀⠐⠀⠀⢀⣠⡤⣿⣿⣷⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠜⣇
⠀⠀⠀⠀⢀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣥⡶⠞⠋⢹⣿⣿⠈⣿⠏⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣭⣯⣿⣿⣿⣿⣿⣷⣾⠿⢛⣡⣤⣤⣴⣾⣿⣿⡷⠢⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠉⠉⢻⣿⣶⣤⣤⣤⣤⣉⡉⠉⠙⠻⠗⠊⠉⠁⠀⣀⣨⠝⣻⣿⣯⣆⠤⣀⣧⣄⣀⣀⠠⠄⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⣯⣯⣿⣿⣿⣶⣦⣤⣤⠀⠈⠁⣀⣀⣼⣿⣿⣿⣶⠒⣙⣛⣻⡓⠀⣠⣶⠶⠛⠁
⠀⠀⠀⠀⢸⣿⣶⣶⣤⣤⣄⣀⠀⠀⠈⠉⠙⠛⠻⢿⣧⠴⣒⣋⣭⣽⡿⠿⠿⣿⣿⣿⣯⣯⣿⡏⠨⠟⠋
⠀⠀⠀⠀⠈⠛⠛⠷⠶⢿⣿⣿⣿⣷⣶⣶⣤⣤⣤⠞⠂⠉⣉⡠⠤⠖⠃⠀⠀⢠⣀⠀⠀⠀⢸⠓⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣧⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⢰⠣⢿⠅⠠⠾⣍⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣿⡂⣐⡛⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣼⡥⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠈⠉⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣂⣠⣄⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⢄⠀⠀⣾⢋⣨⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠑⢄⣸⠓⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⢀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡉⢱⢀⡤⡤⠊⠀⠀⠁⢀⣠⠛⠓⣄⡀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢳⠀⠀⠸⠖⠁⠀⠀⠀⠀⡎⢿⠙⢻⠛⠛⣿⣶⡄⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣭⡠⠂⠀⠀⠀⠀⠀⠀⠀⠃⠾⠶⠾⠶⠶⠷⠝⠃⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢯⡥⠔⢺⠀⠀⡤⠹⠀⠦⠀⠀⠀⠀⡾⠑⠹⡄⠀⠀⠀⠠⠎⠞⠰⡄⠀⠀⡥
⠀⠀⠀⠀⠀⣠⡼⠁⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⢀⡞⠀⠀⢀⠀⠀⠀⠀⠀⠈⢤⠀⣄⣻⠀⠀⠀⢰⣇⣀⢀⣟⠀⠀⠀⣟⠀⠀⢸⠀⠀⠀⡛⠁⣤
⠀⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣀⣴⣤⣆⢩⠀⠀⠀⡾⠀⢹⠀⣿⠀⠀⠀⡟⠀⠀⢸⡄⠀⠈⡇⣀⡿
⠀⠘⢤⡀⠀⠀⢀⡘⠤⠀⠠⠃⠀⠈⠉⠉⠁⠈⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠉⠀⠁⠈⠀⠀⠀⡀⠀⠑⠚⠂⡀
⠀⠀⢠⠃⠀⠀⠀⢘⡧⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠤⠤⠤⠤⠤⠷⠦⠈⠈⠤⣀⠝⣐⠇
⠀⠀⠀⠇⠀⢀⡴⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⢨⡗⠣⠀⠂⠀⠀⠀⠂⠀⠀⢸⣿⢻⡏⣷⢹⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⢂
⠀⠀⠀⠈⢻⠁⠀⠀⠀⠈⠉⠗⠀⠀⢸⠀⠇⢰⡅⠿⠁⠀⠐⢶⠲⠀⢰⠶⠆⠀⡀⠀⠒⠒⠒⢒⡒⠂⠒⢶⠒⠒⠶⡶⠃
⠀⠀⠀⡀⢸⠗⢀⡠⠠⡄⠀⠀⠀⠀⢸⣀⣀⠀⣀⣀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⡏⠉⡡⢬⠉⠉⡛⠀⠀⠀⠱⣤⠀⡇
⢐⣿⣾⣿⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠛⣹⠿⡉⠙⡇⠀⠀⠀⠘⠀⢀⣀⡀⠀⠇⢘⠁⠀⣡⠀⡇⠀⠀⠀⠀⠈⠐⡇⢀
⢹⣿⣿⣿⣿⠀⢀⠀⠀⡇⠀⠀⠀⠀⢸⡆⢄⣀⡀⣰⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠘⠤⡤⠁⠀⡇⠀⠀⠀⠀⠀⢸⣇⡟⡇
⠀⢠⣿⣯⣽⡀⢸⡄⢰⣇⠀⠀⠀⠀⠞⠛⠒⠀⠐⠛⠓⠀⠶⣦⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣹⣿⠃⡇
⠀⠀⠉⠀⢸⠇⠉⠉⠉⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣷⣟⣠⡀⠀⣀⣀⣠⣄⣀⣀⠀⠀⣀⣿⡀⠀⠀⠀⣠⣿⡇⠀⢻
⠀⠀⠐⠾⠿⠿⠶⠶⠖⠀⣶⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⠋⠉⠉⠉⠉⣙⠟⠀⠀⠀⠀⢘⣿⠋⠀⠀⠈⠉⢻⡅⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠒⠶⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣧⣀⣀⣀⣀⡤⠭⠤⠤⠤⠭⠭⠭⠍⠧⣤⣤⣤⣤⣼⣧⣤⣤⡌⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢯⠥⡭⠽⠆⠀⠈⠓⠒⠒⠒⠀⠀⠀⠀⠀⠈⠁⠰⣞⠩⠄⢩⠭⠆⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠐⠲⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⡀⠀⣼⠃⠘⡀⠀⠀⠀⠀⠀⠀⢀⣶⢟⠛⠻⣦⠀⠀⡄⠀⠀⠈⠒⠈⠀⠀⠀⠀⡀⠤
⠀⠀⠀⢀⡉⠩⣿⣷⠦⢀⠀⠀⠀⡗⠉⠉⠁⠐⠛⠒⠒⠂⠈⠃⠀⠘⣿⡆⠄⣰⣿
⠀⠀⠀⠚⠁⠀⢻⣿⡄⢀⡽⠒⠂⠑⠂⠐⢇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠇⠀⠀⠀⠀⢰⠇⠀⢠⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢽⡀
⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠠⠌⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⡄⠀⢀⣾⠁⠈⣷⡄
⠀⠀⠀⠄⠀⠀⠀⢰⡀⠀⣀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣀⠄⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⢿⣿⣿⡆⠀⠁⡟⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠐⡿⡄⠀⠀⠀⣴⠈⠀⠀⠙⣧⠀⠈⣧⣤⡴⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠰⠆⠀⠐⡂⠀⠀⠀⠿⣷⣤⡀⠀⠹⠾⠀⠀⠠⣇⠀⢀⠀⣿⡇⠀⠀⠀⠀⠤⣤⡾⠛⠁⠀⠀⠀⠘⠁⠈⠃
⠀⠀⣃⠀⠀⣴⡀⠘⠁⠁⠀⠀⠀⣓⠷⢿⢿⠇⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠐⠒⠤⣄⠀⢰⣄
⠀⠀⢿⣷⣤⣽⣿⡿⣿⡇⠀⠀⠀⠉⠀⠙⠃⠀⠠⠀⠤⣤⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠈⠛⠋⠁⠙⠛⠛⠁⠀⠀⢠⠀⣀⠠⠄⡄⠀⠀⣀⠔⠛⠛⢉⣿⣿⣿⣿⣿⣿⡿⠿⢶⣄⠀⠀⡀⠘⠂⢰⣦⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⢻⡿⠃⠈⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡯⠄⠀⠀⢸⠐⠷⠀⠀⢀⠀⠀⠀⠀⢠⣤⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⣀⡀
⠀⠀⠀⠀⠀⣰⣧⣤⠀⠀⠀⠀⠀⠟⠇⠄⠀⠆⢲⠄⢠⣴⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣿⣷⡶⠂
⠀⠀⠀⢀⣼⣿⣧⢰⡆⠀⠀⣐⡼⠀⠀⠱⢠⠀⠀⠀⢸⠃⢌⣻⣷⡆⠈⠀⠀⠰⢦⣀⠀⢠⣠⣄⠀⢰⠁⠀⠀⠈⠿⢻⠁
⠀⠀⣀⣾⢿⣿⣿⣋⠀⠀⠚⠀⠀⠀⡜⠀⠆⠀⠀⠀⠸⠀⠈⢧⠹⡇⠀⠀⠀⠀⢸⠃⠀⠈⣿⠋⠀⡏
⠀⠈⠁⠀⠈⠉⠉⠁⠀⠀⠀⠈⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⣶⣶⣶⢶⠶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⠶⣶⣶⢶⣶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⡠⠄⠒⠈⣁⡠⠬⣆⠀⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣠⠴⡠⠒⠉⠑⠢⢀
⠠⢶⡋⢉⡠⣤⣲⠯⠝⠒⢉⣁⡠⠂⠀⠀⠀⢀⠔⢀⣀⣠⠤⠔⠒⠉⠁⠀⠑⢅⠀⠀⠀⠀⠀⠉⠒⢤⡀
⡇⠃⢸⡔⠿⣅⡠⠤⠒⠋⠉⢐⠳⠄⠀⢀⠔⠁⢰⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⡇⠀⣿⠀⠀⡛⠀⠀⠀⡠⢊⠥⣴⣤⠶⠓⡂⠄⡘⢆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⣨⠗
⡇⠀⣿⢀⠀⡇⠀⠀⣸⡖⣁⠤⢼⣷⣉⡑⠀⠀⢈⡊⠳⡄⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣠⠴⠚⣁⢜⣦
⠧⢯⠏⢸⢀⡇⠐⣫⣄⠉⠀⠀⠀⠀⠀⠉⠳⢌⡀⣧⣀⡽⢆⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⠯⣀⠔⡪⠖⠁⠈⠑⠦
⠀⠀⠛⡤⣼⢷⡛⢟⡢⡙⠲⢤⡀⠀⠀⠀⢀⡤⠛⡞⠣⣀⠀⠳⡀⠘⢦⠤⠴⠖⠒⠋⣉⡡⢤⣳⢮⡕⠉⠀⠀⢀⣠⠴⣾
⠀⠀⠀⠀⠘⣾⢍⠓⠬⣲⣕⠄⡈⠑⢦⠊⢁⡠⣾⠷⢄⡈⠳⢄⡙⢶⡇⠅⢀⣒⡦⠭⠒⠊⠁⢀⣠⡟⣀⡤⠞⠉⣠⢔⡩
⠀⠀⠀⠀⠀⠀⠑⠭⡢⢈⠙⠻⢦⡤⠾⢖⡩⢞⠉⠢⣀⠈⠢⣄⠉⠲⣌⠉⠉⠀⠀⠀⣠⠴⠚⣉⢔⣻⣇⣀⠔⣫⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠥⣀⢸⡧⠖⠉⠀⠈⠲⢄⠈⠓⢤⡈⠑⢦⡀⠙⠢⡤⢖⠋⢀⠤⡪⢔⡩⢶⣋⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠉⠳⢤⡈⠳⢴⣇⢀⣊⣵⠿⠚⢉⡤⣚⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠉⠲⢄⡠⠞⠋⢀⡠⢖⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣸⡪⠁⢐⡭⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠔⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⠔⠁⡠⠊⠑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⡱⠋⡠⠊⠀⣀⡀⠀⠑⢄⠑⢄
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⡠⠊⢀⠰⡊⠀⠀⡱⢄⠀⠑⢄⠑⢄
⠀⠀⠀⢀⣌⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠝⡠⠊⠀⢠⣦⣤⣬⣶⣮⣤⣤⣵⠀⠀⠑⢄⠑⢄
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡡⠊⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⡅⠀⠀⠀⠀⠀⠑⢄⠱⡄
⠀⢀⡊⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣣⣺⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠓⠤⠃
⠀⠚⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⢸⠁⠉⠁⠉⠉⠉⠉⢨⠅
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡀⡀⠀⠀⠀⠀⣀⢂⣀⣀⣀⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⡄
⠀⠀⠀⢸⠀⠘⠍⠉⠍⠉⠝⠀⠀⠀⠘⡍⠉⠍⠉⠝⠀⢀⠞⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣞⡔⣦⣴⣤⣴⣶⣶⣦⣴⣴⣤⣤⣤⣦⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤⣼⠆
⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⡂⠀⠀⠀⠀⡇⠀⠀⠀⡃⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠠⠤⠤⠤⠤⠤⢤⠂⠀⠀⣀⢀⡀⣀⢀
⠀⠀⠀⢸⠀⠀⠍⠉⠉⠉⠅⠀⠀⠀⠀⠏⠉⠉⠉⠅⠀⠀⠀⠈⡏⠉⠉⠉⡏⠀⡁⠐⠀⠀⠀⢰⢸⠆⠀⠈⠂⠀⢐⠀⠚
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⢸⠂⠀⠀⡄⠀⢸
⠀⠀⠀⢸⡀⣀⢀⢄⣀⣀⡀⢀⠴⣄⣀⠀⠀⢀⣀⣀⢀⢠⠀⠀⡇⠀⠤⠀⡇⠀⠀⠀⠂⠀⠂⢰⢸⠀⠀⠀⣇⠀⢸⠀⡄
⠀⠀⠀⡜⠁⠀⠉⠀⢀⠈⡈⠉⠀⠀⠁⡑⠊⠁⠀⠈⠀⠁⠀⠀⠓⠂⠐⠂⠃⠀⠀⢀⡀⠀⠀⠈⢸⠄⠀⠀⠓⠀⠂⠀⠂
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠁⠀⠁⠀⠘⠂⠀⠀⠀⠠⡄
⠀⠀⠀⠐⠒⠐⠒⠀⠈⠀⠐⠒⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠀⠠⠀⠈⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠛⣉⣡⣴⣾⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⡿⠿⠛⠋⠉⣁⣤⣴⣶⠿⠛⠋⣁⣤⣴⣿⣿⣿⣿⠿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠟⠛⠋⠉⢀⣀⣤⣶⡾⠿⠛⠉⣁⣠⣴⣶⡿⠿⠛⠋⠉⣁⣠⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠉⠁⠀⣀⣠⣴⣶⡿⠟⠛⠉⣀⣤⣴⣶⠿⠛⠋⠉⣀⣀⣤⣶⣾⣿⣿⠿⠿⠛⠋⠉⠁⠀⣿
⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠀⢀⣀⣤⣶⣾⡿⠟⠛⢉⣀⣤⣴⡾⠿⠛⠉⢁⣀⣤⣴⣾⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⣾⡟⠛⠉⠀⠀⢀⣠⣤⣶⣾⡿⠟⠛⢉⣀⣤⣶⠾⠟⠛⢉⣀⣤⣴⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣷⣤⣴⣶⣿⡿⠟⢋⣉⣠⣴⣶⡿⢛⣋⣡⣤⣶⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣿⣿⠉⠛⠿⢿⣿⣶⣿⣿⣿⣿⣿⠿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠁⠙⠛⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇
⣿⣿⡏⠉⠛⠻⠿⠿⣿⣿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠋
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠿⠛⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠛⠋⠁
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣤⣶⣾⣿⣿⠿⠛⠉⠁
⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠙⢿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡖⠊⠉⢸⡀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⣶⣶⠖⠲⠲⠾⡟⠀⠀⠀⠸⣇⠀⠀⢠⠀⠀⠀⠘⣆⢨⣅⣀⠀⠀⠀⠀⠬⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠠⠉⠀⠀⠀⠀⢻⠀⢀⠇⠀⠀⠀⠀⢹⡂⠉⢳⣴⣿⠿⠛⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢹⠀⢸⠂⠀⠀⠀⠀⠀⠳⣀⢼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣢⣿⠀⠀⣤⠄⠄⡈⠄⠀⠄⠀⢈⣀⣨⣄⣠⣀⣾⠟⠉⠈⠀⠾⣿⣿⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢿⡇⠀⠀⠀⢀⣠⠼⢿⠶⢲⠞⠋⠁⠀⠀⠀⡄⠈⠉⠣⢅⣀⡈⢿⣿⣿⢈⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣟⣛⣀⡨⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠻⠁⢸⠈⠉⠉⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠻⠷⢤⠞⠁⠀⠀⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢻⣦⣘⣊⡠⠌⠉⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠤⠤⠤⠎⠀⠀⠀⣼⠟⣿⠋⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢻⣿⣿⣍⣂⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⢸⠁⠀⠀⠀⠀⠠⠀⠀⣏⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣟⠡⠄⠐⡚
⠀⠀⠀⠀⠀⠄⠀⠐⠛⠢⠤⠤⠤⠴⠚⡄⠀⠹⣾⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢨⣈⣿⣧⡈⠴⠃
⠀⠀⠀⠉⠀⠀⠀⠀⢀⠁⠀⠀⠐⠒⠉⠈⠀⠀⠸⣿⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⣿⠏⠁⣄
⠀⠀⠀⠀⠀⠆⠠⠠⠄⠤⠀⠈⠉⠀⠀⠂⠀⣄⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡘⠀⠁⢀⣾⡿⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠜⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢷⣼⣆⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⢀⣏⠀⣴⢿⠃⡀⢒⠢⢹⣿⣿⠇⢠
⠀⠀⠀⠀⠀⢰⡂⠀⢲⣃⣀⣀⣀⡀⠀⠀⠀⢹⣄⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣰⣿⣿⣿⣷⣏⡳⢬⣝⣥⣾⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠭⠦⠠⠤⠥⠤⠄⢘⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣿⣿⡿⠊⣻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠂⠒⠀⠀⠛⠻⠷⣿⣿⣿⣶⣤⡤⠀⠀⠀⠀⠀⠔⠈⣿⠟⢀⣾⡿⡋⠈⠙⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠂⡀⠀⠀⠀⠀⠀⠁⣀⣿⡿⢚⡣⢤⡀⢀⢠⣿⡿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢾⣿⣿⠰⣈⣉⡷⢹⣻⣿⠏⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⢷⣬⣭⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠛⢿⠟




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢠⠄⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡇⡀⠀⠀⠀⠀⠉⠉⠁⠁⠂⢤⢀⡀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣅⣠⢀⣀⣀⣀⡀⠀⠉⠉⠉⢲⣂⠄⠀⡏⠁⡈⠻⠐⠖⡄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣉⠉⠉⡉⠘⡇⠜⢠⢸⣏⠇⢸⠇⣠⡀⠈⡇⣸⠇⡜⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠏⠉⠓⠲⠞⠋⠀⠻⠟⢸⠀⡇⠰⡇⢸⢿⠀⣼⠀⣶⠆⢸⢣⡇⢸⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠃⠀⠀⢈⠃⠀⣀⣀⠀⣿⠀⠇⠰⡇⢸⡏⠀⡟⢴⡿⠀⡇⢸⡁⠇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⢸⡄⣀⡀⠀⠈⠓⣊⠀⠍⠀⢺⠀⡃⠸⡇⠀⡃⢸⡃⢰⠄⡾⣹⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⡆⠉⠁⠀⠀⢃⣜⠀⠙⡆⢸⠐⣥⠈⡃⠀⡇⢠⣁⣼⠴⠟⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠄⢱⣁⢢⣔⣆⡀⠑⠞⠀⢠⣇⣸⡄⠷⠤⠧⠐⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠄⠀⠀⠀⠀⠈⠉⠉⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠠⠁⠏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠊⠀⠀⠀⣀⣠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⠀⢀⠈⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠒⠈⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠐⠂⠀⠉⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⢀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡟⠛⢷⣄⠀⠀⠀⠀⠀⠘⠄⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂⠀⠈⠁⠀⠀⠀⠀⣾⣿⣿⣄⣿⡇⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡀⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠙⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠈⠂⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢿⣿⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣷⡀⢸⣿⢀⣰⣿⠿⠃⠀⠀⠀⠀⠀⠐⠠⠠⠂⠀⠂⢂⣈⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢀⡀⠄⠐⠀⠉⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠦⢤⡀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠳⣄⠀⢀⠜⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡱⡄⠀⠀⠄⠀⠀⢀⠀⢀⡜⠁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣦⠤⠬⠷⠶⠥⣤⣿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⣀⣀⣀⠀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠖⠋⠉⠀⠟⡏⠉⠙⠒⢤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠀⠱⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⣿⠉⣟⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣿⢠⡇⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⠇⣿⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢻⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠋⠲⠤⣤⣤⣠⣤⡤⠴⠚⠧⠤⠤⠔⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⡀⠀⢀⠔⠙⢦⡀⠀⠠⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠈⢀⡴⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣀⣀⣀⠀⠀⠙⢢⣄⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣤⡖⠋⠀⠀⠀⣰⢋⠅⠂⠨⡳⡄⠀⠀⠈⠙⢦⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⠀⠄⠀⢴⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⡇⠀⠄⠀⢼⡞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⡇⠀⠀⠀⣸⠣⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠓⠚⠛⠓⠛⠓⠒⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠒⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠐⠒⠒⠶⠖⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠒⠒⠒⠒⠒⠒⠒⠺⡇
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠀⠀⠀⠂⠐⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⢀⡀⢧
⠀⠀⠀⠀⠀⣰⠁⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠀⠈⣇
⠀⠀⠀⠀⢰⣃⣈⣀⣀⣀⣉⣉⣀⣀⣀⣈⣀⣀⣀⣉⣉⣀⣀⣈⣉⣀⣀⣀⣈⣉⣁⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢠⠞⡩⠀⠂⢍⢳⡀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣉⣙⠻⢦⡀⠀⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢾⢸⠀⢀⡀⠀⡇⡇⠀⠀⠀⠀⠀⡾⢡⠞⠁⠀⠈⠙⣆⢻⡄⠀⢸⠁
⠀⠀⠀⠤⣀⠠⠠⣀⠄⢸⡇⠀⠀⠀⢽⢘⠀⠀⠀⠀⡇⡧⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⢸⢸⡇⠀⣼⠠⠄⢀⠠⡀⢀⣀
⠀⠀⡌⠀⠘⠀⠀⠉⠀⢸⡇⠀⠀⠀⣸⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⢺⢸⡇⠀⣿⠀⠀⠃⠀⢸⠁⠀⠆
⠀⠀⡇⠀⠀⠀⠀⢿⠈⢸⡇⠀⠀⠀⠼⠼⠶⠶⠶⠶⠧⠥⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⢸⠁⠀⣿⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠀⢼⠸⠂⠀⣿⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠓⠒⠚⠓⠒⠛⠒⠚⠓⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⢲⠶⠶⠶⠖⠓⠛⠒⠒⠒⠒⠶⣺⠶⠓⠒⠛⠒⠚⠓⠶⠾⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⡄⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⣀⣤⣾⣿⡟⢹⡄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣛⣭⠿⣿⡟⣽⢿⢁⢸⡀⠀⠈⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣖⠶⣒⠭⠀⠼⠛⢛⣠⡞⠟⠈⢋⣿⡊⠀⠋⠦⡀⠀⢬⡳⠀⠐⠦⣄⡀⡄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣎⡁⢜⣃⡤⢒⣿⣟⡴⢠⣴⡿⢻⣧⠐⠢⠀⠻⡄⠀⠁⡰⠄⡀⡂⠌⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⠿⠋⠿⣋⣥⣴⣿⠟⡴⢡⠏⣅⢰⠏⠈⠧⠀⠈⠳⠀⣴⠠⡇⠠⣿⣧⣶⣿⣷⡀
⠀⠀⠀⠀⢠⣾⠟⢉⢁⣠⡌⢉⣤⡊⠁⠀⣴⠃⢠⡀⢠⡇⡄⢃⡐⠀⡆⢠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠛⠋⠙⢷⢀⡀⡀⠁⣿⣇⣸⣿⣷⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⣸⡜⣃⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣦⣉⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⣄⣸⣿⣿⣿⣿⣿⣿⣿⡿⠏⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⡿⠟⠛⠛⠉⠉⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⡿⠀⠀⣀⣀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠑⠠⢄⠀⠀⠠⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠛⠛⠻⢶⣤⣀⠀⠖⠒⠀⠈⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣴⠶⠶⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⠋⠥⠤⠄⣀⣀⡀⠀⠀⠀⠈⠙⠻⠿⠾⠿⠿⠛⠛⠛⢛⡩⠝⠛⠉⠁⠀⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠉⢀⣒⠒⠀⠤⠤⠭⠉⢉⣁⣒⠒⠢⠤⢄⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠒⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢢⠑⠄⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⢠⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣀⠒⠒⠠⠤⠍⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⡰⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿⠻⠶⠶⢦⣤⣤⣍⣉⣉⡓⠒⠒⠒⠊⣰⣥⡶⠾⠛⠉⠀⠀⠀⠀⠀⡔⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠐⠒⠒⠂⠐⠀⠂⠀⠉⠉⠙⠿⠿⠿⠿⠿⠯⠥⠄⠀⠀⢀⣀⣀⣀⣀⠜⠀⠀⣰⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡄⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠤⠈⡀⠀⣼⣏⠄⣊⣉⡳⣄
⠀⠀⠀⠀⠀⠀⡴⠾⢍⣴⠾⠅⠅⠈⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣇⣾⡫⠔⠉⠀⠀⢱⢹
⠀⠀⠀⠀⠀⠀⢹⡗⠲⠿⣶⣶⡶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣍⣉⣉⣉⣁⣴⠟⠉⠀⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⣼⡇⠥⢂⡀⠘⢻⣷⣲⠬⣍⣒⠢⠤⢄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠩⣋⠝⠋⠀⠀⠀⠀⠀⠀⢀⠴⣣⠞
⠀⠀⠀⠀⠀⠀⠓⠬⣍⡒⠦⢄⣀⠈⠀⠁⠀⠀⠉⠛⠒⠦⢭⣑⡒⠢⠤⣀⣀⠤⢚⠵⠊⡠⠀⠀⠀⠀⠀⢀⡠⢊⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣍⣑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠉⠉⠐⠶⣖⡚⢥⢐⠩⣀⣠⠔⢁⡠⢒⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣍⣑⠒⠤⠄⣀⠀⠀⠐⠤⢟⣻⣇⠼⠊⡠⠔⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣉⡑⠒⠤⢄⣈⣡⠔⣋⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⠤⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⣧⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⡀⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⠠⠁⡴⣂⠀⠀⠀⠒⠀⠈⣀⣠⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣆⢂⣡⣼⣾⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠄⠀⢠⣀⠀⡞⢉⡓⠆⠀⠀⣸⣿⡇⠀⠀⠘⣠⡀⠈⠁⢀⡌⠎⠀⠀⠀⠀⢶⡾⠀⠀⠀⡏⠀⢸⡍⢹⡻⢆⠘
⠀⠀⠀⠀⠀⠀⠶⠁⠐⠠⣄⡙⠶⣬⡁⢭⡓⠎⠓⠁⠀⠉⠉⠀⠀⠀⠀⠀⠓⠺⠁⠐⠉⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢀⡀⢹⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣉⡛⢦⡙⠂⠙⠓⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡆⠀⠀⢈⠁⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢡⣤⠀⠀⢶⠆⠰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⠈⠀⠈⣁⠁⢰⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠃⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠊⡁⠈⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⡀⠰⠰⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠶⠂⡛⠀⠀⠀⠀⠑⠂⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠔⠞⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢀⡛⢋⡉⠃⣤⠀⡄⠉⠀⠀⠀⠀⠀⣷⠀⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠚⠃⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣩⡄⠀⠠⠎⢁⠀⠆⠀⠀⠃⠀⠀⠀⠀⠀⠛⢁⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡤⠖⠀⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⢄⠀⠙⠃⠶⠂⢠⡔⠃⢤⢘⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡤⠇⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠃⣍⡃⠀⢈⣁⠀⠀⢠⣦⠀⠨⠾⠀⠀⠀⠀⠀⢻⡆⠀⠰⣾⡿⢀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠉⠰⡿⠃⡈⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠰⣾⠗⢠⣶⠀⠘⡇⠀⠀⠀⠀⠀⠃⠀⠛⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣠⡄⠀⢀⡤⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠆⣠⠀⠀⠀⠀⡀⠁⠀⠀⠀⠻⢰
⠀⠀⠀⠀⠀⠀⢀⡴⠃⢉⠲⠤⠚⠁⠈⠘⠒⠊⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⠇⠀⠀⠙⠀⠉⠀⠀⠀⠀⡅⠃⠀⠀⠀⠀⠀⢢⣤⣤⡀⡈⠓
⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠂⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡶⠶⠶⠶⠟⠻⢯⠟⠉⠯⠙⠏⠁⠀⣨⣭⣭⣭⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠠⠄⠤⠐⠂⢈⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠖⠀⠀⠲⠞⠲⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣶⣶⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢠⣶⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡀
⠀⢠⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠴⠒⠁⠀⠀⠀⣾⣿⣿⣿⡟⠀⠸⠿⠿⣿⣿⣷⣄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⢶⣶⣶⣷⣶⣦⣤⣾⣿⣿⣉⣉⣀⡀⠀⠀⠀⠈⠻⢿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⠛⠛⣋⣹⣿⣿⡿⠿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⣿⡇⠀⠀⠀⢤⡄⠀⠀⠀⠀⢸
⣿⣿⣿⣟⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠀⠀⠙⢿⠾⠃⠀⠀⠀⢠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠛
⣿⣿⣿⣯⣍⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣤⣤
⠱⡉⠈⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠉⠉
⣾⣧⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠉⠓
⣿⣿⣧⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢀⡀⢠⡀
⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⡬⣿⣽⠏⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣧⣸⡿⡻⠂⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠒⠒⡿⠀⠀⠀⣠⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⡇⠀⠀⠓⠒⡲⠞⣡⠖⢤⡈⠳⢖⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠇⠀⢀⡤⢊⡤⠊⠀⠀⠀⠈⠢⣄⠑⢤⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠔⣩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡉⠢⣀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡠⢚⡥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠓⢄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣼⢀⠴⣋⠔⠉⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠑⢤⡙⠦⡀⣇
⠀⠀⠀⠀⠀⣠⣟⣡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣌⣻⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⡷⢦⣄⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⡀⠔⠉⠀⢀⣧⡀
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⣿⡇⠀⠀⠀⠉⠓⠾⢿⣦⣴⣯⣤⡤⠤⠤⠴⠋⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣿⣛⣛⣒⡒⠤⢤⣀⠀⣀⣠⣴⣶⣿⣧⣤⣸⣧⠀⠀⠀⠀⠀⠠⠚⠉⠉⠓⠦⣍⡑⠲⡄⠀⠀⡠⠊⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠐⢠⢀⠀⠀⠀⠉⠓⠿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣂⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣌⠳⢄⡀⠀⠀⠀⠀⠀⠀⠁⢈⡢
⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠉⠻⢦⣉⠲⢄⡠⢄⡠⠔⢊⡡⠔⢀⠤⠄⢀⣀⣀
⠀⠀⠀⠸⡉⠓⢄⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⢀⣴⡵⣛⠵⡀⠀⠀⠙⠻⣶⣟⣥⠖⠊⠁⢀⠔⠁⠀⠀⠀⠀⠀⠈⣠⡆
⠀⠀⢠⣦⠑⢆⠀⠑⠄⠀⠀⠀⠙⠶⣄⠀⢀⣾⣿⣮⣽⣿⡧⠀⠀⠀⠀⠀⠁⠀⠃⢀⠔⠉⠀⠀⠀⠀⠀⠀⢠⠞⡱⠃
⠀⠀⠀⠙⢦⡀⠑⢄⠀⡊⠉⣍⣠⣤⣬⣷⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠭⢄⣀⡀⠀⠀⠀⢀⠔⣡⠊
⠀⠀⠀⠀⠀⠙⢦⠀⢐⣈⣛⣯⠽⠛⠋⠉⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠛⠢⠤⢌⣉⠁⠒⣋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠀⡀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠
⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⠃⠀⠀⠀⠀⠀⠀⣧⢀⠀⠁⠄⢀⡈⠁⣴⠆⠀⠊⢀⠰⠀⠀⠄⠉⠀⠀⠉⠀⠀⠐⠀⢀⠘⢀⣄⠀⠈⠁
⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠃⠀⠀⠂⠠⠄⠀⠀⠀⠀⢠⡔⠀⡟⠟⢉⣤⠀⠁⠀⠁⠤⠀⠀⡠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠜⠇⢤⣄⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⣀⠀⢘⠃⠀⠉⣀⡈⠉⠀⠠⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠲⠄⠀⠀⠄
⠀⠀⠀⠠⡄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠘⠀⣴⡎⢹⣿⠆⠀⠀⠁⠀⠄⠀⠈⠐⠀⠐⠛⠀⠀⠀⠀⠀⠀⠚⠀⠐
⠀⠀⠀⠀⠐⠂⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠦⠷⠈⠁⣀⠀⠉⠙⠃⠀⠀⠀⢀⢀⡀⢠⡀⠋⠀⠀⠀⠀⣀⠀⣀⠈⠉⠙⠁
⠀⠀⠀⡀⡀⠀⡀⠀⠀⠠⠀⠀⠋⠀⠀⠀⠊⠀⠀⣠⡄⠀⠀⠀⢠⣶⣤⡹⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡄⠀⠲⠀⠀⠈⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠘⠀⠀⢀⠀⠈⢿⣽⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠈⠀⠂⠀⠀⠂⠈⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⢈⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠈⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠈⠁⢀
⠀⠀⠀⠀⣀⠀⠀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢁⣴⣿⡿⠿⡿⠿⢿⠿⢿⣷⣄⠀⠀⠀⠠⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉
⠀⠀⠀⠨⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⣾⣿⣿⣷⣾⣷⣶⣶⣿⣿⣿⣿⡆⠀⠀⠀⠙⠛
⠀⠀⢠⣄⣀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠁⠁⠈⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⢿⣿⡿⠿⣿⠿⡿⢿⡿⢿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠐⢂⠻⠀⣄
⠀⠀⠀⠠⠆⠈⡤⢀⠀⠀⠀⠀⠀⠘⠀⣶⣾⠁⠀⠀⠈⢿⣤⣦⣿⣼⣵⣶⣧⣼⣾⠟⠀⠐⠀⡀⠀⠐⠀⠀⠀⡅⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠚⣷⢤⣤⢄⠀⠀⣀⠀⠀⠀⡀⣀⡀⠀⠉⠛⠿⠿⠿⠿⠿⠋⠁⠀⠐⠀⠀⠉⠉⠀⠀⠀⠀⠀⣤⢀⠀⠀⢀⣀
⠀⠀⠀⠐⠛⠄⠀⠀⢽⠚⠓⠶⠀⠀⠀⠀⠀⣀⠠⢛⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠄⠀⠀⠘⠃⢀⠉⠀⠛⠀⠀⠸⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠘⠚⠀⠐⠑⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠒⠀⠓⠀⠀⠉⠀⠠⠛⠄⢀⡄⠀⠀⣠⡁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠈⠙⠉⢀⠀⠠⠀⠀⠀⠀⠀⡄⡀⠀⠐⢆⠧⠀⠘⠤⠞⠀⠀⠈⠀⠇⠹
⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⢀⣀⠀⣸⣅⠀⠀⠀⠀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠙⠋⠹⠆⠀⠂⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠒⠓⠀⠀⠐⠢⠀⡀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡅⠀⠀⠀⠀⠐⠂⠀⠒⠒⠀⠐⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⢙⡁⡀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⢀⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⢀⠘⠙⠁⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⣄⠀⢈⠀⠀⢀⠀⠀⠀⠹⠈⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠂⠂⠀⠤⠄⠠⠆⠀⠀⠠⠐⠂⠠⠄⠐⠿⠂⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⢤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣀⠀⠀⠀⢀⣠⣶⡋⠁⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠿⠛⠛⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⠎⠁⠀⣠⣶⣿⣿⣿⣷⣦⣀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣠⣾⣿⣿⣿⠛⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣉⠀⠀⠀⠀⠀⣀⣀⣠⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣠⣾⣿⣿⣿⣿⣿⡿⠆⠈⣩⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣡⠔⠋⠁⠀⣀⣤⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⣾⣿⣿⣿⣿⣿⣷⣴⡖⢾⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣀⣤⣶⣿⣿⣿⣏⣿⠏
⠀⠀⠀⠀⠀⠀⡠⠏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣉⡉⣠⣤⣹⣿⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⢀⣠⣴⣾⣿⣿⣿⣿⣟⣻⣿⠃
⠀⠀⠀⠀⢀⡴⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠈⠙⣿⠇⠀⠀⠀⣀⠴⠚⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠃
⠀⠀⠀⣰⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣖⠛⠿⣿⣿⠀⢀⡠⠞⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡏
⠀⣠⡾⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⡛⣁⣤⣽⡿⠿⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠁
⠀⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠹⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣍⣡⣾⠃
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⠻⣷⣿⣿⣿⣇⠀⠀⠈⠻⣿⣿⠋⢠⣤⡄⠈⣿⣿⣿⡿⣿⣿⣿⣏
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣨⣭⣽⣿⣯⠀⠀⠀⠀⠈⠛⢦⣈⠙⠀⣠⣿⣿⣿⣿⣤⠹⢿⣿⡄
⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣇⠀⠀⠀⠀⠀⠈⢿⣷⣾⣿⣿⣿⣿⣯⣤⣄⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠋⠉⠛⠿⠿⠏⠿⣿⡷⠙⢷⠀⠀⠀⠀⠀⠘⣿⣿⠿⠟⠛⠛⠋⠙⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⡄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣧⣾⠀⠀⠀⠀⠀⠀⢹⣷⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⡇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣧⣜⠁⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⢈⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⠃⠀⠀⠀⠀⠀⠈⢋⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢠⡎⡀⠀⢀⣠⣼⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⢿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣿⣿⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣤⠀⠀⠀⣿⣿⠇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣀⣦⣼⣿⡿⢰⠀⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣄⣠
⣿⣿⣿⣿⣽⣶⣆⡴⠟⠀⠆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠂⢠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣄⡀⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⠀⠀⠂⠈⠈⠀⠠⠁⡀⠀⠀⠤⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠈⠂⠀⠀⠀⠠⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠅⠘⠦⠀⠀⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠚⠚⠀⣂⣀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⣾⣿⢛⠁⠀⠀⠁⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠡⠀⠠⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡀⠠⠀⡂⠀⠈
⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠁⠀⠀⢀⡔⠉⠀⢀⠀⠀⠀⢄⠀⠀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾⡏⠁⠀⠀⠀⠀⢀⣄⠱⡄
⠀⠀⠀⠈⠁⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠉⠹⠿⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⠿⠛⣳⣒⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠐⡇⡾⣫⠾⣧⢤⡄⠀⢠⢞⠿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠋⠉⠓⢡⣠⠁
⠀⠀⠀⠀⠈⣷⡀⠀⢳⡀⠀⣀⢿⡟⠡⡎⠀⡄⡇⠀⢸⡘⠆⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠉⢀⣤⠞⠿⠃
⠀⣀⠄⠂⢀⣿⣿⡆⠉⢷⣶⡏⢉⢧⠀⢻⠓⢣⠇⠀⣾⣿⠆⠀⠀⠀⠀⠀⢰⣀⠀⠀⢀⢁⡀⢹⣄⠠⡟
⠸⡿⠀⠀⡏⠉⠈⢷⠀⠀⠈⠻⡌⣧⢳⣀⡴⠃⠀⣰⡿⢿⠂⠀⢠⣶⣶⣶⠇⠈⠂⠀⣸⠉⠉⢐⠨⢧⣀
⠀⠀⠀⠀⠀⠠⠀⠈⠇⠀⢀⣴⡷⠸⠋⠁⠀⢀⡼⣯⣟⡞⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⣇⠀⢰⡏⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠁⠀⠀⠀⣔⣭⣿⣮⣛⠁⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠉⠳⠚⠷⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢎⠟⡟⡝⣿⡇⠀⠀⢢⠴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡍⢲⢊⠂⡓⡬⢷⡩⣇⠀⡠⠿⠅⢾⣙⣿⣇⢀⣀⣀⣀⣤⣤⠴⣆⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢇⡍⡦⢚⣣⠋⠠⢽⡇⠘⠿⠋⠁⠿⡿⢻⢳⣋⢟⣿⢴⢫⢗⡭⡛⢒⠏⠑⠢⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⢂⡁⣀⡕⠀⠀⡹⠢⠾⡛⣀⠀⠀⠀⠀⠻⠅⢋⡫⣿⢻⣸⠛⠘⣋⣵⡆⠁
⠀⠀⠀⠀⠀⢀⣶⡦⠀⢀⢋⠁⢑⠀⠀⠀⠀⠇⠃⠀⠀⠑⡁⠀⠀⠀⠀⠀⠁⡰⠟⠀⠛⠂⠀⠈⠁⠍⠂
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠸⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢀⠠⠀⠀⠀⠀⡀⢀⣧⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠤⠶⠚⢀⡜⢡⣠⣼⣡⣴⡄⣐⣼⣿⣿⣿⠇⠀⠀⢰
⠀⠀⠀⣦⣴⠎⠁⠐⠖⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣆⡽⢭⠉⠹⠿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⡀⢠⣤⡷⣄
⢰⠀⠀⢸⡿⠀⢀⠀⠀⠐⣧⣆⠀⠀⠐⢰⣷⠀⣦⠹⢄⠨⠀⠀⢠⣿⠿⣿⣿⣿⣦⣤⣴⣿⣿⣧⣿⣿⠇⠘⡇
⠘⣆⠲⣼⠁⠀⢸⣿⣶⣶⣿⣿⣶⣶⣾⣿⣟⣾⢯⡀⠀⠀⠀⢀⠎⣀⡴⠛⢿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣆⠀⠙⠶⠄
⠀⠈⣿⡇⠀⠐⠛⢿⣿⣿⡿⠿⠿⠛⢹⣿⣻⣿⠀⠀⠀⠀⢰⣴⡞⠋⠀⠀⠘⠿⣿⠀⠀⠀⣿⣿⡏⢻⡋
⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⣸⣿⣹⣿⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠠⠙⠆⠀⣰⣿⡏⠀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠈⠙⠛⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠉⠙⠛⠓⠶⢤⡀
⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⠴⠖⠒⠒⠶⢤⣀⠀⠙⢦⡀
⠀⠀⠀⣠⠏⠀⣠⠞⠉⠀⠀⣠⡤⢴⣇⠀⠉⢳⡀⠀⢹⡄
⠀⠀⢠⡏⠀⢰⠏⠀⠀⠀⠀⢯⡀⠀⠙⠂⠀⠀⣷⠀⠀⣷
⠀⠀⢸⠀⠀⢾⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡿⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⣋⣉⣉⣉⣉⣙⠛⠒⠶⢤⣀
⠀⠀⠸⡆⠀⠸⣆⠀⠀⠳⣤⣤⡴⠃⠀⠀⢀⡼⠃⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣤⠖⠛⣩⠉⠉⢉⡭⠿⠯⣍⣛⡲⣄⠈⢳
⠀⠀⠀⢻⣄⠀⠈⠳⣄⣐⠋⠀⠀⣀⣠⠴⠋⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠳⢬⣉⣛⡲⠚⠋⠀⠀⢀⣋⣠⡤⠞⠀⣰
⠀⠀⠀⠀⠻⣷⣤⣀⠀⠉⠉⠉⠉⠉⠀⣀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣦⣤⣀⣉⡉⠉⠉⠉⠉⢉⣁⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣶⣶⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣽⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⢸⣷⣽⣻⠿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⢿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣩⣤⠤⠤⠤⠤⠤⢤⣬⣉⡙⠻⢷⣽⣻⠿⣿⣷⣾⣿⣿⣿⣿⣷⣾⣿⠿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⡞⢩⣤⣖⣀⣀⡴⠛⠉⢛⡶⠾⠙⣦⠀⢹⡿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⢿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠉⠓⠶⠭⣭⣭⣤⣤⣤⡤⠤⠖⠛⠁⣠⣾⡿⣷⣾⣿⣭⣭⣭⣭⣭⣭⣿⣷⣾⠿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⣻⡿⣷⣾⣿⣭⣽⣿⣿⣯⣭⣿⣿⣾⠿⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣛⣿⣿⡿⠿⠿⣿⣿⣟⣻⣯⣷⡿⣻⣿⣷⠾⠛⠉⠉⠉⠙⠛⠿⣽⣿⣾⡿⣟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣻⣿⡿⠿⢿⠿⠿⣿⣿⣻⣯⣷⣾⣿⠏⠀⢀⣠⠴⠶⠶⠦⢤⣀⠀⠙⢷⣿⢟⣿
⠀⢀⣠⠴⠖⠛⣋⣉⣉⣉⣉⣉⡙⠛⠲⠦⣿⣿⣶⣭⣟⣻⣿⠿⠿⠿⠿⠿⠿⣿⣛⣯⣽⣿⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⣿⣿
⣴⠋⢠⠴⠛⣉⡍⠉⠀⣠⠶⠿⢭⣍⣟⢦⡀⠙⢯⣟⣻⡿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⡟⠀⢠⡇⠀⢠⡄⠀⠀⠀⣀⣀⠀⠘⣦⠀⠹⡿
⣷⡀⠘⠲⢭⣙⣓⡒⠛⠁⠀⣀⣘⣁⡤⠞⠃⢀⣾⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⠀⢸⡇⠓⢿⡀⠀⢀⡞⠁⠈⣧⣀⢸⡄⠀⣷
⣿⣿⣷⣦⣤⣀⣉⡉⠉⠉⠉⢉⣉⣀⣠⣴⣾⣿⢿⣟⣿⠿⣿⣿⣷⣶⣶⣿⣿⡿⢿⣟⣿⣿⡄⠀⢷⡀⠀⠙⠶⠋⠀⠀⠰⠏⠈⢸⠃⠀⣿
⣿⣿⣽⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⠿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡿⠿⣿⣷⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡇
⢿⣾⣽⣻⠿⣿⣿⣶⣶⣶⣶⣶⣿⣿⠿⣟⣫⣷⡿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⢿⣿⣦⡀⠈⠓⠶⢤⣤⣤⠴⠚⠁⠀⣠⠞
⠀⠙⠿⣿⣿⣿⣶⣾⣾⣿⣷⣷⣶⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⣀⠀⠀⣀⣀⣤⠞⠁

⠀⠀⢀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠
⠀⠀⣿⣿⣿⠿⣿⣿⣇⠀⠀⠀⠀⣴⡖⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⣤⢀⠢⡄⢠⣤
⠀⢘⣿⣿⡃⠒⢸⣿⣯⠀⠀⠀⣼⠿⠿⡘⠿⠿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠉⠀⠒
⠀⠀⠚⣿⡇⠀⢸⡟⠗⠀⠀⠀⣿⣤⣤⣤⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⢀⣀⣠⡄
⠀⠀⠀⠘⠛⠻⠿⠇⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣫⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿
⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⡟⠋⠁
⠀⠀⠀⠀⣀⣄⠄⠀⠀⣤⡐⠂⠀⠀⢣⢸⠀⢼⡄⣷⣦⡇⢸⢸⢾⠠⣷⢲⠀⡇⢼⡆⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠊⡟⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣀⡀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⡄
⠀⠀⢀⣄⢀⣀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢰⣿⡶⣾⣷
⠀⠀⠀⠉⠈⠉⠀⠉⠀⠀⢴⡄⠀⠀⠀⢠⣦⠀⠀⢀⠸⣷⠟⠚⠛⣶⠗⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠺⢺⣿
⠀⠀⠀⠶⠆⠠⣤⡆⠀⠀⠘⣿⠀⠀⠀⣼⠏⠀⠀⠸⢿⡏⠀⠈⠀⠘⡿⠇⠀⠀⠀⠀⠛⢻⣲⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⣤⣤⣤⣾⡇⠀⠀⠀⠘⠀⢀⠀⠉⠀⠀⠀⠸⠟⣧⣀⣀⣀⣴⡻⠏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢻⣀⣸⠋
⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢸⠿⡧⠻⠃⠀⠀⢀⣀⡀⠀⠿⠿⠃⠀⣀⣀⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠙⠋⠁⢀⣾⣿⣷⠀⠉⠉⠁⠀⢸⣿⣿

⠀⠀⠀⠤⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⡠⡤⢠⡄⠀⠀⠀⢀⡀⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡤⡄⢀⠁⠀⢠⡄⠀⠀⡂⡀⠀⠀⠀⠀⠀⢀⢡⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠇⣀⠀⠀⢠⠢⠀⢄⠕⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⠀⠀⢀⠀⠀⠀⡈⣠⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠤⠄⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠸⠰⢳⣦⠁⠀⠉⠀⠁⠈⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡂⠀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠇⢀⡀
⠀⠐⠐⠀⠉⠛⠒⠆⠀⠀⢠⠀⠀⠀⠀⠈⠀⠘⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⠀⠠⠄⠐⠛⠀⠀⠐⠒⢀⡄⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠
⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠎⠄⠅⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢔⠄⠑⠀⠁⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠉⠁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠄⠂⠀⠐⠠⠀⠀⠠⠤
⠀⠀⠀⠐⠀⢀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠐⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⡂⠀⢠⠅⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠖⠐⠾⠣
⠀⠀⠀⠐⠂⠀⠑⠀⠄⠀⠀⠁⠀⠋⠃⠀⠀⠘⠉⠀⠀⠀⠐⠀⠀⠋⠀⠀⠀⠀⠐⠀⠀⠀⠠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠀⠄
⠀⠀⣤⡄⠡⠄⠀⠀⠀⠀⠀⠀⠂⠀⠐⡐⠂⠀⠀⠀⠈⠀⠐⠀⠀⠀⠒⠲⠒⠐⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠤⣤⠄
⠀⠀⠀⠀⡄⣅⡀⠀⠀⠠⠦⠄⠀⠀⠀⠰⠀⠀⠀⠈⠥⠖⠀⠀⣔⠠⠀⠀⠐⠡⢐⠄⠀⠀⠀⠀⠂⠠⠀⠁⠀⠀⠂⠀⠀⠒
⠀⠀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠒⠀⠀⠐⠂
⠀⠀⠁⠀⠂⠂⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠈⠀⠐⡃⠀⠀⠸⠂⠀⠐⠆⢀⡀⢆
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠧⠀⠀⠠⠄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡈⠀⠉⠈⠀⠱⠎⢠⠃⠈⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠰⣄⡀⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠒⠀⠀⠄⠀⠀⠀⠂⠀⠔⠀⠀⠈⠉⠀⠀⠀⠂⠄⠰⠀⠀⠀⠀⠀⠀⢈⠈⠀⠉⡀⠸⠀⠀⠀⠀⠨⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠘⠐⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠈⠘⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟
⠈⢿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⢀⡾⡷
⠀⠈⣳⣄⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁⢀⡠⠖⢉⣼⣥
⠀⠀⠙⣍⠓⠦⣄⣀⠀⠉⠉⠉⠓⠒⠢⢤⣀⠀⠀⠀⠀⢀⣤⣶⡿⣿⡿⣿⣾⣛⡛⣿⢀⠴⠊⠁⣀⡤⠴⠒⠫⢭⠄⡠⠞⡱⠃
⠀⠀⠘⢿⠳⢤⡀⢨⡭⠭⠙⠛⠒⠒⠦⢄⡈⠙⢦⡀⣴⠿⠋⢡⣾⡿⠿⠟⠛⠉⣷⡟⠁⣠⠖⠋⠁⠀⢀⠤⢖⡁⠀⣠⣾⡇
⠀⠀⠀⠀⢙⣦⣀⠀⠉⣗⠒⠂⠀⠀⠀⠀⠈⠳⢄⠙⢌⠳⢤⡢⣳⣄⡀⠀⠀⢸⡟⢀⠞⠁⠀⠀⠤⠤⣶⠒⠉⠐⣊⠵⠋
⠀⠀⠀⠀⠈⠛⠭⣙⣂⠀⠉⠹⠭⠥⠤⢤⣀⣠⣤⣁⡈⢧⠀⠙⠺⣿⣿⣄⡇⢸⣷⣆⠀⣀⠴⠊⠉⠉⠀⣴⡶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠧⢤⣄⣀⣀⠀⠙⠻⠿⠿⢿⣶⣶⣾⣿⣿⣿⣿⣿⡿⢷⡄⢀⣀⣴⡶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣛⡷⠖⠛⠋⣙⣩⣿⢿⡟⢸⣧⠻⡄⠙⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⢾⣛⣿⠿⠛⠁⠚⠀⠀⠏⠃⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠏⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⠶⠶⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣐⣈⡆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢾⠀⠶⠠⢼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠅⡨⠍⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⠒⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠈⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣒⣒⣒⣒⣒⡒⢆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠄⠄⠀⠀⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠰⠭⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⠠⠀⠐⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠼⠼⠤⠤⠤⠤⠤⠶⠼⢄⣀⣀⣀⣀⡼⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠭⠁⠀⠀⠀⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⡭⣉⣁⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣶⣶⠖⢒⡖⠂⠀⠀⢠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⢄⡤⠤⠤⠦⠤⢬⡦⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣯⣄⡬⣨⣅⢀⣽⢸⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠛⠛⠿⠿⢿⣿⣿⡿⣿⣾⡅⠀⠀⠀⠀⣀⡀⠃⢀⠀⠀⣀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠟⠉⠉⠉⠉⡡⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠒⠤⠤⣤⣤⠴⢋⠡⠀⠀⠀⠀⠀⣀⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠀⠀⠤⠄⠀⣀⣀⡀⢈⠠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠍⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠲⠖⠒⠒⠳⠖⠒⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠃⠀⢲⠘⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠈⠀⠕⠄⠀⡄⠀⢨⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⡆⠈⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡠⠂⣁⣀⡁⠢⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠐⠂⠀⠡⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠰⠡⠂⠘⣶⠈⠂⠐⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡆⢠⢀⣰⣾⢀⣰⠀⠂⠀⠀⠀⠀⠀⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⡇⠈⠀⠨⡍⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠛⠐⠂⢀⠄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠂⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢀⠀⠂⠙⠁⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⢸⠀⠭⠅⠸⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠴⠄⠀⠰⠀⠀⠀⠀⠠⠀⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠂⠥⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠊
⠀⠀⠀⠀⠀⠀⠠⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡨⠋⠀⠀⠈⠉⠀⠀⠈⠫⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢹⠀⠀⡃⠀⠀⠐⠀⠢⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡄⠤⠀⢸⢸⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⢸⠀⠀⠄⡀⠀⡄⠀⠀⠀⢀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⢸⢸⠀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠈⢸⠀⠈⠁⠂⠀⡀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠠⠀⠀⠀⡠⠀⠀⣀⠁
⠄⠂⠀⠸⡀⠀⠀⠀⢬⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠒⠀⠀⡸⠁⠀⠀⠀⠸⠀⠀⠒⠄
⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠆⠉⠉⠀⠀⠀⠀⠀⠀⠈⠩⠱⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡀
⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣾⣷⣤⣤⣤
⣠⡄⠈⠃⠉⠘⠁⣦⡄⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠻⠙⠉⢧⠀⠀⠀⠀⠀⠀⠀⠸⢉⣤⡀⢀⣤⡙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠈⠻⢿⣤
⢸⣏⠁⢐⣀⡀⢈⣹⠁⠀⠀⠀⠀⠀⠀⠠⣯⣁⣤⣤⣤⣄⣽⡆⠀⠀⠀⠀⠀⠀⠇⣾⠟⠃⠈⠻⣿⠸⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀⣀⠀⠀⢸⡟
⡀⠲⠂⠀⠉⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⡠⠃⠀⠀⠀⠀⠀⠰⣿⣷⣀⠀⠈⠉⠀⣠⣾⠿
⢛⠀⣘⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣶⣶⣿⠛⠃
⠘⣴⡇⢠⠄⡤⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⢭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣶⢰⡆⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠠⠃
⠈⠛⠆⣸⡀⣇⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⢉⣻⣛⣼⡉⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣆⢀⣶⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⠁⠀⠉⠙⢿⠟⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⡀
⠀⢀⠀⠐⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣠⣤⣤⣤⣀⣈⠀⠀⠀⠀⠀⢠⠋⣸⣾⠿⠿⠿⣷⣄
⠣⠁⠀⠀⠀⠀⠀⠡⠇⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⣿⢻⡟⠁⠠⠀⠀⠀⢻⡛⣷
⠠⠀⠀⢿⣿⠿⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⢄⠀⠀⠀⡀⠈⢛⡯
⠀⠑⠤⠀⣀⠠⡔⠁⠀⠀⠀⠀⠀⠸⢿⣿⣷⣄⠀⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠉⠹⣷⣄⢿⣞
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠈⣬⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢀⣿⣜⡞
⢠⠂⠀⠀⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢘⠀⡄⣤⢚⠎⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣤⣾⣿⣿⣿
⠈⠉⠺⠭⠶⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠖⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠻⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⡤⣀⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⢀⣀⣀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⢼⣥⢤⡀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣐⢛⡀⠀⢓⣒⠒⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣦⠀⠀⢠⣶⣶⣖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠏⠀⡼⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⢤⠀⠀⠀⣤⠤⠼⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠘⠉⠒⠦⠊⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠓⠤⠔⠋⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠒⠙⠒⠶⠐⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠉⠉⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⣴⠛⠋⠉⠙⠛⠳⣦⣀⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⣷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠾⠛⠻⣤⣀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⣿⠛⠉⠁⠀⠀⠀⠀⠀⠠⡤⠤⢄⡀⠀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡖⢤⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢳⡜⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⢠⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠓⠉⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣃⠀⠀⢘⡓⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠈⡽⠗⠀⠀⠀⠿⠿⢧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣛⠀⠀⢘⣙⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠀⠀⠀⡖⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿