National treasure Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⠄⠀⠠
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠤⠠⠤⠤⠤⠄⠀⠒⠒⠒⠂⠀⠓⢄⠀⠀⠙⢨⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠢
⠀⠀⠀⠀⠊⠉⠙⠢⡀⠀⠁⢄⠐⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠠⠀⠳⡀⠀⠀⠐⡄⠀⠀⠀⢀⣀⠀⠀⠍⠀⠓⠀⡈
⠀⠀⠀⠀⠀⠀⡲⢄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠠⠤⠤⠤⠐⠒⠒⠓⡀⠀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠂
⠀⠀⠀⠀⢀⠔⠀⡰⠣⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠓⠀⠁⠀⡋⠁⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⢠⠊⠀⡜⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⠀⠤⠤⠤⠤⠄⠒⠒⠒⠈⠉⢉⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠰⠒⠒⠒⠒⠀⠉⠉⠁⠀⠀⢀⣀⣀⠠⠤⠤⠐⠒⠒⠒⠈⠉⢹⠀⠀⡠⠊
⠀⠀⠀⠀⠏⠀⠜⠀⣠⡀⠤⠤⠄⠒⠒⠂⠋⠈⠉⠉⠀⢀⣀⡀⠴⠤⠤⠐⠒⠒⡘⠉⣹⡔⠋
⠀⠀⠀⠀⢀⠎⠀⠈⠀⣀⣀⢠⡤⠤⠤⠐⠒⠒⠈⠉⠉⠀⠘⠀⠀⠀⠉⢀⠸⠋⠳⠀⠀⠙⡆⠤⠀⠄
⠀⠀⠀⠀⠈⢆⠈⠉⢁⠀⢀⣀⣀⣤⢤⠴⠒⡶⠻⠙⠃⠀⢰⣁⣎⣉⠁⢻⣀⣤⠀⠚⠀⢪⠀⠀⠁⢈⠅⠀⠒⣠⠐⠀⠀⢀
⠀⠀⠀⠀⠐⠶⠖⡲⠋⠐⠂⠉⡔⢋⠉⣁⣮⣀⠀⠀⠀⠂⡔⠈⠁⠈⠳⠄⠀⠄⠛⠒⠀⡜⠁⠉⠈⠄⠁⣀⠠⠄⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠉⠳⡤⠐⠀⠄⢘⡀⠀⡁⠀⣈⡁⠀⡀⠆⠁⠀⠀⠀⢴⡈⠉⠀⢦⢀⡀⠠⠄⠀⠀⠉⠀⠀⠀⡀⠠⠤⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠛⠀⠐⣴⠄⠀⣀⠈⠆⠐⠀⠀⠀⠀⠅⠒⠊⠁⠀⠀⠀⢀⡀⠠⠐⠐⠁⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⡀⠄⠐⠒⠀⠉⠀⠀⠀⢀⡀⢤⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⡀⠠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠐
⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⡶⣶⢶⢶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⣶⣶⣶⣶⡶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⢖⣏⣑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡞⠛⢛⣢⡀
⠀⠀⠀⠀⠀⠀⣠⣛⢋⣠⣤⣾⣦⣬⣍⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢮⣏⣭⣿⣛⣶⠆
⠀⣰⢦⣤⣀⣚⡟⣿⣏⠈⢹⣯⣭⠴⠟⠻⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟⠛⠷⢀⣰⣟⣛⣻⠶⣭⡖⠲⠞⠫⢭⡀
⠀⣁⠈⠻⠿⠿⢿⣿⣿⡿⠋⠙⢿⡓⠒⠶⠾⢶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⢿⠓⠛⠏⠩⠋⢉⠣⢽⣍⣉⣉⠉⠁⣡⣇⣤
⠀⢀⣤⡄⠀⢀⣄⡀⠈⣡⣀⣿⣆⡀⣴⣦⠀⢩⣤⡀⠀⠀⠀⢠⣤⣴⠄⠀⠀⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠾⠿⠿⠿⠿⠿⠷⠤⠾⠿⠿⠿⠟⠛⠛⠿⠾⠿⠆⠀⠠⣞⣻⣿⣯⣻⡖⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⢖⣟⣻⣷⣶⣾⣿⣽⡳⠂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣿⣿⣿⣟⣿⣭⣾⣯⣿⣛⣟⡾⢿⣟⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⢿⡿⣿⣻⣿⣽⣶⣮⣿⣿⣻⣿⣿⣛⣻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣤⣤⡄⠀⣩⣥⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣛⠂⠀⠀⠀⠀⣠⣷⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣖⣫⣽⣯⣿⣿⣿⠂⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠳⢾⣟⣛⣿⣿⣷⣤⡀
⠀⠀⠀⠠⡜⣿⣯⣽⣿⣿⣯⣤⣝⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠫⡿⣏⣽⡿⢿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣦⣦⣤⣽⣉⣍⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⢚⡛⠋⣩⣉⣹⡝⣡⣀⣈⡹⣦⣀
⠀⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⢛⣉⢉⣉⣁⣤⣼⣿⣯⣽⣿⣾⢧
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡍⣹⣿⣿⢛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⣀⠀⠀⠀⠀⠀⣠⣀⡄⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢴⣶⢞⡛⡻⠿⠿⠟⠛⠿⡿⠚⠟⢛⣛⣻⣿⠳⣔⣶⡶⣶⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠂⢀⠔⢩⣿⡼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢭⡌⠈⠉⠙⢮⡁⢀
⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⣸⡼⣧⣄⣀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⡇⣿⠄⠀⠀⠀⠳⡄⠘⡄
⠀⠀⠀⠀⠀⠒⣸⡅⠀⠀⢸⠰⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡏⠋⠀⠀⠀⠈⠇⠈⠠
⠀⠀⠀⠀⠀⡀⣯⠀⠀⠀⠸⡷⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣇⠀⠀⠀⠀⠀⢼⢠
⠀⠀⠀⠀⠀⠀⢿⠒⠒⠒⠒⢿⢹⣿⡃⠀⠒⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⣿⡞⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠐⠸⡆⠀⠀⠀⠈⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣹⡇⠀⠀⠀⠀⠀⢀⡏⠀⡜
⠀⠀⠀⠀⠀⠂⢠⢻⣄⠀⠀⢀⣽⣎⣥⣶⣀⠀⠀⠀⠀⠀⠀⠀⣰⣻⡟⡟⠀⠀⠀⠀⠀⠀⡼⠡⠰
⠀⠀⠀⠀⠀⠀⠀⢀⢻⣄⠀⡸⣿⠿⣿⡆⣤⡐⡀⠀⠀⠀⢤⡞⣩⣿⠄⠉⠀⠀⠀⠀⠀⡞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠹⣾⠁⣬⠢⣻⣺⣆⠻⠇⠆⣀⣴⢻⡟⣗⣀⣀⡀⠀⠀⠀⢠⠞⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡙⣿⠸⡏⢠⠁⣿⡙⢷⣿⣷⢮⣿⣿⠏⠉⠉⠉⢩⠍⢉⡿⠻⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⣷⣴⣷⣖⣫⣖⣛⠿⣿⣦⣽⣧⣭⣻⣟⡁⠉⣀⠟⣡⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠊⠈⠁⠀⠀⠀⠑⠛⠂⠙⠁⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣀⢠⠤⠤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡤⠗⣺⠿⣾⢋⣼⠽⡛⠛⠛⠛⠷⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⠋⠀⢰⣯⡾⡇⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⢂⣤⢆⢴⣏⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣰⠋⠀⠀⢠⡆⡻⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡾⢒⠏⠔⠉⢈⠽⠻⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡂⢰⠃⠀⠀⠀⠨⡆⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠷⠃⠀⡔⠁⢀⡾⣥⡤⠾⢿⡛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠐⣿⢿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⢃⠀⢀⡄⡀⢠⢟⡿⢏⡠⠖⠉⠡⡔⡄
⠀⠀⠀⠀⠀⠀⠐⢰⠏⠀⠀⠀⠀⠀⣿⢠⡵⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠻⠋⠀⡜⠀⢠⢟⣾⣶⣯⠔⠃⢀⡤⡆⢃
⠀⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⢹⣾⡃⠋⠁⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⢀⡿⢉⡃⠀⠀⢠⠁⠁⡟⣌⡽⠿⠁⣠⡶⠉⠀⣁⣿
⠀⠀⠀⠀⠀⠀⢰⢾⣷⣤⣀⡀⠀⠀⠸⣄⠟⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣱⢻⡍⠀⠀⡎⠠⣸⣿⠋⣀⡤⠞⠁⠀⣠⡾⣏⡿
⠀⠀⠀⠀⠀⠀⠈⡄⣹⡀⢨⡭⠟⠒⠶⣛⡇⣍⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⣇⠀⠀⠀⢸⠁⠀⣿⡷⣾⡿⠀⢀⣤⡾⠋⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⢳⠈⣧⡇⢠⣴⠦⢠⠈⣽⡋⣧⣟⠓⠲⠦⢤⣄⣀⣀⣾⡔⢛⠍⠀⠀⠀⢀⠈⢰⣿⠟⢁⣠⠶⠋⠉⠀⣠⢾⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢹⣿⠀⠋⠀⠈⣙⠿⣧⣹⠘⠀⠀⠀⠀⠀⢉⡿⠺⡇⠐⠤⣤⣀⡀⣟⣶⣼⣳⣶⣿⡿⠀⠀⠠⠊⠁⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⣿⠷⡖⠛⠛⠁⠀⠸⣙⠷⣘⡖⠂⠀⠤⡜⢰⡟⡚⠀⠀⠀⠈⢹⡿⣿⣿⢿⠙⠉⣀⠄⠀⠀⡠⠖⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠀⠀⠁⠀⠐⠀⢽⣄⢻⠱⡄⠀⣘⣳⠇⠌⠁⠀⠂⠠⠀⢨⡇⡀⣿⣨⡴⠎⠁⢀⠠⠊⢀⠜⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠙⢻⠶⠂⠄⢄⣀⠀⠀⠹⡟⡷⣶⡶⣣⣾⣮⣀⣀⠀⠀⠀⠀⠀⡇⠀⢹⠉⠀⡠⠄⠁⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢶⣼⣷⡜⢗⠛⣔⠀⠀⠀⠉⠁⠒⠀⠀⣷⢀⢸⣶⡿⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠙⢿⠛⠦⣤⣤⣀⣀⠀⠀⠀⢹⣀⣸⡏⣇⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠈⠁⠐⠸⣯⣀⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠋⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣻⣿⠀⠀⠀⠀⠀⣾⡿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⢺⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⢤⣤⣤⣤⠴⣶⡶⠶⠶⠶⠶⠶⠶⢶⣶⢶⡶⠶⠶⠶⠶⠖⠶⠖⠲⣶⠖⢛⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⢻⠟⣩⡛⢦
⠀⠀⠀⢀⡴⡿⠟⠉⠀⠀⠀⠀⢀⠒⢋⡴⢋⠁⠀⠀⠀⠀⠀⠀⡰⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠃⠌⠀⠀⠈⢧
⠀⠀⣠⢿⠟⠀⠀⠀⠀⠀⠀⣀⠊⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠊⠀⠀⠀⠀⠀⠠⠄⠀⠀⡄⠁⠀⠏⠡⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡄⡰⠀⠀⠀⠘⠸
⠀⣰⢻⠃⠀⠀⠀⠀⠀⠀⠰⠃⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠁⡁⠀⠀⠀⠀⡇
⢰⢇⠇⠀⠀⠀⠀⠀⠀⢀⠂⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢨⢰⠁⠀⠀⠀⠀⣱
⡾⢸⠰⠒⠀⠀⠀⠀⠀⡘⠀⢀⡃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠸⡀⠀⠄⠀⣀⡹
⣇⣙⢚⡒⡒⣒⠀⠀⠒⠀⠀⠀⠐⠓⠒⠂⠀⠀⠤⢴⣜⣀⣀⡱⠦⠠⠤⠤⠤⠤⠤⠤⠦⠆⠀⠀⠨⠩⠥⠀⣀⢀⡀⠀⢀⣨⣘⡇⠀⠃⣏⡸⠤⣒⣫⠥⠶
⣾⠈⡀⢀⣉⣉⣉⠉⠉⡉⠉⠉⠘⠛⠋⠒⠒⠒⠒⢺⠚⣓⡶⣿⠒⠒⠒⠒⠒⠶⠲⠄⠤⠤⠤⠤⠤⠤⠤⢤⣄⣠⣀⣀⣀⣀⣀⣀⣠⡦⠤⠒⣉⡭⠔⠚⠷
⣿⠀⡗⠈⠀⠀⠀⠀⠀⠀⠄⠀⡆⠉⠉⠉⠉⠉⠃⢺⣾⢿⣧⣜⠒⡖⠒⠒⠒⢲⠖⠲⠶⠄⠀⢠⢤⣤⣤⡤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⢎⠙⠀⠀⠀⢰
⢹⠀⡇⠀⠀⠀⠀⠀⡀⠀⠄⠀⡇⠀⠀⠀⠀⠀⠀⣾⣋⣽⠚⡿⠉⡁⠀⠀⠀⠘⠀⠘⠒⡇⠀⢰⠤⠂⠀⠀⠀⠀⠀⠀⠀⢄⠠⡇⠀⠀⣿⠀⠀⠀⠀⡀⡘
⢸⠀⡇⠐⠀⠀⠀⠀⡇⠀⡆⠀⡇⠀⠀⠀⠀⠀⡏⠛⠉⠘⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣡
⢸⠀⡄⠀⠀⠀⠀⠀⠇⠀⡇⠀⡇⠀⠀⠀⠀⢀⡆⠀⣶⠀⠀⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠸⡿
⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠀⠀⠈⢧⠀⠉⠀⢠⡿⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇
⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠉⢰⠚⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⠀⠀⠀⠀⢀
⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡃⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢹⠇⠀⠀⠀⠀⢰⣀
⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠀⠀⠀⠘⠬
⢼⠀⡆⣰⠀⠀⠀⠀⠀⡀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡸⢠
⠸⣵⣄⣀⣉⣉⠀⠐⠒⠓⠂⠀⠢⠤⠀⢀⣀⠀⠀⣀⢘⠀⠀⠄⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⢀⠞⣡⠞
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠲⠶⠦⠤⢤⣤⣄⣀⣀⡈⠉⠀⠀⠂⠶⠥⢤⣤⣀⣀⣀⣀⠀⡇⠀⠸⠂⢰⠂⠀⠀⠀⠀⠀⢀⡤⣀⡇⠀⢸⣇⢀⡰⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠲⠦⠤⣤⣀⣀⣀⣀⣈⠁⠀⠘⠒⠛⠂⠀⠠⠤⠤⠤⣼⡄⠈⠃⠀⠜⠿⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠲⠶⠤⢤⣤⣄⣀⣀⠀⢀⡤⢶⣷⡶⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡇⣶⣤⠴⣄⣶⣤⠶⣤⡶⡤⣄⣤⢦⠀⡇⠀⡴⣄⡶⣤⡴⣧⡴⡶⣦⠰⣤⡶⣯⡄⢴⡀⢨⣤⠦⡴⢦⡴⢦⢤⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠉⠉⠀⠉⠁⠉⠈⠉⠀⠀⠉⠉⠁⠉⠁⠉⠉⠉⠁⠁⠉⠁⠉⠁⠉⠈⠁⠁⠈⠉⠈⠉⠉⠁⠈⠁⠁
⠀⠀⠀⠀⠤⡄⣠⡀




⠀⣤⣶⡄⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⠾⢶⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⡆⠶⢰⡆
⠐⣿⣿⣟⠁⣺⣿⣿⠂⠀⠀⠯⠽⣴⣲⣴⡦⠦⠶⡄⠀⠀⠀⠠⣭⣶⣮⡦⠀⠀⠀⠀⠀⠀⣀⣻⣧⣿⣁⠀⠀⠀⠀⠀⣸⣃⢊⣤⡁⡘⣫
⠉⣿⣿⡟⠉⢻⣿⣿⡉⠀⠀⡃⢨⡞⣿⢺⣿⢨⠉⡇⠀⠀⠀⣠⡿⢻⡙⣧⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⡄⢢⣉⠄⠢⡯
⠸⣿⠿⠃⠀⠘⠿⣿⠇⠀⠀⠓⠚⠉⠉⠈⠉⠛⠛⠁⠀⠀⠺⡿⢁⣶⣇⠹⡿⠃⠀⠀⠀⢀⣿⣿⣟⣿⣿⡃⠀⠀⠀⠀⠀⠘⢻⣿⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡆⠀⠀⠀⠀⠀⢘⣷⣶⣶⣶⣶⣟⠀⠀⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠀⣴⣶⡆
⠀⢠⣿⣶⣶⣶⣿⡄⠀⠀⠀⠀⢰⣿⠋⠙⣿⠄⠀⠀⠀⠀⢻⢿⡟⣩⣙⣿⣿⠃⠀⠀⠀⢠⣾⣿⠓⢻⣿⣆⠀⠀⠀⢀⣤⣽⣿⣧⣄⣿⣠
⠠⠿⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⢿⡿⣿⣿⢿⣿⠀⠀⠀⠀⢸⢐⡟⣧⡿⣱⣚⡀⠀⠀⠐⢻⣾⠉⣶⡎⢹⣿⠃⠀⠀⠛⠿⠿⠿⢿⣾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠷⠹⠍⠵⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠈⠁
⠀⠀⠀⣀⡀
⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢺⣿⣥⣽⣷⠂⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⣀⣹⣭⣹⣽⣁⡀⠀⠀⠀⠴⡿⣿⢿⡿⣿⠿⡶
⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⣶⣤⣤⣼⣿⣿⣿⣿⣶⠀⠀⠀⠀⠨⣿⣿⣯⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣟⣻⡏⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡧
⢠⡞⣧⣫⣛⣼⢱⠀⠀⠀⠿⠋⠀⠉⠿⠋⠀⠙⠿⠀⠀⠀⠀⢚⣿⣻⣿⡂⠀⠀⠀⠀⠠⡯⢿⠭⡿⢽⠽⡇⠀⠀⠀⠘⠉⠉⠉⠉⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⡀
⠀⢻⢲⣷⣷⡧⠀⠀⠀⠀⢠⣶⣶⣿⡟⠰⣶⣶⡄⠀⠀⠀⠀⠀⢐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⣈⣿⣋⣍⣻⣋⡀
⠀⠘⢻⣿⣿⡁⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢨⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⢸⣿⣿⡅⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡉⢩⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡛⠃⠉⠉⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡯⣀⡀⠀⢠⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣞⣴⣶⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⢙⣿⣿⣿⣿⠿⠛⠋⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣶⠀⣤⡀⠶⣶⠶⢠⡆⣠⣤⢠⣄⢰⠀⣤⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠿⠼⠋⢷⡀⣿⠀⢸⡇⠻⡼⢻⡟⣿⢰⠟⢷⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⠀⢀⡀⢀⣀⡀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⢱⣿⡇⢸⡯⠁⡼⣧⢈⣯⡇⢿⢹⡄⣿⡃⢸⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠃⠙⠘⠛⠁⠃⠈⠉⠓⠋⠈⠋⠈⠃⠙⠘⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠶⠶⠖⠒⠲⠶⠶⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⢁⣀⡀⠤⠤⠤⠄⠶⢲⠦⠤⠤⢤⣀⣀⠈⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢀⡠⡔⠚⠊⠉⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡈⠉⠲⠦⢄⡈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣀⡤⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⡤⠊⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⡈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠨⠓⢄⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠁⢀⠄⢀⣴⣶⣦⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣠⠟⠁⡠⠋⠀⠀⠀⠈⠀⠀⠄⢀⠀⠂⠀⠀⠀⢀⢄⣡⣺⠋⣵⣶⣮⠙⣷⢌⣢⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⢷⡄
⠀⠀⠀⠀⣴⠋⠀⡜⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠷⠉⣀⠙⢮⣍⣥⠞⠀⣈⠻⡷⠦⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠻⣆
⠀⠀⠀⣼⠃⢀⠎⡀⠁⢀⠤⠤⡀⠀⠂⠀⠐⠂⠀⠀⣁⠀⠁⠀⠀⠀⠈⠛⠶⠿⠵⠖⠛⠉⠀⠀⠀⠠⠓⠈⠁⠒⠒⠂⠠⠤⠤⠄⣀⡀⢀⡀⡀⢣⡀⠹⣆
⠀⠀⣰⠇⢀⠎⠀⠁⠀⢘⠀⠀⣱⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⢀⠉⡀⠀⠀⠄⠐⠀⠁⠀⢀⠀⠠⢀⠁⠀⠐⠀⠀⢀⠀⠀⠀⠀⠁⠀⢱⠀⢹⣆
⠀⢠⡏⠀⡜⠀⠀⠀⠀⣿⣿⣿⣿⠁⠰⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠸⠅⠠⠀⠄⠀⠠⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢇⠀⢿⡀
⣀⣾⠀⢠⠁⠀⠀⠀⠀⠻⢿⡿⠟⠀⢠⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣧
⣻⡇⠀⡈⠀⠀⠀⠀⠀⠲⠾⠿⠶⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿
⠉⠉⠉⠉⠉⠓⠒⠒⠲⠤⣠⣀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠴⠚⠋⠉
⠀⣶⣄⠀⣄⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣤⠤⠤⠶⠒⠚⠋⠉⠁⠀⠀⠀⣀⣀⡤
⠀⣿⠙⣦⣿⠀⣴⣦⠀⠶⣤⠤⣠⠀⣀⣀⡀⠀⡉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠒⢒⣚⣉⣉⡉⠉⠉⠉⠉⠉⢉⠁⠀⡀⠀⢀⣤⣤⢠⡄⠀⣦⢰⣟⣽⠇⣿⢖⣀
⠀⠀⠀⠈⠋⠘⠋⠙⠇⠀⣿⠀⣽⠸⣏⢈⣿⢸⡿⣆⣿⠀⣼⣧⠀⣾⠀⠀⠀⠀⠀⠉⣿⠉⢰⣿⣹⡗⢸⡯⠍⠁⣼⣻⣄⢘⣓⣶⠘⣧⡤⠏⠸⠃⠙⠂⠉⠉⠁
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠃⠈⠛⠘⠋⠉⠃⠻⠶⠶⠀⠀⠀⠀⠛⠀⠘⠃⠘⠃⠘⠓⠛⠘⠁⠀⠉⠈⠉
⠀⢻⡒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣸⡇⣿⠀⠀⣿⢸⣷⣄⢸⡏⠋⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⣧⠀⢺⠉⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠹⠇⠻⢤⡤⠟⠸⠇⠙⠿⠇⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠊
⠀⠀⠹⡆⠀⢃⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠲⠦⠤⠤⠤⠶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢛⠛⠛⠛⠛⠛⠛⠉⠉⠉⡟⠉⣽⠃
⠀⠀⠀⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠈⢉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣼⠏
⠀⠀⠀⠀⠹⣆⠀⠳⡀⠀⠀⠀⠀⠀⠀⠦⠤⠔⡖⠀⠄⠀⠀⠀⠀⠀⢠⣶⣶⣦⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⢸⡿⠛⠷⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣼⠋
⠀⠀⠀⠀⠀⠘⢧⡀⠘⢆⠀⠀⠀⢠⠘⢱⣿⣿⠑⣤⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠑⢄⠀⡁⢈⡂⠭⠉⡋⠀⠅⠀⠀⠀⢰⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣸⡅⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠑⠦⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⣀⠔⠋⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠈⠁⠻⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠈⠀⢀⣀⠔⠊⢁⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⡀⠉⠒⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠉⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⠀⠉⠉⠁⠒⠒⠒⠒⠒⠒⠉⠉⠉⢁⣀⣠⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⢯⣍⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡛⢛⣳⠄
⠀⠀⠀⠀⠀⢀⣾⣋⣡⣴⣼⣷⣶⣶⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣰⡿⣯⡭⢿⣟⣷⡖
⠀⡸⢶⣦⣤⣹⣫⣿⣅⢀⣻⣥⠶⠛⠙⢷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⠛⣂⣀⣾⣏⣻⢟⣣⣗⣒⣛⠑⠻⡤
⠀⠤⠄⠛⠛⠻⠿⢿⣿⡋⢁⡹⡯⠙⠛⠛⢛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠿⠉⠉⣉⣘⣂⣐⣚⣿⣠⣄⣀⣉⣭⣯⣶
⠀⢠⣶⣀⣀⣠⣤⠀⠀⣦⣼⣿⣤⣰⣷⣆⢀⣶⡄⠀⠀⠀⣰⣶⣾⣂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿
⠀⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠀⢐⣿⣽⣿⣷⣽⣏⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠿⣿⠿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣴⣿⣿⣴⣿⠿⢿⣿⣾⣏⣑⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣽⣿⣿⣿⣿⠿⣿⠷⣿⣿⣯⣽⣯⣭⣽⣖
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠶⢿⣛⣽⣯⣽⣷⣾⣿⣿⣿⣿⣶⣾⣯⣭⣿⣯⣽⣿⠒⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣠⡿⣿⢿⣿⣿⣄⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠁⠀⢠⣀⣴⣷⣿⣶⣦⡀
⠀⠀⠀⠀⣶⣶⠷⢿⣿⣿⠟⠻⣧⣀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣽⢿⣶⢿⣿⣿⣿⡦
⠀⠀⠀⢸⣮⣿⣿⣟⣙⡛⢛⡿⠾⢿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠦⠽⠿⡾⠿⠉⠛⠷⣄
⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⡾⢿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⡤⠼⠶⠾⠓⠚⢳⣾⣷⣲⣶⡽⣗
⠀⠒⣿⣿⣿⣿⣿⣿⣿⡿⣿⣛⠿⣿⣾⠿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣻⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⢿⡿⠟⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠐⣄⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠂⠒⠂⠁⢀⣈⠉⠉⠉⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⣼⣿⣧⠀⠾⠷⢀⣶⣿⡆⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣄⠘⠿⠟⠀⠀⠀⠘⢿⠟⠁⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠉⠁⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣿⣶⣶⣾⣿⠿⡿⣿⣿⡿⣿⣿⣷⣶⣶⣶⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣎⣛⣼⣙⣬⣼⣿⣰⣅⣯⣂⣇⣆⣅⣂⣇⣆⣅⣂⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠋⠉⠉⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⠀⢀⣤⣤⣶⠶⠓⠒⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠉⠙⠋⠛⠉⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⢠⣅⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢸⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢻⡶⠖⠀⠀⠀⠀⠀⢀⣀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠶⠖⠂⠀⠀⠀⣸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣀⣀⣠⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣦⣤⣤⣈⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢰⡟⢛⡟⠛⣿⢛⡛⡟⢻⡟⣿⢻⢛⡻⢛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣷⣿⣷⣭⣿⣦⣶⣯⣴⣷⣤⣾⣼⣿⣮⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠛⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣠⡄⠀⢰⣶⡆⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠀⠀⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠠⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⣤⣀⠈⢡⣄⠀⢠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠔⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠛⠋⠄⠀⠻⠿⢂⣼⣿⡆⢸⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣀⣄⣀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠀⠀⣀⣤⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣄⠀⣀⠀⠈⠻⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠀⣿⣷⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣠⣾⣧⠀⠀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⢿⣿⣿⠇⠀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠖⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣶⡄⢠⣴⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠚⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣿⠄⣾⣟⣛⣉⡀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣶⡄⠀⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣴⣿⣿⡿⢿⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠸⣿⡿⠃⣴⠈⢛⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠠⠲⠏⠙⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣤⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⢿⡆⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣼⠟⠉⠙⠿⠷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⣿⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣄⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠁⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⠉⠀⣾⣆⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣀⢀⣀⢨⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⣸⣿⣿⠀⠻⠟⠁⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡏⣿⣿⣟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢹⣿⣿⡇⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⣿⣿⠛⠉⠀⠉⣹⣿⣟⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠁⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⡀⡤⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠚⠀⡄⠐⠒⠲⠁⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣴⡶⣠⢤⣀⠠⠯⠤⣠⢶⣾⣵⡖⣤⣄⠤⠤⠴⢶⣾⣿⣦⢯⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠛⢿⣿⣷⣄⠈⢣⡀⠀⠀⠀⠀⠉⠻⣦⡈⠣⡄⠀⠀⠀⠉⠻⣦⢑⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠠⢿⣿⣿⣷⠒⠹⡖⠒⠒⠒⠒⠒⠚⣷⡘⠙⡖⠒⠒⠒⠒⢻⡏⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢘⠋⣿⣿⡄⠀⢀⣞⣻⣿⣿⣧⠀⡙⣄⣀⣀⣀⣀⣐⣘⣧⣄⣼⣆⣤⡤⣤⣬⡷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣀⡀⠀⢐⣻⣿⣿⣿⠘⠖⣿⣉⣽⠷⠀⠀⣸⣿⡶⢾⣄⣰⣷⣫⣽⣷⢤⢤
⠀⠀⠀⠀⠀⠀⠀⢱⣲⡍⠉⠉⢉⣛⠛⢻⣿⣿⣿⠤⠤⠟⠛⠛⠋⠉⠉⢿⣯⣶⢶⡅⠀⠀⣴⠄⣿⠾⢆⠈
⠀⠀⠀⢀⡀⠰⠀⠰⢶⣶⣤⣤⣬⣟⣀⣻⣽⠻⣿⠐⠷⢦⣀⣐⣂⣀⣤⣽⢯⣿⣃⣼⠶⠶⠾⣿⡟⠋⡥⠓⡶⠁
⠀⠀⠀⠀⡀⢶⠛⠠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⠋⠉⠉⠀⠈⣿⣿⣿⢭⠃⠀⠀⠈⣽⡇⢀⣇⣠⠇⣀⡈
⠀⠀⠀⠈⣀⠈⣤⣴⣇⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⣻⠀⠒⠠⠄⠄⠬⠟⠛⠋⠨⡄⡀⢘⣿⡟⠻⣭⡉⢢⠤⠁
⠀⠀⠀⠀⢠⢀⡤⠬⢿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⣿⠀⡠⢠⡁⠀⠀⠀⠀⠀⠀⠓⠎⢠⣿⠇⢰⣦⡙⣬⣦⠒
⠀⠀⠀⠀⢰⠊⣴⠞⣻⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⢀⢀⠫⡆⠀⠀⠀⠀⠀⠀⠁⣁⣼⣿⣴⢼⣦⣽⣿⣹⡄
⠀⠀⠀⠠⣣⣼⠿⢾⡿⣹⡛⡻⠿⣿⣿⣿⣿⣿⣿⡇⣀⢿⣄⣀⣁⠀⠀⠠⣤⠔⠒⠚⠉⣭⠁⢀⣔⣜⣷⠀⠈⠛⠃
⠀⠀⠀⠰⠛⣓⣛⣿⡛⢷⠯⠀⠐⠮⠿⢻⡽⢿⣿⣗⢋⠉⠉⢡⣤⠀⠀⠀⢛⣂⣀⣤⣤⣴⣦⣌⠩⠥⠟⣂⢀⠚
⠀⠀⠀⠀⠉⠙⠋⠹⡿⠟⠛⢻⣿⣿⣿⣶⣿⣿⣿⠧⠓⠃⣠⡶⠶⠾⠿⠛⠋⠉⠉⠉⠉⠉⠉⠤⠂⡐⠲⠶⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠛⠒⠒⠁⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠛⠛

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠑⠂⠀⠀⠘⠋⠀⠀⠀⠀⠀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢶⢶⣦⡄⠀⠀⠀⠀⢰⠀⣀⡀⠀⠀⠀⠀⠐⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⠇⠀⠀⠀⠀⠀⣻⢻⢮⣦⢁⠀⠀⣠⣶⣾⣦⠛⠋⠀⣶⡄⠀⢀⣼⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠉⠃⠀⣼⣿⣿⣿⡿⠁⠃⠀⣉⣁⣠⣾⣿⣿⣇
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢰⠄⠀⢰⠇⠰⢀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣦⣤⣴⣶⣶⣿⣿⣷⣶⣶⣶⣷⣯⢹⡏⠉⢀⣾⣿⣷⠙⢿⠿⠏⠀⠀⠀⢀⣄⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠘⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⠈⣀⡀⣾⣿⣿⣷⡀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠃⠀⠲⢀⠈⠛⢿⣿⣿⠟⠋⠒⠲⢶⡆⠘⢿⣿⡿⠟⠛⠁⠐⣶⣶⣾⣿⣿⣿⣿⣿⠗⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣶⣨⣄⣀⣀⡙⠁⠀⠀⢰⣤⡘⢿⣿⣿⣿⠿⠛⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠑⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢐⣿⣿⠿⠿⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢿⣿⣯⠋⢀⠀⠀⠀⠀⠰⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡉⣩⣤⣤⣤⡦⣤⠶⢒⣶⠖⣒⠒⣒⣒⣻⣿⠯⣿⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠼⡽⣧⠈⢞⠛⠉⠃⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠂⠙⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣟⠃⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⣄⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⢀⡀⢠⣸⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠙⣾⣿⡾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⠈⣿⣿⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢶⣸⣿⣧⠠⡤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠡⠤⠤⠽⠿⠿⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣆⠶⢶⣞⡛⠫⢉⣭⢥⠤⠀⠂⠒⠒⠀⠀⠀⠀⠐⠀⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣽⣦⠙⠖⠒⠂⠀⠀⢀⣀⠀⡀⠤⣀⡴⠶⣀⠠⠄⠰⣿⣏⡙⠳⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢿⣿⣷⣄⠸⣶⣤⣛⠶⣶⠞⢳⣂⡅⢯⣍⣛⣇⣐⡅⣀⠬⠝⠓⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠢⣷⣌⠻⣷⣌⠳⣟⠳⡌⢿⣥⠟⢲⡋⣓⠘⣦⠓⣦⡉⠮⠶⠾⠌⠟⢛⣛⣈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠃⠈⢂⠐⠸⣿⣧⡈⢭⡿⠯⠽⠚⠛⣉⣉⣩⣭⠤⠵⡶⢶⠒⠊⠙⠛⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠂⠀⠀⠙⣌⠀⣻⣿⣶⠒⡞⠈⠙⠉⠈⠉⢹⠛⠷⡀⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⢜⠚⢆⠀⡀⠀⡀⣽⣿⠋⢀⠭⠁⢀⣀⣀⣀⣿⠟⠲⠃⠂⠀⠀⠀⠀⠀⡘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠙⣌⠢⠙⣿⡟⠀⡘⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣴⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⢷⣼⢇⠇⡀⠇⠀⢀⣀⣀⡀⠀⠄⠀⠒⠒⠚⠉⠈⠉⠁⣽⢠
⠀⠀⠀⠀⠀⠀⠀⢤⣀⢶⣒⣲⣤⣴⢂⢤⡀⡀⠑⢆⠀⢠⣽⡾⠁⠸⠛⠉⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠒⠃⠈
⠀⠀⠀⠀⢀⣀⣰⠮⣽⣒⡸⠯⣌⣛⠲⣂⢢⡜⠀⠀⠙⢶⣷⡇⢁⣄⡠⠤⠐⠒⠀⠀⠈
⠀⠀⠀⠀⠬⠝⠪⠑⠚⠁⠛⠃⠓⠘⠓⠈⢀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠛⠋⠉⠉⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣤⣤⣐⣶⣶⣦⡈⢻⡿⠿⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠤⠐⠂⠀⡀⡲⠻⡿⠿⣿⠘⣿⡿⠟⣹⡿⢶⣿⠒⠻⣦⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣷⣎⣷⠀⠀⠀⢸⠛⠳⠶⠿⠉⣿⡟⢠⡴⠌⠱⠞⠁⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠿⠍⠉⡌⠀⠀⠈⢧⡐⣒⣂⠙⠏⣠⡾⢶⣄⠀⠀⠾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⡰⠁⠀⠀⠀⢨⡳⣈⣁⡠⢾⡟⡀⢸⣿⡆⠀⡀⠅⢠⣄⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⡇⠘⢁⣹⣁⣠⣤⣴⣶⣷⣇⡜⠀⠀⡉⣾⠃⣿⣿⠿⠋⠀⣀⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢐⠀⠀⢈⣿⣿⣿⣿⣿⣿⣧⣝⢷⡄⢾⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⡀⢠⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣇⣠⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢌⢍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠔⠒⠚⠉⣉⣉⠭⠽⠗⠢⣌⠓⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⣀⠀⣀⣀⣠⠤⠤⠖⠒⠊⠉⢉⣀⡠⠤⠤⠒⠒⠉⠉⠀⠀⠀⣀⣀⠤⠤⠳⡄⠱⡀
⠀⠀⠀⠀⢀⠎⠉⠉⠲⣄⠀⠙⢍⣀⣠⠤⠤⠖⠒⠉⠉⠁⠀⠀⠀⢀⣀⣠⠤⠴⠒⠊⠉⠁⠀⠀⠀⠀⢸⡀⢱
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠣⡀⠀⠣⡀⠀⠀⠀⣀⣀⡠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠋⠉⢇⠀⠇
⠀⠀⠀⠀⢸⠀⣶⣷⣦⡀⠀⠱⡄⠀⠱⡖⠊⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⢀⡼⠀⢽
⠀⠀⠀⠀⠸⡀⣿⣿⣿⣿⣄⠀⠘⡀⠀⠙⡀⠀⠀⣀⣀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠉⣁⡠⢴⡟
⠀⠀⠀⠠⣞⠣⡈⠓⢝⢿⣿⣆⠀⠱⠀⠀⠸⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⣀⡠⢔⣒⠭⢒⣪⠽⣷
⠀⠀⠀⠀⣇⣷⣌⡢⢄⠙⠻⣿⣆⠀⢣⠀⠀⢡⠀⠀⠀⣀⣠⠤⠒⠊⠉⠀⣀⡠⠔⣒⠭⢐⣪⠥⠒⢺⠃⠀⣰⡏
⠀⠀⠀⠀⢻⢡⣾⣱⢦⡑⢄⠈⠻⠂⠘⡀⠀⠀⡖⠚⠉⠁⠀⣀⡤⠔⣒⠭⢔⣊⠭⠖⠊⣁⡤⠴⠚⢻⠑⣼⣿
⠀⠀⠀⠀⠀⣿⣿⣹⣿⣟⢦⡑⢄⠀⢰⡇⠀⠀⢀⠀⠔⣚⡩⠔⣂⡭⠖⠊⣉⡤⠖⢺⣿⠁⠀⠀⠀⣼⠋⠙⠁
⠀⠀⠀⠀⠀⣬⢿⣟⣿⣿⣷⣽⣦⡙⢮⣳⣀⡠⠞⣒⡭⠔⠚⢉⣠⡴⠚⠉⠀⠀⠀⢸⡏⠀⠀⠀⢀⣿⡤⢷
⠀⠀⠀⠀⠀⠘⢿⣗⣿⣿⣿⢿⣇⢈⠢⠬⠵⠒⠉⠀⠀⠀⡸⠁⣿⡇⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣼⠏⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢯⣸⣿⣿⣾⣿⡧⢠⣤⠀⠀⠀⢀⡔⠋⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⡁⢴⣪⡕⢁⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⢿⣿⣿⣿⡸⣿⡀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⢀⡠⣚⣿⣾⠟⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡋⣿⣿⣷⣯⣧⠒⠉⣧⠀⠀⡀⠀⣿⡠⢔⣪⣵⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠻⣿⣿⣿⠀⠀⢹⠆⣸⠷⢚⣽⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣝⣿⠤⠒⠻⢯⠵⢾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣧⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⡤⠔⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⣿⠔⠒⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠥⣄⣀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⣠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠈⠉⢒⠒⠤⡥⣀⣀⠀⠀⠀⠈⡙⠋⠀⠀⠁⠐⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢂⠀⠀⠀⠀⠑⠀⠀⠉⠉⠓⠒⠤⠄⣀⢀⠀⠀⠠⠄⠉⠲⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠁⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⢒⢶⠁⡄⠐⠒⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⣀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢸⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⢋⣬⠝⣦⢜⡣⠄⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡀⠄⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡼⢴⠚⣥⢻⡞⢰⢃⠴⡶⡟⡏⡯⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠁⠐⠀⢸⠀⠁
⠀⠀⠀⠀⢀⡠⠤⠒⠐⣻⠃⣷⣀⠡⢭⠟⢶⣈⡓⠲⠐⢨⡖⣏⣴⠂⢫⠓⢆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠠⡞
⠀⠀⢸⡗⠈⠁⠀⣔⠋⣹⣻⠁⣏⣩⡊⣏⣀⡿⣀⢼⠋⡴⠙⣡⣯⢐⡂⣠⣡⣷⣢⡤⠛⠏⢰⡄⢸⡄⢀⢡
⠀⠀⠠⠀⢲⠂⠄⠉⠉⠭⠳⢴⢳⣴⢫⠛⣛⢜⠧⡤⢿⠰⢚⣛⣺⢁⣍⣩⣉⡵⢷⠛⣤⣀⣠⢗⣤⡺⣈⠄
⠀⠀⠐⠀⠸⠀⠀⠀⢘⠁⣤⣀⡁⠉⠛⠻⣭⣭⣧⣤⢯⣉⣎⠍⠳⡴⠷⠶⣏⠒⠀⠀⢻⣬⡹⠂⠉⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡔⠋⢠⣶⣻⢝⡿⠹⡩⠧⢤⠟⣒⠚⢳⠳⣲⠚⡶⠛⢫⠍⠀⠀⠄⡖⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣯⠝⠁⠈⠉⠀⠰⡈⡓⠴⣯⣉⡷⡤⠞⢋⣭⠀⠘⠃⠀⠀⠀⡇⢸
⠀⠀⠀⣄⠤⠂⠀⠀⠀⠀⠀⠀⡾⠷⠀⠀⠀⠀⠀⠀⠴⠊⠀⠐⠀⠨⠠⠀⠩⠄⠀⠀⠀⠀⠀⢀⡟⠇
⠀⠀⠀⠈⠀⠄⠁⠀⢐⠀⡀⠀⢻⣷⠀⠀⠀⠀⠀⠀⢰⢃⡆⠀⠀⠀⠀⠀⢸⠩⠀⠀⠀⠀⠀⢸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠸⡅⠀⠀⠨⢯⣵⠀⠀⠀⠀⠀⠀⢭⠇⠀⠀⠀⠄⠀⡏⡘⠀⠀⠀⠀⠀⣨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⡆⠂⡘⣿⢲⠄⠀⠀⠀⠀⠰⣊⢇⡀⠃⠀⠀⣂⡀⠀⡀⣠⠴⡺⢅⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠙⢋⣩⣿⢽⠅⢠⣤⣤⡀⠀⡗⢋⡗⡒⡆⡤⠗⢢⣖⡟⠓⡿⢾⠗⠻⣟⡼⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣉⣾⡗⠘⣶⠊⢰⣄⡩⠖⢋⣽⠀⣥⠠⣍⣠⣗⠕⠹⣖⡳⠾⢫⠹⠏⠵⠶⠾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣏⣠⢉⣉⠍⣳⣙⡤⢼⡶⠛⡗⠐⢳⡶⢻⠛⡾⡗⠐⡿⠹⣖⠛⣢⡘⠓⠖⠦⠤⠵⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣍⣉⡴⠚⣰⣉⠖⠔⢓⡶⢧⣤⢿⠒⢤⠇⢥⡞⠛⢷⠤⢯⢹⢻⡍⡟⡷⣂⠠⠛⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠭⠓⠒⠑⢋⠚⡭⢥⠒⠢⣾⣞⣷⠈⢠⢴⠛⠖⢫⡈⣋⣈⡿⣇⢉⠗⠣⣿⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠦⠤⠋⠈⠀⠙⢿⣿⣡⣨⡧⣾⡟⠧⠤⠅⡀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠃⠀⠀⠀⠀⠀⠀⠁⠛⠀⠉⠀⠀⠀⠀⠀⠓⠒⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⣶⡄⠈⣶⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠈⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠟⢿⣦⡀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⠻⣿⣿⣿⣿⣿⣿⣧⢹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣷⣾⣿⣿⣿⣛⠋⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⢹⡿⠋⢀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⡎⢰⣷⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡍⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡃⠀⠃

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⠿⠟⠛⠉
⠀⠠⢤⣶⣶⣿⣿⡿⠿⠛⠛⠛⠛⠋⠉⠉⢉⢉⠄⠀⠀⠀⠊⠉⣉⠉⠀⠀⢦⠀⣐⠈⠋⠉⠉⠠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠈⠅⠀⠈⠀⠀⡐⠀⠐⠀⠐⡤⠂⠀⠈⠉⣐⠒⠒⠒⠀⠒⠲⡀⠂⠀⠀⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⡀⠀⠠
⠀⠀⠛⠀⠀⢀⠀⢰⠀⠀⢀⠠⣀⡀⠀⠀⠃⠀⠀⠀⠠⠥⠄⠈⠀⢀⠛⠈⠉⠉⠉⡁⠂⠀⠀⠐⠀⢀⡀⠀⠠⠤⠐⠂⡉⠒⠄⠀⠀⣀⡀⠐⠀⠠
⠀⠂⠀⡀⠁⠀⠈⠀⠀⢐⡀⠀⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠀⠐⠓⠂⠂⠀⠀⠀⠒⠃⠒⠒⠒⠒⠐⠚⠓⠒⠂⠀⠀⠀⠑⠐⠒⠂⠀⠀⠐⠉
⠁⡀⠀⠰⠀⠉⠀⠀⠐⠈⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡽
⠀⠀⠀⢹⠀⢰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠁⠊⠙⠲⢤⣀⡈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⡷⠦⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⣻⣿⣛⡻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⣀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣬⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣷⡄⢀⣤⣄⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⠋⠉⢹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⢰⣿⠁⠀⠀⢸⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⣿⡇⠀⠀⢠⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⢻⣧⣀⡴⠋⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⣴⣾⣏⠉⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⡿⠇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠎⠙⠛⠉⠁⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠒⠒⠒⠺⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀
⠀⠀⠀⠀⡀⢤⠴⢧⠤⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠐⠒⠚⠓⠀⠉⠀⠉⠐⠓⠦⣄
⠀⣠⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⡘⠛⠿⠿⠿⠟⠛⣡⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞
⠀⠉⠘⠓⣢⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⣠⣔⣢⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⡒⢄⡀⠠⣤⠤⠤⠐⠾⡇⠈⢹⡄
⠀⠀⠀⠀⣿⣤⠋⠀⠀⠈⠉⠉⠉⢉⡾⣯⠚⠁⠀⠒⠒⠂⢰⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⢎⡹⠀⠀⠀⠀⠀⠀⠈⠑⣾⡏⡄
⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⣎⠻⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢡⠛⣄⠀⠀⠀⠀⠀⠀⠈⢠⢇⡀
⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡀⠀⠀⠀⠀⠀⠀⢺⠀⢣
⠀⠀⢷⢀⡇⠀⠀⠀⠀⠀⠀⣴⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡇⠀⠀⠀⠀⠀⠀⢨⣠⠆
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⠃⠀⠀⠀⠀⠀⢀⡙
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢋⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠋⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠛⢿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠙⠻⣶⣦⣀⣀⠀⠀⠀⠀⣀⣀⣠⣶⠿⠋⠁⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠈⠉⠉⢻⡟⠛⠛⢿⠉⠉⠀⠀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠂⢤⡤⠄⠀⣸⠀⠀⠀⣿⠃⠠⣤⡤⠐⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⢹⠁⠀⠀⢻⠁⠀⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣟⠁⠀⠈⠉⠙⠋⢻⡇⠀⠀⢹⣋⡉⠉⠉⠁⠈⢻⡄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠛⢦⡀⠀⠀⠀⠘⣃⠀⠀⢻⠉⢃⠀⠀⣠⠔⠿⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣀⠁⠀⠀⠙⢢⡄⢀⡼⠛⠷⣤⠆⠀⠘⣷⢾⣁⠀⢐⡇⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⣽⣀⣀⡀⠴⠧⣤⡼⠳⠦⠀⠀⠀⠀⢰⠆⠀⠀⠈⠛⠛⠦⠄⣀⡀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⠀⠘⠁⠈⠉⠈⠁⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⣀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠁⠈⠙⠿⢲⡤⠤⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠒⢶⡶⠶⠶⠶⠶⣶⠶⢶⣶⠶⠷⠶⢶⡶⠒⠒⢉⡴⢦⣤⣄⣈⣽⠋⢻⡷⠒⠒⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡸⠦⠤⠤⣾⠀⣄⣀⣀⣘⠛⠁⠀⢛⣋⣉⣥⠀⣇⢀⠀⢼⣆⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣄⢀⣶⣄⠈⠛⠁⠀⠁⠙⣷⢤⣤⣾⠉⠀⠉⠚⠃⠀⠀⠀⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⣿⠖⠺⣿⠉⢹⡿⠳⣄⠀⠀⠀⣾⠇⣸⣇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢙⣁⣀⠤⠴⡿⠄⢀⣭⠀⠀⢠⡏⢠⠛⠻⣧⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣗⠚⣉⡁⣤⠤⣶⠦⠤⣤⣏⠀⠀⡶⠀⣻⠀⠄⠈⣦⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣉⣀⡀⢰⡏⠀⠀⣸⣧⡄⠸⣄⢰⡋⢹⡷⠀⠀⣦⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡘⣿⠛⠋⠁⠀⠛⠃⠉⠛⠋⠀⠙⣂⠀⠙⠃⠙⡇⠀⠀⢀⠛⠀⠉⠛⠋⠀⠀⠀⠀⠀⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣦
⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⣰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢠⡇⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠙⣤⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠈⢁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠧⠤⠤⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠤⢚⣿⣿⣿⡿⠟⣛⣛⣯⣭⠉⣿⡟⢩⣽⣿⣛⡛⠛⢿⡿⠿⣿⡓⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⣠⣿⣷⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠃⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣡⠤⢀⣀⣀⢀⣀⠠⣦⣉⠷⠋⠱⠞⣡⡤⠄⣀⣀⣀⠀⢠⢤⣜⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣻⡆⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠒⣛⣙⣛⣛⣛⣀⣒⣂⣀⣠⠄⣤⣭⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⣧⠀⡶⣶⣿⣿⣟⣻⠟⣡⠽⠏⠡⠄⠠⠤⠄⠰⠖⠓⣾⣿⣿⣿⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⠞⠿⢿⡆⠙⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠻⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⠀⠀⠈⢫⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢿⡙⠀⠐⢿⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣂⠂⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣠⡄⢻⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡀⡀⠈⡀⣻⡄⢀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠀⠀⠀⠉⢳⣿⣷⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢷⣻⡄⢹⣿⣿⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡄⣿⣿⣇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⡈⣙⣻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢿⣿⣿⣿⠀⣁⣤⣤⣤⠤⠦⠰⠒⢚⣛⣒⡀⡉⠩⠥⠍⠁⠀⠀⠀⠄⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⡉⢰⣓⣀⡈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣿⡿⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣄⠁⢀⣀⣀⢠⣔⣐⠒⢡⣖⡉⠴⢺⣭⣱⣦⡘⢁⡌⠟⠻⢦⣤⣹⣷⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣨⠻⣟⠿⣷⣄⠛⣭⠶⣬⡉⢿⣭⠹⢧⠖⢈⣷⠦⠼⠶⡬⠒⣤⠀⡊⠀⠽⢿⣿⣷⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠑⢼⡷⢈⠿⣷⡄⠻⣏⣳⣆⣺⣶⠚⣙⣧⣨⣍⣉⣿⠉⡻⠄⠓⠚⠛⡓⣈⣉⣩⣥⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠀⠁⠀⠒⣌⠁⢿⢿⣦⡈⠲⠿⠗⠒⢛⣉⣉⣭⣤⣤⡶⠒⠒⣶⠶⠋⠉⠁⠉⠉⠁⡟⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠂⠂⠀⠀⠈⢣⡃⢈⢻⣿⣶⠚⡟⠉⠉⠉⠁⠀⠈⣿⠛⠟⡀⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣑⠛⢄⠀⠀⠀⠀⢀⣿⡿⠛⢀⠩⠅⠀⠀⢀⣀⣠⣷⠿⢶⠃⠀⠀⠀⠀⠀⠀⠀⡞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠳⡍⠲⠈⠻⣿⡇⠀⡸⠀⠂⠈⠉⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⢶⡀⣼⢹⠃⢀⠃⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠠⠴⠒⠐⠒⠚⠛⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⡤⠄⡤⠄⣀⠀⠈⠣⡀⠀⢀⠙⣧⡾⠈⢰⡦⠴⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢴⡏⠶
⠀⠀⠀⠀⠀⠀⠀⣄⡹⢷⣜⢓⣶⠟⢿⡇⣩⣉⡖⢢⠀⠈⢦⣄⣿⢩⡇⠆⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠶⣖⠿⢍⣻⠶⣬⣍⣳⡒⣤⡐⠢⠻⠅⠀⠀⠀⠈⠫⠿⠁⠐⠒⠋⠉⠁
⠀⠀⠀⠀⠐⠊⠉⠁⠈⠉⠉⠁⠉⠈⠁⢈⣁⢀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠄⣀⢀⣀⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡜⢫⣙⠛⠓⠒⠒⠒⠒⠒⠂⠠⠭⠭⠭⠭⠍⣉⣉⣀⣐⣒⡒⠒⠒⠒⠂⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠎⠀⡄⠉⠉⠉⠙⠛⠛⠒⠒⠒⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡝⡠⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠟⡐⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⢀⡞⠁⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡔⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⠤⠴⠶⠶⠒⢲⠿⠿⣷⣿⣖⣤⠄⢀⡄⢲⠀⡀⠀⠀⠀⠀⠐⠂⠠⠤⢀⣀⠀⠀⠀⠀⠀⢠⠎⠈⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⢸⡾⠽⡻⠶⣤⣄⣀⣀⢰⡧⣤⢞⣹⣶⠖⠠⢤⣉⣉⣛⡟⢡⣿⢿⣴⣷⣤⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠌⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⢹⡟⠚⢯⣽⣮⡿⢾⡒⢲⡌⣫⣭⣿⠂⠀⠀⠘⠚⡵⢿⢋⠛⠲⠮⠭⣭⣒⣒⡂⠤⢤⡤⢁⠊⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠱⣀⠀⠈⣹⣟⠛⢿⣠⣿⣹⣿⣿⣷⢮⣴⢴⣆⢙⣿⢹⣖⢲⣤⣤⣤⣈⣹⠿⠿⢶⡖⠷⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⡿⣷⠈⠀⠀⠉⠀⠈⠀⠈⠙⢣⣽⣶⣿⣯⣿⣓⡾⠬⠝⣿⣂⢹⡧⠄⠀⠒⠈⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠒⠀⠀⠀⢷⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠘⠃⣸⠥⠐⠒⠈⠉⠹⡇⠀⡇⠀⢀⡀⠠⠀⢺⡇
⠀⠀⠀⠀⠀⠀⣠⡏⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠊⠁⠀⠀⢀⣀⠤⠤⣿⢸⡇⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣽⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀⠀⠀⠀⠀⢙⠀⠀⠘⠈⠉⠀⠀⠀⠀⢿⣼⠃⠀⠀⠀⠀⠀⢨⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⡀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠁⠀⠤⢀⣘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⠙⠻⣷⣤⣀⠀⠀⠀⠀⠘⠀⠀⠘⠈⠁⠀⠀⠀⢀⡀⢄⣠⣾⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⠢⠽⢤⠆⠀⠉⠛⢿⣦⣄⡀⢠⠞⠢⡄⢀⣀⠤⣖⣮⠷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡇⠀⠀⠈⠙⣿⣿⣀⡇⣹⣯⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⣶⣟⣻⠧⠆⠉⠛⠉⠁⠀⠀⠀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠨⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠦⠤⠤⠤⠤⠤⠤⠶⠲⠦⢤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠶⠤⢤⣤⡤⠤⠤⡤⠤⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⡿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠈⠉⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⡀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠈⠁⠀⣧
⠀⠀⠀⠀⠀⠀⢸⠤⠤⢤⡤⠄⠠⠦⠤⡠⢄⡤⠀⣠⢤⠤⠤⠤⠤⠤⠤⠤⠬⠤⢤⣤⠤⠤⠤⠤⢤⡤⠤⢤⠤⡀⡤⠤⣤⠤⠥⠤⠤⠤⢤⡬⡤⡇
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠁⠉⠉⠉⠉⠉⠉⠁⠀⢸⠉⠉⠉⠉⠉⠉⢹⠉⢩⡍⠈⡏⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⠀⠘⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠂⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⢀⠀⠀⣀⢀⡄⣈⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⣃⣃⣀⣀⣠⣄⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣄⣀⣀⣘⣀⣂⣀⣀⣀⣠⣀⣀⣀⣀⡀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⣵⠶⠿⠛⠛⠛⠛⠛⠛⠷⢦⣍⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣩⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣵⠛⣁⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠙⣆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣭⣶⡿⠿⠿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣦⡀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⣿⣿⢻⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡝⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡇⠀⠀⠀⢹⣷⣼⣿⡇⠿⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⣛⡛⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⣭⠄⠀⠀⠀⠀⠀⠈⣿⣯⣹⡿⠚⠛⢳⣝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⢸⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡣⡎⢠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⠀⣶⣿⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡇⠈⣾⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡅⣿⠀⠉⠋⠈⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡸⠂⠈⢠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡆⢸⠀⠀⠀⠀⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠁⠀⢀⡾⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⣼⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣿⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⢀⡀⣿⠤⠤⠤⣤⠶⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠛⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⡀⢹⢸⣇⠀⠀⠀⠐⢏⠁⠀⠀⠀⠀⠀⠀⠀⠱⡌⠳⣄⠀⠀⠀⢀⠤⠤⠚⠛⠛⠛⠛⢻⣶⢽⡇
⠀⠀⠀⠀⠀⣠⠾⣫⣽⣶⣶⢮⣍⡻⣇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⢦⡔⠚⢻⡄⠀⠤⠤⠤⠤⠄⢸⣿⢸⡇
⠀⠀⠀⠀⣴⢫⡞⢡⠞⠀⠲⢀⡉⢻⣦⢠⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠳⣄⠀⠛⠆⣼⡄⠀⠀⠀⠀⠀⠀⢸⡿⣸
⠀⠀⠀⢰⡏⣾⠀⢈⠀⠀⠀⠀⢳⠀⣿⡌⣷⣌⢻⣆⡀⠀⠀⠀⠀⠀⠀⢶⡋⠁⠀⠀⠹⢦⡀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣾⣧⡝⣦
⠀⠀⠀⠸⣇⢿⠀⠘⣄⠀⠀⢀⡌⠀⣿⢣⡟⠈⣣⠝⣿⣦⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢷⠀⠙⠦⠤⣀⣀⣀⣀⣀⡤⠛⢀⣼⠟⣼⠁
⠀⠀⠀⠀⠹⣎⠷⣄⡈⣛⡿⠋⣡⡾⢋⡾⠁⢰⡋⣾⣿⣿⣿⣷⣦⣤⣄⣀⣀⣹⣦⠀⠸⣇⡀⠀⠀⠀⠉⠉⠉⠉⠀⢠⠏⠙⢷⡝⣧
⠀⠀⠀⠀⠀⢈⣷⠎⣙⣿⣰⣿⠻⡆⣿⠀⠀⠈⠛⠶⢮⣍⡛⢻⣿⣿⣿⣿⣿⣿⠙⠀⠸⣇⠙⠂⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢻⡌⢧
⠀⠀⠀⠀⠀⡟⣵⠟⠋⢙⡄⣟⠀⢿⡹⣄⠀⣀⣠⣤⠖⣫⣴⣿⣿⣿⣿⣿⣿⣿⣾⠄⠀⢹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡜⣧
⠀⠀⠀⠀⠀⣷⣸⡧⠚⢛⠄⢛⠀⠈⢻⡌⢛⣭⣷⣶⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠘⠃⠸⠂⠀⠀⠈⢿⡸⣇
⠀⠀⠀⠀⠀⢳⡙⣷⠞⢛⡆⢠⠀⠀⢸⣇⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡹⡆⠲⣀⠀⠈⠳⠶⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⢹⡀
⠀⠀⠀⠀⠀⢸⡄⢿⡤⠛⣹⡷⠂⠀⠀⢀⠉⠙⠻⠏⠀⠀⠉⠛⠿⠿⠟⠛⠁⢰⡆⠳⣄⣀⣀⡄⠈⡀⠀⠈⠉⠙⠓⠶⠶⡇⠀⠀⠀⣼⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠹⣮⡛⠶⠿⣿⠉⠐⠦⣼⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⢿⣿⣿⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⣿⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⡞⠁⣀⣠⠞⠁⣾⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⡆⠀⡴⠃⠀⠀⠀⠀⠀⣠⠟⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⡁⠀⠀⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⠿⠷⠶⣶⣶⣶⡖⠚⠁⣰⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⣿⣿⣿⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⡇⣾⣿⣿⣿⠀⠄⠀⠈⠁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⣋⣾⣿⣿⣿⣿⠐⠶⠀⠀⠀⣤⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⣾⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢩⣶⣿⣿⣿⣿⣿⣿⣅⣼⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠛⠀⠀⠐⠒⢛⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢿⣿⣿⣿⡿⢟⣭⢸⣇⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣛⣭⠶⠛⠳⣮⡻⢦⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣝⡓⠶⣤⣤⣄⣀⣀⡀⠀⠀⠙⠒⠀⠠⠤⠤⠤⠒⠊⠁⣀⣀⣠⣼⣿⣿⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠭⢭⣍⣛⣛⣛⣛⣛⣛⣒⣒⣚⣛⣛⣛⣛⣫⡭⢭⡜⣿⣿⠟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠻⣮⣡⠞⠁

⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣰⣀⠀⡀⠀⠀⠀⠀⢄⢀⢸⠀⠀⠀⠀⠀⠀⢰⣒⠄⠀⠀⠀⠀⢢⠄⢠
⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡟⢧⣸⡨⢻⡄⠀⠀⠤⠴⢮⣇⡾⣤⠂⠀⠀⠀⠀⠰⣿⠂⠀⠀⠀⠀⠨⢣⢪
⡀⡀⠀⠀⢀⠀⡀⢀⠀⠀⠈⠹⣏⡹⢿⢯⣹⠏⠁⠀⠀⠴⢲⡏⣷⠶⠶⠦⠀⣀⠀⠀⣿⠀⠀⡀⠀⠀⢀⢣⠣
⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠠⠛⠳⢾⠖⠛⠆⠀⠀⠀⠀⢸⠃⠈⠃⠀⠀⠀⠙⢦⣤⣿⣤⠾⠁⠀⠀⢰⡁⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⢀
⢀⣠⣴⣶⠿⠛⠃⠀⠀⢨⢸⠖⠒⠒⠒⣒⠰⠁⡇⠀⠀⢰⡄⠁⠀⠀⠑⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠋⠁⢹⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠛⠃⠀⡇⠀⠀⠀⠙⢦⣶⡾⠟⠋⣛⣢⡄⠀⠀⠀⢈⣇⢠⣾⣦⢀⣿
⠀⠀⣸⡅⠀⠀⠀⠀⠀⢸⢸⠀⠈⠀⠀⠀⠀⡤⡇⠀⠀⠀⠀⠐⠈⠙⣾⣿⣿⣿⡇⠀⠀⢰⣿⡉⣿⣗⣿⡇⣙
⠀⠐⠉⠉⠂⠀⠀⠀⠀⠘⠒⠂⠉⠀⠈⠉⠒⠒⠁⠀⠀⠀⠀⠈⠢⢀⣿⡿⠿⠛⠁⠀⠀⠀⠙⠛⠉⠛⠛⠙⠛

⢧⣤⣤⣤⠴⣶⣶⡄⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⢸⣿⣿⣷⡀⣿⣿⣿⠀⠀⠀⣀⣿⣏⡀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡻⣙⡟
⠀⣿⢿⣿⡵⠾⠿⠿⠆⠀⠀⣟⣛⣛⣻⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⢐⣻⣓⣻⣷⡀⠀⠀⠀⢀⡀⢽⣷⣿⢀
⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⡾⠀⠀⠐⢿⣿⣷⠀⠀⠀⢿⣷⣿⣭⣧⣭⣿⣿⣿⠟⠀⠘⠛⣳⡿⣗⡛
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⣈⣡⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⣄⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀
⢴⣿⣿⣿⣿⣿⠿⠟⠛⠁⠀⠀⢀⣴⣾⣿⡿⠿⠿⠿⠻⠇⠀⠀⢰⣿⣿⣿⣦⠀⠀⠀⠀⠻⢷⣤⣀⣠⣴⠾⠋
⠀⠈⠉⣵⣿⣿⣷⡄⠀⠀⠀⠀⢼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠀⠀⠀⠠⣄⣲⠊⠉⠉⠳⣖⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢤⣀⠰⢆⠶⣖⡒⠒⠒⠒⠛⢲⢫⠻⠤⣗⡝⡆⠀⠀⠉⠙⠺⢿⠋⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡊⠁⣀⣐⣇⠀⠙⠢⣄⠓⠛⠢⡀⠀⣀⣚⣮⣗⣞⣯⣝⣉⠀⠉⠈⠭⣭⣬⣥⣽⣥⡜⡆
⠀⠀⠀⠀⠀⠀⢰⠁⣡⠋⠀⠀⠈⠑⣟⢁⠜⡥⠭⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⠉⢩⠋⠉⠉⠀⠀⠀⢠⢊⠔
⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠀⠀⡴⣡⢫⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡰⢡⠊
⠀⠀⠀⠀⠀⠀⣷⠶⡇⠀⠀⢀⠞⡴⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢣⠀⣠⢋⠎⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡠⢃⠜
⠀⠀⠀⠀⠀⠀⠀⢣⡮⠒⡡⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡄⠤⡤⢗⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡔⠁⠆⡠⢴⠶⠒⠖⢲⡤⠍⢉⣠⠤⠤⣶⣒⣒⣢⣶⠴⠴⠗⠛⠯⠀⠖⢂⡀⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡝⠟⣳⡶⠾⠖⠛⠊⠐⠐⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⠤⠲⣏⡛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣬⣑⠢⣉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⢤⢀⣲⡒⠿⠍⠛⣒⠒⡉⠩⠰⢲⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⠑⠢⢍⡒⠦⣍⣓⣂⣐⡩⠭⠭⣑⠒⠒⢋⡭⢍⣴⠀⠶⠀⢫⠉⢩⠈⠋⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⢀⣼⣹⢷⠉⢳⠀⡇⠀⠀⣞⠋⠉⢩⡄⠿⢺⠀⣾⡇⠶⠀⠘⠙⢄⢹⡇⡦⠈⠀⢸⠀⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⢰⡟⢙⢻⡇⢸⠸⡽⠀⠀⣿⠀⠀⢸⡇⣀⣸⢀⡟⢀⣀⣾⣤⣦⢨⠏⡇⣀⢰⠀⢸⡔⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠳⣈⠙⣚⡼⠁⢸⢀⢧⡀⠀⢹⠀⠀⠘⡇⠉⢹⠺⣧⡜⣕⣻⢗⣎⢇⢰⡇⠉⢸⣀⠼⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠐⠌⠓⢦⡀⠀⢸⠈⢺⠁⠀⢸⠀⠀⢸⡇⠶⢸⠀⠀⣣⠫⣙⣫⢞⡼⠙⢁⣀⠤⠄⠐⠊⠉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡈⠢⡙⢦⣘⡀⣭⠆⠀⢸⠀⢀⣸⡇⠀⠘⠷⢓⣊⡵⠶⠖⠛⢒⣩⠤⠤⠒⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠢⢍⡁⢸⠀⠀⠸⠛⢛⡉⠥⠤⠒⢊⣁⣀⠀⢀⣂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠉⢸⠀⠒⠊⣉⡠⠤⠖⠒⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⠿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣿⡶⠶⠦⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣾⣿⣿⡝⢋⣉⣁⣠⣴⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠡
⠀⠀⠀⡀⠀⠚⠋⢀⠘⠷⠃⠰⣶⣶⣾⣿⣿⡿⢿⠿⠿⠛⠛⠉⠉⠁⠀⣿⣅⡿⣺⣿⡿⠿⢿⣿⡄
⠀⠠⠀⠀⢀⣀⠀⠈⠀⠀⢀⠀⡉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡃⠀⠀⠀⠁⣀⣀⣤⣤⣤⣴⣶⣶⠟⠋⠃
⠀⠁⠀⣰⣿⣿⣿⣶⡄⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣾⣿⡁⠀⢠⣄⠙⢿⣿⣿⣿⣿⣿⠋⠀⠂
⠀⠀⠸⠟⠿⢿⣿⣿⠃⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡤⠈⢿⡁⠈⢻⣿⣿⣿⠏⠀⠂
⠀⠀⣷⡆⠀⠘⠋⠛⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣯⣭⣿⣿⣿⣿⣿⣷⣄⠈⠳⣾⡄⠙⣿⠏
⡇⠀⣿⣷⡄⠀⠀⠆⠀⠀⠀⠿⢟⣛⣛⣭⣭⣽⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣄⣀⣀⣀⣤⡎⠀⠈
⠁⠀⣿⣿⣹⣧⡟⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣛⣿⣭⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢹⣿⣿⣿⠃⠘⠀⠀⣿⠿⣿⣟⣛⣻⣭⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣛⣛⣫⣭⣭⣿⡟
⠀⠀⢂⣽⣿⡟⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢟⣛⣋⣭⣭⣵⣶⣶⣿⣿⡿⠿⠿⠛⠛⠉⠁⠄
⠀⠀⠀⣹⣿⠃⠀⠀⢰⣿⠿⠿⢟⣛⣋⣭⣥⣶⣶⣶⣿⣿⣿⡿⠿⠟⠛⠛⢉⠉⠁⠀⠀⠒⠂⣀⣀⣤⡄⠀⠀⡀
⠀⠀⠀⢨⡇⠀⠀⠀⣶⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⢉⠉⠁⠤⠀⠀⠀⢀⣁⠠⡀⠀⠦⠘⠟⠛⢋⣉⣥⣴⣶⣤⣀⡁⠂
⠀⠀⠀⠈⠁⠀⠀⠰⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀⣴⡶⠶⠟⣛⣋⣭⣴⣴⣶⣶⠀⠀⠀⠈⠽⠀⠀⠈⠙⠛⠛⠷⢦⣄⡀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⣠⣤⣤⡶⠄⠒⠀⣀⣁⣠⣤⣴⣶⣿⠿⠛⠻⠃⠀⠀⠀⠀⠋⠐⠓⠘⡀⠀⡀⠀⣐⢂⣄⣄⠰⠛⠿⢷⣦⣄⡀⠒⠄⡀
⠀⠀⠀⠀⢁⡀⠲⠜⠛⢋⡉⠁⠀⠠⠶⢾⡿⠛⠋⠉⠛⡟⠉⢻⠀⠀⠀⠀⠠⠆⠠⠖⠁⠀⠂⠀⡉⠁⠀⠀⠘⡁⠀⠀⠀⢄⡤⡀⠉⠉⠉⣶⣤⡄⠁⠒
⠀⠀⠀⠀⠈⠀⢦⡀⠙⠻⣧⠀⠀⠀⠠⠈⠀⠀⢀⡠⠂⠓⠒⠄⠀⠀⠀⢠⡀⠀⠀⠀⠉⣛⠁⠀⠁⠐⢂⠀⠈⠃⠠⢤⠀⠈⠰⠤⠀⠈⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠫⢁⠀⠈⠁⠚⠉⣃⠤⢀⠀⢁⠶⠤⠀⠀⠒⣦⠀⠀⢀⡠⠤⠀⢀⣠⡥⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⣤⠁⠀⠀⠑⢁⠀⠀⠀⠀⠒⠀⠉⠉⠆⠆⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⣀⣀⣤⠔⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠉⢳⣦⣀⠸⠆⠀⠀⠀⠀⠁⠸⢷⠆⠀⠔⠚⠳⠶⠒⠀⠉⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠒⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⡀⢰⡆⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣭⣴⣶⣾⣿⣿⣿⡟⠛⣉⠹⣿⣿⢿⣿⣿⡿⢻⡟⠋⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠘⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣯⡀⠉⠀⠈⠙⠋⠁⢀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠇⠙⠿⣿⣿⣿⣿⠀⠙⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⢈⣷⣤⢸⣶⣤⣤⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣧⣤⣀⠀⡿⢿⡿⠀⠀⠂⠀⠀⠀⣿⣿⣿⣿⣿⡟⠛⠋⠀⠉⠉⠋⠁⠀⢀⠀⢸⢹⡿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⣿⣿⣿⣿⣄⠀⣱⠀⠀⠀⠀⠀⢰⣿⠟⠛⠉⠙⠃⠀⠀⠀⢀⣠⣴⣶⣿⡟⠱⠞⠀⠀⠛⠉⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄⣀⣠⣤⣶⣿⣶⣾⣿⣿⣿⣿⣿⣟⠁⠀⠈⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠛⠟⠀⠀⠿⠀⢀⠀⢀⡀⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡙⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠤⠀⠀⠀⠀⠈⠢⠄⠠⣄⠰⠒⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣷⣾⣷⡾⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣿⣀⣀⣤⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠶⠒⠒⠒⠒⠒⠒⠤⠤⠤⠤⠤⢤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠜⠉⠀⠀⠀⠀⢀⠖⡫⠭⠭⢕⡢⡄⠀⠀⠀⠀⠀⠀⠀⣉⠝⡫⢛⣛⣑⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣁⣘⣒⣒⣒⣒⣒⠒⠣⠾⠤⠬⠭⠤⠽⣈⣆⣀⣀⣀⣀⣀⡰⢣⢎⠴⡁⠀⠀⡈⢻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠐⠀⠀⠈⡈⢹⠉⠍⠉⠉⠁⠒⠒⢒⣒⠒⠒⠒⠂⠠⢎⢸⡀⠀⠀⠀⠈⠂⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠨⠄⡠⣔⠧⠀⠀⣀⡀⠀⣀⠀⠀⠀⢎⡡⠀⢀⠀⠈⠱⣝⢄⠀⠑⢄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⠣⠕⡛⠀⣠⠔⠒⠲⠊⠉⡀⠉⠣⠽⠧⣠⠾⠚⢰⣒⠎⠢⡑⣄⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡳⢆⡻⠉⢡⠅⠀⣄⡀⢀⠀⠀⠁⣰⠃⠞⢠⠗⡸⢇⠀⢀⣀⠈⢮⠣⣠⣿
⠀⠀⠀⠀⠀⢀⠄⠠⢀⡀⠀⡉⠀⢀⣀⠀⡠⣄⠤⠄⠀⠊⠀⣀⢀⡈⠳⠐⠒⠚⠀⠈⠁⠂⣄⣀⣀⣑⣤⠃
⠀⠀⠀⠂⠘⡬⠶⠦⠤⠬⠭⢥⣄⣀⣈⣁⠄⢡⣀⡀⠀⣀⠈⠣⠔⠓⠀⠀⠘⠃⢄⠄⢀⣩⠭⢓⣚⣭⡝
⠀⠀⠀⢹⣈⡉⠉⠉⠉⠉⠙⠒⠒⣖⢶⠘⠦⠴⠤⠍⣲⣭⠭⢍⣉⣙⢅⣜⡓⢲⠚⣩⠥⠒⠋⠁⠀⡆⡇
⠀⠀⠀⠈⡁⡨⢄⣀⡀⠀⢀⢀⠔⢆⠠⣛⠤⠤⢴⣺⢉⣅⠀⠀⠀⠥⠔⠀⡇⢸⠀⡇⠀⠀⠀⠠⢰⢹
⠀⠀⠀⠀⣇⢳⠸⠁⢱⠘⡇⠀⠃⠉⠀⠈⠉⠉⠉⠀⠒⠤⠃⢒⢉⠶⠀⠀⡇⢸⠀⡇⢀⣜⠐⣢⣸⠸
⠀⠀⠀⠀⢸⠸⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢠⠀⠂⡆⠄⠘⡟⡇
⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡇⢸⢸⠚⡄⠣⠀⠔⢡⠃
⠀⠀⠀⠀⠀⢣⢹⠐⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⡈⠖⢒⠾⣹
⠀⠀⠀⠀⠀⠸⡘⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢁⡤⢚⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠨⠭⠭⠭⠭⢉⣐⣒⣒⣒⣒⡒⠒⠢⠤⠤⠤⠤⠇⢸⢘⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣉⣀⣀⡀⠀⠀⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠋⠙⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠙⠛⠛⠉⠹⡟⠛⠛⠋⠁⢀⣾⣿⠟⢩⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⣠⣿⡿⠁⠀⣾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠚⢧⡀⠀⠰⣄⠀⠀⠀⠐⢲⡾⠉⠁⠀⠀⢠⣿⡿⠁⠀⣼⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⡄⠈⠳⡄⠀⢸⣿⣶⣤⣤⣾⣅⡀⠀⠀⠀⣾⣿⠁⢀⣼⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣹⣶⣾⠉⠁⣸⣿⣿⣿⣿⣷⣦⣄⣿⣿⢠⢞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠻⣿⡟⠧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣇⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⣀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⠟⠉⠀⠀⠀⠀⠀⣄⠙⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⠀⠲⣌⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢷⣄⣴⣿⣿⣋⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣤⠤⠤⣀⠀⠀⠀⠤⠄⠴⠒⢶⣏⠁⠺⡛⠻⣆⠐⠢⡀
⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠙⣆⠀⠙⡄⠈⠻⣧⡀⣈⣦⠀⠀⠀⢀⣄⡀⠀⠹⣧⠰⠘⡄⠹⣧⠀⠹⡆
⠀⠀⠀⠀⢰⠃⣠⡾⠋⠙⡄⠙⣇⠀⠸⡄⠀⠸⣷⠀⠈⣆⠀⣀⣿⠁⠉⣦⢀⠹⣧⠀⢹⡄⠹⣇⠀⢹⡄
⠀⠀⠀⠠⡇⢰⣿⠀⠀⠀⠸⡄⢽⡄⠀⢳⠀⠀⢻⣇⠲⢸⡞⢧⡀⠀⠶⠀⢰⠉⡾⡌⠁⢧⠀⣿⡄⠀⣇
⠀⠀⠀⠀⣅⡀⠉⠙⠷⣦⡄⣧⠈⡇⠀⠸⡆⠀⢸⢿⠀⠀⣇⣀⣷⠀⠀⠀⠸⠶⠶⠃⠀⠘⠋⠉⠁⢀⣹
⠀⠀⠀⠀⣧⣨⣑⠢⢄⡀⠈⠋⠀⣿⠆⠀⠳⠚⠚⠛⠀⠀⠉⠀⣴⣖⣰⣄⣽⡷⠤⠐⠒⠒⣲⠬⡍⢨⢇
⠀⠀⠀⠀⣭⢸⡏⠙⠲⢬⣑⠢⢄⣾⣀⣀⠤⠤⢤⣖⠒⠒⢢⡭⢽⡇⢰⣂⢲⣏⣉⣯⠘⠛⡄⠀⣿⠈⡇
⠀⠀⠀⠀⣋⢸⡇⣴⢚⢶⠈⢳⢀⡇⠀⠀⣞⠉⠉⣿⠐⠟⢸⣤⠞⡟⢁⡼⢸⣆⠹⣿⣦⡄⡇⠀⣿⠘⡏
⠀⠀⠀⠀⡟⢻⣿⡟⠛⣇⡇⢠⠹⡞⠀⠀⣿⠀⠀⣿⡇⣤⣼⠘⠺⡇⢾⡉⢉⠼⣧⢿⣇⣄⡇⠀⣿⠘⡝
⠀⠀⠀⢀⠇⠻⣷⣑⣖⡼⠀⣼⢸⢣⠀⠀⣿⠀⠀⣿⡇⠀⢸⠀⠸⣽⡿⠒⢲⣏⠀⠘⣧⠁⢧⠤⠟⢀⣳
⠀⠀⠀⠸⣄⣑⠪⡑⢭⡢⣀⣻⢈⣇⠀⠀⡟⠀⠀⣿⡟⠛⢸⣀⣠⢹⠁⠀⠈⣏⠄⠀⣼⠦⣒⣒⣭⣍⣈⡆
⠀⠀⠀⠀⠀⠈⠣⣈⠢⣉⠢⣿⠸⡞⠀⠀⣇⣀⣠⣿⠇⠀⠘⣛⠭⠿⢧⣔⣀⣤⠴⠚⠋⠉⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡑⠦⡀⡇⠀⣀⣉⠭⠤⠐⣒⣉⡭⠤⠶⢦⡤⠭⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⡇⠈⢀⠤⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠒⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣤⣅⢩⣿⣿⣿⣭⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡆
⠀⠀⠀⠀⠀⠚⠃⠉⠉⠉⠉⠀⠀⠀⠒⠀⠀⠀⢼⣿⣿⣿⣿⣮⡻⣿⣿⣵⡽⢃⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⡎⣅⢗⠀⠄⠀⠀⢠⣸
⠀⠀⠀⠀⠀⢰⡇⠀⠈⠁⢠⣃⣄⠄⢀⠄⠀⠀⠈⣽⣿⢿⠫⢠⡛⢉⣭⡿⢁⣘⠋⠀⠀⠀⠀⠀⠀⠀⣑⢀⡉⠚⣂⠀⡀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢠⡠⠀⠐⠛⠁⠀⠀⠀⠀⢸⣷⣎⣈⣷⣤⣳⣿⣾⣶⣯⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⠿⢿⡗⣿⡇⣿⣫⣿⣿⢾⣿⣇⣀⠀⠀⠤⠶⠬⣴⠖⠄⠀⠖⠒
⠀⠀⠠⠀⠀⠑⠀⠈⠍⠠⠐⠫⡷⠀⠀⠐⣷⢿⠍⠒⢧⠤⠖⢻⣷⣿⠿⡿⣯⡼⠉⢷⠞⠹⡶⠀⠠⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢁⠥⠀⢸⠮⠋⣀⡺⠉⣀⢘⣒⢈⠭⣀⡳⣊⣰⠊⠁⣨⣤⡊⠌⣀⡠⠀⡳⠲⠦
⠀⠀⠈⠴⠶⢾⡃⢢⣀⣀⣌⣠⢤⡄⠀⠐⠐⠉⠐⡚⡠⠊⠢⠘⠁⠀⠀⠀⠒⠁⠀⠀⠀⠀⠠⠉⢚⠛⠹⡏⠑⠚⢢⢠⡀⡇
⠀⠀⠈⠀⡀⣐⠓⠲⠉⡯⠱⣁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡫⢃⠸⢿⣗⡃⣤⡤⢈⡀⠁⢁⠂⠤
⠀⠀⠀⡺⠇⣒⠀⠀⠌⠓⣿⠟⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⠃⡆⢘⣒⣤⡋⡁⠳⢄⠑⡺⡎⢀⣄⠂
⠀⠀⠀⠀⠀⢁⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠈⠂⠈⠠⠒⠊⠑⠒
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⡅⠱⠀⠀⠀⠀⠀⠀⠙⢟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡄⠓⠂⣀⠀⠀⠀⠀⠀⢷⠨⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠤⣿⡈⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠋⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢈⡬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⠂⠀⢠⠀⠀⠰⣿⠀⠖⠀⠀⠀⢠⠀⠀⠀⣼⠀⠀⣁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⡾⠄⢀⠇⠀⠐⢰⡇⣀⡀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⢩⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⠛⠛⠓⠒⠚⠓⠒⠃⣼⠀⡍⠀⠀⠀⢀⡇⠀⡌⢸⡇⠀⠁⠸⠂⢻
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣭⣵⣶⣅⣀⣤⣨⡥⠴⠖⠚⠋⠻⣿⣿⣿⣿⡿⢾⣷⣶⣷⣿⣯⡉⠝⠀⢡⢸⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡟⣿⣿⣂⠀⠀⠀⠀⠀⠈⢻⣷⣯⠿⠚⠻⣾⣿⢿⣿⣿⣦⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠻⣇⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣌⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣼⣿⣿⣿⣿⣿⣿⡟⠁⠉⠛⠶⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠉⢉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣦⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣰⣿⠿⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⢀⡠⠴⠒⠉⠁⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢰⡇⠀⠀⠀⢀⢀⣀⣀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⣠⢤⣿⣿⣿⣿⣿⡷⣤
⠀⠀⢀⣀⣤⣤⣤⠀⢀⣾⠏⠀⠀⠀⠀⠀⠳⢤⣴⣾⣿⡿⣿⣿⡇⡜⣈⡧⠀⠀⣀⣤⡖⠋⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠡⡉⠻⡿⠴⣊⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡝⡓⢸⣿⣿⣭⢤⣴⣺⡽⠛⠉⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢳⡀⣠⢞⠟⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠷⠃⣏⠉⠉⠉⢩⣾⠉⠁⠀⠀⣠⣾⣿⡿⡿⠿⠿⢿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⡴⢻⣿⣿⠟⠁⠀⠀⠘⡄⠀⠀⣾⡏⠀⠀⠀⠘⠋⠉⠉⠀⠀⠠⢀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡿⠛⠁⠀⠀⠀⠀⠀⣷⠀⠴⢿⣇⠀⠀⠔⢉⢀⣦⣔⠶⠋⠚⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⡄⢆⠤⠊⠈⠀⢠⡄⢀⠆⡀⢐⣤⣤⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠜⠀⡏⠐⠘⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣦⣴⣶⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⢿⠻⣿⣿⣿⣿⣿⣿⡿⡇⠸⢿⣿⣿⣿⣿⣿⣿⢿⡿⢧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⡈⠉⠛⠛⠛⠋⠀⡇⢸⠂⠉⠛⠛⠛⠉⢁⢨⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡙⠇⠀⠀⠀⠀⢶⣀⠃⠈⣀⡶⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣹⣟⣟⣉⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠙⠛⠛⠻⠟⠛⠛⠃⠘⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢦⡀⠀⠀⠈⢹⠿⠿⠋⠁⠀⠀⣠⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢮⣿⣦⣆⡀⠀⠀⠀⠀⢰⣦⢞⠕⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠿⢿⣶⣶⠶⠞⠫⠐⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣷⣻⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡏⢷⡀⠀⠀⢀⣾⣿⣿⣿⣻⣆⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠾⠿⠿⠿⠿⡷⠛⠂⠰⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡠⠤⠔⠒⠒⠘⠳⠶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠚⠉⠉⢉⡙⠛⢿⣷⣶⣍⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⠛⠿⣿⣿⣶⣤⣀⠀⠀⠂⠈⠀⠀⠀⠀⠈⠙⣿⣧⡀⠀⠀⠀⠀⠈⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠠⢆⠘⠀⠒⠂⠬⢍⡻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⣀⡀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡰⠁⠀⠈⢳⡀⠀⠀⠀⠀⠙⢢⡻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣷⣒⡋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠁⠠⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣤⣶⣶⣿⣿⢿⢿⣿⣿⢻⡉⡀⠿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣈⡁⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⡿⣿⣷⠟⡛⣫⡟⣥⣴⣽⣯⡏⣴⣶⠅⠰⠿⢉⣛⣉⢻
⠀⠀⠀⠀⠀⠀⠀⢁⣻⣟⡛⠿⣿⡟⠏⣻⠟⠟⣙⡻⠿⠛⢋⡑⠉⣥⣎⠀⣭⡝⠒⢁⣿⣿⣷⣶⣾⠿⠛⠋⠉⠀⢰⢼
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡻⠿⣟⣷⣾⣽⡿⡈⠭⢑⠀⢤⠘⠓⠺⢉⣘⣿⣶⣾⢵⠿⠛⣻⣿⣿⡝⣿⠀⠀⠀⠀⠀⢨⣼
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢇⢈⠿⠿⢿⣿⡱⣶⣤⣀⡉⣘⣈⠷⠒⡿⠉⠀⣿⠀⠀⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠂⣈⠁⠀⠀⠀⠹⢧⡛⠻⠿⣿⣿⡼⡧⢼⠁⠀⠀⣠⡀⠀⠙⢻⣿⡟⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠛⢿⠀⠀⠀⠀⠈⠻⣍⣱⣛⣻⣘⡓⠚⠀⡀⠤⢿⠃⠁⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⡀⠀⣿⠀⠀⠀⠀⢾⣷⠈⠁⣿⣿⣽⡇⠉⠀⠀⠀⠘⠁⠀⠀⢸⣿⡇⠐⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠈⠿⢶⣿⣦⣀⣰⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⢰⠄⠀⠀⢸⣿⣿⠉⠀⠀⠀⠀⠀⢸⣿⣹⠂⠄
⠀⠀⠀⠀⠀⠠⢰⠼⣿⡀⠀⠀⠀⠈⠛⠋⠙⠓⠃⠀⠀⢸⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣀⠤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠐⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠐⠈⠁⠄⠀⠠⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠲⢤⣀⡀⠀⢸⣿⣷⡇⠀⠀⠀⣀⡠⠔⠒⠉⠀⠀⢀⣠⣤⢌⡋⠁⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠾⠿⠿⠇⠒⢊⠉⠤⠤⡀⠀⠀⠀⠤⠘⠣⠽⠿⢿⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⠭⠤⠎⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡟⡟⣛⢻⣛⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠶⢚⡝⠛⠦⣤⣄⣀⣠⠤⢤⣄⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⣞⣉⣀⠉⣡⠋⠉⢹⠓⠒⠂⠀⠤⠐⠡⡾⠉⢀⣉⣉⣉⣉⣏⣩⠟⠛⠒⠒⠒⠶⢲⠚⠉⡓⢦⡀
⠀⠀⠀⠀⠈⢦⡲⠿⠿⠿⣭⠭⠿⠷⢶⣖⣲⣷⣶⣦⡦⠤⠤⣤⣤⣈⣀⣀⣩⡉⠉⠀⠐⠒⠚⠒⠊⠥⢁⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠙⠮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠯⠭⠭⠿⣷⣒⡒⢦⠊⢳⡜⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠮⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡱⡀⠑⡄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⢆⠈⢦⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠆⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠈⢢⡳⡀⠱⡀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣻⣶⠂⠉⠑⠒⣺⣋⠁⠲⠦⢤⣐⡤⠤⣀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠑⣝⢄⠈⢷⣼
⠀⠀⠀⡀⠀⠀⢀⣀⡤⠶⣯⡝⠟⠉⠅⠉⠄⠀⠀⠀⡡⠀⣱⠁⠀⠀⠁⢀⠀⠈⠂⠁⠀⠘⣄⣼⣠⣄⣤⢂⠐⠐⠈⢮⢣⣠⠇
⠀⠀⠀⣬⠴⠚⣽⣏⣰⠉⠙⡆⠀⠀⠀⣄⡴⠢⠼⠃⠀⠀⠀⠳⣄⠀⠀⣸⠁⠈⠀⢀⣀⠴⠛⠙⠛⢽⣍⡉⠓⣲⠒⠢⣴⡏
⠀⠀⠀⣇⠉⠀⠒⠒⠠⠍⢙⣓⠓⠠⠤⣮⣀⣀⠀⠑⠄⠄⠀⣤⣉⡙⢛⡈⠶⣤⣀⠈⠛⢤⡀⠀⣀⣈⣯⠴⢛⠡⠒⠉⣠⡇
⠀⠀⠀⣟⠓⢲⡶⠤⠤⣀⣀⠀⠉⠉⠛⢳⢾⠍⡏⣙⣛⡶⠶⠧⢤⣬⣣⣀⣀⠈⠏⠄⠀⢀⣹⡿⠿⣋⠕⠊⢁⡠⢴⣟⢡⡇
⠀⠀⠀⢿⠊⠉⢁⠀⠀⠀⠀⠉⢟⠑⢂⣎⠙⠂⠓⠼⠂⠉⠉⠑⠒⠒⠠⠭⢉⣉⣛⠓⠓⠛⡩⠖⠈⢀⡠⠖⠉⡀⢨⡏⢹⡇
⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇⢙⠈⢆⠀⠴⡸⠏⠐⠒⣦⠤⠤⣀⣀⡀⠀⠀⠈⠙⠋⢀⣠⡖⠉⠀⠀⠸⣤⢻⡇⢸⡇
⠂⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠜⠁⠘⠃⠈⡅⢴⠟⠀⠀⠀⠈⠫⡙⠓⠲⢶⡶⡋⣼⢱⢇⡄⠀⢀⡞⣼⡁⢸⠃
⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠞⠂⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠃⠤⠞⠣⠔⣻⢸⡘⠤⠤⡪⠊⣐⠁⢸⡀
⠀⠀⠀⣸⣄⣀⣐⡀⠀⠀⠀⠀⠀⣥⢤⡆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠉⠈⠁⠀⠀⠠⣂⣸⡀
⠀⠀⠀⣿⡀⠉⠙⠱⣤⣀⡀⠀⡞⣁⣸⠣⡀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠚⠟⣩⠇⢀
⠀⠀⠀⠀⠉⠙⡲⠞⠁⠀⠈⠉⠳⢦⣘⡀⢄⡀⠀⢀⠞⢁⠘⢃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡷⠀⠀⠀⣠⠴⠛⠒⢻⠉⠀⠀⠈⠢
⠠⡀⠀⠀⢀⣤⠦⣀⠀⢀⢀⠆⠀⠀⠀⠉⠉⠉⠉⠛⠮⣄⣘⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⡤⠊⠁⠀⠀⠀⠈⢄⣀⠀⢀⡠
⠀⠈⠈⠈⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠲⢤⣼⠀⢀⠄⢠⢀⢻⠉⠀⠀⠐⠂⠀⡀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢄⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣀⡀⠀⠀⠎⢀⡼⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢯⣍⣉⠙⠩⡓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠛⠶⣶⣀⠉⠻⣷⡄⢈⣦⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⠋⡉⠙⠻⣦⣠⠤⢬⡻⣿⡀⠜⢛⣩⠴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⢭⣄⣀⠈⠙⢧⣄⡀⢸⡁⢠⣄⠹⡥⠶⣫⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠷⣽⡻⣦⡈⢿⣷⢀⣠⡿⢟⣫⢿⣄⣥⡔⡑⠋⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡡⡶⢿⡿⢿⣻⣟⣻⡤⢟⣻⣵⣾⠿⠾⠿⠛⢓⣤⣽⢲⠾⢷⡎⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡁⠹⣦⠙⣶⡟⣹⢇⣶⠟⢇⡁⡠⢧⣄⣲⡧⣾⠋⠉⢣⣀⣾⣏⡉⣸⡵⢂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠙⣆⣸⡿⢫⠞⢁⠀⠵⣠⢼⡍⠤⠊⣽⡏⢁⡛⠦⢴⡿⣏⣸⣶⣺⠿⣻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡾⢋⡴⢉⢒⣨⣿⡻⠉⠈⡿⣶⡖⣻⣿⡢⠙⢢⣼⠷⠒⠛⠉⠀⢘⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣯⣼⣩⠟⢿⢿⣋⣘⣿⡻⠛⢓⢝⠉⢑⣾⣏⣧⣤⣴⣶⣶⣿⢿⣻⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⠿⠿⢿⣦⣇⣈⡟⠉⢳⠊⠙⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⡄⡇⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣦⣤⡀⠈⠉⣷⠊⢉⢳⣧⢤⣾⡋⠻⣿⣿⣿⡿⢻⣇⠘⡇⣧⡇⣴⠁⡴⢾⡃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢘⢿⣿⣿⣷⣶⣾⠓⠀⠠⠧⣀⡸⡇⠈⣿⠛⠙⠀⠀⣿⣤⣷⣧⡗⢰⣘⠀⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⠁⠛⢿⣿⣿⣿⢼⡗⡦⢄⡤⠃⡅⠀⢻⣠⣴⠶⣿⣿⡿⠋⢩⡟⠉⣨⣤⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠍⠉⢛⣿⣦⣇⡈⡟⢿⣿⢸⡇⡇⢸⣀⣠⠵⢞⣫⣿⣿⢿⣉⡟⠉⢢⣏⠉⠉⠉⠀⠀⠈⠻⣎⡶⢀⠀⢰⣶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣋⠀⠁⠘⢻⣿⣿⣿⣯⣗⣾⣻⣞⢷⣛⣫⡵⢖⣻⣟⢛⡤⠐⣯⡀⣸⠷⣾⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣅⣬⡟⢿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⢾⣿⡼⠶⢧⠁⠿⣿⣃⡇⠀⡝⠿⠿⠟⣿⠋⠉⣏⠛⣿⠋⠘⠦⠲⡇⢈⠁⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢆⣹⡞⡁
⠀⠀⠀⠀⠀⠀⠀⣦⣴⡫⠀⢹⡆⠐⠊⢨⡿⠟⠛⠉⠙⠨⠐⠀⠒⠈⠀⠀⠿⡾⣿⢠⣾⡒⢠⣧⢀⠀⠀⠀⠀⠙⢿⠤⢄⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣿⠍⠀⠀⠈⣷⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣙⢿⣦⡷⢞⠧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡷⢀⠀⠀⠀⠀⠀⠈⣷⣆
⠀⠀⠀⠀⠀⣿⣭⠅⠀⠀⠀⠀⣹⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠦⣴⡆⠀⠀⠁⠀⠀⢠⠒⡤⠃⢀⡀⠀⠀⠀⠀⠀⠈⠛⠛⢆⠀⠀⠀⠀⠀⠀⠹⢧⡀
⠀⠀⠀⠀⢂⠿⠁⠀⠀⠀⢀⡐⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⡀⢾⡹⠃⠙⠾⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠈⡠⠁⠀⠀⢀⣤⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠛⠻⠮⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⡼⠶⣶⣴⣎⣀
⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠈⠱⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⡤⠤⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡻⠟⠛⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠲⠶⠶⠶⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠁⠒⠒⠒⠒⠒⠀⠤⠤⣤⠴⠖⠒⠒⠒⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⣳⠶⢒⣶⠶⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣡⠴⠛⣉⣉⣉⠓⢦⡉⠻⡍⠁⠉⠉⠉⠉⠉⠁⠐⠒⠒⠒⣶⠋⣡⠞⢉⡤⠶⢶⠲⠦⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣃⣀⣈⠙⠛⠛⠛⠓⠒⠶⠶⠶⠾⠧⠴⠧⠤⢤⣤⣭⣉⣀⣀⣿⣀⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⢁⡼⡁⠀⠀⠈⠳⡀⠈⠳⡜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠉⠻⡉⠛⠛⠛⢳⡒⠒⢲⡶⠶⠶⠶⠶⠤⠤⠤⣤⣤⣤⣭⣭⣉⣉⣉⣉⣙⣛⣛⠒⠚⠀⡼⠁⡾⠀⠈⢢⡀⠀⠀⠈⢦⡀⢹⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠹⡄⠀⠀⠀⢱⡀⢸⠉⠉⠁⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠄⣀⣀⣉⣉⣉⣉⡉⠉⠉⠳⡌⠳⣄⠀⠀⠙⢄⠀⠀⠀⠑⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠘⣄⠤⢄⣀⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡦⠀⠀⠀⣉⡉⠉⠙⢦⡘⢦⡀⠀⠀⠑⢄⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠠⣦⢤⠴⡎⢹⡼⠁⠉⠉⠉⠉⣁⡒⠒⣂⠶⠶⠶⠶⠤⣤⠤⠭⣾⡁⣀⣀⣺⣀⣹⠀⢀⣀⠙⢦⠙⢦⠀⠀⠈⠳⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢙⠷⢚⣉⡑⠢⡄⣠⠚⠉⠉⠉⠙⠁⠀⢀⣀⡀⠀⠘⠓⣺⢉⡷⣤⠞⡿⠁⠀⣹⣇⣈⠟⠈⠳⣌⠳⣄⠀⠀⠈⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣸⢠⣿⡝⠋⠳⣘⡃⠀⠀⣀⠀⣀⡀⠀⠉⠉⠉⠓⠀⣰⠃⢸⡗⠁⣼⡥⠖⣻⠋⠀⠀⠀⠀⠀⠈⢳⡈⠳⡄⠀⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠲⠤⣄⡤⠏⠘⠚⠲⢄⠀⠐⠋⠁⠀⠘⡓⠤⢄⣈⡆⠀⠀⠀⠐⠺⢥⡀⠈⠀⠀⠋⢀⡞⠁⠙⣆⠒⠒⠒⠒⠒⠠⠽⢦⡙⠶⢣⠏
⠀⠀⠀⠀⢀⠤⠀⠴⠁⣠⠀⠀⠠⡀⠀⠀⣀⣀⢀⠔⠀⣂⣦⣀⣤⠤⠄⠁⠒⠛⠀⠀⢀⣀⣀⡀⠀⢳⠒⠋⠉⠛⠛⠆⠀⠉⠉⠉⠙⢢⣀⣀⣀⣀⣀⣙⣤⡞
⡠⠔⠴⣖⣓⣤⢠⣀⠀⠉⠙⠛⠒⠟⠀⠀⠀⠈⠛⢶⠋⠉⠞⠉⠀⠀⠀⠀⠀⠐⠒⠶⣝⠉⢀⣿⣄⠀⠀⠀⠀⠮⣉⠑⠢⡀⠀⠤⠖⠚⠓⣦⣤⡤⠟⠛⢹
⢇⡀⣀⣭⣉⣉⣉⣉⣉⡙⠛⠓⠒⠒⠒⠒⠶⠶⠶⠤⢴⠋⠈⢳⣬⣦⣀⡔⠐⣖⣢⣀⣀⣛⠁⠀⠀⡠⢄⣀⡀⠑⠨⠉⣲⣃⣠⡤⠖⠚⢉⣁⡤⠴⢺⡇⡿
⠸⡏⠉⡣⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⣶⠒⣶⠚⠙⡦⠤⠾⠤⢤⡖⠙⣞⣁⣉⣉⣉⣉⡉⠉⢻⣄⢀⠟⠛⠒⢲⡞⠋⢉⣠⡤⠖⠛⠉⠁⠀⠀⢸⢀⡇
⠀⣿⠋⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⢳⡞⠁⠀⠀⠀⢀⣵⠞⠳⡏⠀⠀⠉⠉⠉⢹⠁⠈⡟⠛⢳⠀⢸⡇⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃
⠀⢹⣄⣠⡼⠢⣄⣀⣀⢀⣀⣀⡤⠤⣜⠉⢱⠦⠴⢧⣌⣙⠒⠒⣚⣉⡼⢤⡤⠧⣄⠀⠀⠀⢠⠞⠓⡎⠀⠀⢸⠀⢸⡇⢸⡇⠀⠀⠀⣀⡠⠔⠊⢉⡇⣾
⠀⠀⣇⠸⡧⠤⡏⠀⠉⡦⠒⢿⡀⠀⣸⠋⠉⠉⠉⠀⠚⠛⠛⠛⠋⠠⠤⠴⢇⣀⡼⠒⢤⠊⠉⢦⣔⢃⠀⠀⣸⠀⢸⡇⢸⣇⠤⠒⣩⠕⠒⠲⡄⢸⠃⡏
⠀⠀⢹⠀⣷⠀⠑⠤⠒⠓⠤⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠜⠓⠒⠋⠈⠉⠉⠉⣿⠀⢸⠀⢸⠁⢀⡞⣹⠂⠄⠺⣝⢿⣸⠃
⠀⠀⠘⡇⢸⡄⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⠀⡜⠀⣧⠴⠒⠉⢸⡇⣿
⠀⠀⠀⢷⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣿⠀⢸⠀⣸⠴⡇⠀⢇⠀⠀⢀⡼⠁⡿
⠀⠀⠀⢸⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⣿⠀⠹⣄⠈⠑⠚⠉⣀⢼⠇
⠀⠀⠀⠈⣇⢸⡏⠀⠒⠒⠒⠒⠒⠀⠤⠤⠤⠤⠄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⣿⠀⠀⡨⠟⠒⢚⣫⠏⣸
⠀⠀⠀⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠐⠒⠒⠒⠒⠒⠀⠤⠤⠤⡇⠀⣸⠀⣿⠔⠉⢀⣠⠖⢋⣠⠖⠋
⠀⠀⠀⠀⠸⡆⠹⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⣇⣠⠖⢋⣠⠖⠋
⠀⠀⠀⠀⠀⠓⠒⠲⠶⠶⠤⠤⠤⢤⣤⣭⣉⣉⣉⣉⣉⣙⡛⠛⠓⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⣤⣤⣀⣀⣀⡇⠀⣿⠀⠋⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠒⠒⠒⠲⠶⠦⠤⠤⠤⢤⣤⣀⣀⣀⣿⠴⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⣂⣨⡭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢒⣀⣤⣴⣾⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⣊⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣺⡏⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣏⢿⣿⣿⣿⣎⢿⣍⠉⡉⠹⠟⢡⠙⠙⠉⢈⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠸⣿⡄⣿⣷⣿⢿⡿⠛⠃⠀⡔⢁⣐⣀⡴⠞⠋⣀⠆⠌⡠⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠁⣸⡇⣿⣿⣿⣿⠟⠀⣤⠴⢙⡋⢀⣹⡀⢀⠁⠀⠒⢋⣁⣀⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠃⠿⠋⠉⠱⣤⠴⠷⠖⠛⡽⠟⠁⠙⠍⠧⣤⡴⠚⠛⢡⣤⣶⠷⢈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣄⠲⣤⠤⣄⣀⣠⡶⠋⠉⣦⠴⢎⣠⡴⢻⠷⠶⠒⢛⣠⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡈⠲⠀⠉⠙⣶⡄⠀⠱⠶⠚⠤⠄⣛⣠⣴⣻⡿⢟⣿⣋⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣦⡘⠞⠛⣉⣰⡴⠖⣀⣤⣴⣾⣿⡞⢻⡀⠃⣴⢿⣿⢻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠻⣿⣦⣉⣩⣴⣾⣿⣿⣿⣏⣏⣵⣷⣿⠿⠀⠀⠀⠹⠞⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢛⠉⢠⣼⣻⡇⢿⣿⣿⢹⣿⣿⠻⠿⠛⠟⠿⠅⠀⣀⣤⣶⡴⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣄⣮⠢⡀⠿⣿⡇⢸⣿⣿⣿⢹⡟⠀⠰⠀⣉⣤⣶⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⣿⣷⣌⠢⡑⣧⢸⣿⣫⡿⠞⣋⣤⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠿⣿⣿⣿⣷⣝⣿⢘⡋⣥⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠩⠟⠿⣿⣧⡻⠄⠀⠀⠁⠀⠀⠰⣄⡠⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⢽⡗⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠐⠾⠤⣄⣀⡀⠀⠀⠀⡀⠀⠀⠁⢀⡰⠲⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠐⠲⡤⠤⢄⣀⠀⠀⠠⠍⠁⠀⠀⠄⠀⠐⢀⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠑⡒⠠⠄⣀⣀⡀⠒⠀⠘⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⠘⠈⡈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡧⡀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠇⡲⡯⣿⣢⢌⡃⠰⢶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠄⢀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⡀⣠⡿⢸⠙⢍⠰⣩⢓⠸⣕⣛⢛⢳⣻⡏⢓⢠⣄⡀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⡸
⠀⠀⢠⣄⠲⠈⠀⢡⠤⣧⡽⠷⠤⡣⡭⠙⢷⠀⢩⡄⢚⢽⢒⡯⣀⠂⠨⢦⣀⠀⣠⡤⢀⡀⢳⠀⠐⠇
⠀⠀⢸⠃⣀⠀⠘⠧⠴⢷⡃⣳⣒⡶⡾⡶⣻⣗⠙⣤⢋⣤⡮⣧⢒⠴⠷⠿⣓⣿⡁⠁⠘⣻⠐⢣⠰⠘
⠀⠀⢸⠀⡇⠀⠁⠀⠮⢁⠉⠒⠛⢴⣧⡖⠲⡆⠋⡝⠼⡴⠲⣇⣸⣒⣳⢞⡩⠃⢸⡒⠾⣕⠿⢪⠖
⠀⠀⠸⠀⠃⠀⠀⠀⡆⠘⢛⠴⠄⣀⣰⣿⣿⢻⠟⣜⣓⣿⡤⠼⣥⣉⣩⣧⣴⠤⡶⠋⠁⠀⣀⠸
⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠫⣤⣶⣿⠟⠙⠋⠈⢡⠼⣍⣠⡗⢲⣉⣉⠧⠚⠂⠈⡅⠀⠀⠀⢗⢠
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⣉⠒⠀⡛⠋⡎⡁⠔⣋⠁⠀⠆⠀⠀⠀⡷⡎
⠀⠀⠀⠉⠄⠀⠀⠀⠄⠀⠀⠸⣾⠀⠀⠀⠀⠀⠀⠠⢦⠀⠀⠁⠀⠀⠀⡖⡂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠄⠀⣷⣳⡀⠀⠀⠀⠀⠈⣋⡂⠀⠀⢀⠀⢰⢂⠂⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡤⠠⢀⢻⣦⠄⠀⠀⠀⠀⠡⠚⢆⠰⠸⠀⠸⠈⠀⠀⢀⡠⠴⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢥⠘⠒⣈⣾⢾⡗⢀⣠⣄⠀⠀⠓⢊⠂⢦⠤⠼⠴⡤⣾⠛⢵⣿⡾⠿⣟⡸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣉⣫⡟⠘⢶⠞⢰⣈⡩⠔⢃⡿⢸⣍⢼⣀⣽⡪⠙⣶⣓⠶⢋⠹⠏⠵⠤⢖⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢘⣡⠌⣉⡭⣱⣛⡤⢽⡶⠛⡇⠒⣷⠞⠟⢻⢗⠂⢺⠈⣖⠚⣲⡜⠓⠖⠦⢤⠇⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣉⡰⠊⣱⣉⠒⢐⣓⠶⣤⡬⡗⠂⡼⠠⣴⠛⠻⠤⠾⣽⠛⡍⣟⡿⡀⢐⡓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠛⠊⠑⠩⢋⠥⢤⠓⢰⣿⣹⣇⣀⠶⡎⠛⡻⣄⣩⣀⠽⢧⣈⡛⠃⡯⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠦⠔⠉⠁⠈⠙⣿⣯⣤⣿⡹⠟⠓⠤⠂⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⡻⢿⢻⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⣻⡿⠞⢛⠉⠉⠉⠉⠛⠻⢿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⠟⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠓⢌⠳⣽⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠣⠙⣟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠇⡇⢹⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠰⠘⠀⠘⢰⠀⡎⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢏⡎⠁⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠺⣷⡀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠚⣡⠞⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢿⣦⡈⠒⠆⠀⠀⠀⠀⠀⡴⠏⢁⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⠟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡊⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠈⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣇⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣻⠦⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⡍⠇⠨⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣽⣬⣿⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⡀⢀⡀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⡐⠐⠆⠀⠀⠀⠉⠿⢢⡀⠀⠀⠳⡤⠈⠀⠻⢦⡉⠀⠀⠀⠉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⠖⠾⣦⠀⠈⢠⣀⣤⣠⣤⣤⣽⣶⣶⣾⣇⠄⠉⢶⠶⠿⠿⠿⢿⣿⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢃⡼⠃⠄⠉⠉⠁⠀⠀⠀⠀⠈⡀⢀⡉⠓⠦⠄⠀⠀⠀⠀⣴⠟⣵⠟
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠀⠈⠀⠀⠀⠀⡀⠀⢀⠆⠀⠃⠠⠀⢀⠈⠀⠀⠀⠀⢸⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠠⠄⢻⡄⠀⠈⠀⠀⠀⠀⠠⠾⠀⠀⠀⠙⠀⠐⠂⠀⠉⠐⠲⠁⡀⠀⠀⠉⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠈⢿⣾⣾⠴⠌⠋⠃⠀⠄⠊⠀⠂⠀⠐⠈⠀⠀⠀⠀⠀⡇⡀⠀⡤⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⡿⢁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⠉⠵⡀⠐⡂⠀⠀⠘⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣴⡏⢼⢃⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠒⢀⣀⡴⡂⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⢢⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡏⠉⠫⢍⠋⠄⢢⢠⠀⠀⠀⠂⠀⠀⠘⠈⠋⠁⢀⣠⡁⡀⠀⠀⢀⡁⢠⠐⠁⠀⠀⠰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣅⣾⡗⠀⠀⠀⠈⠒⢄⣘⣤⣨⡄⠀⠀⠀⢠⡄⠀⠀⠀⠈⣷⡞⠌⠀⢸⡇⠈⠀⠀⠀⠈⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡅⣿⣷⡻⣀⠀⠀⠉⢺⡏⠀⢤⠀⠀⠀⠀⢸⡇⠀⡀⠀⢠⡌⡀⠠⠂⣸⠇⢀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢲⢿⡏⠉⢚⢵⡶⠠⣼⠋⠀⣤⠀⠀⠀⠀⣾⡇⠀⠇⢀⠀⠈⢁⠀⠀⢻⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢺⣇⠀⠀⠈⠁⠀⡿⠐⠀⠄⠀⠀⠀⠀⢹⠁⠀⠀⢸⢈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⡷⠀⡀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⠺⣤⠤⠀⠰⣿⠀⡐⠀⠀⠈⠈⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢻⣙⠫⣄⠀⠈⠐⠦⣧⠀⠀⡀⠀⠀⠀⠀⢿⠄⢰⠀⠀⠀⠀⠀⠀⠀⢘⢀⣀⣤⣤⡤⢶⣘⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣧⣀⠀⢀⣟⡇⢁⠇⠀⠀⠀⠀⢺⠀⢀⢀⠀⡀⣀⣠⡤⠴⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣿⡇⢀⣆⣤⣤⡶⠦⠴⠆⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠄⠀⠋⠁

⣠⠖⠛⢉⣉⣉⣉⠉⠉⠉⠉⢳⢀⠋⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠙⠛⢳⡄⡴⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⣄⡀
⡟⢤⣸⣁⣀⣀⣀⡽⡆⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⣾⣉⣉⣉⣱⣠⠽
⢧⠀⠀⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠸⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸
⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⣄⠀⠀⠀⠀⢀⣠⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠑⠒⠶⠦⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠒⠒⠛⠦⣤⣄⣤⡤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣶⡷⠶⢶⣶⣦⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⣿⣿⠓⠒⠀⠀⠈⠉⠛⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠃⠀⠨⠽⢶⡆⣤⠠⠿⠿⠿⢓⣴⡮⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠬⠇⠂⠀⠀⠀⠠⡫⠋⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡞⠉⠁⢿⣶⠀⠀⠀⠠⠠⠔⠡⠀⠀⠀⣿⠃⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⢰⠁⠀⡆⠈⢿⣦⠀⠀⠀⣹⣦⣄⡀⠀⠀⢩⣘⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡀⠀⠘⣤⠀⢰⠀⢠⣿⣿⢿⣿⣿⣿⡿⠋⠉⢉⡉⠛⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠸⡁⠀⠘⣆⠏⣼⣿⣤⣿⣿⠟⠀⠀⠀⢄⡈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠙⣄⠀⢻⣼⣿⣿⣿⠟⠁⠀⠀⠀⠐⢤⣹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠆⢸⡿⠛⠋⢁⠀⠀⢀⣀⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠴⠊⠁⠀⠀⢠⠈⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣀⠀⠀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣹⣟⣻⣿⣝⢋⡹⣿⡻⢿⡿⠟⣿⣟⢹⠟⣯⣩⣭⣭⣉⣉⡓
⠀⠀⠀⠀⠀⠀⢀⣾⣪⣿⣛⣟⣿⢷⣿⡿⣿⣟⣃⣉⣳⣾⠛⣟⣸⣿⣍⠻⢿⠿⣻⣟⡁
⠀⠀⠀⠀⠀⠀⢾⢅⡿⠿⠿⢿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠿⣾⠁⠿⠿⠿⠿⠶⠾⠟⣟⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠒⠒⣢⣧⣤⣤⣤⣩⣭⣭⣿⣛⣿⡶⢿⣿⣿⣿⣿⣿⣿⣯⣹⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣶⢸⣿⢻⣷⣾⠣⣖⠻⣿⣛⢿⣷⣿⡛⠫⠍⠓⠛⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⣯⣿⣿⢸⡇⠁⢠⣤⣿⣿⡷⣾⣿⣯⣐⣲⢶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠒⠻⠟⠜⢷⣾⣻⣿⣛⣽⣧⣨⣿⣿⣿⣻⣥⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠨⠉⠙⣿⣈⣿⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠹⡄⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡞⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡁⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠧⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⣀⡤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⢄⣀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⣣⡴⠎⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⢀⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣤⡴⣧⣤⣤⣤⣤⣤⣤⣤⣾⡧⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⢰⡅⣀⣀⣤⡄⠀⠀⠈⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠸⢄⣘⣿⡷⠃⠀⠠⢄⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⢸⠇⠀⠲⠖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠸⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⢀⡇⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠷⠶⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣼⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⣨⡟⢻⣏⠉⢹⡇⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠁⢹⡏⠙⢧⣈⡇⡤⠀⠀⠀⠀⢰⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡻⡇⠀⢀⠸⡇⠀⠀⢹⡟⠀⠀⠀⠀⠀⠈⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡸⠁⡇⠀⠸⠀⡇⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⢸⠀⠸⡀
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇⠀⡇⠀⡄⠀⡇⠀⢰⠁⠀⠀⠀⠀⠸⠀⠀⠈⣇⠀⣿
⠀⠀⢀⣀⣹⣿⠿⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿⠀⠀⡇⠀⠇⠀⡇⠀⡜⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠀⢸⡆
⠀⣴⣶⣿⡇⢉⣀⣀⠘⣆⢰⠛⢦⡀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣠⢺⡇⠀⠀⡇⢀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠈⣧
⣷⢿⡟⠁⠀⠻⣄⡈⠓⢾⡿⠀⠠⡈⠳⡶⠖⠋⠁⠀⢀⡴⠊⠁⣼⠁⠀⠀⡇⢸⠀⠀⡇⣸⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⢸
⠙⢦⡀⠀⠀⠀⢠⡙⠆⣼⠃⠀⢀⡇⠀⢇⡀⣠⡴⠖⠉⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠘⡄
⠀⠀⠹⣦⠀⠖⣩⠁⠀⢥⣤⡴⣏⠑⢦⣜⠉⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⡏⠀⠀⠀⣇
⠀⠀⠀⠈⢣⡄⠁⠀⠀⠀⠀⡤⠈⠙⢦⣈⠑⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣄⠠⢄⠀⠈⠀⠀⠀⠀⢻⡟⠃⠸⠷⠀⠀⢠⡏⠀⠀⠀⠀⡟⢧⣄⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡜⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠈⠙⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⡇⠠⢤⣀⣀⡞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⢨⡉⠛⠳⠦⢤⣼⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠈⠓⠶⢀⣀⠀⠀⠉⠛⠛⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠴⠖⠚⢙⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠤⠤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠏⠀⠉⠙⠛⠓⠶⠶⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠶⠶⠶⠖⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠶⠖⠀⠉⠉⠁⠀⠀⠀⣠⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠋⠀⢘⣧⡿⠷⠶⠶⠶⢀⣤⡤⠤⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠒⠒⠉⣉⣉⠁⢀⣠⣤⡤⠶⠚⠉⠹⣤⣤⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣑⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠙⡇⠀⠄⠀⢀⠰⢀⠸⠘⢀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠠⠀⢁⣀⣡⣁⣀⣀⣀⣠⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠙⢤⣤⡤⡴⡦⠦⢖⠖⡳⡳⢓⢗⠖⡲⡲⢒⠖⠦⡲⡚⠫⢋⠝⠙⠙⡡⡫⢋⠌⠜⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠊⠊⠐⠐⠁⠃⠊⠈⠀⠁⠀⠈⠀⠀⠁⠈⠈⠀⠀⠁⠀⠀⠀⢀⢄⠀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠄⠔⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣂⡀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠒⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠻⣇⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠚⠈⠉⠐⠛⠒⠶⠦⠦⠦⠤⠤⠤⠤⠤⠤⢀⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠠⠀⢠⢠⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⣀⣂⣀⣠⢡⡤⠦⠶⠶⠴⠶⠦⠶⠷⠧⠦⠦⠦⠦⠦⠦⠆⠤⢬⣬⡤⠄⠬⢬⣄⣄⡘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢄⠤⠢⢂⠄⡪⢐⠄⠂⠄⡨⠔⠠⠀⠄⢂⠤⠀⠄⡂⠄⡠⠑⠨⠀⠌⡂⠕⡠⠐⠠⠂⠄⠠⠐⠔⠡⠀⠠⠂⠄⠪⠂⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠖⠂
⠀⠐⠀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⠿⢿⣿⠟⠛⠛⠛⣿⠿⠿⠟⠛⠉⠉⠛⢻⣿⣿⠿⠛⠛⠻⠏⠉⠉⠉⠙⠋
⠀⠈⠛⠻⠟⠛⠟⠋⠡⡀⢀⡃⢀⢀⡀⠀⠰⠅⢀⣀⣀⡠⢰⣂⠀⣀⠙⠐⡞⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠄⠂
⠀⢰⡄⠈⠀⠀⠀⡀⠰⠀⠀⠀⠀⠋⠀⡄⠀⢐⡆⠀⠀⠀⠄⡀⡷⠤⠤⠤⠴⠀⡀⠀⠀⡐⠀⠀⠀⠐⡀⣀⠀⠄⣀⠈⠀⠀⠔⠀⠄
⠀⠀⠀⠀⠄⠁⠀⠃⠀⠐⠈⠑⠢⠀⠠⠁⠀⠀⠀⠉⠉⠉⠤⠀⠁⠀⠀⠀⠰⠆⠤⠤⠤⠤⠠⠷⠤⡀⠉⠀⠀⢆⠤⣁⡀⠈⠓⡐⡀⣀⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠐⠳⢦⣄⡀
⠀⠃⠀⠠⠀⢠⠀⠀⠀⠀⠀⠈⠙⠛⠚⠒⠓⠚⠒⠒⠚⠛⠓⠚⠛⠛⠛⠛⠛⠛⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠈⣇⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠘⠠⠉⠳⢦⣄⡘⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢧⡀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠹⡓⠶⠶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣄⡀⠠⣶⣶⣦⣄⡠⢠⡶⢶⣶⣄⡠⠶⢶⣦⣄
⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⣤⡙⠣⠄⠈⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢠⣿⣿⠟⡿⢿⣿⣿⣿⡄⠤⠀⠀⠀⠈⢦⠀⠀⠠⠶⠀⠐⠀⠀⠀⠀⢣⠀⠀⠀⠈⣆
⠀⠀⠀⠀⣻⣿⣽⣻⣉⣾⣿⣝⣿⣿⡄⠀⠀⠀⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⣿⣯⣽⣛⣻⠯⣾⣷⣿⣿⣧⠀⠀⡄⠀⠀⢸⡆⣂⣄⡀⣀⠂⣤⠀⡀⠰⠿⠶⠗⠂⠘⢛⣛⠂
⠀⠀⠀⠀⢿⣻⣿⣿⣧⣿⣿⣿⣿⣽⡟⠖⣚⠛⠀⠀⠈⠉⠉⠀⠀⣼⡞⢚⠛⣇⡀⠀⠬⠤⠴⠶⠒⣟⠂
⠀⠀⠀⠀⠸⣿⣿⢿⣷⣟⠉⠛⣻⡿⣿⣮⡭⠴⠶⠖⠒⠒⠒⠋⠉⢻⡟⣴⡌⢣⣤⣤⣤⣤⠶⣶⡶⠶
⠀⠀⠀⠀⠀⢿⠛⣾⣹⣿⣿⣻⣿⣷⣿⣟⣣⣤⣤⠤⡶⠶⠒⠾⠛⢿⣎⢿⡟⠈⠀⢁⢠⣁⠂⠍⠃⣂⡴⣁
⠀⠀⠀⠀⠀⢸⡄⢈⢁⢀⣽⣷⠊⢿⣽⠉⠁⡟⣥⡄⠀⢸⠶⠀⠄⠘⣮⠏⣧⠄⢤⠀⠘⠫⣥⣼⢠⣿⡏⠕
⠀⠀⠀⠀⢀⠀⡄⠀⡛⠉⢾⡄⢀⣼⡿⠀⠀⣉⠋⣠⢦⡯⠀⠀⠀⠀⠈⠀⠡⡴⠁⠈⡠⡲⢬⡧⡎
⠀⠀⠀⠀⠆⠀⣧⢀⠻⣄⢾⣿⣾⡿⢿⠀⠠⣿⣼⢁⣞⡀⠀⠀⠀⠑⠀⠀⠀⢻⣬⠛⠹⣗⣿⣿⣇⠁
⠀⠀⠀⠀⣠⣿⣿⣾⣿⣫⣽⠿⢿⣴⣯⡅⠀⡿⣵⢿⣽⣆⠒⡄⡜⠓⠈⠋⠂⣰⣿⣶⣿⢿⣯⠷⠦⠠⣀
⠀⠀⠀⡀⠀⠀⠸⢿⣞⣿⣿⣮⣿⣿⣿⣇⠤⣿⣯⣾⣶⣿⣿⡿⠿⢿⣂⣿⣯⣿⣟⣿⣿⠏⢁⡤⠂⢀⣀⣰
⠀⠀⠀⠪⠂⠸⠀⠀⠉⠛⠿⣿⣿⣿⣿⠀⢚⡸⣟⡯⠉⣑⣤⡮⠵⠟⠚⠛⠉⠉⠁⠰⠄⠀⡩⡁⠀⠡⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣠⠿⣇⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠈⠉⠒⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠢⠄⣀⠀⠀⢸
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠘⠂
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⣀⠀⠉⠉⣿⣿⣶⣴⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⢹⣿⣿⣿⣿⣿⡏⠔⠈⠙⡄⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠛⢿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣮⣴⡇⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠱⠟⠀⠀⢸⠉⠉⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢸⣿⣿⣿⣿⣿⡄⠘⡏⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠙⠻⢾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠛⠿⣟⡛⠁⠀⢹⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠙⢦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠂⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠄⠈⠀⠉⠀⠀⠒⠛⠐⠒⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⢲⣶⣶⣦⠲⠲⣶⣶⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⠀⠃⠄⠐⠀⠀⠁⠈⡂
⠀⠀⠀⠀⠀⢁⢠⡄⠤⠤⠤⠤⠤⠒⠂⠀⢀⡀⠀⠀⠀⢰⣿⣿⣦⣿⣟⡻⣿⣿⣿⡻⡫⣻⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢡⠤⣌⠀⠐⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠠⠄⠀⡔⠀⠀⠀⢀⠀⠀⠀⢹⣿⣿⣿⣿⠼⠿⣌⣿⣿⣾⣿⠋⢤⡶⠛⠁⠀⠀⠀⠀⠀⠀⢠⠀⢸⠕⢢⠑⠆⡇⠀⠀⠀⠒⣾
⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⡆⠰⡰⣦⡚⠀⠜⠀⠀⠀⢀⡾⠿⠟⣏⠂⠢⡄⢠⢶⡿⠃⣴⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠆⢘⣈⡥⠆⣇⢀⠀⢠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⠐⠘⠛⠀⠀⠀⠀⠀⠀⢸⣿⣯⣅⣠⡧⣠⡜⢿⣷⣿⣿⡿⢿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⠿⢿⢿⢸⣿⡇⣹⡏⣹⣿⣿⣾⣿⢬⣀⠀⠀⠀⠠⠖⢋⡀⡒⠰⠀⠀⡖⠂⠂⠠
⠀⠀⠀⠀⠐⠐⠲⣀⠈⠊⠐⠀⠭⢧⡦⠀⠀⠀⠠⣜⣻⠛⠠⠄⣀⡄⠦⢟⣧⢸⣯⣿⡟⢋⡨⠉⢺⠴⠛⢒⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠠⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠶⠐⡁⠀⢒⣮⠲⠃⠀⣔⠋⠉⠉⠧⠠⠚⡛⠀⢦⠜⠂⡠⠞⠁⠐⢀⠰⠂⢋⠀⢁⠀⢦⣤⣦⡀
⠀⠀⠀⠄⣀⣀⡠⠄⡀⠁⣐⠀⠦⢀⠀⠀⠀⠀⣉⣙⠶⢈⣩⡅⣘⢗⡉⣰⠛⠈⠉⠉⠉⡉⠆⠉⠑⠚⠁⠀⠂⠛⢨⢗⠾⡶⠳⣡⡰⠍⠀⠀⢰
⠀⠀⠀⠌⠁⠁⠙⡆⢈⡶⠶⡮⢲⠎⡙⠇⠀⠀⠀⠀⠀⠀⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠀⣰⢌⣄⢣⠁⡇⠀⠰⠄⠷⠸
⠀⠀⠀⠠⣰⡆⠎⠀⠉⠀⡨⢃⣈⣔⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠶⡆⠉⢌⡚⠿⢥⢀⢳⡓⠰⢄⢂⠀⠤
⠀⠀⠀⠈⠓⢁⣥⠀⠀⡆⠈⣻⠯⡊⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⡌⠱⠇⡨⠍⠁⠈⢀⡀⠹⣀⠈⠵⠦⢀⡠⠄⠁⠈
⠀⠀⠀⠀⠀⠀⣐⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠒⠌⠁⠀⠁
⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠻⣟⠟⣻⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠤⢄⡀⠒⠀⠒⢖⠒⠈⣉⡉⠉⠉⠉⠉⠁⠐⠢⣄⠈⠐
⠀⠀⠀⠀⠠⠖⠚⠮⡍⠁⠉⠻⣉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠘⢦⠀⠀⠀⠙⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠀⠱⡀
⠀⠀⠀⠐⠁⢀⠤⣀⠈⠢⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣧⠀⢀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠂⢐
⠀⠀⠀⠁⠀⠀⡠⠊⢱⡄⠀⠀⠀⠈⠤⠤⠴⠶⠖⠒⠒⠒⠋⠉⠉⠉⠁⠀⠀⠈⢒⠀⢀⡀⢉⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⠀⡩⠛⠁
⠀⠀⠀⠀⠀⡴⠁⢀⠎⠐⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣁⣀⣀⣠⡇⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠞⠀⣠⠋⠀⠀⠈⠠⠂⠀⠀⠉⠀⣀⣀⣀⡀⠤⠤⠤⠤⠶⠒⠒⠒⠊⠉⠉⠉⠁⠀⣀⣰⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠄⣰⠃⢀⠌⠀⡐⠓⠒⠋⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠖⠒⠒⠒⠉⠉⠁⠀⣸⢀⠀⡠⠊⠁
⠀⠀⠀⠂⢸⠁⢀⠎⠀⣄⡠⠤⠤⠤⠶⠒⠒⠒⠉⠉⠉⠀⠀⠀⢀⣀⠠⠴⠤⡤⠴⠒⠒⢒⠉⢉⣹⡔⠊
⠀⠀⠀⠐⠈⣠⠋⠀⠔⠀⠀⠀⣀⣀⡀⠤⠤⠤⠖⠒⠒⠊⠉⡉⠁⠸⠀⠀⠀⠚⠀⡱⠞⠿⠦⡀⠀⠙⡆⢀⡀⢀⡀
⠀⠀⠀⠀⠁⢡⠀⠚⠒⠉⠉⠉⠀⠀⢀⣀⣀⡤⠤⣴⣶⡲⠟⠀⠠⡴⢂⡚⠉⠂⢺⡄⢀⡤⠀⠾⠀⢔⠉⠀⠒⠠⢓⡀⠅⠤⣀⠂⠤⢀
⠀⠀⠀⠀⠀⢠⣄⣤⣤⠾⠩⡀⠿⢋⠭⠗⠛⠠⣤⠁⠀⠀⠀⠤⠐⡩⠙⠋⠛⣢⡜⠉⠉⡱⣤⡤⠀⠺⠗⠑⠒⢢⡙⠀⠀⠀⣉⡠⠤⠖⠈⠑
⠀⠀⠀⠀⠀⠈⠑⠦⣅⠀⢀⠔⢀⠉⡆⠋⠐⠂⠉⠓⡄⠀⠀⢀⠐⠄⠀⠀⠈⢀⠠⠐⠀⢤⠁⠀⠈⠓⡀⠀⠀⠠⠖⠒⠉⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠠⠀⠈⠯⠀⡀⠈⠀⢲⡄⣐⡀⢀⠖⠑⠀⠐⠌⠀⡠⠐⠀⠈⢉⡗⢀⣀⠰⠁⠐⠒⠈⠉⠀⠀⠀⢀⡀⠤⠄⠒⠉⠉⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠠⣄⠈⠀⠀⡠⠿⠑⠀⠤⠄⠚⢀⣉⠄⠀⠀⠁⠈⠁⠀⠀⠀⢀⣀⠠⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⢀⠃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⣀⠠⠄⠚⠀⠩⠁⠀⠀⠀⠀⣀⡠⠤⠐⢒⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣠⠀⠤⠒⠊⠉⠀⠀⠀⠀⠘⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⡀⠀⠐⡒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⢻⣟⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⠈⠡⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠧⠤⡤⣼⣈⣷⢦⠦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⡴⢦⣬⠶⣾⠶⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠋⠀⠀⠉⠀⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡵⠀⠀⠀⠜⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣠⠄⠀⡠⠚⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⡛⠉⢹⠉⢳⡤⠊⠀⡸⠀⠈⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣄⡀⠀⠀⠀⡜⠀⡀⢸⠠⣿⡿⠦⢠⠇⠀⠀⣇⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣾⠇⡰⠦⠾⣄⠔⠁⢸⣷⣾⣧⠐⡎⠀⠇⠀⢸⠀⢀⠀⢠⡄⠂⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⣿⡟⣻⡜⠀⠀⠈⠛⠋⠉⡿⠀⢸⡇⠀⡀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⠀⢿⡆⠀⠀⠀⠀⠸⣋⣼⣿⠁⠀⠀⠀⠀⢀⣼⢁⣀⣾⡀⠀⢷⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⢸⣷⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀⡞⣿⢛⣋⡟⠀⠀⢸⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡏⢠⡞⠀⣿⠀⢸⡟⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⡠⠻⡭⡤⠤⢸⡇⢀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⣿⠀⠈⠁⢹⡀⠀⠀⠀⡜⠀⠀⠀⢀⡄⠀⠀⢠⠄⠀⣼⣷⠈⢇⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣠⣤⣄⡀⣿⠀⠀⠀⠀⢳⠀⠀⡜⠀⠀⠀⣠⡞⠀⠀⢠⡟⠀⠀⣿⣿⠀⠘⢧⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣅⠀⢤⣬⡁⣿⡀⠀⠀⣤⠘⣧⡼⠁⠀⢠⣾⠋⠀⣀⠀⡞⠀⠀⢸⣿⣯⣤⡶⠦⣅⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⠓⠀⠀⠀⣿⡇⢸⠀⠉⠀⣹⠁⣀⢶⡿⠁⠀⠀⠛⠀⢰⠀⠀⣿⣿⣇⣀⡰⠖⠋⢡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠛⠛⢀⣿⡇⢸⠀⠀⠀⢣⠞⠡⠋⣀⣀⣄⠀⠀⠀⢸⠆⢸⣿⡿⢋⡉⠀⢠⠾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡀⠀⠘⣶⣿⠐⠚⠛⠉⠁⣤⠀⠀⢠⠟⠀⣿⠟⠀⠉⠁⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⡏⣇⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣶⡴⠋⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠐⢾⣿⣄⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣿⠁⠀⠀⠔⢋⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⡇⠀⠀⠀⢸⣿⠉⠉⠀⢰⡧⡤⠀⠀⠀⠀⢠⣤⠀⡀⠚⠿⠿⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣶⠟⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠞⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⢸⣁⢿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⢸⡷⢿⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡿⣦⣿⣤⣤⣄⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠀⠆⢸⢹⡉⠉⡹⢿⠉⠉⣭⡿⠋⠉⠉⠉⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⠸⡇⠀⣷⠘⡆⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣘⠂⢳⠀⣿⠀⢣⠀⠀⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣾⣧⣤⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⠚⣉⡷⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⣏⣉⣲⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢀⣤⢄⣀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠁⠀⢀⡼⠋⠀⠀⠀⠀⢎⣱⠀⢳⡃⢨⣽⣿⣒⠦⠭⢍⣓⠒⠦⠬⣗⣎⣦
⠀⠀⠀⠀⠀⢀⣠⠴⠒⢋⣼⣏⣽⣭⣷⠀⠀⠀⠹⣷⡀⣿⢦⡎⢣⡞⢿⣸⡏⠁⠀⠀⠤⢉⣉⠑⠒⠬⠭⣙⡒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠈⢩⣯⡆⣼⣧⣀⣠⣴⡿⠃⠀⠀⠀⠘⡝⠟⠈⠃⠈⠁⠘⢱⡏⠁⠀⠀⣴⣿⣿⣷⡀⠀⠐⠠⠤⢍⡉⠒⠂⠭⢍⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⢻⣿⡇⠀⠀⠀⠀⠀⢹⡴⠰⠀⠷⠐⢂⣻⣤⣠⣄⣰⣯⠐⣽⣿⣿⣄⠒⠤⠤⣀⠀⠀⠀⠓⣲⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⣠⣤⣤⠴⠚⣍⡙⠭⠤⣬⠭⡗⠋⠀⠀⠀⠈⣿⠿⢿⣿⣿⣿⣦⣤⣤⣤⣤⠀⡐⢲⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡏⢠⢍⣙⡒⠲⠤⣔⣢⠦⠄⠡⣉⣠⠤⠄⠀⣟⣙⣖⡦⢝⡻⣿⣿⣿⣿⣣⣀⣠⠏⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣷⠃⡗⠢⠤⠈⠉⠒⠲⡟⠉⠓⣶⡦⢤⣞⣻⣒⢀⣐⡶⠂⣤⡟⠚⠏⡩⠀⠘⣦⣎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⡄⠀⠀⠈⢱⣶⡱⡶⠆⠀⠀⢈⡛⢉⢽⣰⣸⣁⠒⠪⢿⣼⣓⠲⠤⣬⢧⡿⣴⠖⠒⣒⡛⣻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣟⢛⡁⢶⣷⠖⠉⣼⣿⠀⠀⠢⣤⣴⣳⣌⠶⡬⡧⠬⠀⠀⢸⢴⡯⣩⠀⡏⢸⡷⢯⡉⠁⢤⠇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⣄⡰⠁⠈⡞⡯⠄⠑⢠⠋⠀⠀⢾⠃⢸⡹⠀⠀⣀⡀⡹⢶⠓⠾⢠⠇⣾⣽⠾⡤⠒⢺⢸⠃
⠀⠀⠀⠀⢀⣴⡻⠭⠷⡿⢿⡇⠈⢸⣀⡓⠤⠄⣹⡀⠋⠀⠈⡹⡷⠣⠤⠀⠀⠉⣱⠾⡦⡇⢸⠀⡧⢼⠛⡅⠂⡏⡼
⠀⡀⢠⣴⣋⠁⢧⣴⡞⠓⢼⡆⠀⢸⡀⠉⠙⠒⠬⢿⣖⣒⡩⢼⣉⠀⠀⢤⣄⡀⠙⣾⢳⡇⡟⢸⢤⣞⣻⢉⡹⠟⠳⡄
⠀⠀⠀⠉⠉⠓⢲⠗⢑⠀⠘⢣⠐⡀⠻⡗⠒⠀⡀⠀⠀⠉⠑⠒⠤⣍⡱⠛⢄⠙⣄⠙⢻⣭⣷⢺⣯⡽⠥⣾⠄⠀⠔⠉⠧⡄
⠀⠀⠀⠀⠀⠀⢯⡐⠌⡓⠓⠂⠑⢬⡂⠈⣳⣤⡉⠉⢀⠴⢞⡉⠉⠉⠣⡀⣨⢶⠯⣍⣁⣸⣉⣉⡤⢤⠶⠇⣀⡀⠐⠚⠁⢈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⣷⠚⣩⠌⢒⡂⢨⣩⠯⣀⣀⡻⡿⢤⡈⠹⢆⠀⢀⣯⣼⣫⣤⠴⠋⠉⠁⠀⠀⣏⣠⣄⠄⡀⣀⠠⠜⢀⣀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠚⠦⢾⣉⣩⠟⠳⣄⣱⠀⢈⡶⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠮⠖⠢⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⢀⣔⣵⣿⠟⠋⠉⠀⠀⠈⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⣼⣿⡏⠀⣀⡴⢚⣿⠟⠛⢿⣟⠲⣄⡀⠈⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⣾⡉⠀⢾⡇⣄⣽⡆⣿⠆⠈⣹⡆⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣇⠀⠙⠶⣬⣷⣮⣭⣴⣏⡤⠞⠉⢀⣿⣿⠇
⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⡀⠀⠀⠀⣀⣠⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣮⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠏

⣿⡯⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⢀⡀⣀⣤⡤⢠⣄⣀⠀⠀⢠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠩⠍⠲⢝⠩⣲⢄⠀⠀⠀⢘⠆⠱⠎⠰⠄⠤⣤⣬⣵⣀⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠖⠁⠀⠑⢶⢂⢟⡤⠆⠀⠈⠉⠁⠀⠙⠛⡾⠃⠉⠀⠀⠚⠋⡘⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⡠⠣⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢲⠀⠀⠔⡕⡵⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡇⠀⠀⠀⠀⢊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢊⢎⡞⠁⠀⣀⣠⠤⠠⠀⠘⡉⠉⠑⠶⡿⠤⠀⡀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⢣⣾⣴⣶⢯⡬⠗⠞⡣⣤⡀⠀⢈⣄⣠⠆⠰⠃⠛⢓⠢⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢝⠫⣏⣉⠋⢘⠶⢦⣀⠀⠀⢹⣇⣁⣠⡇⠀⡴⢣⡀⣀⣼⣳⠉⠷⠶⢒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⢍⡢⢍⡒⠬⣂⣰⣟⣀⣴⣈⠰⠯⠤⢟⣚⠻⠾⢖⣛⣡⡄⣀⡴⠒⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢈⣒⠬⡙⢲⠕⠒⣲⡦⠽⡍⢉⣰⠒⡾⢃⠙⣈⣷⠀⡇⠉⡇⠀⡗⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⢰⣯⡿⠇⠾⢾⠀⠖⡇⠀⠀⡇⠀⢸⢠⡏⠉⣠⠀⡘⣆⡇⠛⡇⠀⣷⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠺⢨⠼⠉⠈⢁⡜⠉⣢⠀⣥⠀⠀⡇⠘⢹⢸⣀⡟⣿⣾⡳⠃⣇⠒⣇⣀⡿⢸⡟⠗⠶⣤⣄
⠀⠀⠀⠀⠀⣠⠄⢲⡊⠴⠉⠁⠳⠂⢦⣀⣠⠉⠈⠑⣀⣀⠦⣬⣷⠒⢧⡶⢸⠀⠉⣏⠷⡾⡳⠶⢃⡀⢭⠋⠁⠚⣧⣠⣦⠿⠸⠗⠂⢤
⠀⠀⠀⣀⡎⠀⡐⠉⠉⣦⠀⡴⠳⠀⠀⣨⠀⠉⢲⠞⠉⠨⠃⠀⣸⣻⢹⣧⠼⣞⡟⠛⠻⡟⢑⣬⠕⠶⠋⠐⡆⠠⢿⠕⠠⠐⠀⢰⣶⠾⠠⠤⡀
⠀⠰⣁⣀⣀⣿⠆⠀⢰⠀⠀⠈⠎⠳⣊⣉⡽⢾⠹⠄⣀⢀⣼⡛⠛⠡⠀⠬⠄⢼⣄⣀⣠⠿⠷⠞⠛⠆⠠⢴⣤⣤⠏⠦⢄⣠⣤⠾⠓⠤⠤⠴⠃
⠀⠀⠀⠉⠀⠘⢤⣠⡾⠲⡀⠀⠀⣰⣟⣩⢦⡄⠀⣀⡇⢀⡹⣁⡈⠀⢦⣀⣀⣠⣇⣈⣧⣤⣤⠾⢧⣀⢀⣠⠟⣇⡀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⣄⠁⠀⠈⣻⠒⣎⠁⢀⣰⣄⠁⠀⢈⡿⡏⠁⠀⠉⡾⠋⠁⠀⠈⠉⠁⠀⠰⣀⠀⣀⡼⠃⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⢩⠋⠉⠉⢱⡖⠛⠁⠙⠲⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠦⠤⢄⣀⡠⠀⠀⠀⠐⠲⠶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁
⠀⠀⠀⠀⠰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠑⠒⠠⠌⢀⡀⠀⠀⠀⠘⠶⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠐⢄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⡀⠂⠀⠀⢢⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣰⢦⡀⢄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⠳⠎⣜⡿⢺⣐⠾⢟⡛⢆⣀⡀⠀⠀⠀⠀⢣⠁⠀⡸
⠀⠀⠀⠀⠰⣄⠀⠀⠀⡤⠐⠁⡠⢒⢵⢓⢂⢫⢙⣂⢀⠄⡿⢳⠇⠀⡈⣦⣀⠤⠄⣄⢃⠀⢃
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠐⠀⠉⣩⠉⠾⣔⣛⡯⢟⠤⣶⣲⣫⡞⡍⢍⡵⠛⣠⢤⣼⡬⣆⠘⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⢀⣩⣷⣾⡾⣕⣲⡤⢿⣝⡹⣮⢤⡼⠒⠁⡁⠇⠀⠀⢈⠀⠀⡄⠀⠀⠃⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⠏⠈⠁⠐⠚⠢⢯⣲⠤⠓⡡⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉⠀⠀⡀⠀⢨⡗⠀⠀⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠒⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⢿⡄⠀⠀⠀⠐⡁⠀⠀⠀⠠⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⣄⢸⣿⡆⠀⠀⠀⠠⡣⣉⡀⢠⠀⢀⡤⣚⣥⣯⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣐⣺⠋⢥⠮⣍⡢⠄⡸⢒⣰⣀⣾⠟⣤⡯⢗⠸⠿⠤⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢤⠩⣨⢾⡕⢻⣛⠃⢸⡞⢋⠿⣀⡛⢲⣌⣷⣈⣂⠂⢈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠺⠥⠷⡠⢒⡪⢙⣿⣤⠃⣸⠧⢬⠈⡛⣼⡒⠯⢢⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠈⠈⠉⠻⣯⡴⠶⠟⠣⠈⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡇⠙⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣈⣉⠙⠒⠲⠮⣭⣟⣛⣶⣦⣿⣷⣶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⣭⣛⡻⠿⢷⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠷⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣶⣶⣬⣍⣛⡛⠿⢿⣿⣿⣿⣥⣄⡈⣥⣶⣶⣶⣾⣿⡿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣭⡙⣿⣿⣿⡿⣣⣿⡟⣿⡟⣷⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡍⣶⣿⢿⣷⢹⡇⣿⣿⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣇⣿⣿⡜⣿⡜⣿⣿⡿⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣇⣿⣧⣿⢟⣵⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠙⢿⣿⣿⣿⣿⣿⡏⣥⣭⡸⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠦⣤⣀⠒⠛⠠⠤⢖⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠻⣿⣿⣿⣷⣿⡷⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣟⠉⠑⠒⠒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠰⣾⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠘⣾⢻⣿⣿⣿⣷⣤⣀⡤⣤⠀⠀⠉⠉⠒⠒⠤⠤⣠⣀⡀⠀⢀⣠⣤⣔⣲⡿⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣭⣟⡻⢀⣩⡙⣶⣶⣦⣤⣌⣙⡓⢺⡏⢉⢻⡟⣿⠏⣯⣶⣿⣿⡿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣟⣛⡻⢿⣸⣿⣧⣾⣭⣭⣛⣻⣿⣿⣿⣷⠄⢸⠿⣋⣼⣿⣿⣯⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⡿⢿⣿⣿⠃⠀⠉⢛⣿⣛⡻⠿⣿⣿⣿⣿⣿⠀⢰⣾⡇⣿⣿⡿⢟⣫⣵⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣸⣷⡒⠀⠙⢦⣉⡠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣇⣿⣿⣿⣿⣿⢿⠟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠲⠤⣄⣀⡀⠈⠉⠛⠛⠿⠿⣿⣿⣀⣼⣿⡿⢿⣿⣿⣿⣶⡿⢸⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣍⣒⣶⠟⠉⠀⣟⣿⣶⡙⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⡽⠟⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠴⠶⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠎⠁⡈⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠂⣠⠀⠀⠀⠀⡁⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡈⢠⡿⠀⠁⠀⢀⣐⣐⣀⣀⠀⠃⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢁⣩⣬⣶⣾⡿⣛⢿⣿⠿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢚⢻⣿⠛⣿⣿⡇⢱⡯⣿⡄⣸⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⡨⢿⡾⠛⠁⠈⠠⡴⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⣠⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢀⠇⢸⡀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⡰⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠁⠀⠀⢄⠀⠀⠀⢣⡀⠘⣆⣾⡷⠶⠶⠦⠼⠇⠠⡄⢠⣄⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠲⣌⠣⠀⠀⠀⢷⡀⠘⣿⣶⠀⠀⠀⠀⠀⠘⣀⣞⠁⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣧⠀⠐⡈⣷⡄⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡭⠻⣿⠃⠀⠀⠸⡟⢺⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡴⣿⡐⡶⠶⡄⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡇⣿⣷⠀⡀⢐⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢂⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣿⣿⣰⢻⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢈⣀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢿⣿⣿⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠠⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⡷⣺⣿⣿⣿⡿⠟⠉⠀⠀⠱⣄⠀⠀⠀⠀⠐⣰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠿⠛⠁⠀⠄⢉⡉⢤⣠⡘⣾⠷⠀⠐⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣻⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⣾⠿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣾⡿⣷⣷⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣩⡖⠋⠁⠀⠘⣧⢳⡀
⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠙⠛⠃⠒⣂⠤⠤⣄⣹⣼⡇
⠀⠀⢰⣿⡏⢹⠿⣦⠀⠀⠀⠀⠀⠀⠀⠺⠤⣀⣀⠀⡠⢤⢄⠊⠙⠳⣤⡀⠀⠉⠉⢢⣀
⠀⠀⢸⣿⣇⠛⠀⠘⢷⡀⠀⠀⠀⠀⡠⠀⠈⠁⠀⠀⢰⣶⠀⠇⠀⠀⠘⠉⠑⡶⣶⠒⠚⠃
⠀⠀⠈⣿⣷⠀⠀⣶⢈⣷⡀⠀⠀⠺⢤⣀⣤⣤⣤⢤⣸⡁⠀⠀⠀⠀⠀⣀⡠⠋⣸⣿⣶⠶
⠀⠀⠀⠹⣿⣧⢠⡇⡀⠸⣇⠀⠀⠀⠀⠀⠀⡇⠀⠂⠙⠤⣤⠖⠀⠀⠀⠀⠀⣸⣿⣿⠁⣠⠞⠃⣠⣄
⠀⠀⠀⠀⠹⣿⣿⣡⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠐⠂⠂⠚⢣⣀⠀⡠⠔⠁⣼⣿⣿⡷⠀⠘⢀⡈⠘⠉⢪⣦⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠦⣄⣼⣿⠿⣿⢳⠀⠐⡿⣿⠢⣤⡈⠉⠈⣙⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⣤⠖⠆⠀⠀⠀⠀⠀⡰⠛⣿⣿⡿⣿⣾⣤⠤⡷⠺⠀⠐⡖⣶⠖⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣲⣡⡆⠸⣿⣿⣿⡖⢚⣣⡎⠉⣸⡆⢺⠭⠀⠀⠤⠄⠀⠀⣀⣀⣀⡎⠀⠉⢙⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠉⠀⢰⣿⣿⣿⠋⠈⠋⣠⠞⠉⡇⣴⣶⠀⠀⢴⢠⠀⠀⠠⡤⠄⠀⢨⡯⠏⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠤⠠⠊⠀⠀⠀⡇⠀⠉⠀⠀⠈⠈⠀⠀⠀⠉⠁⠀⠈⠉⠁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠋⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠌⢤⣄⣀⡀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡖⠒⠓⣷⣶⣾⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣾⣿⣿⣧⡀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠛⠻⣿⣧⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⠿⠿⠿⠟⠛⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠐⠦⣀⣀⡀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⠀⠀⠀⠀⢀
⠀⠀⢄⠀⠀⠀⢠⣄⣤⠤⠛⢻⠶⡛⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⣀
⠀⠀⠈⢣⡀⠀⢸⣃⣀⣀⠀⣸⠨⣁⣄⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⠄⠘⠛⢻⣿⣿⣿⣶⢹⣿⣧⡀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠺⣵⣶⣟⠿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣧⣄⣤⡶⣿⣟⢈⠉⣄⣀⣴⣿⢦⡀⠀⢻⣿⣿⣷⣤⣟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢛⣻⣷⣶⣶⣷⠁⠒⣿⡿⠧⣾⠃⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣭⣙⡡⣇⠤⠖⠺⠀⠀⠀⢀⣀⣉⣭⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⣿⣿
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⢮⣛⠉⠙⠙⣶⢻⡄⠀⣤⣤⣴⣖⣈⣻⣷⠞⠃⢹⣿⣿⣿⡙⠛⣿⣿⢻⣆⣐⣽⡿
⠀⠀⠀⢸⠛⠙⠉⣽⣿⣿⢃⣶⠈⠃⣰⡀⢿⣬⣇⣀⣿⣿⣿⣿⣏⣹⣟⡘⣉⢸⣿⣿⣿⡿⠿⢿⠃⠘⣿⣿⡏
⠀⠀⠀⠰⠀⠀⠠⣡⡞⠁⢸⣿⣄⠀⠀⠈⠺⠟⠃⢀⣠⣄⠀⠀⠀⠁⠐⠀⠀⢈⣯⣯⣏⣀⣀⣾⣇⠈⣽⣿
⠀⠀⠀⠀⠀⠀⣦⡼⠷⠿⢾⣯⠢⠄⠘⠛⠛⠈⢴⠛⢿⣿⣿⠿⠿⠿⠟⠛⣙⣸⣿⣿⡿⠿⠿⣯⢷⣶⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠘⠓⠔⠒⢺⣿⣥⣀⣠⡐⠛⠛⢶⣷⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⣀⣠⣤⣟⢎⣿⡟⣽⣧
⠀⢰⣅⡀⠀⠀⠈⠉⠉⠛⠛⠛⡛⠛⠛⠛⠻⠶⢶⣿⣿⣿⢿⣿⣿⣶⣿⣿⣿⡿⠿⠿⣿⣿⢿⡿⠟⣿⠗⠘⣿⡀
⠀⠈⠉⠉⠁⠀⠀⠀⠈⠛⠚⠛⠛⠚⠒⠒⠒⠀⠒⠂⠚⠒⠒⠶⠂⠀⠲⠾⠋⠁⠐⠒⠲⠖⠛⠓⠚⠛⠚⠛⠛⠃
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣒⣒⡒⠒⠒⠀⠀⠤⠤⠤⠤⠾⣤⣭⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⡯⠻⣶⢉⠍⢍⠍⠈⠝⡘⠐⡐⡒⡒⠢⠄⡤⠭⣥⠬⣉⣉⣀⢒⡒⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣼⠀⠁⣸⠠⡢⠢⠊⢊⢄⢔⠔⡐⡐⡡⡐⡠⡠⠠⡈⠠⢀⢂⠆⢨⢏⡏⣓⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠃⡀⠂⠼⠀⠁⠑⡁⠡⠢⢀⢊⢜⢔⢌⠔⠔⡐⡑⡠⡡⠣⢢⡚⡛⣾⣥⡏⣷⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠈⢠⠃⠎⠀⠀⠀⢸⠀⠁⠀⠀⢀⠀⠀⠀⠊⠀⠂⠈⠊⢈⠌⠄⢐⠱⠋⣼⢁⡇⡞⣨⢿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠌⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⣼⣸⢳⣏⢸⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣲⣏⣉⣘⠯⢚⠅⢢⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠘⢰⢟⡟⣾⣗⢸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⣬⣿⣛⣿⣤⣼⣇⣿⣤⣤⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢃⡏⡼⣾⣿⣃⣿⡇
⠀⠀⠀⠀⠄⠀⠀⠀⢰⡊⣿⣹⣏⠙⣶⡶⢋⣧⣿⣍⣩⣿⢭⣿⣈⣿⣧⢮⢿⣧⣭⣦⣀⣀⠀⡀⠀⣀⣠⠀⠀⠀⠀⠀⠐⠎⣸⣽⢷⣿⣁⣞⡿
⠀⠀⠀⠀⠀⠀⣀⣴⣹⣿⠩⣧⣼⣛⣟⢿⠿⣩⣿⣉⣉⢦⣴⣋⣩⣏⣹⠶⣾⢧⡼⢿⣾⣷⣯⡼⣷⣯⣤⡗⢤⣀⠀⠀⡘⡰⠅⠙⣿⢿⣋⠜
⠀⠀⣀⠤⠒⢩⣖⣿⣧⣛⡿⢿⣭⣥⣴⣷⠾⢿⣁⣉⣧⣾⣧⣤⣏⣉⣿⣶⠿⢾⣻⣿⣿⣯⣽⠓⣾⣤⣚⣿⣟⠑⣶⠮⣤⠝⠴⠵⠘⠋⠁
⠀⠀⡉⠄⠒⠂⡦⠠⠭⠍⣐⡛⠛⠡⠤⢯⣵⣟⠋⢻⣍⣩⠟⢻⡜⣿⣉⣵⣾⠞⢻⡷⢾⡛⡟⣿⠻⣭⣿⠿⢟⡫⠷⠺⢾
⠀⠀⡇⠀⣾⡖⠒⠺⠤⢧⣤⣊⣙⡉⠀⢒⡖⠀⠭⢉⣙⡛⠛⠿⢭⣅⣫⣝⣻⠿⢿⣉⣷⠿⠗⢛⣉⢭⢖⣚⣽⢴⡟⣧⡹
⠀⠀⠀⠁⣿⡇⣀⣀⡀⠀⠀⠀⠉⣩⣗⣬⡆⢰⡶⠷⢤⣀⣈⡏⠀⠒⡖⠈⠭⠡⣬⣥⡖⢊⣩⣿⡶⣾⣽⣱⣤⣿⡇⣟⣿⡄
⠀⠀⠀⠆⣿⡇⠑⠾⠧⠔⠂⠀⠙⠛⠿⣿⣷⢸⡧⠀⢀⠀⡀⠉⠉⠓⠒⠦⢄⠉⢸⠍⢽⡫⢉⣾⡏⠝⡈⣩⢉⣫⡉⡏⠫⠁
⠀⠀⢀⣀⣿⡇⠉⠐⠂⠀⠠⢀⡖⢒⣟⠓⢃⢸⡇⠀⠀⠀⠀⠽⠥⠒⠀⠀⢼⢀⢘⣥⣽⠷⠽⣿⡛⡛⣛⡙⣿⡛⡃⡛⢛
⠀⠀⢰⡇⢹⡗⠂⠠⠤⠀⡀⢀⡁⣼⣿⠄⡾⣻⡏⠀⠀⠒⠒⠂⠤⢀⡀⠀⢸⠈⢐⡡⣿⢂⣈⣼⢿⠟⢟⢝⠟⢛⡏⡿⢿⣇⣀⡀
⠀⠀⣸⠡⠘⠻⠦⢤⣤⣀⣀⣀⡉⠐⡖⠚⠁⣿⡏⠐⠌⣶⣒⣀⣀⡀⠀⠀⢩⠘⠀⠛⣾⡩⡡⡣⣠⡢⠾⠛⣺⣿⣧⢎⢺⡗⠂⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠉⠑⠒⠶⠨⡄⣀⡒⠉⠩⡍⢛⣡⠀⠀⣿⢧⠀⣀⠀⠀⠀⠀⠀⠉⠀⠲⢚⢈⣅⢯⢼⢔⢟⣝⣵⣵⡯⣿⡛⣻⣫⡿⣋⠉⠉⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠿⠏⣄⣀⣀⠄⠀⠒⣠⠩⢤⡶⢤⣄⡴⢈⢰⠃⣿⣶⣯⠴⡺⡹⣇⣴⠛⠛⠉⠁⠀⠀⠐⠒⠒⢂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠲⠦⠥⣘⣠⢀⠔⡌⢺⣱⣅⣡⢿⢖⠛⠋⠉⠠⠤⠀⣀⣐⠈⠉⠥⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠾⠶⠯⢭⣅⣀⣐⠂⣡⠬⠅⠀⡒⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃