Catchphrase Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠔⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⢸⡀⠸⠞⡆⠀⣿⠁⠠
⣀⠀⠀⠀⠀⠀⠀⣿⡙⣷⣾⣔⠰⣀⣀⡤⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⠠⠘⠷⠖⡂⢘⡀⠻⠀⠈
⠘⣿⡙⠲⣗⡢⠿⡭⣬⣥⠤⠤⠒⠉⣿⠁⠐⡿⠓⠀⠀⠀⠀⢰⣿⣇⢀⢠⢤⠀⠁⠉⠃⣀⣐⣿⣾⣦⡄
⠉⠻⣯⡀⣇⡽⠾⡴⣤⡴⠶⢶⡖⣛⣿⡤⠞⠁⠀⠀⠀⠀⠀⣻⣿⡃⠀⡿⡉⣴⡆⣶⣪⣿⠁⠈⣻⣿⠇
⠀⠀⠈⠙⠛⠉⡷⣶⣶⡖⢲⡮⠽⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠀⢑⣯⡙⠊⠝⢱⠘⠒⠈⢿⠟⠁
⠀⠀⣠⡄⣶⠒⠰⣤⢠⡖⢠⢠⢰⡂⣯⡈⠧⣄⠳⠁⠀⠀⠀⠀⠈⠁⠀⢸⠜⠅⠀⡼⢉⡀
⠀⠀⠈⠁⣏⡀⠰⠻⡜⣧⠼⣦⠀⣼⠻⡴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣸⣺⣇⣯⣏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠉⠉⠙⠇⠀⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⡠⠄⢨⣯⢰⣄⡆⡂
⠀⠀⠀⠀⠀⠀⠑⠒⡘⠉⡉⠈⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⢻⠁⠀⣰⠎⠀⠀⠀⠀⠀⠀⠀⢸⡯⣉⣿⢩⠭⠉⢩⣭⣏⣹⣁⣤⣸⣁⣀⣼⢗⠰⠶
⠀⠀⠀⠀⠙⢣⡄⣀⡀⡉⡅⡐⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⡤⠈⠀⠀⡀⠀⠀⠀⣤⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠂⠁⡤⠒⢠⡀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣀⢳⣀⣤⠈⡏⢠⡖⢀⡯⡄⢀⠁⢸⡇
⠀⠀⠀⠀⠀⢰⢰⣼⣇⣭⣻⢡⣖⣿⣤⠀⠀⠀⠀⠀⠀⢸⡇⠳⠋⠚⠛⢊⣌⣨⡍⡩⡅⠃⠓⣁⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠋⢙⣉⡥⡅⡴⡆⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠑⠌⠿⠘⠙⠧⠶⠥⠴⠐⠇⠀⢸⡇
⠀⠀⠀⠀⠀⢀⠈⢻⠸⠷⠧⢟⣡⣦⣆⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠃
⠀⠀⠀⠀⠀⠈⠹⠲⡶⡶⢶⠶⠞⠛
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡟⣆⡠⣤⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡩⢄⡏⣣⣊⣑⢊⠃⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⡀⠑⣞⢁⣉⠉⠉⠛⣡⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⢸⣧⡾⢦⠀⠀⡴⣤⣒⡯⢆⣐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠃⢀⠘⣘⠛⣠⠀⢀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⢿⠸⣇⢸⣶⡶⡡⠾⡕⠾⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡗⢻⢸⡋⠀⠳⣄⠸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠉⠈⠪⠶⣉⠀⢀⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢟⣁⣈⣬⣷⣥⣴⣯⣤⡱⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠀⡴⢢⣠⠀⡀⣾⠁⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⣾⣶⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⡄⣾⠉⣸⣿⣠⡇⢻⡾⣾⢹⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⢉⣿⣁⠘⠁⢉⠨⠽⡅⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢛⣿⡿⠿⠿⢿⠛⠛⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⢠⢠⡀⢀⡤⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡆⢠⠂⠘⠶⠴⠾⠿⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡆⣇⠀⢸⡎⣧⠄⣿⠂⠀⣾⣇⡤⢲⠆⠀⠀⠀⠀⢯⡴⠀⣷⢸⡤⢀⣀⡀⢿⣄⠀⡠⣇
⠀⠀⠀⠀⠀⠐⠭⣉⣹⣾⣇⠻⠐⠊⠃⠘⡁⠙⠂⣠⣿⣟⠭⣁⠀⠀⠀⠀⠀⠀⢠⡌⣹⢻⣁⣿⠼⣥⠞⣏⣸⣤⣿⡀
⠀⠀⠀⠀⠀⠀⢠⡎⠔⣫⡥⢤⡀⠀⠉⠉⠁⣤⡤⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠠⢤⡿⣋⠀⣍⢠⣤⣴⣦⠨⠄⠈⣽⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠘⠉⣱⡴⣿⡾⡞⢦⡠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⢨⣍⣬⡀⣤⡀⣤⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠶⣶⣾⣾⠀⢉⣻⠆⢐⡖⠁⢰⣶⡯⢃⣤⣄⣠⠀⠀⠀⠀⠀⢀⣀⣉⣘⣥⣿⣵⣯⣧⣼⣬⣁⣀
⠀⠀⠀⠀⠀⠀⢴⣾⣿⣷⣄⡙⡏⢁⣤⣿⠰⣟⣀⣩⣿⣿⣿⠿⠛⠀⠀⠀⢀⣠⣼⣿⢿⡯⣿⢿⣿⣿⣿⣿⡿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⣏⣻⠿⢷⡶⢤⣤⡶⠟⣿⣥⠋⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣷⣄⣴⣿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡀
⠀⠀⢰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡇
⠀⠀⢸⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⣀⣤⣤⣄⠀⠀⣤⣤⣤⣄⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⠉⢻⣿⡀⢰⣿⠏⠙⣿⣧⠀⣿⡏⠙⣿⡇⢸⣿⠉⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⠀⠘⣿⡇⣿⣿⠀⠀⢸⣿⡀⣿⣷⣴⣿⠇⢸⣿⣷⣶⠀⠀⣄⡀⠀⢸⣷
⠀⠀⢸⡇⠀⣀⣶⠀⣿⣿⠀⢸⣿⠇⢻⣿⠀⠀⣼⣿⠀⣿⡟⠉⠁⠀⢸⣿⠀⠀⠀⠀⡟⠛⠀⢸⣿
⠀⠀⢸⡇⠀⠉⠃⠀⢿⣿⣶⡿⠏⠀⠈⠿⣷⣾⡿⠃⠀⣿⡿⠀⠀⠀⢸⣿⣶⣶⡶⠀⠁⠀⠀⢸⢹
⠀⠀⢸⡇⠀⢀⣤⣤⣔⠲⣄⠀⡠⠴⣦⣄⠀⣠⡶⢲⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣴⠖⠆⠀⢸⢸
⠀⠀⢸⡇⠀⠊⠸⠀⠈⢇⣿⡟⠀⠀⢸⡿⢰⣿⡡⠃⠀⢀⠀⠀⠙⠇⠀⡀⠀⣰⣿⡣⠊⠀⠀⢸⢸
⠀⠀⢸⡇⠀⢄⢠⠈⡇⠸⣿⡇⣀⣤⠞⠁⣼⡏⠀⢀⡾⠁⢸⣿⠀⣰⠏⠈⠂⣿⡯⠒⣦⠀⠀⢸⢸
⠀⠀⢸⡇⠀⠈⠉⠛⠐⠁⠿⠃⠀⠈⣷⠀⢻⡇⠀⣸⡇⢀⢾⡏⢠⣿⠀⠀⡰⣿⢤⡞⠃⢰⠀⢸⢸
⠀⠀⢸⣷⠀⠹⢏⠀⠀⠀⢎⠈⣃⣼⠟⠀⠈⠳⠔⠁⠉⠁⠈⠷⠊⠙⠧⠐⠁⠈⠀⠻⠔⠁⠀⢸⢸
⠀⠀⢸⣿⢠⣤⠀⣤⡄⢠⣤⠈⢁⣤⣤⣄⠀⢀⣤⠀⠀⠀⣤⡄⠀⣠⣤⣤⡀⠀⢠⣄⠀⢠⣤⢸⢸
⠀⠀⢸⣿⢸⣿⠀⣿⡇⢸⣿⠀⣾⡏⠀⣿⡇⢸⣿⡇⠀⢰⣿⡇⢀⣿⠃⢹⣷⠀⢸⣿⡀⢸⣿⢸⢸⡆
⠀⠀⢸⢸⢸⣿⢰⣿⣧⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⣿⠀⣼⣿⡇⢸⣿⠀⢸⣿⠀⢸⣿⣇⢸⣿⢸⢸⡇
⠀⠀⢸⢸⠈⣿⢸⡿⣿⢸⡏⠀⣿⡇⠀⣿⡇⢸⣿⣿⣦⣿⣿⡇⢸⣿⡀⣸⣿⠀⢸⡟⣿⣾⣿⢸⠘⡇
⠀⠀⢸⢸⠀⣿⣿⡇⣿⣿⡇⠀⣿⡇⠀⣿⡇⢸⣿⢻⣿⡏⣿⡇⢸⣿⠛⢻⣿⠀⢸⡇⠸⣿⣿⢸⠀⡇
⠀⠀⢸⢸⠀⣿⣿⠁⢸⣿⡇⠀⠹⣿⣴⣿⠃⢸⣿⠈⣿⠁⣿⡇⢸⣿⠀⢸⣿⠀⢸⡇⠀⢿⣿⢸⠀⡇
⠀⠀⢸⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣠⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⣰⠟⠁⠈⠙⢳⡄⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠠⡀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢘⠀⠔⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢡⠕⡲⢤⡖⠒⠀⢢⠀⠀⠀⠀⡻⢖⣖⣂⠉⠨⠟
⠀⠀⠀⠀⠨⡎⢲⣴⣴⣄⠧⡄⠀⠀⠀⠀⠀⠀⠠⠈⣯⣿⡀⠷⢂⠀⠀⠀⠀⠀⢀⣨⣍⣩⣤⠄
⠀⠀⠀⠀⠀⠐⢤⣇⣤⢱⣾⠀⠀⠀⠀⠀⠀⠰⠐⢀⢐⣄⣠⣀⡐⠓⠀⠀⠀⠀⢦⠉⣯⡭⠍⠔⠂
⠀⠀⠀⠀⠀⠂⠻⢾⣿⣿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠸⢀⢀⠀⡸⠀⠀⠀⠀⠀⠀⢳⡰⢦⣭⣠⢖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠚⠚⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⢴⡀⣀⠀⢀
⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⢷⣼⠀⢠⣀⣤⣌⡄⠀⠀⠀⣀⠠⠿⠿⠆⠯⠙⡀
⠀⠀⢀⠀⠷⠀⠘⠂⠀⠒⠠⠺⣀⡠⠂⠀⠀⢧⢸⠘⠮⠏⠒⠄⠮⢇⡀⠀⠀⠻⢤⠀⣯⢡⡄⡆⠃
⠀⠀⠈⢀⣶⢆⣸⣓⣂⣉⣉⣒⡠⠁⠀⠀⠛⠶⠶⠖⠀⠛⡓⡛⣃⣋⡀⠀⠀⠀⠘⠒⢉⡘⠃⠃
⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣦⣿⣤⣤⣴⣶⣖⣆⣰⡇⠀⠀⢰⠘⣤⡲⢖⣤⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠘⠛⠋⠉⠉⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠙⡾⠋⡏⠀⣷⢸⠀⡷⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⢀⠄⠀⠿⣤⣴⣼⣾⣴⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠚⠁⠈⠂⠋⡀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣩⠉⢷⢸⢀⡀⠀⠀⠠⣶⡄⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢐⢺⡟⠓⠓⣠⡐⠟⣻⡛⢺⡇⠘⡓⠒⠆⠀⠀⠀⠀⠀⠀⠉⢀⡐⢺⡾⡟⢰⣟⣿⠁⡽⡿⢠⡏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣬⢿⡷⠆⠺⣟⣡⣜⡇⣴⣦⣸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡾⠁⠳⠞⡉⠈⣉⡀⠓⠊⠙⠳⡿⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⢾⠦⣠⠦⠨⠵⠀⣀⣉⠀⠀⠀⠀⣀⠀⠀⠀⠚⢹⣏⠰⠆⡾⠆⠿⠻⠻⠟⠈⠃⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⢿⡷⠒⠙⠙⠟⠋⠀⠿⠳⠇⠀⠐⡍⠙⠛⠛⠛⠀⠿⠟⠀⠀⠀⠈⠁⣠⠰⣶⢶⣶⢠⣷⡌⢺⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⡿⠟⢷⠟⠳⠛⣿⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣥⣬⣥⣭⣥⣥⣬⣤⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠚⠛⠊⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠿⣿⢽⡿⣿⣯⣝⣿⢻⣿⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢡⣶⠆⢻⡋⢳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡟⣧⢲⣴⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣥⣤⣬⣭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠈⣰⣃⠀⠈⢈⣀⠀⣄⣠⠆⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣾⡇⢸⡵⣾⢡⢴⠤⠀⠀⠀⠀⠀⠀⠀⠀⣰⣜⢦⢂⡉⠘⠆⠲⠜⠉⠓⣪⠼⣞⣉⡛⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢉⣝⣓⠘⢁⣘⡊⢠⣄⠀⠀⠀⠀⠀⠀⠐⡖⠛⠛⠹⠿⢧⣤⣤⡤⠤⡤⠤⣦⠶⣿⠿⡟⠉⠙⡳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡏⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⡂⠀⠀⡁⠀⢸⡏⢡⡟⠙⣧⠙⢲⡏⠀⣇⣀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣽⣳⣤⢤⣴⣤⣶⣿⡤⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⣇⣈⠚⣁⠈⠳⠞⠃⣀⣘⣣⣤⠌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣨⡉⠉⣁⣠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠋⠉⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠲⣶⣴⣤⣦⠀⢴⡴⠶⠛⣠⠠⠤⠤⣤⠄
⠀⠀⠘⠳⡍⠉⣿⣓⣶⣞⣶⣅⢒⣶⢲⣉⢩⡭⡇⢀⣘⡡⠄
⠀⠀⠀⠚⠒⠒⠛⠙⢻⠉⠸⠋⠘⣛⣀⣙⡠⠥⠉⠁⢰⠂⠀⠀⠀⠀⠀⣠⠞⣳⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⣠⢆
⠀⠀⠀⠀⢙⣶⣤⣴⡦⠉⠛⠻⠛⠚⡃⣀⢀⣠⣤⢰⡛⢥⠀⠀⠀⠀⠳⡧⠶⠋⠀⠀⠸⣸⢀⠀⠀⢤⣾⣿⠿⢿⣿⣇
⠀⠀⠀⠀⠀⠆⠀⣾⣵⣀⡏⢹⣈⡟⠙⣇⡌⠻⣜⣋⢳⠀⠀⠀⣠⣄⠈⡏⠙⣦⣾⠡⣿⠃⣬⢰⢲⢠⠉⣦⣜⠆⡰⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⡄⢰⣟⡂⠺⡁⠻⣍⠓⠲⠋⠀⠀⠀⢸⣿⣦⣌⣶⣿⡽⢷⡙⣖⡙⣚⡗⣉⣚⠁⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠷⠎⠓⠂⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠿⠁⠀⠀⠀⠙⠁⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠠⢸⠃⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣼⢃⣀⣀⡴⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣰⣧⡶⣆⠀⢰⢦⣴⡮⢅⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢷⡟⢿⣹⡏⠀⢷⠟⠁⣶⠆⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡿⣇⣼⢶⠫⢚⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣉⢁⡌⠁⠈⠉⠀⠀⠀⡾⡇⢠⣆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠚⡡⡀⢠⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠋⣼⠁⢀⢀⡄⡀⢀⠀⣿⡃⢻⡏⠀⠀⠀⠀⠀⠀⢠⡼⡇⢠⡯⢱⣶⢠⢸⣁⣁⢾⡇
⠀⠀⠀⠀⠀⠀⠀⡔⢡⡏⣰⣟⡿⢻⣇⣾⡾⢏⣹⠟⣧⡄⠀⠀⠀⠀⠀⠸⣥⢧⠼⣧⠏⠿⠻⠞⢫⡟⠘⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠉⠉⠁⠈⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⢿⣶⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⢸⣿⡇⠀⠀⢸⣿⣦⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣶⡄⠀⢻⣿⡇⣼⣿⠃⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢸⣿⣿⣿⣧⣤⣤⣀⠀⠀⢁⣤⣶⣶⣦⠀⠀⠀⣾⣿⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠙⠻⠁⠀⢸⣿⡿⠋⠉⠀⢹⣿⣷⡀⣾⣿⠋⣰⡿⠀⠀⠀⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣼⣿⢹⡇⠀⠀⢀⣼⣿⡇⠀⠀⠀⣸⣿⣿⢻⣿⡿⠛⠉⢀⡾⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢠⣿⡟⢸⡇⠀⢠⣿⠉⣿⡇⠀⠀⣰⣿⣿⠏⠈⢿⣿⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⣠⣤⣤⣄⣀⣸⣿⠇⠀⠀⣸⣿⡇⢸⡇⠀⠈⢿⣄⣀⣀⣴⣾⣿⠟⠁⠀⠀⠀⢈⣉⠉⠀⠀⠀⠀⠀⢠⣶⣆⠀⣸⣿⡀
⠛⠷⣤⣤⣭⣿⣿⠓⠒⠲⣿⣿⡷⠾⠃⠀⠀⠀⠈⠉⠙⠉⠉⢰⣦⣴⡟⢻⣦⢸⣿⡆⢀⣴⠶⣶⡄⢸⣿⣿⠀⣽⣿⣇
⠀⠀⠀⠀⢀⣿⣿⠀⠀⢀⣿⣿⠇⠀⠀⢠⣶⣿⠟⣿⣆⠀⠀⣿⣿⡟⣀⣼⡿⢼⣿⣿⡿⠁⣀⣿⡟⠙⣿⣿⣦⣿⣿⣿⣤⠴⠒
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠀⢀⣴⣿⣿⡏⣼⣿⣿⠀⣼⣿⣿⣿⣽⠿⠁⠈⣿⣿⣧⣞⣵⡟⠁⠀⠀⠉⠉⢀⣿⣿⣇
⠀⠀⠀⠀⠸⠿⠃⠀⠀⢸⣿⣿⣴⠟⢹⣿⣿⣿⠇⣿⣿⣰⡇⣿⣿⣇⠀⠀⠀⠀⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⣰⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⠀⠈⢿⣿⡟⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢸⣿⣿⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⢀⣴⡖⠰⠂⠘⠿⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣄⡤⠔⠚⠉⣁⣠⠼⠛⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⣀⣛⠋⠀⣾⣼⣠⣿⣖⣲⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠻⣟⠯⢭⣁⣀⠀⢀⡤⠖⠚⣛⡶⠒⠀⣠⠴⣋⠽⠛⣁⣼⠿⠯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠳⡀⠀⠹⣝⠻⢟⠛⠛⠲⢶⡶⠾⣿⣛⠛⠋⠙⠛⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠼⠂⠀⠈⠓⢤⣑⣄⠀⠀⠈⠓⠲⠶⠽⣦

⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡤⣿⠋⠁⠀⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠋⠀⣿⠀⢀⣴⠏⠀⠀⠀⠀⢠⡶⠋⡁⠀⡀⠀⡀⠀⣠⡄⢸⠁⡠⢀⠀⠀⣀⠀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠟⠁⢀⣀⠀⠀⢰⠟⢀⡾⠁⡼⢁⠀⠈⢰⡷⠃⠼⣏⣴⠏⣑⢰⡇⣸⡇⣰⠇⣠⠖⠆
⠀⠀⠀⠀⠀⠀⠴⣿⠉⠑⡆⣴⠏⠀⠀⠀⡟⢀⡾⠁⣼⠃⣼⠀⡇⢸⣀⠎⡀⣿⢺⡀⠀⡾⢡⠟⣰⠏⣼⠃⢁⠠
⠀⠀⠀⠀⠀⠀⢠⡇⢀⠜⠁⣇⡠⠂⠀⠀⠓⣹⠃⠀⠻⠀⠻⡴⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠞⢠⡏⠀⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠔⠀⠀⠀⠀⠀⠀⠀⢆⣀⡼⠃⠀⣠⠖⠀⠀⠸⠃⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣄⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠄⠀⠀⠉⠀⠀⢠⡏⠐⠀⠀⠀⢰⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡄⠀⠈⢀⣰⠞⢣⠀⠀⢿⠀⠄⣾⠀⠀⣠⣄⠀⠀⢸⡁⠀⠀⢀⡀⠸⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡇⠀⠀⢰⡇⢀⡞⠀⠠⠹⣇⢰⡇⠀⣼⠧⠊⠀⠀⢻⠀⠀⣰⠣⠀⠀⢷⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⣇⢀⠁⢸⠃⢸⠇⠀⠁⠰⡗⢸⠃⢰⡏⠀⡀⠀⠀⢸⡆⠀⣿⠀⡠⢁⢼⡄⡜⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠹⠆⠀⠈⠁⠸⠄⠀⠄⢸⠇⠸⡠⠙⠧⠊⠀⠀⠀⠀⢧⡀⠘⠋⢠⠂⣼⠁⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠛⠋

⠀⠀⠀⠀⠀⣤⡶⠿⠿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠸⠟⠀⠀⠠⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠐⠤⠊⠀⠈⢆⡀⠀⠀⠀⠀⠀⠠⡤⠤⠶⠦⠤⡤⢤⣄
⠀⠀⠀⠀⡨⡉⠩⠉⠅⠄⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡼⢻⠼⢴⡟⠛⠂⠈⡆⠀⠀⠀⠀⢀⣯⣛⣝⣗⣀⣇⡠⠟
⠀⠀⠀⢈⣖⠷⡤⣤⢴⣀⢿⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢹⣧⣿⡀⢣⢤⠀⠁⠀⠀⠀⠀⠀⢀⣀⣏⣉⣡⣤
⠀⠀⠀⠀⠙⡄⢹⠍⡉⢞⣨⡌⠀⠀⠀⠀⠀⠀⠀⢔⣁⡀⡘⠍⠉⠩⠤⠠⠵⠦⠀⠀⠀⠀⢂⡘⠟⠿⢛⡚⠛⣢⠇
⠀⠀⠀⠀⠀⣬⣌⣶⣤⣮⣧⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠞⠓⠚⡳⡗⠀⠀⠀⠀⠀⠀⠈⡍⢀⡚⠹⠝⠈⣁⣤
⠀⠀⠀⠀⠈⢈⠛⡝⢿⠿⣟⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⡃⡔⣰⢂⢔⡁⡂⠀⠀⠀⠀⠀⠀⠹⣾⡛⠷⠶⢞⣡⠇
⠀⠀⠀⠀⠀⠀⠈⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡴⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⡆⠠⡡⠀⣀⡍⢃⠀⠀⠀⠀⠀⠀⢈⣷⣺⣯⢱⠏⠷
⠀⠀⠀⣧⠀⠀⣷⠀⠀⣼⠀⢀⡎⠀⠀⡄⠀⠀⠸⡏⠙⡏⣏⠤⡍⣯⡍⢩⢾⠀⠀⠀⠀⠀⣿⠸⠁⠦⠤⠄⣉⣀⣴
⠁⠆⠐⢉⠑⠪⣄⣈⡉⢤⣨⢉⣑⠒⡞⠀⠀⢀⣠⠑⠴⠃⡈⠋⣁⡀⣑⣉⢉⡥⠀⠀⠀⠀⠃⣷⠀⣿⠃⣿⡇⡏
⠀⠈⢄⣟⣋⣴⣯⢥⢥⣤⣤⣬⣝⡜⠀⠀⠀⠀⠙⠛⣛⡋⡋⠀⠅⠭⠭⠬⠬⢅⡀⠀⠀⠀⠀⢈⠍⠡⠤⠁⠡⠁
⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⡷⠺⣤⣴⢶⣼⣴⣞⣳⣀⣸⡅⠀⠀⠀⣧⠸⣤⣎⠗⢇⡦⠀⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣁⣸⣟⣛⣋⣉⣓⣀⣿
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⢀⡀⠀⠀⠀⠀⠀⢀⢀⢀⡠⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠍⢳⣾⣶⣖⣊⠂⠁⠂⣀⡌⠓⣗⣲⣾⣴⠟⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠹⠁⢻⡄⢋⠏⢻⠈⢁⣤⠈⣽⢁⡟⠁⠈⢸⡏⢯⠓⣰⠋⢹⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢠⣷⡟⠀⠀⠀⢩⡏⢙⣿⢿⣄⣴⡄⠀⠀⠈⣷⣿⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢿⣷⡌⠁⢀⡀⠀⠀⠛⠛⠋⠘⠛⠛⠀⠀⢀⡀⣫⣥⠄⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣽⡿⠀⣸⡏⠛⠷⠀⢠⣾⣷⡄⢀⣶⠿⢟⡇⢹⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠭⢿⣥⡀⣿⡷⢾⠆⠀⣿⣅⣸⢷⢸⡏⠀⠈⠁⢨⣿⣴⣾⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⡇⣿⣇⣀⣶⢠⣿⡍⢸⣿⣼⣷⣀⣜⡆⢰⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣴⡿⠃⣿⣛⠉⡁⣠⠀⢠⣄⣄⣀⡉⠉⣙⡀⠿⣿⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠿⡟⢉⣤⡶⠦⠈⠀⠇⠿⠂⠟⠛⠘⠸⠇⠈⠩⢥⣤⣄⡁⠘⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣞⣀⠛⢋⣴⠃⢰⣴⢠⣶⣰⡖⣲⣶⣲⠶⣴⠀⠲⣬⠻⠇⣘⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡿⠻⢿⣾⡁⢋⡈⠈⠙⠙⠛⠋⡁⠋⠁⠈⠋⠈⣃⡉⣴⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣷⢸⡇⠀⠀⠀⢠⣿⢻⣧⠀⢠⣾⠿⢟⡇⣿⣿⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣠⣴⣿⠇⣼⡇⠀⡀⠀⡾⣧⣼⣿⠀⣿⡇⠀⢀⠀⣿⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠋⠁⠀⣿⣧⣼⡇⠠⠿⠿⠹⠿⠆⠹⣿⣶⣽⠀⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡄⠀⠀⠀⠉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡉⠀⠻⣿⣦⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠛⠁⠀⣴⣾⣿⠟⢋⣽⣿⣶⠚⠓⣶⣿⣿⠟⢋⣩⣷⣦⡈⠙⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⣼⣟⣋⡤⠞⣉⣿⣿⣿⡇⢸⣿⣋⣤⠞⣋⣽⣿⣿⣧⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠆⢻⣿⣿⣷⣾⣿⣿⡿⠋⠁⠈⠙⢿⣷⣾⣿⣿⣿⣿⡟⠰⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠛⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⣀⡤⢤⡄⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢠⡾⠁⠀⠐⠃⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⢻⣧⡀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠀⠀⣰⣿⠏⢀⠔⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⣀⠔⢺⣷⡄⠀⣠⡀⠀⠀⣰⣿⠧⠊⠁⠀⣠⣾⠟⠉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⢀⣴⠀⢀⠔⣶⢀⠔⠁⢠⣾⡿⠀⣼⡿⠁⠀⣰⣿⠇⠀⠀⢀⣾⣟⣡⠴⠊⠀⣠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣿⡇⠀⣾⡏⡴⠁⣼⡷⠃⠀⣰⣿⠟⠀⢰⣿⠁⠀⣠⣿⡏⠀⠀⠀⣼⣿⠁⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⢸⣿⠇⢸⣿⡜⠀⣼⡿⠁⠀⣰⣿⠃⠀⢠⢻⣏⣀⠔⢹⣿⠃⠀⢀⡜⠹⣿⣄⡤⠞⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠏⠀⠸⡿⠀⠀⠙⠃⠀⠀⣿⡇⠀⡰⠁⢀⣭⡅⠀⠸⣿⣦⡴⠋⣠⣤⠴⠤⢤⣄⣀⣠⣿⠇⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠊⢀⣴⢋⡔⠁⠀⠀⠈⠉⠀⠈⠁⠀⣀⣠⢀⠀⣸⣿⠿⠤⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣔⠲⠂⠀⡿⢉⣟⣠⣧⣖⣋⡇⠀⠀⢠⣾⡡⠊⣀⡀⠀⠀⣠⡴⠖⠂⢠⣾⠋⢀⡴⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⢸⡇⠀⢸⠃⡾⠁⡟⢿⣀⣰⠀⠀⢀⣾⢏⡴⠋⠉⢻⣆⣴⡿⠁⠂⠁⠘⣿⣦⠀⠀⣿⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⢀⣠⣀⠀⠀⠀⠀⠀⠩⠤⠴⠚⢹⣿⠎⣠⠔⢀⡽⠡⣿⠁⠀⡰⠀⠀⢸⣿⡇⢸⣿⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡴⠋⣸⡿⠀⣠⠖⢩⣿⡇⠀⠀⠀⠸⠏⠀⠓⣰⢏⠀⠀⠙⠷⠞⠁⠙⠲⠞⠋⠀⠈⠛⠒⡁⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⠞⠀⣰⣿⢣⠞⠁⣠⣿⠏⠴⠖⠲⠶⠤⣄⣴⣟⠥⠊⠀⠀⠀⣀⡀⠀⠀⣠⡶⠀⢀⣾⠂⣼⡿⠊⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⣰⣿⡷⠃⠀⣰⣿⠏⣠⡴⠒⣄⡀⢀⣾⢋⡴⠶⣶⢀⣴⠟⠉⡹⢀⣾⠟⠀⠀⣾⠃⣼⠏⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⢠⣿⡿⠁⠀⣼⣿⢃⡾⠋⠀⢠⡿⠃⣾⡟⠁⣀⠴⢣⣿⡧⠔⠊⡀⣼⠏⠀⠀⣼⡏⢰⣿⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣿⡿⠁⠀⣸⣿⢃⣿⠁⠀⢀⡿⠁⢸⡿⠛⢿⣤⠀⠸⢿⣄⡤⠊⠀⡟⠀⢀⠞⣿⢀⡿⠀⠉⠉
⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⠃⠀⠀⣿⡏⠘⣿⣀⠴⢻⣇⡠⣿⡇⠀⠀⠻⣷⣄⣀⣀⣠⠔⠀⠙⠖⢋⣀⠬⠼⣟⠉⠉⢹⡄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⢿⣧⠀⠀⠀⢀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠉⠉⠉⣁⣠⡤⠖⠋⠁⠀⠀⠀⠈⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠈⠳⣤⣀⣀⣀⣀⣀⣤⡤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⣄⡀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠉⠄⠀⠀⠀⠂⠐⡂⠒⠀⡀⠀⠀⣀⣀⣀⡁⠀⠀⠁⠃⠄⠖⠂⠀⠀⠸⠐⠀⠂⠹⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⠂⠀⠀⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠾⠛⠛⠛⠻⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢐⠀⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠄⠠⠀⠀⠀⠄⢩⢈⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠐⣀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠒⠚⠐⠒⠀⠀⠀⣒⢀⣀⣁⡁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠅⠄⠀⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠠⠄⠀⠀⠀⢰⠖⠔⠀⠀⠀⢀⡄⠀⠀⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⠐⡄⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣨⠓⠶⠶⡄⠀⠀⢲⣿⣷⣾⢓⡆⠀⠀⣴⢠⡀⣤⣄
⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠳⠀⠐⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠉⠒⠶⠖⠂⠀⠀⠀⠙⠻⠟⠛⠃⠀⠘⠛⠛⠛⠋⠁

⠀⠀⢠⠄⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⢤⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠
⠀⠀⠶⠀⠀⡰⠖⠀⠀⠀⢉⠈⢀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⣴⣷⣶⡀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣴⣄⠀⠀⠀⠀⣄⠀⣠⡷
⠀⠀⠀⣀⣀⣤⣤⠀⠀⣶⠀⠀⢸⣷⡀⠀⢠⣿⠀⢰⡟⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠁⣰⢖⡆⠀⣿⠻⠋⡿⣰⡶
⠀⠀⢻⣏⠉⠀⠀⠀⠀⢹⡆⠀⢸⡏⠻⣆⢸⡇⠀⢸⡇⠀⠀⣷⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣿⠟⠁⠀⠙⡓⠲⡴⠋
⠀⣀⠈⣿⡖⠃⠀⠀⠀⠘⣷⠀⠸⣷⠀⠙⣿⡇⠀⣼⠇⣀⣴⠃⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⢳⠘⣦⠉⡗⣾⡽⢂⡄
⠀⠠⠄⠘⣿⡀⠀⣀⣀⣤⣭⡤⠴⠷⠶⠶⠾⠦⣤⣽⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠴⠋⠀⠉⢀⠉⠙⡓⠋
⠀⠀⠀⠀⢽⠟⠛⣭⣭⣿⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠛⢛⠛⠛⠛
⠀⠀⠀⣤⣾⣷⠀⠀⣀⣿⣄⣤⣶⠆⠻⣿⣤⢼⡇⢰⣿⡀⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠤
⠀⠀⠀⠉⠛⣱⡾⠋⠉⢸⡇⠀⠸⣦⡴⠟⠀⠈⣻⣿⣽⡷⠶⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢰⡆⣶⠒⠀⣶⢲⣀⣾⢳⡀⣀
⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣰⠟⠋⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡿⠟⣿⣸⡇⣿⠒⠀⣿⡟⢸⣿⢸⡗⠿⠟⠃
⢰⣦⣤⣄⡀⠻⢦⣤⣴⠟⠁⠀⠀⠀⠀⢀⠙⣶⠛⠛⠁⠀⣠⣴⣶⣦⣶⣶⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⣃⠛⢒⠀⠛⠙⢠⠛⡎⠁⣠⡄
⢸⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⡇⠾⠇⠀⠸⠇⠿⠖⠾⠧⠀⠘⣿⣿⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠸⠿⠶⠻⡦⠀⠶⠛⠛⠋⠀⠛⠛⠃
⠸⣿⠿⢿⣿⣿⠟⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠸⣿⣿⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣿
⠀⠹⣶⣿⡿⠃⠀⠈⣻⠶⣏⡸⡟⣸⣟⣿⣹⡟⣷⣾⡷⢷⣧⠀⠸⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⣱⣿⣇⣤⣀⣤⣤⣤⣄⣤⣠⣟⡷⣀⠄
⠀⠀⠈⠉⠀⠀⢠⠎⢹⠀⠈⠀⠀⠀⠁⠈⠀⠀⠀⢀⡴⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠟⠧⠿⣥⠯⠼⠥⠇⠛⠙⠛⢋⠾⠃
⠀⠀⠀⠀⠀⠀⠘⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣰⣄⠀⠀⣠⡄⣰⣦⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⣽⡿⠓⠒⢿⣟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣿⣿⣿⣿⣿⣷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠲⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⡄⢠⠆⠈⡙⢿⣷⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠠⠆⠙⠳⣤⢤⡞⠁⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⢠⡄⠀⣰⠈⠁⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡁⠱⣆⣤⣀⡍⣀⠀⢀⣀⡀⣀⡀⠀⠻⠋⢳⣀
⠀⠀⠀⠀⠀⠈⣭⡀⣀⣠⣤⢨⡄⢠⡄⠀⢠⡄⣠⡌⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⠏⢿⣀⣿⢁⣯⡼⣧⠏⠿⢿⣇⠴⠈⠉⢹
⠀⠀⠀⠀⠀⠀⠉⠁⠈⢀⣉⣈⡉⣄⢠⣄⠨⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⣁⠀⡾⠁⠸⠅⣴⠞⠰⢢⡀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⣰⠏⠉⣷⠀⠈⠉⠀⠀⠁⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠰⣷⡟⢰⡇⣿⠀⠀⠘⣧⢀⣤⡐⠒⠉⠁
⠀⠀⠀⠀⢻⡄⠞⢩⣤⣀⡀⣠⣤⢀⣤⡄⠀⢸⣟⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡟⠀⠀⣿⢹⣾⡇⣿⠤⣤⠀⢸⡟⠉⢷⡀
⠀⠀⠀⠐⢾⣷⡖⠉⠉⣿⢀⣯⡟⣉⣷⡿⣶⡉⢿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠻⠟⠘⠇⠘⠶⠋⠀⠀⠀⠀⠈⠉⢣⣄
⠀⠀⠀⠀⠀⠉⠳⠆⠀⠈⣭⡉⠛⠉⠹⣤⠟⠁⠘⣇⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣴⡄⢶⡆⠴⠶⠶⠶⠄⠆⡀⠐⠓⠓⠛⠒⠀⠻⠓⠛⠛
⠀⠀⠀⠀⠀⠀⠴⣯⣁⡸⣿⠉⣠⢤⡄⠰⡷⣶⠆⣿⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣶⡵⠖⣦⡻⢿⠃⣞⢁⣟⣾⢀⣇⣤
⠀⠀⠀⠀⠀⠐⠂⠀⢩⡇⠹⣄⣧⡸⢇⡼⠃⢿⡤⢿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⠴⠛⠁⠈⠋⠉⠉⠉⣩⢿⡟⠁
⠀⠀⠀⠀⠀⣴⠒⠾⣯⡶⣶⣺⢶⣶⡾⣿⡟⢫⣿⢿⡥⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⣀⡀⢀⣀⢠⠋⠀⣸⠃⠀⣀
⠀⠀⠀⠀⠀⠸⣃⡤⡋⢴⢿⡟⡻⣶⢶⣾⣟⣿⡷⠉⠻⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠉⠀⢸⡉⠁⣿⡏⣀⡙⢛⣉⡀⢐⡡
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⢁⡷⠘⡏⣿⠉⠈⢻⣷⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠠⣞⢻⡖⢻⣅⠜⢏⣰⣿⣋⣼⠛⣿⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠣⠴⢀⠏⠦⠶⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠁⠀⠀⠀⠈⠁
⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠒⠛⠋⠉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠙⠛⢿⣿
⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⠐⠒⠛⠉⢁⠀⠀⠀⢶⡄⢀⡀⠀⢀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠉⣓⠀⠘⢿⣿⡆⠀⠀⣰⠋⠀⠀⢰⠞⢋⠟⢁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⢸⣿⣿⣿⣿⣿⣿⠟⡀⡀⠀⠀⠀⠈⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠛⠃⠘⠁⠀⠈⠁⠀⠀⠀⠀⣠⣶⣶⣆⠀⢦
⠀⠈⣿⣿⠛⠀⠚⠁⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⢀⣾⣿⣿⣿⣿⣷⣾⣿⡆
⠀⠀⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡄⠐⠂⠀⠠⠀⠀⠀⠀⢀⣾⣿⣿⠟⠋⠉⠀⢸⣿⡷
⠀⠘⣿⣿⡿⠿⣿⠛⠉⠁⠠⠀⠒⣶⣶⣾⣿⣿⣿⠟⠿⢿⠟⠙⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠁⠀⣰⠆⢈⣿⡇
⠀⠐⣿⣿⡄⠀⣹⡆⢠⣶⡆⠀⠀⠻⠋⢉⣹⣿⣿⡧⠀⢤⣄⠞⠁⠀⠀⠠⠀⠀⠀⢀⣾⣿⣿⠁⠀⢀⡀⠿⢠⣾⡿
⠀⠀⣿⣿⣠⣂⠈⢿⣿⡻⠃⠀⠀⠀⠀⢻⣧⡹⣆⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⣀⢼⣿⣿⡇⠀⣄⠀⠈⣠⣿⣿⠃
⠀⢠⣿⣿⣿⡏⠀⠀⠙⢧⢀⠴⠔⠂⠀⠈⠻⣧⣨⣤⠤⠞⠃⡀⠁⠰⣮⡉⣻⡿⣿⠰⣿⣿⠁⠀⠘⣿⣿⣿⣿⡏
⠀⠈⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣟⡗⠀⠀⠠⡼⠃⣸⠀⠀⠀⠙⠓⣿⠀⣿⣿⠄⠀⠀⠈⣹⣿⣿⢳⣄⡀
⠀⠀⣿⡄⡀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠘⠿⣯⣬⡿⠛⠀⢠⡏⠀⠀⠀⠀⠀⠈⠀⠻⠻⡆⠀⠀⠀⣹⣿⡇⡌⢏⡿
⠀⠀⣿⣇⢃⠀⣴⠀⠀⠀⠀⠈⢠⠈⠀⠄⠀⠐⠒⠃⠀⠀⠀⠘⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⢠⡇⠜⡡
⠀⠀⠁⠹⣼⡄⣸⠀⠠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⣠⣴⠀⠊⣴⢨⡏⢁⣾⠇
⠀⠀⠀⢠⣝⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⣼⣧⡾⢣⣿⣵
⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡟⠀⢀⣿⣟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠖⠀⡀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠸⢧⡀⢸⠇
⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠙⣦
⠀⠀⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠉⠀⠀⠈⠃⢀
⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠟⠳⡶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡇⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣞⣀⡀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⠀⠟⠸⠀⠀⣠⠀⣼⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠂⠀⠀⠀⠀⣠⠄⣡⣾⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠈⠑⢶⣦⣤⣤⣤⣤⣤⣶⠟⠁⠀⠀⠂⠀⠀⢀⣴⣾⡿⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠈⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡠⠀⠀⣴⣿⢇⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢈⠙⠿⠏⠋⢀⠀⠐⠀⠀⠀⠆⠾⣧⣿⣿⣿⣿⡻⣿⣿⣿
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠟⠛⠛⠛⠛

⠀⣀⣐⣠⡤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡒⡛⣂⠀⠀⠀⡆⢰⡀⠀⠀⠀⠤⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠣⠩⠤⠤⠤⠀⠀⠀⠈⡐⠂⠀⠀⠀⢃⣉⢉⡍⡙⠀⠀⠀⠻⢌⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⢀
⠀⠐⡄⢥⡤⢶⠀⠀⠀⠒⠐⠒⠂⠀⠀⠙⢉⣨⣉⠕⠀⠀⢐⢂⠀⠀⠀⠀⠀⠀⠠⠆⠄⠀⠀⢀⣀⢤⠛⡛⠃⠃⡯⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢡⡆⠠⣤⢫⡏⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⣠⣤⣄⠀⠀⠀⡞⠠⢞⠀⠀⠀⠞⠒⠧⣀⣉⡄⠁⠁
⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠥⠖⠒⠀⡀⠸⡙⣋⣉⣠⣀⡀⠀⠀⠁⣡⠼⡓⠀⠀⠀⠀⠄⢀⠀⠄⢀⡀⡀
⠀⠀⢀⡀⣀⡀⠀⠀⠘⠘⠒⢚⠖⠁⠀⠑⣂⠂⣉⢞⠁⠀⠀⠤⠼⠤⠥⡆⠀⠉⠘⠉⡊⡙⠀⠀⠀⠈
⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠀⠉⠈⠀⠀⠐⠲⢦⠞⠁⠀⠀⠀⠓⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⢿⠀⡀⠀⢀⠠⠠⠠⢀⠀⠀⠀⠀⠀⠀⠠⡶⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄
⠀⠀⠀⠀⠀⢀⣤⣼⣦⣧⠄⠱⣀⣤⣴⣄⡄⠀⠀⠀⠀⠀⠀⢨⠷⡻⡽⠶⡙⠓⠀⠀⠀⠀⠀⠀⠀⢀⠄⣁⣀⣄⡀⡀
⠀⠀⠀⠀⠀⠈⢇⠀⣧⠣⣎⡿⠐⢶⢀⣎⢇⠀⠀⠀⠀⠠⡿⡐⠀⣶⠒⣀⢈⡀⡇⠀⠀⠀⠀⠀⠀⠘⠋⢉⡉⣍⡛⠁
⠀⠀⠀⠀⠀⣴⣌⣓⣁⣤⠀⣴⣤⣰⡥⡄⡾⠋⠀⠀⠀⠀⠂⣇⣸⢿⠈⣿⠀⡇⠃⠀⠀⠀⠀⠀⠀⡖⠦⡦⠲⠴⠐⢺⠆
⠀⠀⠀⠀⠀⠈⡍⢍⡉⠈⠀⠐⡐⠐⣒⠂⠒⣤⠀⠀⠀⠀⠀⢀⠀⢐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠟⠂⡁
⠀⠀⠀⠀⠀⠈⣟⣘⣳⣶⣛⣶⣿⣿⣭⣧⣤⣷⠀⠀⠀⠀⡇⢘⣶⣥⣛⣴⡆⢠⣆⠀⠀⠀⠀⠀⢠⡀⣀⡋⠛⣋⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠈⠁⠘⠉⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠄⣇⢘⡟⣄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣧⣶⣿⢧⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠤
⠀⠀⠀⠀⠀⣠⡄⠀⢈⡁⠲⠦⠈⢠⣄⣄⡀⠀⠀⠀⠀⠴⠂⡶⠒⠁⠈⠈⢩⠀⠀⠀⠀⠀⠀⠀⢻⠦⠐⠘⠟⠲⠧⠆
⠀⠀⠀⠀⢐⣿⡏⠰⣟⢡⠄⣤⣾⠉⢙⣿⡇⠀⠀⠀⠀⠀⠸⠥⠼⣾⠾⠚⠞⠀⠀⠀⠀⠀⠀⠐⠒⣂⣐⣒⠀⡂⠀⠒
⠀⠀⠀⠀⠈⠻⡿⠀⡹⡍⠉⣉⠎⠁⠘⠏⠁⠀⠀⠀⠀⠀⣠⠀⠉⣉⢉⠛⢁⡀⠀⠀⠀⠀⠀⠀⠈⡩⣏⣍⠩⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢁⣰⢧⣦⡀⠀⠀⠀⠀⠀⠀⠻⠐⣁⡼⡟⣗⢸⠇⢸⡆⣩⠀⠀⠀⠀⠀⣶⡀⡍⣉⡀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠚⠻⠋⠙⠥⠂⠀⠀⠀⠀⠀⠀⠰⣌⠐⠐⠐⠒⢒⡰⠂⠀⠀⠀⠀⠀⠀⣹⣓⠓⠃⠿⣛⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⢀⡀⢀⡀⢠⡀⢀⡄⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢀⡀⢦⢼⣀
⠀⢻⡁⠀⠀⡇⢸⡑⣼⠁⡿⢀⡇⠁⠀⠀⠀⠀⠀⠀⢀⠵⣿⣥⣬⠐⠁
⠀⠈⣧⣠⢤⣽⠚⠓⠚⠲⠿⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⡴⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠠⣷⣈⡽⡧⢴⢸⡦⢶⣴⣐⡷⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⢻⢄⡤⠀⠀⠀⠀⠀⠀⠀⣴⣥⠄⠛⠳⠴⡬⠓⠮⣟⡯⠚⠳⣷
⠀⠀⢸⠁⢀⡇⠀⠉⢠⢚⡹⠁⠀⠀⠀⠀⠀⠀⠀⠙⠙⢻⠿⠸⠃⢧⢉⡉⠀⠀⠀⢲⠚⠛⠻⠿⣧⣤⣤⠤⡤⢼⣥⢾⡿⢿⠙⠓⡲
⣿⣿⣶⣯⠉⠀⠴⠀⠆⠧⠦⠘⣿⣿⣿⡇⠀⠀⠀⠀⠈⣨⠉⠁⠈⠉⠈⠉⠀⠀⠀⣤⣃⣀⡀⡇⠀⣽⠉⣾⠉⡧⠙⢾⠁⢸⣠⣉⣀
⠹⣽⡟⠁⣲⢄⠦⡴⣤⢤⡤⣶⠘⠿⠛⠁⠀⠀⠀⠀⠈⡿⠿⢶⠶⠶⠶⡿⠂⠀⠀⠀⠀⠈⠉⠧⠵⠩⣤⣈⣈⡁⣤⠬⠶⠞
⠀⠀⠀⠨⠇⠀⠀⠀⠀⠀⣚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠴⡖⠟⠁

⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣶⣀⣀⣀⣀⡀
⠀⠀⢠⣿⠋⢀⠀⠁⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣈⠳⠒⠁⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠉⣿⣦⠖⣴⣄⠘⡏⠙⣿⠆
⠀⠀⠀⠆⢒⠖⠂⢀⢐⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠏⡯⢧⢶⣟⠛⠃⠀⢙⠄⠀⠀⠀⠀⠀⠐⠚⠓⢿⣾⠶⠖⠑⠛⠁
⠀⠀⣏⢽⡀⠀⡀⡀⠀⡗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠑⣿⣸⡇⠀⢣⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣴⣼⢦⡆⣶⣷⠀⢀
⠀⠀⠻⡌⠹⠼⠯⢟⡎⢣⠧⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠟⠙⠉⠋⢈⡀⣑⣤⡀⠀⠀⠀⠀⠀⢀⡀⠃⢹⣏⣍⣉⢉⣵⠇
⠀⠀⢀⠺⡌⣇⣴⣆⢷⣺⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⡭⢭⡶⣆⣱⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢁⣘⠋⠧⠇⢀⣀⣶
⠀⠀⠰⠔⣷⣾⣷⣿⣯⡿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⣀⢀⢠⠂⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣛⠻⠾⠾⢛⣡⠏
⠀⠀⠀⠀⠄⠹⠛⡎⣙⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣸⡉⢟⣴⣩⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁
⣶⣶⠶⠶⠶⢶⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⣶⠚⠀⢣⢸⡷⣸⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⢡⣼⣃⣋⠜⠁
⠀⠀⠀⠀⠀⠀⠘⣇⡀⣬⠤⠧⠽⠤⠷⡁⠀⠀⠀⠀⠀⠀⠀⠱⢭⡭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⢷⡨⢯⠰⣐⠦⢆⣦⡸⠂⠀⠀⠀⠀⠀⢀⣄⣤⢻⣀⡄⣀⠀⠀⠀⠀⢠⣬⡅⡛⠳⠴⠼⠂⢽⡻⣞⠳⣶
⠀⠀⠀⠀⠀⢀⡀⠫⣀⡼⠀⠀⠰⡫⠏⣀⣀⣀⡀⠀⠀⠈⠉⣛⣛⡘⣹⡇⣩⠀⠀⠀⠒⡋⠉⠿⠳⣶⡶⣢⣖⢶⡶⡿⢛⠉⡙⠒
⠀⠀⠀⠀⠀⢻⣿⣿⢇⠀⠘⠀⠁⠉⠁⠸⣿⣿⠇⠀⠀⠀⠀⣴⣦⡀⣀⣀⣠⡀⠀⠀⠚⠦⠤⣼⠀⡸⠀⢇⣸⠃⠹⠁⢸⠖⠶⠆
⠀⠀⠀⠀⠀⠈⠛⠁⡼⠛⠋⠛⠛⠙⢹⠇⠉⠁⠀⠀⠀⠀⠀⠸⣍⡉⢉⣩⡝⠁⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠐⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠁⠙⢷⡄⠀⠀⠀⠀⠀⠀⢀⡆⣀⠒⠒⠀⠀⠘⠄⡀⠀⠀⠀⠀⠀⢨⢻⣭⢩⣭⡉⡿⠉⣳⠆
⠀⠀⠀⠀⠀⠀⠀⣘⠂⡒⠂⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣽⣳⡺⢋⠙⠀⢈⠆⠀⠀⠀⠀⠈⠊⠁⣿⠿⠗⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡅⢧⡄⣤⣤⡸⡧⡀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣿⣼⣇⡈⠶⢂⠀⠀⠀⠀⠀⠀⢀⠐⡾⣟⣶⣼⣿⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠙⡄⣭⢉⡙⣇⣥⠁⠀⠀⠀⠀⠀⠀⣬⠤⠠⠭⣄⠈⢄⠀⠰⠲⠂⠀⠀⠀⠀⠐⠷⠄⢺⣶⣶⠂⠚⢁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣭⣦⣷⣽⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠉⠒⢉⢷⠃⠀⠀⠀⠀⠀⠀⠀⢻⣔⠶⣤⣤⣤⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠟⡟⠻⡭⠇⠀⠀⠀⠀⠀⠀⠀⠐⣈⣡⣇⡖⣄⣪⡐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⡀⣰⣷⢠⣦⢰⣿⠀⠀⢀⡶⣳⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠀⠀⣾⣿⣷⢿⣿⠈⠿⢾⣿⠤⠄⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⡏⢹⡿⢸⣿⢠⡞⢹⠏⠀⠀⠀⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⣦⠀⠀⠀⠀⠀⠈⠉⠁⢾⣠⣿⣀⡠⠤⠴⠒⠒⠒⣾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣴⡆⣠⢛⣶⠀⣠⣶⠀⣴⢦⣶⠀⣿⠃⢠⣶⡄⢠⡤⡀⣸⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠯⢞⣿⠏⠁⣾⠏⢀⣿⢋⣼⣯⣾⡏⢸⡏⣠⣿⢋⣠⢻⣷⠊⣿⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡟⠀⠀⠛⠒⠉⠉⠁⠀⠀⢿⡠⠋⠉⠁⠈⠉⢰⢉⣿⠀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢪⡾⢻⠀⣰⡟⡇⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠇⢠⣿⡰⢡⡤⠀⣴⠶⢸⣿⠉⣠⡍⣤⢀⣄⠀⣤⠦⣾⡆⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠦⢴⣿⢥⣄⣼⡟⢁⣿⠟⡠⣿⡆⠈⣿⡖⣿⢠⣿⢻⡿⢸⣏⢜⣿⢋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠼⣿⠈⠓⠋⠙⠊⣴⢹⠇⡼⢻⡇⠙⠁⠉⢸⡧⢃⡔⢹⡏⠸⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⣀⠀⠀⢀⠀⢈⠁⠀⠳⠛⠀⠀⠀⠀⠀⢀⢘⣒⣋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⢠⡟⠙⡇⣰⠟⡇⣿⠀⠀⣿⠀⠀⠀⣿⣆⢰⣿⢸⡏⠉⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠒⢂⢸⡇⠀⣄⣿⢶⡇⣿⠀⠀⣿⠀⠀⠀⣿⠻⠟⣿⢸⡗⠒⠀⢐⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠈⠷⡾⠋⠿⠀⠇⠿⠤⠄⠿⠤⠄⠀⠿⠀⠀⠿⠸⠷⠶⠄⠈⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⡀⠀⠀⢴⡞⠲⡄⢠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣷⢾⣧⢨⣿⠂⡶⢺⣿⢟⣿⡔⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡏⣾⡇⡼⣟⣠⠃⢸⡿⢸⣿⢰⡿⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢿⡼⠁⠈⠀⠀⠀⠀⠀⠀⠘⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠀⣎⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣄⡀⣿⠀⣟⢶⢰⣶⣶⣶⡆
⠀⠀⠀⠀⠄⣠⢄⣤⣀⣨⣌⣡⣀⣤⠄⠠⠀⠀⠀⠀⣤⠈⣥⢀⣠⡀⣉⢩⡝⡈⣠⠄⣘⣡⡀⠀⠀⠈⠋⣤⠭⣤⢭⠬⠬⢭⡅⠩⢁⣴⡀
⠀⠀⠀⠀⠀⢙⡦⡟⠻⢿⡸⢻⠗⢿⡄⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⡐⢻⣸⣇⡇⣿⠀⣿⣞⠁⠀⠀⢠⡯⣹⢓⡟⣸⣇⡟⠛⢱⢷⣯⢉⡅
⠀⠀⠀⢤⣦⡄⣦⣀⢦⡶⣤⠶⣴⠶⢄⣰⡀⠀⠀⠀⠉⠀⠉⢁⠌⠀⣌⢡⡍⠀⣭⠁⢉⡈⠁⠀⠀⠘⠓⢛⣞⣲⠛⠃⠛⠀⢙⡻⡙⣛⡂
⠀⠀⠀⣈⢋⢁⣉⢉⣈⣋⢈⢃⣙⠈⡈⢛⢁⠀⠀⠀⢰⣿⡆⣿⢰⡄⡇⢸⣷⡀⣿⠀⢾⡷⠀⠀⠀⠈⡏⡽⠆⡄⢩⡉⢁⣤⢤⠰⠹⡸⡃
⠀⠀⠀⠟⠸⠆⠿⠸⠿⠘⢾⠸⡸⠻⠲⠻⡾⠀⠀⠀⠀⠀⠀⠈⠉⠠⠥⠌⠈⠡⠍⣠⣤⠀⠀⠀⠀⠘⠓⠻⠒⡂⠀⡀⠀⡀⠐⠐⠛⠒⠛
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⡀⢰⣷⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠛⠛⠀⠀⠀⠀⠀⠺⠿⠸⠳⠹⠿⠨⠗⠚⣾⠸⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣴⠀⡤
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢧⠸⡇⣿⢠⡇⣰⠃⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⣀⠠⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢤⡳⣌⣧⢳⣿⣸⣱⢣⢞⡡⠆⠀⠀⠀⠀⣄⡀⢺⣠⡇⡀⣠⡄⣿⡀⠀⢠⡀
⠀⠀⠘⣧⠘⡇⠀⡞⠀⢻⢸⡃⣟⠰⡏⣿⠀⠀⠀⠀⢠⢀⢩⣈⢬⣨⢌⣠⢁⣥⡉⣠⠄⠀⠀⢀⡤⣟⢁⠸⠈⠃⠃⢻⡇⡋⠷⠀⢾⣿⢇
⠀⠀⠀⡜⣧⣁⢰⡆⡀⠀⢀⡦⢸⣆⣠⠌⠁⠀⠀⠀⠸⠼⠤⠻⠤⠻⠯⠹⠼⠷⠃⣬⠗⠀⠀⠈⠳⣆⣞⣴⣆⣴⣎⣴⡾⣵⣾⢸⣆⣶⣯⡆
⠀⠀⠀⣙⢿⣃⠘⢂⣓⡁⠀⣃⠘⣑⠛⣀⣀⡀⠀⠀⢀⡤⣾⢩⠀⢢⡤⣃⡤⣱⢺⣼⠆⠀⠀⠙⠚⠉⠉⡉⠉⠁⡉⠉⠉⠉⠉⠈⠉⢉⠁⠉
⠀⠀⠘⣿⠋⣿⢶⣿⠉⣿⣼⣿⣆⣿⢸⣿⣉⠁⠀⠀⠘⠒⠓⠛⠊⡿⠘⠙⠛⢨⡆⠛⠂⠀⠀⠀⠀⠀⠘⠛⢐⡭⠒⠋⠉⠙⢪⠢⠤⠜
⠀⠀⠀⣿⣋⣿⠜⣿⣀⣿⠑⣿⠹⣿⠈⣿⣉⡀⠀⠀⠀⠀⠀⢰⠒⢷⣲⢶⡶⡆⠀⢠⡦⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠠⠠⠜
⠀⠀⠀⠁⠈⠁⠀⠀⠉⠀⠀⠁⠀⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡇⠀⢀⠄
⠀⠀⠀⣀⠀⣾⡇⣲⡀⢀⠁⣀⠀⢃⡀⢱⣆⡀⠀⠀⢠⡌⠟⢿⢱⣄⢠⢀⣤⣄⣦⣬⡋⠀⠀⠀⠀⠀⠄⣀⠢⠁⠀⠀⠀⠂⠔⣀⠤
⠀⠀⠸⣏⣱⠏⣼⠉⣻⣾⢟⡿⣰⡿⢋⠎⡾⣡⠀⠀⣤⡀⠀⠀⢸⡏⢹⠻⣗⡁⣿⢾⠀⣤⠀⠀⠀⢀⣀⢠⣉⣀⡀⢒⡀⢂⢁⠠⣀⡀
⠀⠀⠀⠟⠙⠋⠛⠚⠁⣛⠈⠛⠙⣳⠋⡀⠛⠁⠀⠀⠇⣿⡆⢛⡴⢦⣰⢄⡦⣠⢶⡶⣸⠏⠀⠀⠀⣸⣟⢸⣿⣟⡇⡏⢿⢹⣾⣑⡯⡟
⠀⠀⢠⣰⣆⡥⣾⢱⡆⣷⢘⣷⡆⣿⠠⣀⣆⡤⠀⠀⠀⣿⣰⠻⠧⠞⢿⣾⠵⠿⠺⠷⠿⠎⠀⠀⠀⠙⠛⠙⠛⠛⠋⠛⠛⠛⠘⠋⠓⠛
⠀⠀⠐⠛⠛⠂⢙⡿⠃⠿⠸⠻⠇⠛⠀⡛⠛⠃⠀⠐⡓⠈⡁⣀⠀⡂⠺⡋⠀⣀⢀⡀⢐⡃⠀⠀⠀⡇⣴⢎⡱⣧⢼⠀⠘⢇⡙⣼⣿
⠀⠀⠀⠀⠄⠐⠀⠀⠉⠉⠀⠀⣀⠀⣺⣿⠇⠀⠀⢰⣧⣿⣿⡏⣽⣧⣿⣿⣿⡏⠹⣏⠻⣍⠀⠀⠀⠀⣴⣤⢠⠤⣬⣤⢤⣤⠠⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠙⠉⠁⠉⠁⠁⠛⠈⠋⠈⠋⠀⠀⠀⠀⠉⠁⠘⠂⠋⠙⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⢀⣇⣀⠀⠀⣀⢀⡀⢀⣀⡀⢀⠀⡀⠀⠀⢀⠀⡀⢀⠀⢀⣄⢀⣀⣀⢾⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠀⣿⡈⠀⠀⡟⢸⡇⣟⢹⣇⣼⢀⡇⢀⠀⢸⠀⡇⢸⢰⠋⣶⢸⡏⢹⢸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⣷⠀⠀⠙⢻⡗⠙⠛⠉⠙⠋⠙⠁⠀⠘⠛⠙⠋⠈⠛⠛⠛⠁⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡟⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⣀⠀⠀⢤⡄⠀⢠⡄⠀⢀⣀⣀⢀⣀⣀⣀⡀⣤⡄⠀⣀⡀⣀⡀⢀⣤⠀⢀⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⠀⣟⠈⢷⠀⢸⣧⠀⢸⣇⠀⡏⠉⠙⠈⠁⣿⠀⠁⢸⠀⠀⢽⠀⢸⠁⢸⣷⡀⢀⡇⢀⡞⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢰⡏⠳⢼⡤⢼⠹⣆⡏⢿⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⢸⠀⢸⡏⣧⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⠘⡇⠀⣸⠀⢸⠀⠿⠁⢸⠀⡷⠆⠀⠀⠀⣿⠀⠀⢸⡶⠶⢿⠀⢸⡀⢸⡇⠸⣾⠀⢻⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣡⡼⠀⠙⠶⠋⠀⠸⠆⠀⠀⠘⠃⠷⠤⠗⠀⠀⠿⠄⠀⠸⠧⠀⠾⠆⢸⡇⠸⠇⠀⠻⠀⠈⠳⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠖⣗⣤⠀⣠⣄⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⣀⡟⢸⣇⣷⠏⣸⠁⣇⣠⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⣼⣇⠠⣴⡤⠀⣠⡶⢦⡀⠀⠚⠛⡟⠙⠛⢶⣄⠀⠛⣿⠋⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣿⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⠀⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢻⡀⣿⠀⢰⡟⢧⣄⣸⡇⠀⣠⡧⠀⣠⡾⠃⠀⠀⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣼⠃⢸⣷⡇⠀⢸⡇⠀⠉⠛⡿⠛⠉⣿⠛⣿⠀⠀⠀⠀⣿⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠀⠘⣿⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠘⣧⠀⠀⠀⣿⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢿⣄⣠⡾⠁⠀⠀⣿⠀⠀⠘⠷⡤⠀⣿⠀⠀⠈⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠉⠀⠀⢀⠀⠉⠉⠀⠀⠀⠺⠿⠂⠀⢀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠳⣼⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⢸⡟⢳⡿⣷⢸⢯⡇⠀⠀⡇⢸⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡌⠷⠟⠋⠘⠷⠃⠀⠳⠛⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠶⣖⠛⠛⢛⠛⡛⠛⠛⣛⣛⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣱⢷⣿⠄⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢹⢶⡋⡷⡿⢤⠀⢸⠇⠀⣨⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠿⠞⣷⡚⡚⣻⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠼⠧⠲⣿⣷⣓⣲⡖⠒⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠾⢇⠗⠓⠛⠚⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢽⣉⠉⡉⣉⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⣘⣒⠙⠀⣙⠠⢤⡤⢤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢧⡟⣾⢹⣧⣇⢹⣽⣿⡁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡷⡟⠛⡇⡗⠴⠷⣧⢤⣼⠄⢗
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡆⡘⠃⢙⣞⢁⣉⡉⣉⠁⣅⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⣦⠀⠦⠆⠰⠆⢰⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⠷⠄⢸⡾⠻⣛⡹⠁⡚⣋⣁⣀⠀⠀⠀⠀⠀⠀⠀⢞⠠⡼⣼⣆⡤⣀⡀⢄⡠⣼⣃⡈⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣠⢦⣇⡈⢉⠁⣀⣠⣶⠖⣿⠀⠀⠀⠀⠀⠀⠤⠟⣀⠧⠋⠛⣃⢨⣯⡉⠉⠁⠛⡀⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⡛⠹⠷⢿⠽⠽⠛⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⢹⠇⠙⠋⠉⠀⠻⠁⠘⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣤⣤⠼⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣾⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢾⣀⡺⠧⠴⢒⣒⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⢿⡼⢿⡾⡸⢏⣠⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⠛⠓⠻⠛⠛⣿⠞⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⠄⢠⣦⣤⡄⡄⠈⠰⠀⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠶⠧⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠱⣮⠉⢰⣵⡦⡠⣶⢴⠘⣿⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡦⣶⣆⢲⣴⣰⡆⣞⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢀⢸⡏⡇⠀⣿⢸⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠝⠦⡁⠭⠠⠭⠄⠡⠶⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⣉⠙⠁⣅⡄⠙⠚⣒⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣋⡉⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⠆⠀⣀⣤⣁⡀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠚⠋⠛⢛⣛⠒⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠘⣲⣷⣇⠼⠙⢇⠼⡴⠃⢀⣧⠀⠀⠀⠀⠀⠀⠀⠘⠣⡖⣾⡟⣆⠀⢿⣾⣶⣦⣴⢄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢾⣱⢿⠿⡗⠒⠛⢽⡇⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠁⠀⠀⠀⠈⠀⠀⠙⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣾⠉⠀⢰⢧⠈⡟⠁⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠼⠂⠀⠀⠀⣿⠤⠀⡾⠼⡄⣇⠀⠀⠀⠈⠲⡜⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⠤⣨⠊⠉⡥⠄⢡⠭⠀⣤⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠰⢄⢸⠀⠀⡇⠀⢸⠀⠀⡇⣱⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡋⢣⠄⠀⣠⣤⡀⠀⠀⠀⠸⡤⠻⣄⡄⢏⣡⠸⣉⡄⡗⣥⡀⠀⠀⠀⠀⠀⠀⠈⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⣿⠇⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠤⢼⣿⠤⠀⠀⠀⠀⠀⣠⣤⠈⠀⠀⢀⡀
⠀⠀⠀⠀⣴⡖⠆⡀⠀⢠⣶⢾⣿⠞⠀⣄⢠⣤⡀⣼⡿⣿⢰⣷⢠⠊⣿⠀⣼⡇⠀⣴⡆⢠⣤⠀⢿⡿⠀⠀⣰⣿⢻⠂
⠀⠀⠀⢸⡿⠀⣼⣿⠀⣿⡇⢸⣿⠀⠀⣿⡜⢧⡧⣿⠟⠁⢸⣿⠇⢸⣿⠀⣿⡇⠀⣿⠃⣼⣿⠀⢨⣿⡆⠀⣿⡗⠋⢀⠄
⠀⠀⠀⠘⢧⠜⠘⣿⡔⠻⡦⠛⣿⣀⡼⢹⣧⡞⠀⠻⢦⠤⠛⠟⠀⠸⡧⠞⠹⣇⡜⠹⠞⠁⣿⣠⠛⣿⠀⡜⠻⣇⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⡦⡄⡤⠄⢠⢦⢠⠤⠀⠀⣤⠀⠤⠶⠀⠀⠀⠈⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠀⠀⠀⠀⠀⠀⠀⣇⠇⣇⡀⢸⢀⣿⣀⠀⢀⣏⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠀⠀⠟⠦⠯⠤⠸⠋⠘⠥⠄⠜⠀⠹⠸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠶⣄⠀⣼⣧⠀⠀⠘⣿⣿⠆⠀⢀⣼⣧⠀⠀⠀⢀⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠘⠁⣼⣿⡄⠀⢺⣿⣿⡗⠀⢠⣿⣯⡁⠐⠜⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⡟⠓⠐⠛⢻⡟⠛⠂⠾⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠤⠬⠅



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀⠀⡀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣠⣤⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⣞⡳⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⠙⠫⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠐⠁
⠀⠀⠀⠀⠀⢀⣴⠞⣿⡟⣀⣀⠀⠀⣠⣤⢄⡀⠀⣾⣆⡤⣤⡄⢰⣆⠤⣦⡄⢰⣦⠀⣾⡆⠀⣠
⠀⠀⠀⠀⢰⡿⠁⢰⣿⠋⣼⡿⠀⣰⡟⣡⣾⠇⣠⣿⢃⠄⣾⠏⣽⣇⢤⣿⠃⣿⣃⢼⣿⢁⣴⠏
⠀⠀⠀⠀⠈⠃⠀⠘⠃⠰⣿⣁⠔⠉⠉⠘⠯⠊⣾⠏⠘⠛⠉⢰⡿⠈⠋⠁⠀⢉⣡⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣾⠃⠀⠀⢀⠔⢉⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣦⠤⢄⡀⠀⠀⠀⢿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠘⠤⠞⡁
⠀⠀⠀⠀⠀⠀⠀⣸⡄⣀⢧⠑⠦⡍⢆⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⢀⣠⠤⠒⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢬⣳⢄⣉⣾⠄⡜⢁⣹⡒⢢⡌⠳⡄⡀⢋⣀⠴⣋⠜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⢉⣥⣲⠃⢸⡀⡁⢾⣫⡤⠓⡤⠁⢸⠛⢒⣈⣁⡀⠆⢀⠘⠛⠓
⠀⠀⠀⠀⠰⠾⠖⢀⠀⠀⠐⠛⠓⣫⡆⠀⠉⡍⠓⠚⠁⡠⠅⠀⢞⣶⡈⣿⢉⠉⠀⠘⠲
⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⢀⣴⠏⠺⠙⠦⡀⠉⠙⠂⠚⠁⡠⠊⠀⠑⠂⠻⠈⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡄⠟⠞⠳⠇
⠀⠀⠀⠀⢀⣰⣁⠰⠗⣆⣀⡲⠒⣸⢞⡥⠿⢴⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠂⠀⣠⠆⠀⠀⠀⠀⢰⢄⣹⡋⣭⠉⣩⣩⣉⣏⣩⣹⣏⣉⡽⡒⠶⠶
⠀⠀⠀⡤⠾⠗⢶⣿⣀⣄⣀⣀⣀⣮⣥⣾⡷⡞⠓⠶⠤⠀⠀⠀⠙⢢⢀⣀⡘⠇⡸⠃⠀⠀⠀⠀⠀⢸⠀⡀⢠⠀⠀⠀⡄⠀⠀⢰⠀⢻⡿⢿
⠀⠀⢀⣌⣂⠀⠈⠀⠉⡟⢡⡞⢱⡌⢳⡟⠀⣀⣀⣃⡀⠀⠀⠀⠀⠀⠀⢠⢒⣠⡄⢀⡆⠀⠀⠀⠀⢸⠈⣧⡜⣦⣶⡈⢇⢼⣃⢾⣶⡠⠀⢸
⠀⠀⠀⠀⠉⠉⠻⠌⠔⣁⣈⣓⣚⣀⣨⠥⠤⠃⠀⠉⠁⠀⠀⠀⠀⢰⡷⣼⠴⠽⠴⠶⠷⠂⠀⠀⠀⢸⠀⠉⢈⡡⣤⢶⢶⢠⡲⠀⡈⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣞⣇⡾⢃⣀⠀⠀⠀⠀⢸⣤⣠⣤⣥⣬⣤⣤⣭⣭⣭⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣥⣀⣩⣬⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⢖⣟⠛⣛⠛⡛⠛⢻⡿⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢰⡄⠠⡴⡄⢰⣂⠀⢠
⠀⠀⠀⠀⠘⡀⣧⠗⣡⣟⣟⣇⣘⣃⣠⡼⠛⠁⠀⠀⠀⠀⠀⠀⢰⡏⣷⢞⣿⠆⣏⣹⠀⠀⠀⠀⠀⠀⢠⠐⣂⣼⡟⣇⡀⠇⠳⠸⠧⢄⣿⣾⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠉⢻⠷⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡹⣏⣍⣍⡽⠋⠀⠀⠀⠀⠀⠀⠀⣠⠪⢛⡭⣄⠀⠘⠚⢁⡤⠈⠉⠑⠋
⠀⠀⠀⠀⠀⠀⢲⡴⣾⢳⣴⢸⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⣭⣤⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⠉⣹⠶⠷⢿⠓⠄⣠
⠀⠀⠀⠀⠈⣢⢀⠃⢙⣞⣉⣋⣉⢉⣱⠾⠀⠀⠀⠀⠀⠀⠀⠀⣦⠷⢶⣿⣰⣶⣡⣸⡅⡇⠀⠀⠀⠀⠀⢈⣫⣿⠋⢠⡼⢃⡼⢃⡀⠿⣯⣴⣶⣶
⠀⠀⠀⠀⠀⢉⣛⠂⠸⡟⣛⣩⠧⠌⠉⢠⡄⠀⠀⠀⠀⠀⠀⢀⠉⣿⡆⠰⠄⠰⠀⠿⡆⠀⠀⠀⠀⠀⠀⠿⠿⢿⡷⣶⣦⣉⣛⣉⣶⣾⣻⠟⠉
⠀⠀⠀⠀⠀⠘⣷⡰⣽⢧⢤⣄⡤⣴⠛⣹⠃⠀⠀⠀⠀⠀⠀⣨⢰⣳⣷⣶⠦⡴⣤⠸⣣⠨⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣯⣉⣽⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣌⣍⣉⣉⣤⠞⠃⠀⠀⠀⠀⠀⠀⠈⠉⣶⠁⣤⠰⡶⠗⢠⡄⢠⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡉⡿⠟⠿⠿⠿⠻⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣥⢿⣆⠸⠿⠋⠀⠀⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⢿⣷⣄⣠⣴⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⣀⣀
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢦⡀
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⠀⡀⠀⠀⠀⢀
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠠⢀
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠹⣿⣿⣿⠟⠉
⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⣤⣄⣀⠀⠀⠀⠀⢀
⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠈⠉⠛⠻⠷⠶⠶⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⠀⢀⣤⣀⡀⠀⣼⣿⣇⡜⠙⠃⣴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢀⡾⣾⠃⢸⡟⠠⣿⡟⣽⣿⣠⣿⣿⡟⣇⠀⣸⢿⣋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢸⡁⠙⠀⢼⡇⢠⠟⠛⠁⠈⠁⢿⣿⡇⠘⠲⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠸⣧⡞⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⣤⡴⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠠⠀⠀⣴⣶⡄⠀⠀⢀⡀⠀⠀⡀⢀⣿⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣴⣆⣼⡟⢻⠎⠳⢾⣿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠁⠀⠀⠸⣿⣿⡾⢹⡟⠈⠁⠀⠀⠀⠈⠛⠃⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⣠⢤⠀⠀⠙⠛⡱⣹⠇⠀⠀⠀⡀⠀⣠⡂⣠⣶⣆⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡼⠁⢸⡇⣠⣦⠐⢷⠋⠀⣀⣀⠀⣿⣰⠏⣿⠃⣿⡟⢈⡿⠷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⡿⢸⣿⡋⢠⣯⢃⣾⣫⠟⢦⠻⣿⠀⠉⠀⣿⣧⡾⠁⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠟⠁⠘⠇⠙⠼⠉⠋⡀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⡀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣴⣤⠀⠀⠀⢠⣀⠉⠀⠀⠀⡀⢤⣄⢀⣤⠀⣀⣀⡈⠛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⢶⡄⣠⣅⠀⣼⣧⢋⡷⣿⣿⣏⣾⢼⣟⣸⣧⠎⣿⣷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⣾⢰⣷⠛⠷⢿⡟⠚⠀⢹⣿⡉⠀⠀⠉⠀⠁⠀⢿⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠏⠀⠀⠀⠸⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠿⢿⣿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣏⣀⣀⣀⣸⣿⣀⣀⣀⣿⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⠙⠻⣷⡄
⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⣿⡇⢀⣤⣶⣦⡄⠀⠘⠿⠀⠀⠀⢠⣴⣶⣤⡀⢸⣿⠀⠀⣠⣄⡀
⠀⠀⠀⢀⣠⣼⣿⣿⠏⠀⠀⠀⣿⣷⣾⣿⡀⣸⣿⣶⣶⣶⣶⣶⣶⣿⣇⢀⣿⣷⣾⣿⠀⠀⠈⠛⠿⣶⣾⣷⡀
⠀⠀⠸⠿⠋⠁⠈⠉⢀⣤⣤⣄⣤⣤⣤⣽⣿⣿⣥⣤⣤⣄⣠⣤⣤⣬⣿⣿⣯⣤⣤⣤⣠⣤⣤⡀⠀⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣦
⠀⠀⠀⠀⠀⣠⡾⠟⠀⣴⣿⡿⢻⣿⠟⠿⠿⠿⠿⠻⣿⡟⢿⣿⠿⠿⠿⠿⠿⠻⣿⡟⢿⣿⣦⠀⠻⢷⣄
⠀⠀⠀⣠⣾⠟⠁⣠⣾⠏⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠻⣷⣄⠈⠻⣷⣄
⠀⢠⣾⡟⠁⣠⣾⠟⠁⠀⣿⡇⢸⣿⡀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⠀⠈⠻⣷⣄⠈⢻⣷⡄
⠐⠛⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛⠛

⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡟⢻⣿⠟⢿⡿⠛⠛⠛⠛⣿⡿⢿⣿⠛⠛⠛⢿⣆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠇⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⢀⣿⠇⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⢷⡾⢿⠷⠿⡿⠶⠶⡶⠶⠿⡿⡿⣿⠶⢶⡾⠿⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣿
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⢠⣶⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⣾⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⢠⣿⣿⣷⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⢸⣿⣿⣿⡆⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⠀⣼⣿⣿⣿⡇⠀⢛⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⣇⣿⡿⢿⣿⡇⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢸⣿⡟⣹⣿⣿⣼⡆⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⠠⣽⣿⡿⢟⣵⣿⣿⢿⣿⢿⣿⡇⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⠀⣶⣶⠎⣽⡇⢀⡇⢸⡇⢸⣿⣷⠀⣠⣾⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠶⣦⣇⠀⣿⡏⠀⣿⠇⢸⠇⠼⠃⢸⣿⣿⣰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⢹⠀⢹⣧⣄⣩⣦⣼⣷⣶⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣛⣛⣠⣾⣷⣴⣿⣿⡏⡀⣿⢻⣿⠇⣸⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢁⣿⣌⣩⣴⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⢛⡟⢻⠀⢸⣿⡿⣟⠛⣯⠉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢸⣿⠃⢸⡇⣿⠀⢼⣿⠀⣿⠀⣿⠀⣿⡯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠸⢿⣄⢈⡀⢛⡀⠸⣫⣧⣽⣤⣿⣦⣩⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠛⠉⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣉⣩⣝⣩⣿⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⢺⣿⡷⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⢠⡟⢰⣶
⠀⠀⠀⠀⣤⣤⢶⣿⡇⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠠⣿⣯⡳⡄⠀⠀⠀⠀⠀⠀⣧⣸⡿
⠀⠀⠀⠀⠀⣟⠛⡛⠀⠀⠀⠀⠀⠀⠀⣯⠟⠀⠀⠀⠀⠀⡐⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠧⠓
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⣩⣯⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡶
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢻⣿⣭⡟⠂⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⢀⣾⣿⡧⢄⣀
⠀⠀⠀⠐⠚⠋⣷⠀⠀⠀⠀⠀⠀⢀⢄⢐⡇⠀⠀⠀⠀⠀⣤⣄⢘⡇⠀⠀⠀⠀⢀⣘⣿⣿⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢸⣟⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣠⣆⣀⡀⠀⠀⠀⠀⠀⠀⣰⣿⣆
⠀⠀⠀⠘⣿⣿⣾⣄⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠙⡟⡟⢻⡿⣧⠀⠀⠀⠈⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠛⠛⠻⡀⠀⠀⠀⠀⢀⡜⢹⠀⠀⠀⠀⠀⠀⠀⣷⠏⠀⠘⢧⠀⠀⠀⡇⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢠⣄⢀⣿⡀⣀⡄⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠠⠤⢿⡿⠤⠄⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠘⠛⣿⡟⠃⠀⠀⠀⠀⠀⢀⣹⣷⡄
⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠈⠋⢁⣾⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠲⠶⣆⡟⢸⡆⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢰⣶⠀⠀⠀⣰⡿⢷⡖⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠘⣿⣷⣾⣿⡆⠹⢿⣿⠟⣿⡇⠀⣿⠟⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣤⣤⡀⢿⡏⠻⢹⣷⠀⢼⡟⠀⣿⡇⣸⣟⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠨⣿⣤⣿⠇⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⢴⣿⡁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣲⢆⠈⠉⠁⠀⠀⢠⣶⠆⠀⠀⠀⢀⣴⠄⠀⠀⠀⠀⠀⠙⠛⠱⣖⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣾⠃⠸⠀⠀⠴⠒⢰⣿⠿⠆⠤⠶⠖⠛⠁⢠⣶⠆⠀⠀⠀⠀⠀⠇⠘⣷⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⢀⣴⣿⠏⠀⢀⣴⠞⣷⠀⢠⣿⣿⠊⠉⣷⡀⠀⣠⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣻⣷⠚⢁⣾⡟⠀⣠⣿⠷⠚⢁⣰⣿⡿⢁⣀⣼⡿⠒⠋⠹⣿⡀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣠⠖⠁⢸⣿⠀⢸⣿⣠⠞⢿⣇⣀⡴⠋⣼⡿⣗⣡⡾⠋⠀⣤⠖⠒⣿⣷⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣯⠀⢀⣾⠏⠀⠈⠉⠁⠀⠀⠉⠁⠀⣼⡿⠁⠈⠀⠀⠀⠐⣇⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⣿⠀⢹⣿⡇⠀⣽⣿⡏⠀⣿⣿⠿⠿⠿⠀⢸⣿⠟⠻⣿⣦⠀⣿⣿⠁⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⠈⠀⠈⣿⣧⠀⣿⣿⠀⠀⣿⣿⣴⡆⠀⠀⢸⣿⣄⣠⣾⡿⠀⢻⣿⡆⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠿⠀⠀⠀⢿⣿⢠⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠟⢿⣿⡁⠀⠘⢿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⡀⠀⢸⣿⣿⣿⠃⠀⠀⣿⣿⣤⣶⣾⡂⢸⣿⠀⠈⣿⣧⠀⠀⣤⣤⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣴⣿⡀⠀⠈⠿⠿⠛⠀⠀⣀⣭⣭⣭⣭⣉⠀⠘⠛⠀⠀⠿⣿⡄⠀⣿⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠛⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⢀⣀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣻⣇⠘⡇⠀⣸⣿⣶⣤⣈⠉⠁⠀⠀⠀⠀⠀⠀⣿⣷⠀⠘⣿⣿⠀⢸⠃⣸⣟⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠾⠓⠀⣰⣿⡟⠙⢿⣿⡆⠀⣼⣿⣿⣿⣿⡄⢸⣿⣇⠀⣿⣿⣇⠀⠚⠷⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⣿⡿⠁⠀⣾⣿⡇⢠⣿⣿⣀⣹⣿⣇⠀⠻⢿⣿⠟⣿⣿⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣤⣾⣿⡟⢀⣾⣿⠿⠟⠻⣿⣿⡄⠀⢠⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠉⢀⣾⣿⡿⠀⠀⠀⢻⣿⣷⡀⠈⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⣀
⠀⠀⠀⣀⠘⠔⠃⢁⡁⣤⠀⠀⢿⡛⢛⡃⠀⠀⠐⠋⠓⠀⠑⠊⠀⠀⠀⠀⣛⣛⢛⣺⣟⠃⠀⠀⠀⠣⠼⠅⠠⠶
⠀⠀⠀⢀⠈⣛⢈⣬⡅⠄⠀⠀⢘⣈⣛⠁⠀⠀⠈⣀⢀⢁⣁⣀⠀⠁⠀⠀⣭⣉⢜⣻⣄⡆⠀⠀⠶⠷⠶⠶⠶⠾
⠀⠀⠀⠈⣉⢈⠤⠤⠄⠖⠀⠀⠘⠛⠉⠂⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠰⠌⠀⠀⠀⠀⢀⠀⡄⢀⣀⠀⠀⢀⣀⣀⣀⣀⢠⡀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠠⠜⠂⠂⠃
⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⣆⠠⡄⡀⣁⠉⡀⠀⢠⣉⡩⡭⣭⣸⣥⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠆⠂⡁⢠⠅
⠀⠀⠀⢠⣿⡿⠛⣿⣷⠀⠀⠀⠀⠠⠤⡤⠀⠀⠀⠈⠉⣧⣴⣤⠍⠈⠀⠀⢈⡈⠐⡉⣉⠀⡀⠀⢀⡉⡍⠠⠤⠃
⠀⠀⠀⠘⠙⠁⢀⣉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣄⡠⡨⠤⠄⠀⠀⢠⣿⣶⣯⣿⣻⣿⢿⠀⠀⠀⠀⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠆⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⠦⠀⠀⠐⠒⠀⠀⠀⠂⠀⠖⠐⠀⠀⠀⠓⠒⠐⠒⠂⠀⠀⠒⠲
⠀⠀⠀⠀⠀⢽⢿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡀⠀⠀⠀⢀⡀⣀⢀⢀⡀
⠀⠀⠀⣀⣸⠿⢿⣿⣻⡀⡀⠀⢀⡀⠀⠀⠀⠀⠘⠂⠋⠃⠀⠀⠀⠀⠛⠁⠀⠻⠃⠀⠀⣤⣭⣍⣭⡥⣭⡥⣠
⠀⠀⠀⠛⠛⠘⠛⠛⠛⠛⠃⠀⠀⠁⠀⠙⠀⠀⠈⠀⠀⠃⠈⠀⠀⠀⠀⠺⠷⠀⠀⠀⠀⠀⠀⠈⠦⠶⠆

⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠰⠀⢀⡀⠴⢂⣂⣀⡀⡤⣤⣤⢤⢀⣀⣒⠀⠧⢀⡀⠀⡈⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠄⢀⠤⢐⡩⣄⠶⠟⠛⣈⡉⠉⢩⠩⡉⠉⠭⡉⠓⠛⠿⠶⣢⢍⡒⠤⡀⢀⠊⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⠤⢊⣵⡮⢓⠩⡐⢀⠀⠅⡐⣰⠀⠈⣆⢁⢨⡄⠁⠀⢀⡔⠂⠀⠉⠚⢵⣦⣑⠤⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢝⡡⡮⠓⠁⠀⡅⠂⠁⠉⠀⣐⠂⠀⢡⡀⢟⡈⢛⠁⠀⡠⡁⠇⠀⠀⢀⡋⠂⡈⠚⢵⢌⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡲⢋⡐⠅⠐⠀⠀⠀⢁⠠⡀⠡⠬⠅⠊⠁⠈⠁⠙⠒⠛⠼⠠⡇⣆⠀⠕⠁⣤⡑⠋⠀⠙⢷⣎⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⣞⠎⠀⡀⠉⠊⠱⠈⠀⡰⠞⠁⢀⠠⠐⠂⠀⠈⠀⠀⠉⠀⠐⠂⠄⡀⠉⠚⠀⣃⠊⣀⡀⠁⠉⢢⠱⣣⡕⡄
⠀⠀⠀⠀⠀⠀⠀⡠⢡⡳⠁⠈⠀⠐⢈⠂⢰⡌⠃⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠁⢮⡀⠐⠈⠒⠂⢀⡜⢞⡌⢆
⠀⠀⠀⠀⠀⠀⡰⢡⡳⠁⠀⠈⠢⠀⣀⠠⠀⡄⠍⠉⡀⣠⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠉⠨⢀⠃⢡⡊⠄⢀⠁⠔⡈⢞⡌⠆
⠀⠀⠀⠀⠀⠀⢁⣿⠁⢪⠀⡁⠌⠀⠀⠁⡠⠇⠀⠀⠐⠮⠀⠀⠀⠀⣄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠷⠂⠀⠀⠸⠄⠈⠖⡀⠡⠳⠈⢩⠌⣿⡈⡀
⠀⠀⠀⠀⠀⠈⢸⡇⠰⠀⠁⠑⠠⠤⠀⠠⠁⢀⢀⠀⠀⠀⠀⠀⠀⠠⡀⢠⢆⡠⡆⢠⢀⠄⠀⠀⠀⠀⠀⡀⡀⠀⠆⠈⣥⠂⠁⠀⠀⠆⢸⡇⢀
⠀⠀⠀⠀⠀⠃⣿⠀⠄⠲⡤⢂⠰⠂⠀⠇⠀⠘⠘⡀⠀⠀⠀⠀⠰⠋⠇⢀⠸⠄⡟⣸⠀⠀⠀⠀⠀⠀⢀⠃⠁⠀⠰⠀⢨⡄⠂⠀⠈⣉⡀⣿⠸
⠀⠀⠀⠀⠀⢐⡖⠀⠠⠔⡠⠁⢄⠀⢰⠀⠀⠀⢁⠡⡀⢰⠀⠐⣝⠏⡅⣤⠃⠀⡏⠌⣾⡍⣠⠀⡇⠀⠌⡠⠀⠀⠀⡆⠀⡤⠠⢠⠀⠦⠀⣓⡂
⠀⠀⠀⠀⠀⠨⠧⠀⠀⠀⠈⠁⠈⠀⢸⠀⠀⠀⠈⠂⡁⠰⠀⣓⠤⢇⡂⠁⣀⠀⣇⢂⠣⢄⡄⢠⠆⢈⠔⠁⠀⠀⠀⡇⠀⠀⠀⠐⠠⠀⠁⠼⠇⠁
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠚⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⢃⠀⠀⠀⢠⠀⠀⠀⠀⠠⡀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠦⣀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠆⣟⡀⠀⠚⠒⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠉⠉⠉⠉⠉⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠱⠤⠇⠀⢀⣳⠸
⠀⠀⠀⠀⠀⠀⠸⢧⠀⠀⠤⠙⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⡀⠀⠀⢀⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡤⠒⣄⠀⠀⡼⠏⠆
⠀⠀⠀⠀⠀⠀⢆⢫⠆⠀⠘⠂⠃⠀⡄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠒⠒⠁⠀⠰⡝⠠
⠀⠀⠀⠀⠀⠀⠈⠎⢯⢄⠀⠀⠀⢄⡜⠗⣀⠀⠐⠄⡀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⢀⠤⠂⠀⢀⣄⡀⠀⠀⠀⠀⣠⡝⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⢻⣦⠀⠒⢘⢃⡮⠏⣦⠀⠀⠈⠂⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠁⠀⣀⣴⢉⠫⣵⣗⠄⠀⣴⡟⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⣵⣄⠈⠁⣀⠐⠁⠚⠓⠄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣧⡀⣸⠏⢚⢢⠑⡬⠋⣠⣮⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡫⣶⣄⠉⠾⣀⠾⢃⢠⡗⠃⠀⠤⢤⢀⣀⣀⣀⠀⠠⡄⡦⠀⠈⠁⠑⡈⠟⣠⠞⠀⣠⣾⢝⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡑⢷⡦⣀⠉⠀⠌⠀⠀⡀⠂⢀⠈⠁⣠⢠⠀⠀⠀⢠⢀⢶⠌⠶⠗⠀⣀⢴⡾⢊⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⡓⠯⣖⡤⣀⠀⠈⠓⠒⠁⠦⠴⠦⠃⠀⠘⠒⠋⠉⣀⣤⣶⠝⢊⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠡⢉⡛⠻⠵⠶⣦⣖⣶⣴⣰⣶⠶⠦⠟⢛⡉⠍⠒⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣐⣂⣀⣀⣀⣒⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸⠁⠀⠀⠀⡟⠐⢄⣀⠐⣿⠀⢀⣀⣀⣀⡀⠀⠀⠣⠤⠄⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⣧⢸⠀⠀⠀⢸⡇⡆⠘⣏⠀⣿⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⣿⢸⠒⠀⣴⢺⡇⡇⢀⡘⡇⣿⠀⡔⣲⢸⣧⣤⠀⡴⠂⣰⠶⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠈⠙⠃⠉⠈⠉⠁⠙⠁⠉⠉⠋⠁⠈⠋⠉⠋⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠒⠊⠁⠒⠊⠑⠒⠉⠐⠒⠉⠒⠊⠁⠂⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⢸⠳⡀⡏⠁⡏⠁⢹⠁⡇⠀⡇⡟⠁⣾⡄⡇⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠈⠛⠲⠤⠄⠀⠀⢸⢰⠃⡇⠀⡇⠀⢸⡀⣏⠀⡇⡇⠀⢿⢷⡇⠀⠀⠀⠒⠚⢋⣁⣀⣤⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡤⠖⠂⠀⠀⢸⣣⠀⡗⠀⣟⠂⢸⡇⣿⣦⡇⣟⠂⢸⠘⣧⠀⠀⠀⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣀⢁⣠⡉⢀⣌⡁⣀⣄⠈⣨⣀⢈⣡⡀⢀⣄⡀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣀⠤⡀⠀⠀⣠⡀⢀⣤⣀⢠⡄⢀⣄⠀⣀⣠⠀⣠⣄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠙⢾⣤⠃⠀⠀⢿⢸⡌⡷⠊⣾⠀⢸⠸⢸⡏⢹⠸⡇⠉⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⢀⣤⣀⣀⠉⡀⣷⠄⢸⣸⡇⣿⡀⢿⡀⢸⠐⠿⡇⢸⠀⠹⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠶⠃⠘⠚⠇⠛⠀⠸⠞⠘⠇⠇⠸⠤⠸⠛⠀⠇⠸⠇⠦⠟⠀⠀⢈⢂⡡⢄⡂⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠀⣠⡀⣀⢀⢠⣄⠠⡖⠰⢢⠄⡂⠒⢒⠀⣿⠀⡀⢺⠆⠀⠀⠀⠭⢠⠁⡀⢀⡈⢒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡒⠦⠴⣒⠂⠀⠀⡇⡇⡻⣸⢸⡌⡆⢈⠛⠁⠀⠈⠒⠀⠀⣿⢀⡆⢸⡇⡀⢀⡀⠔⢀⠑⠤⡤⢒⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠬⣡⡩⠤⠅⠀⠀⠷⠇⠇⠹⠸⡷⠁⢸⠳⢠⢶⠆⡴⢲⠀⡇⢸⡇⢸⠁⠉⢹⠀⠀⠀⠠⠁⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣁⡒⢒⣉⡂⠀⠀⢖⠂⠼⠄⢂⠢⠀⠈⠀⠈⠒⠉⠉⠉⠃⠙⠉⠋⠈⠛⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⠤⠥⣄⡀⠀⠀⠀⠀⠀⠀⣶⡄⣶⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⣀⠀⣀⡙⡋⢀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠀⡖⠀⠻⡄⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⠀⠀⢠⡇⠀⣏⢣⢸⠁⠇⢹⠀⡇⣾⢽⡆⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠙⡆⠀⣷⣠⠄⣀⢤⠀⡟⢸⡏⢀⣤⡄⢠⢼⡇⠀⣿⢸⢸⠀⠀⢸⠀⡇⣷⢘⡇⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠳⠂⠦⠇⣠⠏⠣⠴⠳⠚⠆⠧⠞⠧⠞⠭⠔⠛⠾⡇⠀⠿⠺⠸⣤⠆⢸⠀⡇⢻⣼⠃⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠶⠒⠶⠶⠶⠶⠶⠖⠒⠲⠶⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣝⢿⣿⣿⣿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣛⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣯⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣄⣠⣴⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⣠⠄⠀⢸⠂⠀⠀⠀⣅⠀⠲⢀⢀⡆⠐⢀⠀⠀⠓⡄⠀⠱⠀⢠⠄⠀⠀⠀⡄
⠀⠀⠀⢁⢀⡄⠤⣹⡀⠈⠁⠈⢠⡄⡎⢷⠂⣄⠄⠈⠀⠆⠀⠢⠀⣆⠘⠁⠚⢠⠃⠀⠀⠀⠀⠀⠀⡠⠈⢇⠀⠳⠃⢀
⠀⠀⠀⠈⠑⡀⠀⠠⡽⠀⠸⠀⠀⡀⠀⣼⠀⠁⠀⠠⠐⠆⡀⠀⠠⠿⡄⣌⣦⡈⢄⡘⠂⠔⠐⠁⠀⠠⡀⠁⡄⢸⡀⠈
⠀⠀⠀⠠⡄⠠⠀⠻⣷⡶⣄⠀⠀⣜⣻⠇⢰⠀⠀⠀⡠⣠⢥⣀⠀⠀⡄⠙⠋⠙⣆⠠⣴⠀⠀⠀⠡⡁⠁⠀⡇⢁⣄⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠙⢦⣀⣿⡏⠴⠅⡈⠁⣰⠇⠀⠀⠈⠑⠪⡁⠠⠂⠀⡄⠳⡙⡴⠆⠄⡤⠠⡊⢀⡆⢈⠈⠂⠁
⠀⠀⠀⠀⠀⠙⠂⢈⣠⡀⠀⠈⣿⣿⠀⠀⠀⣶⠾⠷⠦⢀⠀⠀⠀⠀⡈⠷⣄⠠⠤⠑⠉⣄⠓⠒⠀⠠⠟⠉⠡⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⣄⠀⢹⣿⠀⣠⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⠀⠁⠰⣾⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⣿⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣀⣀⣈⡓⡷⣶⢴⠤⠴⠷⠻⣉⣡⡐⠒⠒⡲⠆
⠀⠀⠈⣑⠆⠀⣏⣥⣶⣵⣷⡌⣽⡍⢯⠇⡆⣿⣀⠨⠕⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⣉⠉⠁⠁⠏⢀⣛⣀⣩⣥⠬⠤⠒⠁⠀⣰⡃⠀⠀⠀⠀⠀⡜⢉⡗⠀⠀⡠⡄⠀⠀⠀⠀⣦⣤⣳⣀⡀
⠀⠀⠀⠀⠨⡓⠶⣿⠂⢈⢉⡉⣉⣩⣤⠤⣴⠶⣸⣎⠒⠀⠀⠀⠀⢹⣗⣋⠀⣀⣀⣧⠇⡄⠀⠐⠿⠛⢛⡛⠿⣧⡀
⠀⠀⠀⠀⠈⠀⠐⠧⠳⡼⠀⢧⣼⢁⣻⢚⢈⡯⠥⣸⠀⠀⠀⢶⣧⠈⢇⣈⡷⣧⣺⡏⣸⡇⣯⣇⢇⡽⢾⣛⡜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡀⣿⣏⢈⣶⠈⡷⠈⢉⠁⠀⠀⠀⠘⠻⣷⣶⡿⠿⠙⠳⠈⣶⢠⡞⠲⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⡉⠈⠉⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠃⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡾⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⢃⣄⠀⠀⣀⢀⣠⣖⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣼⣷⠿⣷⠋⢾⣹⠗⣤⡄⠀⠀⠀⠀⠀⠀⠀⢻⠙⣾⢿⠈⣧⡴⢣⢟⢭⠶⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⠛⠀⠛⠃⠀⠈⢡⣤⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠈⠛⠁⠤⢔⡉⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠿⠃⡿⠀⠀⢀⠀⠀⠀⣿⡏⠶⡟⠀⠀⠀⠀⠀⠀⢀⣸⠀⣠⡖⣤⡄⣀⡎⡐⢨⣿
⠀⠀⠀⠀⠀⠀⢀⠔⣸⠇⣼⢿⡿⣾⣰⣧⣿⢛⡶⣇⡀⠀⠀⠀⠀⠀⣿⣘⣇⣿⣰⠻⡷⢧⠿⣻⠅⢿⠄
⠀⠀⠀⠀⠀⠀⠘⠶⠛⠚⠙⠋⠀⠛⠁⠁⠈⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠓⠢⠀⠐⠈⠙

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠖⠈⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠶⠀⠀⢀⠲⠆⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠂⠄⠂⠆⠀⠀⠀⠆⢡⠀⠀⠀⠀⠀⠀⠀⢁⢈⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸⠆⠃⠐⠰⠂⠀⠀⠐⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠨⠄⠄⠀⠀⠀⠀⠀⡠⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠀⠀⠀⠀⠙⠀⠁⠃⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠐⠊⠀⢀⠘⠃⠀⠀⠀⢠⣲⣀⠂
⠀⠀⠀⠀⣈⣉⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠏⠁⡀⠀⠀⠀⠀⢰⠀⠁⠃⠀⠀⠀⢴⣆⣀⠀⣰⠶⠀⠀⠀⠀⡀⢂⠀⢠⡄⠀⠀⡀⣀⠀⢀⠈⠁
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡈⡁⠀⠀⠀⠀⠀⡀⠁⠀⢀⠀⠀⠀⠀⢀⠒⠀⠒⠴⠀⠀⠀⠀⡀⠃⠂⠀⠠⠀⠀⢠⡄⡀⠀⡠⣄⠀⠀⠀⠀⠉⣉⠁⠀⠀⠀⠀⠀⠠⠀⠤
⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠄⠈⠀⠠⠀⠀⢠⣴⣦⣤⣄⡀⠀⠀⢀⠁⡀⠈⢈⠀⠀⠀⠘⠂⠄⠈⠀⠋⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠘⠛⠛⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣐⠀⠲⡀

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣫⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣟⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠙⠁⠻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠇⠉⠉⠀⠀⠀⠀⠀⢰⣦⠘⠿⠿⢿⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⣠⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢛⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠓⠒⠛⣻⠆⠀⠉⠉⠉⠉⠙⠚⠓⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠛⠲⢤⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢠⠏⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡼⠁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡰⠋⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠖⠋⠀⣀⡴⠞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣀⢦⢠⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⣀⠀⢀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠛⠓⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡛⠛⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢠⣶⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠇⠀⠀⣠⡾⠛⢦⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠟⠻⡄⠀⠀⣤⠞⠀⢀⣴⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠴⠛⠁⢀⡴⠛⠁⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⠀⠀⣯⠀⠀⡇⠀⠀⢳⣀⣠⠾⢀⣆⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣀⠀⣀⣀⡤⢾⠃⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠱⡀⠀⠀⠀⠀⢀⡠⢦⢾⣄⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠉⣉⣥⣤⠞⠀⠀⠀⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⠙⠓⠶⠖⠛⠉⠀⠀⠀⠉⠓⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣴⣟⠥⠖⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡆⢠⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⠉⠁⠀⢀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠛⠟⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢳⡄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⠀⢸⠂⠀⠀⠀⠀⠀⢰⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢿⡀⡴⠖⢶⠀⣖⠲⢸⣧⡶⠂⣴⣖⣦⠘⢿⠋⢸⡶⠲⣄⣴⠖⢶⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠾⠃⠳⠤⠾⠐⠮⠽⠚⠇⠹⠆⠻⠭⠧⠀⠸⠀⠸⠷⠴⠋⠻⠦⠾⠻⠇⠿

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡼⠁⠂⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠸⢇⡀⢀⣿⠀⠠⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⢸⡇⠀⠀⢀⠀⠀⢀⣶⠀⣀⠀⢀⠔⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣧⠀⢹⡇⠀⣼⠃⣵⡆⢸⡇⠀⣿⡄⠁⠀⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⢸⡇⢠⣿⢀⣿⠀⠜⢧⠌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡿⠀⠀⠳⠂⠙⠃⠘⠞⠀⠀⠠⢻⡇⠀⠀⡠⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠠⠄⠀⠀⠀⠀⠠⠀⣾⡆⠀⢀⡂⣼⠃⢀⠌⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠡⠞⡉⠀⢀⠀⢰⡆⠀⢠⡅⢠⣿⠀⢀⡂⠙⠁⢀⢋⡤⢠
⠀⠀⠀⠀⠀⠀⠀⣿⢇⣀⡀⢅⡄⠈⡄⠈⢷⡄⣿⠃⢸⡏⠀⣾⠃⣼⠃⢢⡟⠀⠀⢠
⠀⠀⠀⠀⠀⠢⠐⣿⠇⠸⣷⢺⡇⠀⠇⢠⠚⡇⢹⣄⢿⡇⠀⣿⣠⢿⡌⢸⡇⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡆⣞⡏⠈⠓⠊⠀⠘⠘⠁⠀⠁⠸⡇⢸⠀⠁⠈⠁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⢀⢀⠌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡁⠈⣿⣈⡷⣧⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣰⣥⠆⠨⡏⡙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠋⢉⡄⢠⣿⣧⡥⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠃⠀⠹⠞⣏⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠆⡅⢀⣴⡿⢿⠁⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡋⡻⠟⠀⠇⢱⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣷⣆⢲⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢹⣿⣯⣤⡂
⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣀⣄⣠⣎⣧⠀⢀⠀⣀⠤⠄⠀⠀⠸⠞⠚⣉⡿⣠⢆⠑⠢
⠀⣠⣾⣾⣵⣶⣶⡴⠾⠷⠾⠻⠿⢽⠛⢳⢞⡔⣠⣤⡴⣯⣤⠄⢀⣦⡐⠂⣠⣛⣀⣀⣐⣀⡸⠿⠶⠒⠖
⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⡀⡀⠠⠄⠀⠀⠈⠈⠈⠉⠁⠈⠓⠒⠚⠛⠂⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠆⠐⠤⠄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠤⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁⠀⠈⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠄⠀⠀⣐⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠰⢠⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠆⠀⠀⠑⠉⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠷⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠘⣆⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⡂⠀⠀⣠⠒⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⠿⢄⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡩⠛⠉⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⠃⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠰⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡆⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢰⡀⣀⣀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣤⣀⠀⠈⠉⢩⣇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⡈⠁⠡⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⡄⠀⠀⠘⣽⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡧⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢎⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠏⠀⠀⠀⣣⡴⠒⠒⣷⣄⠚
⠀⠀⠀⠀⠀⠀⢀⡿⡷⠤⠤⠔⠀⠒⠺⠀⠀⠀⠀⠀⠉⠺⣢⡤⣊⡡⠌⣉⣲⠕⠁⠀⠀⠀⢰⠁⢀⣶⠖⠛⠺⡆⢃
⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⡠⠆⠤⣀⠼⠀⠀⠀⠀⠀⢸⣢⡜⢁⡀⠀⠀⢹⠈
⠀⠀⠀⠀⠀⢠⠃⢸⠃⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠒⠋⠀⠀⠀⠀⠀⠀⢀⠉⠨⣵⡀⢀⡠⠞⢀⡆
⠀⠀⠀⠀⠀⠈⠒⠺⣄⣀⣆⠀⠀⣀⣀⡠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⠉⢛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠉⠁⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⡀⠈⣽⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠶⠦⣄⠀⠀⢀⠴⠿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⣉⠀⣠⠛⠉⠀⠈⠑⢄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⡔⠁⠀⠀⠀⠈⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⢏⡠⠟⢡⠂⠉⠢⡀⠀⠈⡆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠋⠀⠀⠀⠘⢦⠀⠙⣄⡤⠆
⠀⠀⠀⠀⠀⠘⠒⠀⠀⠄⠐⠒⠦⠤⠤⠤⠤⠤⠀⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠤⠤⠤⠤⠴⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠶⠶⠾⢗⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣷⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⡿⢿⣿⢿⣿⡏⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⢰⣿⠋⢸⣿⣿⣿⡆⢿⣿⡇⢸⣿⠁⣾⣿⡗⢸⣯⢸⣿⣷⣦⣾⡿⢟⣛⣿⣭⣿⣛⡻⢷⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⢸⣿⣦⣸⣿⣿⣾⣷⣼⣿⣇⣼⣿⣦⣿⣿⣧⣼⣿⣼⣿⣿⡿⡫⠖⣉⣤⣶⣶⣶⣮⣝⡳⣭⡻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⠏⠞⣴⣿⣿⡿⠟⠛⢿⣿⣿⣿⣎⠻⣞⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡌⣸⣿⣿⠋⣠⣾⣿⣿⣿⣿⣿⣿⡆⢻⣸⣿⡿
⠀⠀⠀⠀⠀⢀⣿⣿⢸⣇⢸⡇⢸⣿⠃⣼⣿⣯⣿⣿⣿⣿⢰⣿⣿⡇⡇⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣼⣷⣼⣿⣷⣽⣿⡟⣡⣾⣿⡇⣾⣿⣿⡇⣇⠸⣿⣇⠸⣿⣿⣿⣿⠛⠛⣿⣿⡟⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢸⡙⣯⢻⡟⣻⣿⠻⡟⣿⣿⢡⣿⣿⣿⣿⡹⣆⠹⣿⣷⣽⣿⣿⣿⣶⣴⣿⡿⢡⢏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⣿⢸⡇⣿⡿⢸⡇⣿⡟⢸⣿⣿⣿⣿⣷⣜⢷⣌⡛⠿⢿⣿⣿⠿⠟⣋⠴⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣼⣧⣿⣼⣶⣽⣧⣸⣇⣻⣿⠈⠿⣿⣿⣿⣿⣿⣷⣮⣟⣳⣶⣶⣶⣚⣫⣵⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃






⣶⣶⢶⢶⢷⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⢿⣶⣶⡾⢶

⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠜⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠄⠒⠒⠚⠁⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠠⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠣⠀⠀⠐⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠯⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡳⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⡂⠀⣀⣔⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠚⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠉⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠁⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡅⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡇⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠣⢴⡖⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢀⠂⠊⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡈⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⣤⠔⠐⣧⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⠤⠄⠀⠐⠒⠇⠀⠀⠀⠀⠙⢷⡤⢎⡠⢌⣁⡖⠋⠀⠀⠀⡜⠀⣰⠖⠛⠳⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠡⡔⠒⠠⡤⠃⠀⠀⠀⠀⣹⣶⡁⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⢇⣀⠶⣇⣠⠴⠅⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡖⠗⠀⠀⠐⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⡀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠦⢄⠀⢀⠠⠾⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠀⡠⠊⡁⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠁⠂⠀⠀⠀⠀⠑⠒⠤⡀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⡥⠚⠁⢆⠈⠠⡄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠨⠤⠤⡌⠀⠀⠀⠀⣳⠀⠼⠖⠋
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⠉⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢈⠁⠀⠀⠀⢀⠀⠀⠸
⠀⠀⠀⢀⠀⠀⢀⢀⠠⡇⠀⠀⠀⠠⠀⠠⡀⢠⠃⡄⠀⠂⡀⠀⠂⠔⠀⠀⠀⢄⠀⡄⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⢆⣠⣄⢀
⠀⠀⠀⠀⢸⣇⠔⣿⣼⣀⡉⢳⣄⠀⠀⠺⣀⣺⡼⢶⣰⢙⡷⡀⠀⠀⠀⣰⣶⣶⢶⣷⣶⣷⣶⡆⠀⠀⢠⣤⣬⣭⣥⣼⣤⣶
⠀⠀⠀⠀⠀⢡⢴⡏⠃⣈⡉⠻⠏⠀⠀⢘⣁⡃⠈⠀⠈⣀⣃⠁⠀⠀⠀⡻⠟⠛⠛⠿⠛⣛⢛⠉⠀⠀⢸⣟⣿⣿⠿⠿⠿⠿⠁
⠀⠀⠀⠀⢰⠏⡾⠀⠁⠄⠀⠀⠀⠀⠀⠫⠈⠛⠛⢈⠛⠂⠓⠀⠀⠀⠀⠳⠜⠺⠀⠞⢉⢎⠝⠀⠀⠀⠀⣤⠬⣐⣀⢛⠐⠂
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⡴⠁⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠠⠶⠶⢆⡀⠀⠀⠀⠀⠳⠤⠒⠁⠱⢿⣋⡠⠀⠀⠀⠀⢣⠴⠶⢼⠦⠖⠶⠀⠀⠀⠀⢣⣡⡤⠠⠯⠦⢤⠄
⠀⠀⠀⠀⢀⣤⢤⣤⣼⣤⣬⣤⣤⡄⠀⠀⡠⢔⡄⣄⣅⡠⠘⢳⠀⠀⠀⢀⠈⣀⣆⣀⣄⣰⡖⠀⠀⠀⠠⠔⠀⢀⣠⣆⣤⣠⡀
⠀⠀⠀⠀⢸⣷⣿⣼⣯⣿⣿⣼⣿⡇⠀⠈⢓⢿⣉⡧⠂⠈⠉⠁⠀⠀⠀⢠⠂⠀⠋⢉⠌⠀⠅⠀⠀⠀⠀⠐⡩⠍⠥⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠢⡄⠀⡀⠀⠀⠀⠀⠀⢀⠔⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⢀⠁⡀⡀
⠀⠀⠀⠀⠀⠀⡞⠁⠀⢹⣄⣀⠀⠀⠀⠀⡎⠀⢀⣤⣅⡀⣠⠀⠀⠀⠀⠀⠈⠂⣀⠤⠦⢶⡀⠀⠀⠀⠀⡐⠈⠶⢋⣉⢁⠇⣀
⠀⠀⠀⠀⠀⠀⢧⡀⢐⣋⣠⣶⠰⠀⠀⠀⠙⢄⠃⡼⡁⡤⡟⢁⠇⠀⠀⠀⠀⠆⠈⣰⠀⢪⠂⡀⠀⠀⠀⠰⡉⡀⢌⣠⣤⢦⡀
⠀⠀⠀⠀⠀⠀⠄⡹⡇⠩⠥⠓⠂⠀⠀⠀⠀⠘⠅⠁⠙⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠃⠁⠈⠂⠀⠀⠀⠀⠀⢹⠃⣠⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠉⠉⣿⣿⣿⣿⡏
⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃
⠀⠀⠀⠈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣄⡉⠀⠈⠛⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠲⢤⡀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⣀⣴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣛⠃⠀⣀⡐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⡄⢤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣶⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢠⡀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⠧⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⣁⣤⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢈⠻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⢠⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠉⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠱⠟⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⣁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⣿⣿⣿⣿⣿⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠒⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠋⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⠿⠿⠛⠟⠿⠛⠻⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠛⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠛⠿⠛⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡼⢠⠎⣷⢀⡷⣺⠂⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠋⡀⢙⠒⡁⢚⠀⡛⠀⡀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣾⣿⣽⣾⣯⣿⡿⡇⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢙⡋⠋⠛⠛⠃⠀⠓⠚⠛⠛⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣦⢠⣦⢀⣴⣶⣄⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⢸⣿⢸⣿⠸⣿⣮⡉⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⢸⣿⢸⣿⢰⣦⣹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⡁⠈⠛⠛⠛⠈⠛⢛⣋⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⢹⡆⢀⣀⡀⠀⠀⠈⣿⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⡞⢱⡏⣸⠇⠀⠀⣰⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢁⣁⣠⣤⣤⡥⡤⠌⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠩⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠁




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣷⣶⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠟⣻⢿⣿⣿⣿⣿⣍⠛⢿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⠋⠀⠀⠁⠁⠀⠁⠙⠉⠈⠀⠀⢻⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠐⢶⢿⠇⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⠏⠁⠀⠀⠀⠚⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡠⠒⠒⠦⡠⠔⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⡀⣿⠖⠣⡀⢾⣶⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠐⠀⠢⡀⠀⠀⠀⠀⠀⣀⠴⠦⠤⠒⠀⠲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡂⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⢉⡁⠀⠀⠀⠁⠀⠀⣰⠋⠉⠳⣄⠘⣄⠁⠀⠀⢰⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠇⠀⠀⠀⠦⣀⣀⣀⣠⠤⠔⠁⠀⠀⠐⠇⠂⠀⢹⠰⠀⠆⢸⡄⠈⢢⠀⠀⠀⠂⣔⠁⠀⠀⣠⠐⠈⢁⠀⠈⠑⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡸⠁⠀⢈⣷⠆⣀⠀⡼⠀⠀⠌⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡠⠤⣤⠄⠒⠒⠀⠄⠀⠀⠀⢀⡎⠀⠀⢈⠞⠁⠀⢀⠎⡀⠂⠸⢀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠻⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠃⠀⠀⠀⢸⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⠀⠀⠈⢆⠀⠀⠀⢨⠆⠀⠀⢇⠃⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠑⢄⣀⠆⠀⣂⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡠⠤⠼⠳⠆⣀⠀⠈⠣⡀⣛⠓⣄⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⡔⠤⠀⠑⢄⠀⠈⢢⡄⡀⠀⠀⠀⠀⢀⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠰⠀⠀⣠⡰⠀⠀⡀⠀⠀⠀⣀⠀⠀⡏⠀⠀⠀⢠⠖⢇⠑⠦⠈⡀⠀⢀⠀⠀⠙⠀⠡⠀⠀⣂⠤⠓⠈⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢀⠇⠀⠀⣠⠙⠁⠀⠀⠀⠀⢸⡇⡀⠀⠀⡏⠀⢸⠳⣄⠑⠑⠠⠌⣉⢢⢀⠒⠒⢄⡔⠁⣠⠦⡺⢒⣤⡀⠀⠤⠈⢳
⠀⠀⢀⠄⠂⣁⣀⠀⢠⢤⣀⡼⠒⠒⠪⡏⠀⢠⡀⠀⠀⠀⠘⢣⡀⠀⠀⠇⠀⢸⠀⠀⠙⠂⠤⢈⢐⠊⡉⢣⠀⠀⢣⠁⢂⡡⠔⠁⡇⠙⢄⡀⠀⢸⠂
⠀⢀⠇⠀⠀⠇⠀⢳⠎⠀⡈⠀⢡⠀⠁⠘⣴⠁⠐⣄⠇⠀⡅⠀⠙⠒⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠘⠁⠀⠀⠀⠀⡇⠀⠈⠑⠒⠁
⠀⣼⠀⠀⢀⡀⢀⡎⠀⠰⠁⠀⠈⠀⠀⠰⠁⠀⠀⢈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⡔⠉⠀⠈⠉⠁⠉⠉⠠⠀⠀⠀⠇⠀⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇
⢠⠁⢸⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⢸⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⡇
⢸⠀⢇⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡄⠀⠀⠀⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠌⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⡌⠀⠈⡷⠦⣄⠀⠀⠀⡤⠞⢆⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⡸⠀⠀⢈⡟⠒⠊⠀⠀⠈⡄⢸⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠳⣤⣀⡞⠀⠀⠀⠀⠀⠀⠘⣸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⣒⠢⠄⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠤⣚⡥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⢹⣼⡄⢸⡜⡆⢻⠀⢰⣧⣴⠖⠀⠀⠀⠀⣀⠠⠤⣬⠹⠲⠶⠆⠘⣞⠋⠵⡄⠀⢰⠂
⠀⠀⠀⠀⠀⠈⠐⢶⠿⠏⠉⠁⠥⠴⢈⠀⠚⠻⠵⠂⠀⠀⠀⠀⠠⠇⠨⠷⢾⠗⣿⠂⡿⠘⠆⣓⠗⠐⣊⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠊⢯⠓⢢⣤⣴⡿⢅⡀⡀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣤⣬⠀⠲⠶⠺⠏⡉⣀⣀⢴⠳⡀
⠀⠀⠀⠀⠀⠀⠐⢶⡾⠎⢈⡽⠂⡶⠃⠰⣾⣣⣤⣤⠀⠀⠀⠀⠀⠀⠃⠸⠷⣤⠏⢇⡟⢹⣥⣙⡾⠍⡇
⠀⠀⠀⠀⠀⠀⠾⢿⣿⡦⣬⣄⣋⣓⣛⣤⡾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣿⡇⢹⠌⠷⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣽⣏⣉⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⡀⢀⢠⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣉⢫⣰⡶⢹⠹⠺⠸⠓⠄⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⡃⠡⡰⡦⠞⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⡿⣠⣄⣠⠞⡥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⢿⣴⢿⡇⠧⣬⣇⡞⣻⢸⢉⡀⠀⠀⠀⠀⢤⣸⠋⣿⡟⣏⣹⠇⠀⠳⠾⠋⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠘⠥⠎⢉⠉⠈⠉⠁⠀⠀⠀⠀⠀⣈⡉⢠⠌⠀⠈⠉⠀⠀⠀⣰⣳⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⠲⠾⠇⢸⡄⣠⡀⠀⠀⠀⠀⠀⠘⠾⢁⡿⠀⠀⠀⡀⠀⠀⠀⣿⡏⠚⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠷⢸⣧⣦⣠⡖⠰⣄⠸⡇⠉⠁⠀⠀⠀⠀⠀⢠⠆⣼⠇⣸⠿⡿⢾⣇⣼⣧⣿⢋⡷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠋⠓⠈⠙⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⡶⠛⠞⠛⠛⠁⠘⠋⠀⠁⠈⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠐⠀⠀⠠⠆
⠀⠀⠀⠀⠀⡀⠀⠀⡆⢀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⣴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⣤⠄⠀⠀⠀⠀⠀⢀⣾⡷⠿⣷⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣂⠸⠛⣤⣾⣀⣀⠀⣸⣿⠁⠀⠰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣀⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠙⠿⠹⣿⡝⠿⠿⠖⠀⠉⢀⠀⣰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠢⣤⣐⠏⠀⠀⠀⢀⣤⡴⠦⡀⠀⠀⠀⠀⠀⠀⢿⡤⠉⡅⠁⠙⠧⢀⠀⠠⠀⢠⠞⢀⣁⣾
⠀⠀⠀⠀⠀⢰⠷⣠⣴⡶⢦⡴⠞⠀⠈⠉⠀⠀⠀⢀⡿⠑⢆⠀⣷⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⢴⠂⠀⠀⠀⠀⠀⠀⢘⣯⠁
⠀⠀⠀⠀⠀⣰⣾⢧⣏⠀⠀⠴⠄⠀⢠⣠⣠⣤⠀⣾⣿⣤⣉⣵⣇⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠐⠀⠀⠘⠓⠀⠀⠀⢁⣀⠁
⠀⠀⠀⠀⣼⡿⢏⠀⠹⢗⠀⠀⠈⢻⠀⢀⣿⣷⣾⣿⣾⣷⣭⡿⣿⣿⣦⠀⠀⠀⠀⠃⠀⠀⠱⠇⠀⡀⢰⣦⠀⠀⠀⠈⢻⠀⠄
⠀⠀⢰⣧⡈⢷⡄⠙⠛⠋⠁⠀⠀⠘⠀⣤⣿⣿⣿⣿⣶⡇⢹⣿⣿⣿⣿⠃⠀⠀⠀⠰⣄⢀⡄⠀⠀⠀⢸⠃
⠀⠀⢸⡟⠁⢀⣿⣄⠀⠀⠀⠀⡄⠀⢸⣿⣿⣿⣿⡛⠉⣧⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢈⣤⡄
⠀⠀⠀⣷⣿⠏⠉⠉⠀⠀⠀⣘⣃⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠁⠀⠻⠟⠛⣿⣿⣿⣿⢿⣿⣿⣿⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⡄
⠀⠀⠀⣼⣿⣿⣿⣿⠏⠛⣿⣿⡇⠀⠀⠀⠀⠀⡿⡿⠋⠉⠀⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠟⢻⢃⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⣷⠶⣶⠶⠶⢿⠷⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡿⢋⣿⣿⡏
⠀⠀⢸⣿⣏⢀⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⡟
⠀⠀⠘⠿⢿⡿⠿⣿⣿⡿⠋⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⣹⠿⠂
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣀⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣐⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠙⡏⢻⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣽⡿⠗
⠀⠀⠀⠀⠀⠀⢠⠇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠁⠀⠸⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⠉⢷⡄⠀⢾⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠹⢾⣿⣯⣿⣿⣠⣤⡀⣰⡟⠀⠈⢿⣄⠈⠷⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⢶⣛⣽⣟⣀⠀⠀⠈⠻⢧⡴⠋
⠀⠀⠀⠀⠰⣾⠊⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠈⠁⠲⠀⠀⣾⣟⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⠀⢢⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⣀⣼⡆⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⠀⠠⢘⡈⠁⠁⡉⠒⠀⠀⠀⠀⠀⠀⠀⠈⠣⣆⣰⡧⠄⣒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠃
⠀⠀⠀⠀⠀⠀⠠⡾⠛⢿⢺⠗⣃⣘⢲⠶⠿⠓⣺⡇⠀⠀⠀⠀⠀⢠⡄⠸⠏⠎⠓⠃⠫⢆⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡤⠤⢶⣷⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸⠆⠳⡧⠿⠘⠹⣄⠼⠜⠧⢀⠀⠀⠀⠀⠸⡇⡌⠀⣿⡝⣩⣤⠠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢤⠀⠶⠌
⠀⠀⠀⠀⠀⠀⣶⣤⣭⣍⣤⣄⠀⣶⢶⢰⣶⢲⢰⠾⠋⠀⠀⠀⠀⠀⡆⣧⣀⡟⡇⢸⢿⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⢰⠳⠶⠶⠲⠶⠆⠏⣿
⠀⠀⠀⠀⠀⠀⠈⡉⠍⡉⠡⠥⠤⠔⠐⠂⢒⠒⠐⠆⣤⠀⠀⠀⠀⠀⡇⣀⠂⠀⠲⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⢐⠀⠟⠁⡉
⠀⠀⠀⠀⠀⠀⠈⡳⠚⠷⣤⣴⠶⣶⢧⣿⣞⣻⣂⣘⣯⠀⠀⠀⠀⠠⡇⠸⣤⣞⡿⢏⣰⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠖⠗⢖⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠁⠪⠟⠟⠗⠋⠹⠃⠸⠿⠀⠀⠀⠀⠀⠀⠀⠁⣾⡿⠧⠐⠿⠶⢶⣶⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢀⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠸⣄⣸⠘⡦⢸⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⢿⢯⠿⣿⣶⣃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⣉⣡⠘⢂
⠀⠀⠀⠀⠀⠀⣠⣦⠀⢀⣀⠐⠲⠶⠀⠀⣴⣴⣄⠀⠀⠀⠀⠀⠀⠖⠒⡶⠒⠁⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠦⠿⠹⠿⠻⠿⡾
⠀⠀⠀⠀⠀⢰⣿⡏⠀⣾⠋⣤⢄⡤⣺⡋⠉⣻⣿⡇⠀⠀⠀⠀⠀⠀⡸⡡⠼⢶⠷⠷⠾⡞⠂⠀⠀⠀⠀⠀⠀⠲⠒⠓⠂⠒⠒⠐⢚⠓⠛⠂
⠀⠀⠀⠀⠀⠘⠿⣿⠂⢑⣇⡙⠊⠛⡎⠃⠂⢿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠉⠛⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣛⡿⢛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠞⠀⢠⠏⣁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⢸⠋⣀⣶⢾⠀⣾⡀⢹⡅⡒⠂⠀⠀⠀⠀⠀⣤⡘⠘⠛⠓⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣀⣼⣿⡸⢿⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠺⠃⡋⡘⣂⣋⣃⣘⡃⠉⠁⠀⠀⠀⠀⠀⢻⣷⣸⣰⡟⣧⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣅⣀⣁⣀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢠⣴⢥⡄⠁⠀⠉⣷⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⡀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢤⠀⠀⣠⠤
⠀⠀⠠⠀⢀⡔⠀⢀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠘⠂⡀⠀⠀⠈⠀⠀⠀⠀⠀⠆
⠀⠀⠁⡠⡌⠀⠀⠁⢠⡇⢸⢧⠃⠀⠀⠀⠀⠂⠀⠀⠠⠀⠜⠁⠒⠀⠈⠒⠚⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠀⠀⡇⠀⠙⠂⠀⠴⠀⠇⠘⠘⠄⠜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀⠂⠀⢐⠈
⠀⠀⠀⠀⠰⠲⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⡀⠐⡶⠀⢸⢠⠂⢐⠄⠁⠄
⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢐⢠⠆⡄⡤⣰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠈⣧⠀⢸⡌⠣
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⢀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⢀⡀⣀⢀⠀⠀⠀⡀⠀⠀⠀⡀⣤⠣⠤⠃⠀⠚⠀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠷⠊⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⣤⡔⠁⡎⠸⣴⡇⠀⡀⢀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠐⠀⠀⠀⠈⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠑⠚⠁⠥⠏⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠐⠈
⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣶⢰⣿⡍⠘⣿⠀⣠⡄⠀⣰⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣴⠟⣹⡇⢸⣿⠃⠀⣿⣴⣿⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⡴⠛⠁⠀⣿⠃⢸⣿⠤⢤⡟⢸⡏⣰⢫⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠛⠉⠀⠀⠀⢸⣿⠀⢸⡇⠀⣼⠁⢻⡿⠃⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⠇⠀⠘⠷⠞⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣴⣴⣶⣶⣶⣶⠶⢶⣶⣶⣶⣶⠶⠖⠲⠶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣾⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠉⣀⣤⣤⣈⣿⢋⣠⡴⢒⣶⣤⠈⢿⣥⣤⣤⠀⢠⣤⣤⣤⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⠀⢸⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⣿⣿⣋⣉⣀⣤⠀⣽⠀⢸⣿⣿⡟⢀⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⠀⢿⣿⣿⡿⠟⢁⣾⣿⡄⠘⠟⠋⣠⣾⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣤⣤⣤⣴⣾⣿⣿⣿⣿⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⣰⠖⣶⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢤⡿⠆⠀⣼⠇⠀⢀⡀⠀⢀⡉⠀⢿⣠⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡟⠁⡤⠀⠀⣸⠁⠀⡜⣿⠀⡠⢻⠃⢠⡾⠀⠀⡼⠻⣦⣴⠞⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⡟⢀⠞⢰⡇⡴⠁⢿⡴⢻⣇⡴⠋⠀⣼⠋⢹⡆
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠷⠊⠀⠈⠛⠀⠀⠈⠀⠀⠉⠀⠀⠀⠘⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠴⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⣠⡞⠁⠀⠼⠃⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠰⣿⡀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⢀⣼⡿⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⠙⠳⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⡠⢤⣤⡀⠀⢀⠈⠁⠀⢀⣾⣟⡡⠚⠁⠀⣠⣴⠟⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀⠀⠀⢀⣀⠀⢀⡠⣤⠀⡰⠊⢀⣼⣿⠃⢰⣿⠇⠀⢀⣾⡿⠉⠀⠀⢠⣾⡟⢁⣠⠞⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⢀⣾⠏⡰⠋⣴⢯⠎⠀⢠⣾⡿⠁⢠⣿⠏⠀⠀⣼⣿⠁⠀⠀⢠⣿⡟⠋⠉⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠈⣿⣿⠀⣼⡟⡜⠀⣼⣿⠃⠀⢠⣿⠟⠀⠀⣼⡿⠀⢠⢾⣿⡇⠀⠀⢀⠾⣿⡀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⣿⠏⠀⣿⡟⠀⠀⢿⠃⠀⢠⣿⡏⠀⢀⠎⠈⠻⠖⠁⢸⣿⡇⠀⡠⠋⠀⣉⣛⣛⠁⠀⠀⢠⣾⠄⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣠⠾⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠿⣧⡤⠋⢀⡴⢋⠇⠀⠀⠻⠿⠛⠁⠖⠛⠉⠉⠉⠙⠓⣶⣿⡏⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⢁⣀⣀⠀⢠⡤⢤⡄⢀⡄⣠⣤⡀⠀⠀⠀⣴⠟⡠⠋⠀⠀⠀⠀⠀⣀⣀⠀⣠⣴⠖⠁⡆⣰⣿⠏⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠸⣄⠁⠀⣼⠁⡼⠳⣾⡾⠓⠚⠁⠀⠀⣼⡿⢊⡤⠶⣦⡀⢀⣴⠟⢁⠔⠰⣿⣇⠀⠚⢡⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠐⠤⠟⠀⠀⠃⠘⠃⠘⠉⠳⠴⠋⠀⣠⢴⣿⡱⠋⢀⡀⢼⡗⣾⡟⠀⠀⠀⠀⠹⣿⣦⠀⣸⡟⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠴⣶⡄⠀⠀⣀⢤⣄⠐⠒⠉⠀⣼⡿⠁⡖⠁⡠⠟⠀⢿⣇⢀⡔⢁⡀⢀⣿⡟⠀⢿⣇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢃⡔⠁⣰⣿⠃⣠⠚⢁⣼⡿⠀⣀⡀⠀⠘⠃⠀⣩⠟⣳⠀⠀⠈⠉⠉⠀⠀⠉⢉⡁⠀⠀⣀⠉⣡⣄⡠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢧⠎⠀⣰⣿⢇⠞⠁⢀⣾⡟⠑⠋⠉⠉⠙⠒⢲⡾⠗⠊⠁⠀⠀⣀⣤⢤⡀⠀⣴⡿⠃⠀⣾⠏⣰⡟⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢰⣿⣿⠋⠀⣠⣿⡟⣠⡶⠋⠙⣦⡄⢠⣿⡥⠚⠙⡿⣠⡾⠋⢀⠜⢀⣾⠟⠀⠀⣼⡏⢰⡟⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⢠⣿⣿⠃⠀⢰⣿⠏⣼⠏⠀⠀⣼⡟⢠⣿⣋⡠⠤⠊⢱⣿⠗⠚⢁⡄⣸⠏⠀⠀⣼⡟⢠⡿⢧⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣾⣿⠃⠀⢠⣿⡟⢸⡏⠀⠀⣴⡟⠀⣼⡏⠙⢻⣦⡀⠈⠻⠦⠔⠋⡀⢿⠀⢀⠜⢹⡇⣼⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⡇⠀⠀⢸⣿⠃⠸⢧⡤⠊⢿⡤⠜⣿⠃⠀⠀⠙⢿⣦⣤⣤⣤⠞⠁⠈⠓⣁⡤⠴⠒⢻⣅⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠘⣿⣆⠀⣀⠀⢀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠉⢁⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠳⣤⣤⣄⣀⣤⣤⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁