Freak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⢀⣀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣤⣤⡶⡾⠛⠟⢉⣋⣻⢩⡩⠁⢓⠹⠋⠙⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣟⣏⣷⢄⠸⠟⠫⢛⡸⡉⠘⢛⣠⣽⢠⣡⣰⣤⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⠍⠹⡟⠁⠐⠈⠳⡀⠛⢈⣥⡌⠉⣩⣬⣿⡿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢨⢸⠀⠀⠾⢷⣾⣦⢡⣂⣀⡀⣀⣆⡈⠁⠋⢉⣌⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⢬⢦⣶⠷⠾⠋⠉⣿⣿⣾⣲⣿⢣⣸⡶⣶⣾⣿⣛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⣿⠿⢉⢀⠀⠀⠂⠙⣟⡙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡍⠀⠀⠘⣷⣷⣦⡀⠀⠟⠀⢀⣿⡙⢣⣴⣦⠀⠀⠍⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡄⠀⠀⠈⠙⠛⠓⠂⢺⡀⣿⢣⣿⣤⣺⣡⣴⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣵⢠⡀⢀⢀⡤⢤⣿⠀⣻⣧⢙⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢹⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣍⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠋⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢁⣋⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠘⠀⠈⠁⠐⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠶⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⡆⠀⠀⠀⡀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣴⣎⡉⠀⠀⠀⠘⣧⣀⠀⠈⢧⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄⠀⣀⠀⠀⣾⡠⠋⠀⠀⠙⣆⠀⢠⡞⠀⠈⠙⢦⡈⣧⡤⠤⣤⠴⠋⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⠁⠉⠀⢸⠁⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠈⡇⢸⡆⠀⠀⠀⢰⡆⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠀⠀⠀⠀⡆⡆⠀⠀⠀⠀⠀⢈⣷⠁⠀⠀⠀⠄⠀⣽⠀⣧⠀⠀⠀⣼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⡇⠃⠀⠀⢀⣤⣶⢾⣿⠀⠀⠀⣠⣾⠷⣿⠀⣿⠀⠀⠀⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⡏⡆⠀⢀⣾⣿⣧⣾⠟⣇⠀⣼⣿⣿⣷⣿⠀⣿⠀⠀⠀⡿⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠁⠀⠀⠀⢠⡇⢷⠀⢸⣿⣿⣿⡿⠀⢻⣠⣿⣿⣿⣿⡟⢠⡇⠻⡀⠀⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⢀⡴⠃⢸⡈⣧⣸⣿⣿⡟⠳⠆⠈⠻⣿⣿⣿⡿⠁⡾⠀⠀⠹⡄⡅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡦⠐⠣⡀⠀⠀⢳⡈⠻⠯⠍⠀⠀⠀⠀⠀⠈⠉⠋⢀⡼⠁⠀⢠⠁⠐⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠘⡄⠀⠀⠳⣄⠀⠀⠰⡶⠶⣦⣄⠀⢀⣴⠏⠀⠀⠀⣘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⡇⣀⡀⠀⠈⠷⣄⢠⠀⠀⠀⠘⢆⠈⢸⡇⠀⡠⠊⠉⠢⠀⣧⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠸⡀⠀⠀⠋⠀⠈⠣⡀⠄⠈⣿⠀⢠⠀⢀⣸⠀⢀⣷⠒⠳⡄⠀⠈⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣇⠀⠀⠲⠀⠀⠀⡁⠀⠁⠙⠒⣾⣷⣾⣿⠀⢸⡇⠀⠀⠁⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡿⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡇⠈⢉⠇⢰⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⣠⠏⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⡇⠀⠀⠀⠀⠀⠀⠰⣇⠾⠛⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠖⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⡄⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠁⣄⢀⡃⠘⢦⡀⠀⢠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⢠⡿⠀⡇⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠎⠀⠙⠿⣄⡀⠻⣄⡌⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⡘⠃⠀⠃⠀⠀⠀⠠⡅⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢦⠚⠛⢦⣤⣈⡙⢦⠈⠁⢰⠁⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⠇⢀⠀⠤⠄⠀⠠⠤⠥⠖⣰
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠂⠉⠥⠀⠠⠤⠃⠀⢹⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡾⠀⢸⡀⠀⠀⠀⠀⠒⠒⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⠃⠀⠀⠉⠉⠙⠲⠤⠤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠹⢦⡀⣤⢤⠤⠤⠤⢤⣤⣀⣀⡾⠀⢠⡏
⠀⠀⢀⡴⠶⢶⡥⠠⡈⢳⡀⠀⠀⠀⠀⠀⡴⢊⣑⡎⠀⠀⢿⣀⠀⣰⠇⡎⠀⠀⠀⠀⠀⠀⣸⠇⠀⡌
⠀⢠⡟⠀⠀⠈⣷⠀⠘⡀⢻⡄⠀⠀⢀⢎⠔⠋⠉⣣⠀⣠⠗⠬⠤⠥⠚⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⡎⠀⠀⠀⢸⡇⠀⠀⠹⣄⠙⠶⢖⣡⠊⠀⢀⣴⠇⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⡆
⢰⠆⠳⢰⢁⠄⠳⣄⠀⠀⠈⠓⠒⠋⠁⠀⠀⢙⡷⠶⠤⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣁⣀⡈⠂
⠀⠀⠀⠻⠀⠛⢶⣬⡳⠶⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠉⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢡⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⣄⣠⡇⠀⠀⠀⣀⣹⣦⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⢼⣇⠀⣀⣀⣙⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠
⢾⡆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⣷
⠀⠙⣇
⠀⢠⣿⣷⡀
⣠⣶⠞⣿
⣿⣽⣿⣿⣆⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠁⠄⠠⠂
⠀⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣰⣾⣡⣾
⢃⡀⣿⣿⠁⠸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⣿⡿⠁
⠈⢳⣿⣇⠀⣠⡿⣿⣿⣿⡿⠛⠛⠳⠾⣶⣶⣤⣶⣶⣶⣶⣿⣷⡆⠀⣶⣶⣶⣶⣶⣦⣤⣤⣶⠶⠶⠶⠶⠶⠶⢾⣿⣿⣿⣿⣿⣯⠜
⠀⣸⣿⣿⣶⣿⢷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠁⠟⣿⠀⣿⣿⣯⡿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢟⢠⣿⠏⣾⣿
⣴⣿⣿⣿⣿⡇⣼⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⡀⢻⡇⠘⠛⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⣰⣟⣹⣿
⣿⣿⣿⣿⡟⢣⡀⢷⡇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢡⣿⢷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣿⣾⣇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠚⠀⠀⢀⡠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⣈⣥⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣾⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡙⠳⡆⠀⠀⠀⠀⠀⠀⠂⢀⣾⣿⣿⣀⡦⠀⠀⠀⠀⠀⠀⣠⡟⠉⣠⣴⣿⣿⣿⣿⡟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⡔⢹⡄⠀⠀⠀⠀⠀⠒⠋⠛⠻⠟⠟⣇⠀⠀⠀⠀⠀⢀⡟⣠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣄⠀⠀⠀⠀⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⣠⣤⣿⣴⣦⣄⣀⣦⣠⣤⣤⣦⣶⣤⡀⠀⣤⢠⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣹⡿⣯⡿⣿⣿⣿⣿⣹⣽⣿⣿⣾⣿⠿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⣹⣿⣿⣿⣿⣿⣧⣿⣾⣿⣿⣿⣿⣿⡽⠿⣩⠙⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠉⢿⣿⣟⣿⢹⣿⣿⡏⣹⣿⡖⠉⠀⠀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⡀⠀⠀⠀⠉⠈⢛⡛⠻⠟⠞⠛⠃⠀⠀⠀⠀⠙⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⢦⡀⠀⠀⠀⠀⠀⠀⣴⠂⣿⣾⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣴⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣶⣇⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⠀⠀⠀⢰⣾⠀⠀⠀⢰⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣙⢀⠤⣤⣾⣿⣦⣤⣳⣾⡧⠘⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣷⣤⣠⣀⢀⠄⠀⢀⡀⢰⣼⣿⡿⡿⠿⣿⣿⠟⠀⠀⠈⣿⣿⡿⠿⠿⣿⡇⠀⠀⠀⣻⣷⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⣹⣿⡗⠀⣾⠆⣸⡏⠀⠀⣀⠀⠃⠀⠀⠀⠸⣿⣿⣿⣶⡆⡀⠀⢀⣴⣤⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣰⣋⣸⣿⢀⣿⠄⣿⣥⣤⣼⣿⣦⢲⡆⠤⠙⠶⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣾⠋⠉⠙⠿⣿⣇⣴⣿⡿⠑⠛⠿⠻⣿⡃⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜
⣿⡟⣿⣿⣿⣿⣿⡏⢀⣿⠏⠀⠀⠀⣀⣶⣦⡀⠀⠀⣀⠻⣿⣿⡟⠀⠀⠀⠀⠀⠹⠅⠀⠀⠀⣸⡆⠀⠹⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿

⠀⠀⠀⠀⣀⢠⢠⡘⡇⡀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⠿⠔⣾⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⢀⠰⡶⠀⣀⡀⠀⠀⠀⠀⠀⢠⣦⢠⣄⡿⢇⢀⣀
⠀⠀⠀⢠⡍⢊⣥⣤⣤⡙⠻⢀⡄⠀⠀⠀⠀⢠⡟⣁⣀⠀⠀⠈⠫⢅⣄⠀⠀⠀⠀⠘⡛⠄⠁⠁⠀⠈⠑⠢⡿⠃⠀⠀⢀⣀⠰⡦⣋⣬⣬⣅⠫⡞⢛
⠀⠀⠾⢗⡂⣾⠡⣶⣤⡉⢧⠹⠄⠀⠀⠀⠀⠙⠀⠧⠟⠀⠀⣏⡷⣬⣍⠀⠀⠀⠀⠀⠼⠛⠋⠂⠀⠀⢴⣶⠎⠀⠀⠀⠈⢉⣙⢰⡟⢠⣶⣌⢳⡌⢫⢤⡄
⠀⠀⠀⢸⡗⣌⠳⣬⣭⡥⠞⡘⠶⠀⠀⠰⠠⣶⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣻⡀⢿⡄⠻⣿⠀⣿⢸⠯⠁
⠀⠀⠀⠀⠀⢺⡝⠲⢰⡶⢸⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠦⠤⢠⠄⠀⠀⢠⣤⣤⠴⠠⣦⣾⠀⠀⠀⠀⠀⢀⣥⠗⣌⠻⠷⣦⠾⢃⢼⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢄⠐⠀⢁⠃⠀⠀⢀⣽⢱⢫⣝⢦⡱⣂⠀⠠⣉⢙⠃⠉⠉⣀⠀⠀⠀⠀⠀⠀⠉⠈⠩⠟⣰⡀⠦⢹⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠈⠀⠀⠀⠀⠸⠮⠳⠭⠴⣓⠁⠀⠀⠀⢬⡁⠉⠈⠁⠀⠀⠀⠀⠀⡀⡒⡰⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣤⢰⡦⠾⠒⠚⠦⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠎⠈⠀⠀⠀⠀⣦⠉⠛⠲⠀⠀⢠⣄⠀⠀⢲⣙⠂⠁
⠀⠀⠀⠀⠀⠀⢠⡦⠉⠀⠀⠀⠀⠂⠀⠱⡦⠄⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⠘⠿⢶⣤⣄⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠂⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⢎⣀⡷⠀⠀⠀⠀⠀⢀⡴⠀⠀⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⡄⣁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠶⠀⠰⢽⠆⠀⠈⣟⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢰⣷⣦⡀⠀⠀⠦⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠉⠙⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡄⣙⠁
⠀⠀⠀⠀⠀⣰⣿⣇⠲⣤⣄⣀⣠⣠⣴⡞⣹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣇⣀⣤⡄⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠻⠈
⠀⠀⠀⠀⣼⣿⣿⣿⡄⠈⠙⢻⠛⡋⠂⣠⣿⣿⡿⣧⠀⠀⠀⠀⢠⠼⠂⢖⠀⠀⠀⢛⣅⡀⠈⠻⠿⠻⢿⣿⡷⣦⠀⠀⠀⠀⠀⢀⣄⠀⣶
⠀⠀⠀⢰⡏⢿⣿⣿⣿⣦⣤⣀⣀⣠⣾⣿⣿⡿⢁⣿⠀⠀⠀⣙⢰⢵⢈⢂⠁⠀⠀⢠⡿⠏⠙⢦⣤⢤⣀⣤⣠⠮⠀⠀⢠⣄⢻⣆⠫⠤⠏⠴⠟⢀
⠀⠀⠀⣼⣿⣄⠉⠛⠿⠿⠿⣿⣟⢡⣿⡿⠛⢁⣾⣿⡇⠀⠀⠀⠎⢖⠥⠘⠂⠀⠀⠀⠁⠐⢋⡶⢶⠶⣶⠾⢟⣷⠀⠀⠠⡌⠉⠀⠀⠀⠀⠀⠀⢥⠄
⠀⠀⠀⣿⣿⣿⣿⣷⣶⣄⣀⣿⡿⢿⣸⣷⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛⠃⠉⠀⠿⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⡔
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⢙⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⠀⣤⠀⣦⢈⣿⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠐⢠⠠⡄⠀⠘⠁⠀⠀⢀⣴
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣺⣿⣿⣿⣿⠿⠛⠁⠀⠀⣠⣙⡣⠈⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⣆⢥⢎⠀⠀⠀⠀⠀⡴⣤⣄⣀⣠⣴⣦⠀⠀⠀⠜⠙
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠙⠉⡁⢠⣤⡀⠀⠀⠠⡰⢶⡂⠀⠀⠀⠓⠙⡑⠂⠀⠀⠀⠰⣄⠀⠉⠉⠛⠛⢁⡧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⡉⠓⠢⣀⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣠⡤⠤⠤⣶⣿⠆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⣷⣿⡆⠀⠉⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣆⣰⢄⢀⠀⠀⠀⠀⣀⠀⠀⠈⢠⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⡁⠈⢷⡤⠤⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣉⡀⠀⠈⠣⡤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠗⡀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⢶⡤⠔⢆⡀⠀⠀⠀⠰⠘⠶⠃⠀⣤⢤⠱⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⣼⣶⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⡀⠀⢀⣿⣿⣿⣿⣄
⠀⠀⠀⢶⠋⡔⣚⢦⡉⢞⠁⠀⠀⠀⠀⠀⠀⠙⠋⠴⠂⠀⠀⠀⠧⡀⠀⠀⣠⡿⠋⠹⢆⠜⠁⠈⡆⠀⠀⠀⠀⠀⠀⣨⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠞⡀⢧⠻⠆⣷⠸⠃⠀⠀⠀⠀⠀⠀⠀⣴⠄⡀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢀⣴⠚⢻⡿⣿⣿⣿⣿⣿⣿⠻⣧
⠀⠀⠀⠀⠴⢢⣙⡚⡅⠋⠉⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⢀⡀⢀⠠⠐⠛⠛⠀⠀⡄⣌⡝⠀⠀⠁⢠⣿⠀⠞⢨⣿⡄⠃⢸⣷⡄
⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⠀⠀⠘⠒⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠛⠉⠉⠛⢧⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠙⠋⠛⣷⡀⠀⠀⢀⣀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⢰⡄⢀⠀⠀⡠⠀⠀⢀⠠⣀⡙⠳⠶⠏⠥⢀⣀⣨⣿⣻⡆
⠀⠀⠀⠀⣤⡾⡟⠁⢀⡴⠓⠋⠋⠁⢀⣶⣄⣤⣥⢪⣵⣆⣀⢠⣤⣧⣶⣿⣷⣿⣤⣤⣶⣶⣾⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣾⣿⣿⡄⠀⣾⢁⠀⠀⡀⣙⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣷⣜⣛⣥⣦⣄⠶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⣠⣴⡶⠿⠛⠛⢿⣿⣶⡄
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣜⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⡏⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠟⠁⠉⠁⠀⠀⠀⢀⡼⠋⠀⠀⠀⠙⣿⠟⠀⠀⡰⠋⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢹⡇
⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣆⠀⠀⡀⠀⠀⣠⡟⠀⢀⡰⢦⣢⠀⣿⠀⢀⡼⠁⠀⢀⣾⣿⣿⣿⣥⣮⡙⠃⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⢀⣴⣿⣿⠿⣿⣉⠻⡅⢸⡇⠀⢀⡟⢠⣶⣾⣿⣿⣍⢳⠙⣄⣾⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡜⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠻⣆⠈⠙⠂⣿⠈⢫⣴⠟⣠⣿⣿⣿⣏⢙⣿⠀⡿⣿⡏⠀⡸⣹⣿⣿⣿⣿⢿⣿⣿⣿⠃⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣬⣿⠁⠀⠀⠘⡆⠀⢠⡟⠀⠀⠛⠹⠟⠘⣿⠻⠿⣿⠟⢠⡷⣿⡇⠠⣿⣿⣿⣿⣿⡟⠰⢫⣿⠏⠀⣴⠏
⠀⠀⠀⠀⠀⠀⣠⡿⠛⠋⣽⣾⣿⢰⣾⣷⠀⠘⣦⠘⣷⠌⠀⠒⢦⠘⢄⡀⠀⠉⡡⠔⠋⢀⣿⣿⣄⠀⢹⣿⡿⢺⠇⠀⠀⠃⣠⡾⠋
⠀⠀⠀⠀⠀⢀⣿⠆⠀⣀⣽⣿⣿⣧⣁⣩⠤⠚⠁⢰⣿⠶⠛⠛⢶⣄⣀⠈⠉⠁⠀⠀⠐⠋⠉⣿⣿⢰⣿⠁⠀⠈⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⠆⣠⣼⠋⠀⣀⣤⣤⣍⣉⣉⣉⠀⠠⣄⠀⠀⠀⣨⡿⠛⡁⢀⣤⠶⠾⠋⠁
⠀⠀⠀⠀⠀⠈⣿⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣀⠀⣿⣿⣿⣿⡿⡿⣿⡇⠀⣿⠀⢤⣾⣿⣷⣴⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣶⣄⠉⠁⠘⢿⣿⣿⣿⣿⢏⣼⣿⣷⠙⢿⣿⣟⠓⠀⠛⢁⡾⠛⢧⣤⣉⠛⠉⠉⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣤⣀⣸⣿⣿⣿⣿⠀⠛⠛⠛⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠈⠙⠻⣶⡄⠀⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⣿⣿⣿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⣷⠀⠀⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⢄⣤⣲⣿⠁⠀⡏⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢀⣴⠳⣄⣠⡶⠇⣴⣾⣿⣿⣿⠻⣍⢿⠀⠀⣤⡴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠰⠀⠈⣿⣿⣷⣤⡀⠀⣾⣿⣿⠏⠉⢿⠀⠈⡏⡆⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠇⡟⢸⠟⠳⣾⠟⠈⠹⣇⠀⠘⣧⠀⣿⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣥⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⢀⠀⠓⢿⣰⡀⠀⣿⠀⠀⠀⢹⡆⠀⢹⣾⣿⠀⢠⣾⠃⠀⠀⠀⠀⣠⠞⠉⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣄⠀⠛⢧⣄⣽⣇⠀⠀⢸⣿⣴⣿⣿⡟⠀⠀⢻⣆⠀⢀⣴⠊⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⢹⢿⣿⣤⣶⣾⣿⣿⣿⣿⡇⠀⢠⡼⠃⣠⠞⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢀⣸⣿⣿⣿⣿⣿⣿⣻⡿⣿⢁⡀⢹⣧⠞⠁⠀⠀⠀⠀⣰⡿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⣿⣿⣿⢿⢻⣿⣿⠓⠉⠛⠿⠦⠾⠛⠁⠀⠀⠀⠀⠀⣶⡿⡀⠀⠉⠢⡀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⣿⣿⣿⣿⠃⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣣⠇⠀⠀⠀⡽⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠐⢿⣿⣿⣿⡇⠀⠀⠈⠉⠓⠂⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⡿⠿⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⣿⢿⣿⣷⣢⡄⡀⠀⠀⠀⠀⣀⡀⢠⣶⣿⣿⣿⡟⠋⢁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⠻⣿⣿⣾⣯⣿⣄⣹⣯⣽⣷⣿⣿⣿⡿⠉⠀⠀⣸⠠⣄⡀⣶⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⣶⡿⢷⠾⠛⠛⠻⣧⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠈⢉⡉⢺⣿⣿⠇⠀⢀⣀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠛⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣄⣀⣀⣠⣶⣿⠟⠙⠿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⠔⢀⠤⢒⣩⣴⡿⠓⣉⡥⣒⣭⣶⣷⠖⠛⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠏⠐⠋⠙⠉⠉⠁⠚⠋⠙⠋⠷⣿⠏⣵⡾⢟⣃⡉⠀⣲⣯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣰⠿⠛⠂⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠘⠓⢾⣿⣿⣿⣟⣯⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡡⠔⠂⡄⠀⠀⠀⠀⣠⠖⠋⠉⠙⢢⡀⠀⠀⠀⠀⠀⠈⠘⣿⡿⠛⠥⢔⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡾⠃⡼⠁⠠⠀⡇⠀⠀⠀⢸⠇⠆⠀⠀⠀⢀⡇⣀⣀⠀⠀⠀⠀⠀⠐⣿⣾⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡙⣦⡾⠁⢸⡇⣀⣠⠾⠃⠀⠀⠀⢻⡀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⣠⠞⠉⢁⡀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⣿⠇⠀⣨⡿⠛⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⠾⢿⠃⠀⢀⣠⡄⠀⢀⠞⠁⢀⡴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢨⠀⢷⡟⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⣹⡆⠀⠀⠰⢿⡟⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠂⢈⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣥⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟⣋⣁⢠⠀⡀⢤⣴⣿⣿⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⠀⠀⠀⠀⣀⣤⣾⣿⡿⠏⢹⡞⠁⢰⠃⣠⣇⣠⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⡿⠿⢿⣷⣶
⠀⠀⣠⣶⣿⣿⣿⣷⣿⣿⣧⣤⣤⣶⣾⣿⣿⣿⣿⣤⡦⠾⣷⠶⠟⠛⠛⠙⠿⠀⣿⣿⠇⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠈⠉
⢠⣾⣿⣿⣿⡿⠿⠛⠛⣿⣿⣿⣿⣿⣿⠟⠯⢿⡟⠉⠀⠀⡥⠀⡄⢀⡀⠀⣳⣠⠿⠃⠀⠀⠀⢀⣼⣿⠃
⣿⣿⣿⡿⠋⠁⠀⠀⠀⠈⢧⠀⠀⠀⠉⠑⠂⠼⠧⣤⣇⣀⣁⣠⣥⡤⠿⠚⠉⠀⠀⠀⠀⠀⣠⣾⠟⠁
⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠉⠒⠂⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠉
⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣤⣀⣤⡤⣴⣶⣶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⡆⠀⠀⠀⠀⢀⣀⣤⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⣾⡆⠀⢶⣦⡀⠀⠀⣼⡏
⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡏⠉⠿⢿⣿⢛⣿⣷⡶⠟⢷⣾⡏⠉⢹⡟⢛⣿⣁⣴⣿⠟⠁⠀⠀⣿⡇
⠀⠀⢸⣧⣤⣶⣶⡤⠠⢴⣿⣿⣿⣿⣿⠟⠻⣿⠷⡶⣿⣿⣷⣤⣾⣷⣾⣿⣉⠁⣀⣤⡄⠀⠀⠿
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠘⣧⠠⠿⠛⠻⠶⠿⠷⣦⡼⠟⠛⠁⢸⣿⡏⠈⠙⢷⣿⣛⣿⠄⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⠙⠉⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣈⣛⣤⢠⣄⢠⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡿⡿⣿⠼⠿⠾⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⠒⠛⠛⠉⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠙⠍⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⠀⢀⠤⡰⠊⠀⢀⠋⠀⣠⣤⠄⠤⢤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠘⠀⠀⠠⠀⠠⡞⠠⠀⠀⢀⠈⡇⠀⠀⡀⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⢠⠐⣦⡑⠀⠁⠀⠐⠂⠀⢠⣮⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⢱⡗⠓⠒⠐⠀⠀⠁⠈⠛⢉⡠⠞⠤⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠟⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⢰⡾⠃⠀⠀⢀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠘⢃⣀⠑⠀⠀⣿⠂⢠⠄⣤⠄⠀⠀⠐⠚⢻⠛⣇⠘⠊⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠙⠒⠀⠋⢀⠀⠀⠰⠆⠂⢐⠄⡀⠀⠁⠈⠓⠈⢙⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠀⠀⡀⠠⠚⠀⠉⠀⠀⠀⠀⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡠⠜⠒⠂⢉⣉⣙⠛⠲⣄⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡠⠒⣡⠔⠒⠋⠉⠉⠉⠉⠛⢦⡈⠳⢤⡊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠔⢉⡴⠋⠀⠀⠀⣠⣶⡖⠒⣤⡀⠀⠉⠢⣀⠙⣎⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⠏⠀⠀⠀⠀⠸⣿⣿⠯⠼⢻⠇⠀⠀⠀⢸⣷⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠳⢄⡀⠀⠀⠀⠈⠁⠐⠒⠉⠀⠀⣠⠔⢋⠄⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣈⠓⠠⢍⠓⠶⠤⣤⣀⣀⣠⡤⠔⢊⣡⠔⢁⠔⠊
⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⡀⠀⠀⠐⠒⣀⠈⠀⠄⠘⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠨⣍⠉⠉⠙⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⢀⡟
⠀⠀⠀⠙⣧⠀⠀⠀⠛⠒⠒⢶⠂⠤⠤⢤⣤⠀⢻⡗⠒⠶⢤⠄⠀⠀⣦⠀⠈⡇⠀⠀⣰⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠸⣇⡤⠒⠉⠀⠀⠘⡇⠀⠀⠀⠀⢀⡴⢻⡆⠀⣿⣠⠞⠁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠤⠬⢷⡤⠤⠤⠤⠬⢿⠤⠤⠤⠤⠤⠤⢧⠤⠤⢤⡴⠿⠴⠶⢷⠒⢻⡟⠒⠒⠒⠚⣻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⡀⣠⠟⠁⠀⠀⠀⠸⡆⠀⢷⠀⠀⠀⢀⠃
⡀⣀⡀⣀⣀⣀⢀⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠟⡁⣀⣀⣀⣀⣀⡀⠉⢀⣸⣆⣀⣀⣈⣀⣀⣀⢀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⡖⠀⠤⡀
⠀⠀⡾⠙⢏⣹⠟⠀⠀⠀⠀⠐⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡇⠀⠀⠀⠀⠉⢳⣤
⠀⠀⠁⠈⢲⠗⠄⠀⠀⠀⠀⢀⠓⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁⠀⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠉⠀⠀⠀⡀⢤⠀⠈⠂⠀⠈⢡⣀⡀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠁⣀⢀⡀⣀⡸⠁⡰⢄⡠⠄⣠⣼⡄⣠⣤⣤⡠⢤⣽⣤
⠀⠀⠸⠀⠀⠀⠀⢏⣟⡿⢿⠿⠼⢻⠿⠃⣤⢺⣟⠀⠸⠋⣽⡟⠁⠈⠉⢘⠋⢸⡇⠂
⠀⠀⠉⠉⠋⠉⠉⠉⢸⣧⣤⣶⣶⢠⣄⣀⠉⢩⣄⠀⣠⠎⠉⢭⣘⡆⠿⡿⠿⣿⣷⠀⠀⠀⠀⡀
⠀⠀⠠⢰⠀⠀⠆⠀⢿⣿⣯⡛⠉⠙⡌⠠⣚⡲⠙⠂⠁⢒⣁⡠⠸⠃⠀⣠⣾⡿⢻⠀⠀⠀⠀⢀
⠀⠀⠀⣸⢀⢀⠀⣀⣸⠙⢻⣿⣦⡊⠁⠠⢹⡍⣬⣠⣤⠁⠿⠫⠂⠀⣾⣿⣥⣀⣸⡆⠀⠠⠀⠘⡆
⠀⠀⠉⡏⠉⢹⠀⢀⣴⣶⣿⡿⠿⡁⠀⢠⠀⢹⣿⣿⣿⠈⠀⢠⠂⠀⠊⠉⢻⣛⣿⣿⣦⣤⣢⣇⠁
⠀⠀⣠⠁⠀⣼⣉⢹⣿⣿⡗⢠⠒⣁⡀⣾⠁⢸⢿⣿⣿⠀⠀⢾⡄⠤⠠⠙⠉⡸⠟⢻⡍⠉⠙⢿⡇
⠀⠀⡽⠀⠰⠇⠀⠀⠈⠉⠓⠰⡟⠛⢃⣿⡁⠀⠾⣿⣿⠰⠄⢸⡷⠋⠉⠉⠀⠃⠐⠚⣧⠂⠒⠀⠉
⠀⠀⠇⠀⠹⠤⠤⠴⠾⠤⠾⠶⠶⠿⠶⠿⠓⠐⠒⠻⠛⠒⠒⠘⠛⠛⠚⠒⠒⠓⠚⠛⠛⠒⠒⠛⠓

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠈⡇⠸⠇⠸⠀⠚⠆⠛⠄⠀⠨⠄⠃⠇⠿⠀⠖⠆⠀⠀⠘
⠀⠐⠶⠒⠲⠾⠟⠒⠀⠋⠒⠐⠚⠓⠚⠙⠋⠉⠃⠈⠉⠃⠉⠙⠋⠉⠛⠉⠛⠊⠍⠛⠚⠛⠒⠺⠓⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠒⠒⠀⠐⠂⠀⠀⠁⠁⠃⠀⠀⠈⠀⠀⠃⠀⠀⠰⠤⡀⠀⠀⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠠⡢⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠔⠄⠀⠀⠀⠐⠉⡃⠀⠁⠀⢀⠀⠀⢐⡄⢠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠘⠛⡁
⠀⠀⠀⠀⠀⠀⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡀⢀⠁
⠀⠀⠀⠈⢀⠈⠁⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⢰⠂
⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⢤⠠⢔
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠋⣷⠹⣀⠸⠼⠢
⠘⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠚⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⢀⠀⢰⣄⠀⠀⠀⠀⠀⠀⢠⡞⢣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠩⠉⠀⠀⠀⠀⠘⠄⠊⠓⠓⠢⢻⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⡿⢿⣿⣤⣤⣴⣼⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠶⢾⣿⠛⣛⣧⣤⡴⢾⢯⡕⣾⣤⡶⠀⣶⣶⣹⢶⣬⡿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢭⣿⣤⣀⣴⣶⠭⠝⠙⣉⡉⠹⢶⠎⢑⡾⠙⠛⣿⡄⢸⡿⣿⡀⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣖⡶⣶⣶⠶⣶⡖⢿⣾⠋⢿⡏⢃⣴⠯⣄⡼⣟⣛⠷⠼⢟⠘⠳⣿⠷⠶⠿⡟⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣿⣿⣰⡮⠽⢿⣿⣦⣾⡳⣎⣿⣿⡼⠛⠙⣃⠤⣶⠾⢛⣯⡝⣶⣟⢿⣿⣗⣶⣦⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⢿⣿⡛⢳⠀⠘⣿⣇⠘⣿⡍⠋⢉⣀⣶⡟⢛⡞⢳⡏⡟⢘⣿⢸⣟⠿⣿⣿⠟⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣼⣿⣧⡀⠀⡰⠟⠛⠋⣀⣤⣶⠀⢿⣥⣷⢎⡿⢾⣧⣷⣿⣿⣼⣿⣦⡙⠏⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⣀⣘⣇⣀⣼⣶⣴⣿⠉⠀⢸⡆⣴⣿⡿⢹⣏⣾⣿⣿⣷⠋⠨⢉⣃⡻⡤⢫⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣿⣿⡏⠉⠉⠉⣹⣟⣿⣿⣿⣷⣀⣼⣿⣿⣁⣀⣠⣿⣿⣿⣿⣟⣛⣛⣋⠙⣛⣻⣬⠃⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⡄⠻⣷⠀⠀⠀⢿⣿⡿⠟⠉⠙⣿⣿⣿⣿⣿⣿⣿⡿⠏⠘⣿⣿⣿⡟⠙⠛⠋⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⡘⣧⡀⢠⣿⣿⣧⡄⠰⢶⣿⠿⠟⠻⣿⡿⢿⡶⠶⠀⣤⣭⣿⣇⠀⠀⣀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣌⢳⡜⢿⣿⣿⠇⠛⠚⠁⡀⢀⣴⡯⠄⠀⠙⠒⠂⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⠃⣼⠟⠁⠀⠠⠠⣤⡜⣷⢩⡟⣴⠆⡄⠀⠀⠀⠉⠻⣇⢀⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⣿⣿⣿⣿⡟⢸⡏⠀⠀⠀⢰⣷⡌⣷⣿⣿⣿⣱⣿⢿⠀⠀⢶⣄⣠⣼⣿⣿⣿⣿⣿⣿⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠉⢻⣿⣿⣷⣾⣥⣄⣶⣀⣸⣾⡿⡿⣯⣿⣿⢿⣷⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣯⢇⣿⣿⣿⣧⣼⡿⠛⠛⠻⣿⣿⣿⠿⠿⠮⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣛⡆⠀⣴⡿⠿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣾⣷⣖⣿⣿⣿⡇⢼⡿⠛⢸⠏⣿⣦⠐⣢⣛⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣍⠋⠙⠛⠛⠛⢶⣼⣿⣿⣿⣿⣽⣟⠢⠿⢿⡿⣿⣿⣿⡿⣿⣿⣦⣤⣴⡿⠛⠛⠛⠛⠛⣩⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠐⢠⣾⢿⣿⣿⣿⣿⡁⡽⢷⣶⣶⣾⠷⠙⠁⡱⣝⢿⣿⣿⣿⣷⡄⠐⠂⣄⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠶⠤⢾⣿⣼⣿⣿⣿⣿⡟⡁⠀⠙⠁⠋⠀⠀⠀⣿⣮⣻⣿⣿⣧⣿⣧⣴⠾⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⣷⣶⢀⣌⣩⡉⠻⠿⣤⣶⣤⣤⣾⣏⠀⢀⣼⣇⠈⠝⣋⣡⣤⣤⣶⣶⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣬⣍⣙⡻⢼⣷⣦⣤⣉⣙⣛⠛⠛⣃⣉⢩⣤⣾⡛⠿⢓⣋⣩⣭⣵⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠈⠉⢹⣿⣿⣦⣍⡓⠸⠏⢿⡅⠸⡯⠿⠞⢛⣡⣴⣿⣿⡏⠉⠀⠀⣿⣿⣿⣿⠇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡏⠛⠻⠶⣶⣶⣶⣶⠾⠟⠛⢹⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣛⣛⠛⡋⢩⠩⠭⣥⠀⠒⠶⠦⠾⣧⢿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣛⡛⠋⠉⠉⠉⠀⠐⠒⣰⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠁⠤⠀⠀⣢⡀⣾⣺⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⡇⠀⠀⠀⠀⠀⠀⣬⣿⣙⣿⣭⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⣿⣿⣿
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡜⠀⠀⠀⠀⠀⢸⣿⣿⠩⡍⠋⡇⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠙⠈⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⠀⠀⢹⠆⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠀⠘⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠖⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣀⡀⢀⣤⣶⣾⣿⣿⣿⣿⣯⣭⣛⣛⠻⢿⣄⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⡶⠎⠓⠀⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⣠⠤⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣀⢸⣿⣿⣿⣿⣿⣿⣷⣵⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⣶⣄⠀⢀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡾⠳⠆⠀⠀⠀⠀⣾⣿⣿⣿⠊⣿⠁⢸⣿⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠐⠌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠘⠷⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢘⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣀⡀⠀⠂⠀⠒⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢨⣧⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠫⠄⣑⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠐⣿⣿⣿⣿⢿⣿⣿⣿⠟⠉⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡛⠶⣌⠁⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢸⣿⣿⣿⣿⣷⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⣠⣼⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡌⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣾⣿
⢸⣿⣿⣿⣿⡽⣿⣿⡿⢿⣦⠀⢀⠀⠘⣿⣷⡆⠀⠀⠐⠀⠀⠀⠀⡌⠉⠉⡉⠍⢙⡟⠛⢛⡛⣛⡛⠛⡛⠛⢿⠿⣿⣿⣿⣿⣿
⢸⣟⣿⣛⣿⣿⣿⣿⣿⣮⣿⣧⡄⠀⠀⢹⣿⣿⡖⢀⠀⢰⡄⠠⠆⣀⣀⡤⡄⢉⣿⣟⣻⣾⣿⣿⢷⣿⣐⠹⣾⣷⣾⣿⣿⣿⣿

⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⠆
⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣣⠞⠁
⠀⠀⠸⡄⠛⠦⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠖⡉⢘⠉⡆⠀⠀⢀⣀⣤⣤⡤⠶⠶⠚⠛⢋⡔⠁
⠀⠀⠀⠙⠲⠴⢦⠀⠀⠈⠉⠙⠛⢶⣄⠀⣀⠞⢀⣴⠥⣾⠀⢰⣠⠶⠋⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠙⢉⡏⣠⢎⠀⠀⠈⣆⣸⠉⠂⠀⠀⠀⠀⢀⣞⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣴⠟⠀⢨⡇⢰⡯⠹⣿⠀⠀⣤⠀⠀⢀⣾⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⡷⣄⠀⠀⣰⠃⢠⣿⢰⡇⢸⣻⣷⠘⡆⠀⣿⠳⣄⣸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠜⢦⢾⡇⠀⢄⢩⡾⠀⠘⡧⠉⠄⢹⡄⢿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣼⠃⠀⠀⢳⠀⠀⠘⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠃⡌⠳⢤⡾⠃⠀⠀⠀⢸⠷⡤⠞⣽⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠰⠤⠞⠁⠀⠀⠀⠀⠀⠓⠤⠵⠛⠀⢀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠘⠓⠲⠀⠀⠖⢲⡆⠈⡋⠉⡆⠐⠚⢹⡏⠀⢀⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢨⠋⠀⠀⠀⠀⠀⡆⢸⠀⢀⣀⣸⣧⣤⣷⣶⡇⠀⡄⢸⠁⠀⠀⠙⠃⠁
⠀⠀⠀⠀⠀⠐⢁⠄⠀⠀⠀⠀⠀⢩⠈⠀⣿⣿⣿⣿⣿⣿⣿⣧⠐⠀⣼⠀⠀⠀⠀⢬⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡀⠛⣬⠛⠁⠈⠀⠈⢛⣉⠀⠀⡿⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣀⣀⠀⢠⠤⢼⠁⡆⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢔⡃⠀⣀⣀⣀
⠀⠀⠀⠀⢼⠁⠀⠀⠞⢓⠒⢆⠀⠀⠰⣒⠀⣉⣠⣀⣨⣀⣈⠀⠋⠙⡇⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠆⠾⠃⠉⢊⠥⠋
⠀⠀⠀⠀⡐⠀⠀⣠⠂⠘⠁⢉⣀⡀⠈⠉⠉⣀⣀⣀⣀⡀⠈⠉⠉⠚⠃⠀⠀⣀⣀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠴⠒⠢⠤⣆⠀⠀⠀⠒⠀⠠⢤
⠀⠀⢀⠜⠀⠀⡰⠁⠀⣄⠀⠠⢤⡀⠉⠆⠀⠁⢙⢩⠉⠉⠃⠀⠀⣀⣀⡠⠞⠁⢘⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⣀⠀⠀⠀⠀⠀⠛⣁
⠀⢀⠎⠀⠀⢸⠅⠀⡠⢬⡉⠁⠈⠱⡉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠏⣠⡠⢻⠀⠈⠓⢤⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣭⢡⡔⠀⠀⠀⢋⠁⠈
⠀⡎⠀⠀⠀⠦⠠⠲⠈⢦⡈⢦⡀⠠⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠄⢁⡀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⣐⡠⢾⠀⠴⢌⡀
⢸⢀⠔⠀⠀⠀⢸⠀⠆⢀⣀⡂⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⢚⠥⠁⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢘⡄⠀⠀⠁
⠘⡏⠀⠀⠀⠈⠈⠀⠈⠛⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡝⠁⢀⠞⡹⠀⢀⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠟⡼⠊⣰⢀⠎⠀⠀⠀⠈⣆⠀⠄⠀⠀⠀⠀⡰⡰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡤⠦⡌⠀⠀⠀⠀⢀⣻⠁⠀⠉⠐⠠⠾⠊
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡼⠟
⠸⠊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠁
⠀⠀⠀⠀⠀⠙⢢⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⢤⡀⢀⣀⣠⠚⠋⢄
⠀⢠⠂⠀⠰⠂⣁⢀⠜⠩⠤⣤⠀⢱⡀⠀⠀⠀⠀⠀⢰⠋⢀⡴⠶⢀⡟⢁⡤⣈⠧⣀⠜
⠀⠈⠔⠓⠢⣤⠬⠜⢦⢀⠰⠥⠐⠊⠈⠒⠢⠤⠦⠤⠼⢦⠤⢵⠀⡇⠈⠘⠴⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠤⠄⢀⣤⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⣑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠀⠀⠖⠛⠉⠉⠁⠀⠁⠀⠀⢈⠀⠀⠱⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢈⣴⠁⠀⢸⠃⠀⠀⠁⠀⠀⠀⠀⡞⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⣧⠆⠀⣞⠀⠉⠒⠀⠀⠀⠀⠀⢈⣻⡤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠓⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢏⠈⣉⠵⢄⡀⠀⠀⠀⢀⡲⠴⠦⠤⠄⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠝⠂⠠⠀⣀⡀⣀⣀⣀⣀⣀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠤⠀⡀⡀⠀⠀⢠⠀⠀⠁⠃⠇⡇⠾⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠁⠓⠀⠈⠉⠈⠈⠁⠀⠁⠁⠉⠛⢈⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣀⠠⠤⠤⡄⠄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⣠⢪⠕⠊⠀⠀⠀⠀⢀⡖⠃⢀⣐⡉⠀⠈⣉⠁⠀⠀⢈⡁⠀⠀⠀⣄⣀⠀⠈⠹⡉⠉⠒⢎⠵⣆
⠠⠰⠘⠂⠀⢄⠀⠒⠄⠀⣠⠈⠀⢀⡈⠀⠀⠀⢀⠀⢀⣀⣀⣀⣄⣀⡀⠀⢘⡀⠀⢀⣀⡉⡀⠀⠈⢁⠘⠤⡀
⠀⠀⠀⠀⢀⢾⠄⢂⠆⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⡅⠉⢹
⠀⠀⠀⣠⠃⠀⠠⠚⠂⠈⢉⠀⠄⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠰⡁⠆⠐⠄⡠⠤⠌⠁⠐⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⡸
⠀⠀⠀⠈⠁⠒⠚⠀⠀⠈⠉⠁⢧⡠⢤⠤⠤⠤⠴⠤⠤⠀⠀⣠⣀⣀⠀⠠⠤⠤⠤⠤⠤⠦⠤⢄⣀⣁⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠋⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠢⣄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⡆⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⠿⣤⣤
⠀⠀⠀⠀⠀⠀⣴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠛⢻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡏⠙
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣀⣀⣀⣠⡤⠚⢁⠀⠀⠢⠌⣤⣀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠈⢫⠀⠀⢀⣥⢤⣄⠀⢰⠉⠉⠻⣎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡃⠀⢸⠁⠀⠉⢧⠸⣄⠀⠀⢸⡀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡆⠀⢸⡀⠀⠀⢸⠂⠈⠓⠒⠉⠹⣄⠀⠈⠛⠶⠶⠤⠤⠤⠶⠖⢛⡗
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣿⠀⠏⠳⠤⠤⠞⠀⠀⠉⠉⠉⠉⣿⢷⡀⠀⠀⠀⠀⢀⣤⣴⣶⠭⣤
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠾⠃⠀⢳⡀⠀⠀⠀⠀⢀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠌⣻⠀⢠⠀⠀⠀⠀⢀⠖⠒⠲⠦⣤⠄⠀⠀⣧⠀⠀⠀⠙⠻⢭⣉⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠟⠶⠛⢦⡀⠀⠀⢸⠟⠷⠒⠶⠁⠀⠀⠀⡿⠀⠀⠰⣤⣄⣀⡼⠛
⠀⠀⢀⡠⠞⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⣧⡠⣴⠚⠓⢦⢰⠇⢀⡤⠞⠉⠀⠉
⠶⣞⣉⣀⣠⣴⡾⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣾⠃⢀⣤⣤⣤⠞⡿⠞⠉
⠀⠈⢉⡭⠟⠁⠀⢀⣠⢺⠏⠀⠀⡆⠀⠐⣧⠀⠀⠀⢳⠀⡇⢀⡏⢀⣀⠀⢠⡇
⠀⠀⠙⠶⠤⠴⠖⠋⣠⠏⠀⠀⢰⢻⡀⠀⢻⠻⣤⣤⡴⣇⣧⡼⠷⡟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⣀⠴⠃⠀⠛⠶⠾⠃⠀⠀⠀⠸⡿⢆⡼⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢀⡤⢄⡸⠉⡦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠳⡌⠛⠰⡷⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⣀⠴⠲⣤⠔⠃⠀⠠⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣏⣀⣀⣀⣨⡏⠛⠛⠞⠁⠀⠀⠹⡖⠉⠉⠳⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡤⠴⠿⠁⠈⠉⠉⠉⠁⠀⠀⠀⠲⢤⣠⣄⣠⡷⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢀⡼⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⢿⣠⠞⢉⣠⠾⡇⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⠓⠲⠇⠘⠃⣴⠯⣄⠸⠟⢹⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⣀⡴⢻⠟⢀⣀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠹⠇⢠⠤⠶⣄⣀⡼⠀⠀⡏⠀⢀⣠⡤⠴⠚⠋⠉⠀⠀⠀⠀⣠⠞⢻⣶⣁⢞⡻⠟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠈⠉⠀⠀⠈⢹⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⡽⣆⠁⢈⢷⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⡴⠾⠋⢛⣁⡀⠘⣧⣌⣇⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⢟⢉⣹⣇⣠⠴⠚⡩⠝⢻⠄⠹⣆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⢁⡴⠋⠀⣠⠏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡴⣫⠋⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⡴⠉⣩⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠚⠁⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⠔⡛⠲⡌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡘⠛⣠⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢬⡉⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢀⣶⠭⠦⣤⡀⠀⣤⠶⠶⠶⣭⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⣀⠀⣾⠁⢀⣤⡀⢻⣾⠁⢠⣶⣖⡌⣿⠶⠛⠛⠻⢿⡆⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⡼⠛⠻⢿⣿⠀⠿⣷⠟⢸⡿⣄⠀⠛⠛⣡⡟⠀⠠⣾⡆⢸⣿⣠⣤⡤⣀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢠⡏⣠⣤⡀⠀⣿⣦⣄⣀⣴⠟⠱⠍⣛⠛⠛⠉⠻⣤⣀⣀⣠⣾⢏⡉⠉⠻⣶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠸⣇⠙⠿⠁⢠⡿⠦⣭⡉⠄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⡿⣯⠙⠛⠀⠀⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢶⣤⠶⠛⠀⠀⠀⠠⡄⢦⠀⠀⠠⠶⢀⡆⠀⠀⠀⠀⠣⡙⠷⣤⣤⠾⠋⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠖⢁⣠⣤⡤⣤⡀⠀⠀⠀⠀⢀⣩⣤⣄⠀⠀⠀⠀⢀⡔⢦⡄⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠰⡋⠘⠛⠩⡇⠀⣾⢻⣄⣀⣤⠶⠋⢡⡆⠈⢳⣄⣀⣴⠏⠀⠀⠁⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣨⠋⠐⠗⠀⢸⠃⠀⠸⠂⠚⠁⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠸⢤⣀⣼⠛⠒⠋⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉
⠀⠀⠀⠻⣤⠁⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉
⠖⠓⠤⠴⢦⠃⠄⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁
⠀⢰⣠⠄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠤⡀⢀⡀⢀
⠀⠉⠀⠀⠘⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢡⠈⠄⠑⠆⠸⠄
⠀⠀⠀⠀⠀⠀⠑⣄⠢⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⢸⠀⡔⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠱⡀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⡰⢃⡎⠀⠀⠀⠀⠈⢷⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡱⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢊⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣀⠀⠀⢀⣢⣤⣄⠀⢀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣼⡏⠈⠉⠛⠛⠋⢁⡀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢀⣀⣤⡾⠋⠑⠓⠦⣤⣤⠞⠋⠉⠓⠂⠚
⠀⠀⢠⡖⢋⣩⠤⢊⡩⠷⠖⣦⣤⣭⣶⣶⣤⣤⣤⣤⣶⣶⣿⣿⢶⡞⠻⢟⠋⢍⠑⢦
⠀⠀⠈⢷⣏⠀⢰⡋⠀⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢈⡇⢀⣧⡴
⠀⠀⠀⠀⠙⠓⠊⠳⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠋⠁
⣿⣿⣿⠿⢿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠁⠒⠂⢄⠀⡀⠤⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠠⢀⠀⠀⠄⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓
⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⢉⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠃⢀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡐⠑⠃
⠀⠀⠀⠀⢁⠀⠀⠈⠋⠀⣌⠈⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣆⠈⠁⠀⢀⠂⠀⠀⠀⠀⠀⢀⠀⢡⠀⠀⠀⠺⢫⠀⠀⠀⠀⠄
⠀⠐⠺⡇⠀⠁⠀⠀⠐⠈⠀⠉⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣉⡀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠩⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⡦⢀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠁
⠀⠀⠀⠀⠰⡄⠈⠀⠃⠀⠐⠃⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠘⢫⡉⠉⠉⠉⠙⠉⠉⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠒⠂⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠐⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠈⠉⠉⠉⠉⠙⠃


⠀⠀⠀⠀⠀⠀⣠⠶⠶⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠂⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠄⠀⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠤⡀⠀⣀⠠⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⣀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⡠⠀⠐⠛⠛⠂⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⣀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄⠱⢂⣅⣠⣌⢠
⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡀⠀⡀⠻⣿⣿⡿⢃⠻⢿⣿⠟⢠⠀⠠⢤
⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠔⠉⠢⠄⠤⠒⠁⠀⠀⠀⠀⠀⢱⠀⠀⠐⠀⠈⠉⠠⠊⠑⠤⠀⠔⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⢊⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢤⣠⠔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢒⣀⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢃⣠⠒⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⡀⠀⠀⠈⠙⠓
⠀⠀⠀⠉⠍⠉⠛⠿⣷⠶⣴⣦⢶⣿⠛⠉⠁⢸⠀⠀⠀⠀⠀⠀⡞⠉⢠⣤⡄⠀⠇⢤⣤⠀⠁⢱⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠲⠊⠀⠀⠀⠠
⠀⠀⠀⠀⠘⢄⠀⠀⠀⣠⠆⠑⢄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⣀⠦⢀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⢀⠛⠃⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⢀⡠⠊⠒⠤⠀⠤⠊
⠀⠀⠀⠀⠀⠀⣀⡄⠒⠈⡇⠀⠈⠉⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢶⣄⣀
⠀⠀⠀⠀⢀⡞⠐⠒⢲⠐⠃⠀⡎⠁⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠸⡑⡄⠤⠘⠒⠒⠒⠓⠒⠚⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⠆
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣀⡥⠞⠓⠉⠉⠉⠉⠉⠛⠲⢤⡀
⠀⢤⠀⠀⠀⠀⢀⡸⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠱⣄⣠⠖⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢂⡠⠊⠀⣠⠞⠉⠀⠈⠉⠲⡄⠀⠀⠀⣠⡏⢳⠀⠀⠀⠀⠀⠀⠀⠹⣄
⢀⠜⠀⠀⢠⠃⠀⠀⠀⠀⠀⣪⢽⡃⢀⣾⣿⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⡌⠀⠀⠀⢸⠀⠀⢀⣠⣶⣿⡦⢋⢕⣵⢻⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⡇⠀⠀⠀⠸⡄⠔⣉⠴⠋⣡⢖⣵⠛⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢣⠀⠀⠀⠀⠱⠊⣁⠴⢊⣴⡏⣠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠈⢆⠀⠀⢁⡤⠊⢙⣠⠋⣠⠟⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠳⠔⢏⢴⠐⡟⢡⠜⢁⣴⠋⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠃⠒⠋⢀⣙⣦⡀⠀⠀⡔⠐⠁⠀⠁⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠒⠒⠙⠦⢈⠄⠀⠀⠀⠀⠈⠠⢨⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢀⡄⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠠⠀⢠⠏
⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣁⡤⠴⠊⠁⡰
⠀⠀⠈⠀⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⠤⠤⡶⠁⢀⣠⣒⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠁⢢⠀⠀⠑⢀⠣⡀⠀⠀⠀⠈⠉⠓⠒⠲⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⠁⢀⠀⠀⠡⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠔⠒⠒⠀⠀⠀⠀⠀⠀⡤⠔⠀⠂⠀⠃⢈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆⠉⣀⡄⠄⠀⠤⠄⠒⠀⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠻⡉⠉⣙⡛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣠⠤⠤⠤⠤⠤⣝⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢰⢄⠀⠀⠀⠀⠀⣀⣠⢶⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣄⠉⠉⠉⠉⠉⠉⣠⡎⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣿⣿⣶⣶⣶⣾⣿⠟⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⣼⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣃⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣶⣄⣀⡀⠀⠀⢀⣀⣠⡶⠋⠀⠀⠙⠲⣄⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠉⢉⣩⠟⠁⠈⠙⢦⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⠔⠊⠉⣄⠈⠉⣉⠓⠲⠤⢤⣤⠤⠶⠚⡿⠁⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⣷⡀
⡞⠲⠧⠴⣇⣀⣸⢧⡤⣾⣀⡀⢰⡖⠶⠶⠖⠒⠛⠛⠋⠉⠑⠄⠀⠀⠀⠀⠀⣠⡼⠋⢀⡠⠔⠋⠀⠀⠀⢹⡄⠀⠀⠤⠚⠀⢷
⠓⠶⠦⠤⠭⢭⣥⣄⣀⣀⣀⠉⠙⢷⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⡆⠭⣗⡶⢫⡀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠀⠀⠉⠒⣽⡛⡻⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⢸⡇⢸⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⠀⠀⡇⠈⠳⠞⠁⠀⣀⣀⣀⣀⡠⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢻⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢸⠇⡼⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡿⢀⡇⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⠤⠤⣤⣤⣤⡤⠤⠖⠚⠉⠀⠀⠀⢀⡾⣅⣼⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣰⠋⠀⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠙⠓⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢩⠟⠛⠛⠛⠛⠛⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣖⠒⠶⠧⠿⣤⠖⠛⠲⣿⠀⠀⡀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⣼⠶⠋⠁
⠀⠀⠀⠀⠀⠀⡼⠁⣰⠽⣄⠀⠀⡾⠁⣠⢤⡀⢹⡙⠓⠻⠦⠤⠼⠷⠖⠛⠲⠶⠤⣤⣤⣀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠙⠋⢙⠗⢾⡇⠐⣇⡼⠃⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡔⢢⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠋⠀⠘⣇⠀⠉⠀⢀⡾⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⢤⠤⢼⡇⠘⢦⠊⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣷⣄⣀⣀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠾⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠶⠶⢦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠲⣄⠀⠀⠀⠀⡠⠄⠐⠻⡄⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢰⡇⠀⣤⣄⠀⢹⠀⣀⠤⠤⣿⡸⠞⠀⣸⠀⠀⠀⣠⡄⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢠⡶⠋⠉⠉⠉⠳⣌⢧⣀⠛⠋⣠⡟⡾⠁⠀⠀⠀⢻⣄⡈⠀⠀⢠⠔⠛⢧⠀⡀⠀⠀⠀⠘⣷⡀
⠀⠀⢠⡿⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⣶⠶⡄⢹⣄⡽⠛⠛⢿⣘⣇⢰⣯⡆⠀⢀⡇⠀⠀⠀⠻⡸⠃⣐⣧⠃⠀⠀⠀⠀⠘⣧
⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠿⠿⠃⢸⡟⠀⢠⣄⠀⢿⡙⣦⣉⣀⣴⠟⢠⠞⠉⠉⠻⣏⠛⠋⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⣼⡇⠀⠀⠀⠀⠀⡈⣷⡀⠀⠀⠀⠀⣠⢿⢳⣄⠀⠁⣠⠟⠀⠁⠛⠉⠁⠀⣏⠀⠀⠛⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠚⠿⠶⠶⠖⠛⠁⠈⢧⣉⠛⠋⠁⣴⠊⠉⠉⠳⡄⠐⢝⣦⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⣿⠆⣹⡆⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⣷⠀⣠⡶⠾⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⢻⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⢿⡟⠆⠀⠀⠀⢸⡏⠁
⠀⠀⠀⢸⡆⠘⠧⠖⢶⠤⣀⠀⣀⣤⡶⢦⣄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣼⠇
⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠈⡏⠁⢸⠁⠀⢈⡷⠶⠶⡾⡋⠉⠉⢹⡷⠦⣤⡤⠶⡟⠉⠙⢳⠶⢤⣤⣤⣤⡤⢶⠾⠃⠀⠀⢠⡟
⠀⠀⠀⠀⢹⡄⠀⠀⠈⠓⠒⠧⡤⣼⠀⢀⡾⠤⣀⣴⠀⣆⣀⣀⣸⣇⠀⠀⢀⣠⢿⠀⢠⢿⣄⣀⣨⡧⠤⠤⠼⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠶⠃⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⣀⣤⣤⣤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⣤⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡛⠛⠛⠉⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣼⣤⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠠⡄⢀⡾⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⠊⠀⣼⠃⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⡆⠀⠀⠀⠀⠀⠀⠀⣸⣏⡰⠂⣸⠇⢠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⢻⡄⠈⠀⢻⣄
⠀⢀⣦⠀⠀⠀⢀⣠⡴⠛⠉⠀⢹⡆⠀⠀⠀⢢⡀⠀⠀⢀⠻⣖⠀⠀⠀⠙⠳⠶⠀⠉⠛⠶⠶⢤⣤⡀
⠀⠸⣏⠛⠾⠛⠋⣡⣄⣀⢀⠀⠄⢿⡀⠀⠀⣀⣷⠀⠀⠈⢦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢰⣦⡀
⠀⠀⠙⢦⣄⣀⣸⡃⠀⠉⠛⠻⠶⠟⠁⠀⠀⠀⣼⣇⠀⠀⠀⠙⠷⣤⣤⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⠞⢹⡇
⠀⠀⠀⠀⠈⠉⣻⠀⠀⠀⠀⠔⠂⠀⢀⠀⠁⣱⡟⠘⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣌⣀⠀⠠⠀⠀⠀⠈⠀⢐⡿
⠀⠀⠀⠀⠀⣶⡏⠀⠀⢎⡶⠟⠛⠿⠶⠶⠾⠋⠀⠀⠘⢦⣌⠀⠐⠀⠀⠐⣉⣤⣮⡱⠀⢿⠉⠙⠳⢦⣤⣠⣉⣤⡾⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠷⠛⠉⠀⠈⢷⣤⣸⡄⠀⠀⠀⠀⠁⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⡉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠋⠙⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⡿⡏⡌









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣩⣩⣭⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⣏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡖⠀⠀⠀⠀⠀⠀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠟⠋⠉⠉⠉⠉⠛⠻⠿⠿⠿⠿⢻⠉⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠒⠢⣀⣠⠔⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋⠁⠀⠀⠔⠈⠲⠄⠀⠀⠀⠀⠒⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠉⢻⣿⣿⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠑⠁⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠠⠐⠢⣄⠤⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠈⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠘⠀⠀⠈⠋⢰⡀⠀⠀⠛⣠⠁⣸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠈⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠠⠂⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠤⢀⠀⠀⠀⣠⠄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠒⠈⠀⠊⠀⠀⠀⠀⠉⠋⠉⢀⠀⠁⠀⠖⠊⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⣄⠠⠚⠀⠀⠘⠂⠈⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠐⣀⠔⠋⠈⠀⠙⠀⠀⣀⢀⣀⣀⡴⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⣠⠎⠀⠀⠀⠀⠈⡆⠀⠀⠀⠑⠤⣸⡀⠀⢀⡤⠞⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡐⣤⢀⣀⠀⣀⣀⣠⠴⠚⠁⢀⡀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠠⠢⠐⡇⢹⢸⠀⠀⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⠙⠆⠜⠙⡄⠐⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⣀⠀⣀⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠘⠘⠀⠀⠀⠀⠃⡇⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠁⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠠⠃⠈⠁⠀⠰⠃⠜⠠⠠⡄⠀⠀⠀⠀⠀⠘⠄⠀⠀⣀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⣠⠶⠛⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣀⡀⠀⠀⣀⣀⣀⠀⠘⣷⠞⠁⣀⣀⠀⠀⣀⣀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣼⠟⢻⣿⣷⣾⣿⣿⣿⣧⠀⠸⣇⣾⢋⣿⣿⣾⣿⣿⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠀⢀⣀⣀⡀⠀⠀⠀⢻⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⡇⠘⠻⣿⣿⣿⣿⡿⠋⠀⠀⢀⡿⠀⠀⠀⠴⠞⢻⠷⣦
⠀⠰⠟⣿⡉⠉⠁⠀⠀⠈⢧⡀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠁⠀⢀⡾⣧⡀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣼⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠙⠶⣤⣄⣀⣀⣠⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡛⠛⠛⠛⢛⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣷⣾⢡⡶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠶⡟⣻⣦⣿⣧⣙⣲⣯⠶⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠐⠦⣼⣸⢿⣵⠊⠁⠀⠈⣙⢯⣔⡃⠄⠀⠀⠀⠀⣷⢀⡟⠀⡀
⠀⠀⠀⢀⡀⠈⢷⣠⡟⠀⠀⢸⡛⣦⢼⡿⢋⡀⠀⠀⠀⢊⡥⢄⢻⡿⠗⠀⠀⠠⣀⣸⣾⡶⠋
⠀⠀⠀⠀⠉⢓⣬⡟⢻⡄⠀⠐⠒⣨⡿⠁⢾⣿⠀⠀⠀⡋⣶⠀⠘⣷⠀⠀⠀⠀⠈⣿⡼⠋⠛⠒
⠀⠀⠀⠀⠛⠛⢉⡿⠛⢷⢦⡀⠀⣿⡇⠀⠠⠤⠂⠒⠀⠘⠒⠁⠈⣿⡆⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠣⡱⣄⠹⡅⡄⠀⠀⣶⡟⠿⣷⡄⠀⣼⠿⠁⢀⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠳⣝⢌⡲⣜⡛⠾⠟⢛⣠⡾⢋⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠘⠋⠹⣿⠮⠵⡶⢾⣁⣴⠏⢉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠘⢧⣼⠁⠀⠙⠃⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠉⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⣩⢿⣖⠒⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⣴⠏⠸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣰⡟⠀⠁⠀⠀⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠢⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣰⣶⣄
⠀⠀⠀⠀⠀⠀⠸⠿⠶⠋⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣏⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⡿

⠀⢠⣤⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⢀⡤⠾⠀⠀⠀⣀⡀
⠀⠋⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠉⠙⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠰⠀⠀⠀⠀⠆⡀⠀⠀⠄⠀⠠⢶⢶⣤⣀⣠⣀⣠⣤⡶⠟⢻⠉⠀⠃⠀⠀⣀⠙⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠘
⠰⠀⠸⠇⠀⡇⠛⠀⢀⠂⠀⠀⠀⠀⠀⠉⠉⠩⡁⠈⠁⢀⠌⠀⠀⠇⠀⠀⠿⢠⠙⠁⠀⠌⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠈
⠀⠑⠀⠀⠔⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠀⠀⠀⠀⠀⠀⠉⠓⠀⠙⢒⠀⠀⠀⢠⠴⠒⠀⠀⠐⠢
⠀⢀⣠⠾⠛⢁⡐⠳⣄⡀⠀⠀⢀⣤⣤⣀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⠂⠀⠀⠐⠂
⠐⡏⠁⣤⠄⠨⠄⡤⠀⠈⡃⠀⠀⠀⠀⠉⢻⡆⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠠
⠀⢳⠀⠀⠀⢠⡄⠀⠀⢠⠁⠀⠀⠖⠒⠬⣆⠀⠣⠘⠃⠀⠄⠀⠀⠡⡀⠀⣠⡄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠁
⠀⠀⠑⠄⠀⠃⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⣀
⠀⠀⠀⠐⠶⠐⠁⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠂⡘⠛⠦⣄⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠀⠀⠀⠠⢡⣶⢦⡄⠁⣴⣶⠆⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋⠀⠀⠴⠱⠆⠀⠀⡉
⠀⠐⠀⠀⠠⠇⠀⠀⠐⠖⠀⠀⠀⢀⠤⣿⡷⠀⡄⠭⢿⠟⠀⠀⠀⠆⠀⠒⢀⡆⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⣀
⠀⣰⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠤⠄⡀⠀⠙⢛⡢⠀⠀⠀⢘⠩⠅⢀⠀⠈⠍⢃⠀⠀⠀⠀⠨⠀⠠⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠟⠛⠷⠆⠀⠀⢄⣀⣒⡂⠀⠄⠀⠀⠀⠀⠀⢀⠂⠀⢀⣤⢶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣀
⠃⠀⠐⠛⠀⠰⠀⠘⠛⠉⣀⠂⠀⢠⠀⢉⡀⠀⣣⣼⡁⣰⡇⠀⠘⡀⠀⠈⠛⠸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠈⠣⣙⣋⠼⠓⠉⠉⠚⠀⠀⠀⠻⢄⢀⡴⠋⠲⠤⠚⠁⠀⠀⠀⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠒⠤⣤⡐⠂⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀⠀⠈⡠⢊⢀⣤⠾⠛⣩⣭⣴⣄⣀⠀⠙⢦⣤⣄⡀⢸⣷⠄⢢⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⢸⠃⡵⠃⢸⡀⠀⠀⣾⣯⣻⡽⠷⠦⣄⢻⣿⣿⢸⡇⠀⢀⠱⢄⠀⠀⠀⠀⠠⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⢸⠿⢀⣀⣌⣿⣿⣶⣿⣟⠉⣤⣴⣫⡿⡇⠁⡇⠀⠈⢷⡄⠀⠀⢹⠀⠀⠀⢀⡠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⣻⣿⣿⣿⣿⠟⠉⠉⠀⠈⢠⣷⣿⡿⠁⠀⢰⣼⢷⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢿⠇⢉⣭⡀⠐⢠⠶⢦⡀⠀⠹⠉⠀⢀⣀⡬⠗⢸⣤⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⠀⡞⢁⡙⠆⠈⢠⣤⢣⠀⠀⢁⡼⢿⣿⡖⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢃⢼⣯⡄⠀⢸⣿⣾⠀⠀⠀⠐⠃⣯⣴⣿⡿⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠄⠸⠿⠃⠀⠈⢋⡁⠈⠑⡄⠠⣽⣿⣿⣿⣦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠰⣶⣀⣀⣠⣿⠃⠀⡠⠃⡜⢺⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠈⠙⠛⠉⠁⠀⠀⢀⣴⠃⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⡠⣤⣶⣿⡏⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠏⠻⡿⠁⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢀⣀⣀⠀⠀⠀⠙⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡁⣠⠋⠁⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⢀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠈⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⠌⣼⠁⠀⠀⠀⠀⣀⣠⣤⡴⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⢠⡟⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢸⡇⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⢀⣠⠴⠒⠛⠉⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣔⠁⢱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠹⡇⢠⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣧⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠘⠋⠛⠛⠋⠁⠈⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⢢⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⣤⠀⢸⠀⡇⢠⠘⡀⡏⠀⠀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⣦⣾⢐⣾⣆⣦⣸⣤⣷⣤⣇⣴⣀⠃⠆⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣹⣶⣾⣿⣿⣿⡿⠿⠛⠛⠻⣿⣿⣿⡿⢿⣿⣷⣯⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣸⣿⣿⣿⠟⠋⣁⡄⠀⠀⠀⠀⠈⠉⠁⠀⣀⠈⠻⢿⣿⣷⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡈⢿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⠁⠀⣠⣴⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣄⠉⠈⣿⣿⣷⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣿⣿⣿⠀⣾⠷⠒⠲⢿⣷⣄⠀⠀⠀⠀⣡⠶⠶⠬⣽⡄⠸⣿⣿⣊⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⡏⢰⣿⣷⠀⢻⣿⠀⠀⠄⢰⡇⢰⣿⣷⠀⡇⠀⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢬⣙⣋⣠⠿⢿⡇⠀⠀⠀⠙⠢⢭⣥⠴⠁⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠲⣿⣿⣿⠗⠀⠈⠉⠀⢚⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡹⣿⣿⣿⣀⣶⠂⡄⢠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⢿⣿⣿⣿⣿⣷⡞⢉⣻⡇⠀⢀⡀⢦⠀⠀⠀⠀⣤⣿⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⠛⠛⠻⢿⣧⣤⣾⠿⠞⠀⠀⡀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⣿⣿⣿⡟⠀⢀⣠⣤⣤⣯⣤⡀⠀⠀⠀⢡⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡋⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆⠀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⡿⠁
⠀⠀⣠⠤⣀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠈⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇
⠀⠀⠙⣄⠈⣧⠀⢻⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⡁⠈⣯⡙⣿⣿⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦
⠀⠀⠀⠘⣷⠛⢧⡈⢳⣌⣷⡄⠀⠀⠀⠀⠀⠀⢸⣇⢻⡆⢀⡼⣟⣻⣿⣿⣿⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣾⢠⡟⠀⠀⠀⣠⠖⢳⡄
⠀⢐⠛⢦⡈⢷⣀⢻⡄⣯⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⡏⠙⢦⠀⠀⠀⢰⠏⣼⠁⠀⢠⠞⢁⡴⠋
⠀⠀⠳⣶⠻⣌⠻⡄⠳⣼⣾⣿⠀⠀⠀⠀⠀⠀⠀⢻⡇⢿⣿⠧⠴⠾⢽⣿⠇⠀⠀⠀⢸⡇⠀⣸⠀⠀⢀⣾⢰⡏⢀⡴⠋⣠⠟⠀⢀⣠⢤
⡶⢦⣄⠘⣆⠘⢆⠹⣦⣿⠏⠻⣇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠴⢶⣶⢷⣶⡾⠃⠀⠀⠀⣸⠁⢻⡏⠀⠀⣾⣃⠿⢰⠋⣰⡿⠃⣀⠀⢁⣴⠞
⠙⢶⣟⢷⡌⠳⣤⣿⡿⠋⣠⣴⣿⡆⠀⠀⣠⡶⣤⠀⠹⣆⠀⠀⠁⢀⣿⠁⠀⠀⠀⢠⣟⣀⡟⢀⣤⠾⠋⠉⣷⣟⣸⢋⣤⢚⣵⠞⠋
⠀⠀⠙⢷⡙⢦⣿⣿⠁⢀⢸⠟⠸⣇⠀⢠⠏⢀⣸⡇⠀⢹⣷⣶⣶⡞⠁⠀⠀⠀⢀⣾⣏⠙⠛⢿⡇⣶⡀⣄⠻⠈⢷⡎⣠⠾⠋⠀⠀⣤⣄
⠀⠀⠀⠈⠳⣦⠛⠃⣴⣿⢧⡖⣰⣿⢀⡟⣰⣿⠟⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠈⢷⣿⣿⣾⣧⠀⢸⡿⠟⠐⣚⣧⡤⠶⠋
⠀⠀⠀⠀⠀⢹⣇⡴⠙⢿⡞⣰⠟⢻⣾⣿⣿⠏⠀⠀⠀⢸⡇⠈⢿⠀⠀⠀⠀⠀⢠⣿⠤⠀⠀⠀⢸⡿⠛⢉⣠⣹⣮⡿⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢿⡶⠂⠈⢿⣿⠀⠀⠠⣿⣿⠀⠀⠀⠀⣿⣧⠀⣿⠀⠀⠀⠀⠀⢸⣟⠓⠀⠀⠀⣀⣠⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⠈⣿⠀⠀⣰⣿⡟⠀⠀⠀⠀⣿⡋⠀⣿⠀⠀⠀⠀⠀⣸⡿⠛⢣⣄⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⠈⢣⣸⣿⣿⠇⠀⠀⠀⢰⣿⡇⣼⠇⠀⠀⠀⠀⣴⠋⠀⠀⣨⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠈⢻⡿⠃⠀⠀⠀⠀⢸⣿⣇⣿⠀⠀⠀⠀⣰⠏⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠠⣿⣠⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡋⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠃

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣤⣤⣤⣤⠀⢀⣤⣄⣠⣤⣤⣤⡀⠀⠀⢀⣤⣄⣠⣤⣄⣀⡀⠀⠀⢀⣤⣤⣀⣤⣤⡀⠀⢀⣤⣤⣠⣤⣄⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⡈⣿⠀⠉⢹⣿⣿⠁⠙⣿⣿⡆⠀⠈⢹⣿⣿⢁⡉⢻⡷⠀⠀⠉⣽⣿⣿⣿⡅⠀⠀⠈⠉⣿⣿⡇⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡿⢿⣿⠀⠀⢰⢾⣿⣿⣦⣾⣿⣏⠀⠀⠴⢾⣿⣿⣾⣿⠄⠀⠀⠠⢶⣿⣃⣿⣿⣧⠀⠀⠠⠾⣿⣿⡿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠈⠁⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠈⠋⣸⣶⢀⢀⣾⡟⠋⠙⣿⣿⣆⠀⠀⠀⣿⣿⣇⠀⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠛⠀⠀⠀⠘⠿⠟⠻⠟⠋⠻⠿⠟⠁⠘⠿⠛⠻⠿⠟⠛⠇⠘⠛⠛⠿⠃⠻⠟⠛⠿⠛⠀⠛⠛⠛⠛⠋⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⢀⣀⠀⣀⣀⡀⢀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠛⢻⣿⠀⠚⢻⣿⣿⡟⠒⠻⣿⣿⡟⠃⠀⣴⣾⡟⠛⢿⣷⣦⠀⠚⢻⣿⣿⡋⢰⣿⣿⣷⠈⣹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣦⣜⠛⠀⢀⣸⣿⣿⣇⣤⣬⣿⣿⡇⠀⣼⣿⣿⡇⢠⣼⣿⣿⡇⢀⣤⣿⣿⣇⣾⡟⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⡍⠙⢿⣿⣿⡄⠉⢹⣿⣿⡏⠉⠙⣿⣿⡇⠀⠙⣿⣿⡇⠈⢸⣿⣿⠇⠀⠁⠹⣿⣿⣿⠇⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⣤⣾⣿⠟⠁⢠⣼⣿⣿⣧⠤⣴⣿⢿⣧⡄⠀⠘⠿⣿⣶⣾⠿⠋⠀⠀⠀⠶⡿⠿⠟⠀⠈⠿⠟⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡆⠐⢀⠀⡄⢠⠀⢄⢠⠀⡄⣀⠀⢠⠀⠠⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⠀⠈⠀⠇⠨⠀⠆⠘⠀⠁⠈⠂⠈⠰⠄⠰⠀⠇⠸

⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢲⠀⡄⠀⠂⢀⠀⡀⢀⠀⠀⡄⠀⣀⢀⠀⢀⣆⢀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠰⠆⠰⠀⢤⠀⠀⠇⠈⢰⠀⡇⢰⠀⠰⠁⠀⠀⢨⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠄
⠀⠀⠀⠀⢱⠀⢀⠈⠢⣄⠀⠀⠀⠀⢢⠀⡄⠠⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠅⠠⢀⣤
⠀⠀⠀⠀⢰⠇⠠⠁⠀⡌⣠⡀⠀⠀⢰⠆⢡⠀⣤⠁⢨⠀⡗⠀⡄⢸⠀⡅⣄⠈⠘⡄⠀⢀⣔⣡⠁⠀⠁⠀⢜
⠀⠀⠀⠀⠀⠀⢠⡠⣢⡘⣿⡏⣦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣾⣿⢫⣴⠆⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⡇⠈
⠀⠀⡀⠀⠀⠀⠀⠂⢲⣿⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⡷⣿⣿⡗⠀⠄⠀⠀⠀⢀
⠀⠀⠀⢢⠰⣤⡀⠀⠀⠙⢿⣿⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣷⣿⠛⠁⠀⢀⣤⠶⡄
⠀⠀⠀⠀⡆⢈⠝⣦⡀⠀⠀⠹⣿⣻⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⣿⣻⠟⠁⠀⢀⣴⣏⡁⠐⠂
⠀⠀⠀⠀⣁⠀⠐⢿⣿⣄⠀⠀⠘⣿⡷⡝⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢫⢿⣿⠃⠀⠀⢠⣿⣿⠇⠀⢈
⠀⠀⠀⠀⠄⠈⠰⡺⠿⣿⣆⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⣠⣿⠿⠿⡧⠈⠠
⠀⠀⠀⠀⡅⠈⠂⢑⣿⣿⣿⣷⣶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣾⣿⡿⣿⣏⠐
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠚⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⡛⠂⠀⠀⠀⠠
⠀⠀⠀⠀⢐⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢻⠀⡥⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡄⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠢⡀⢠⣿⣿⢿⣯⡿⢯⡻⣬⣆⣀⠠⢤⡂⣸⠀⢆⠐⠦⠄⣀⣦⣵⠞⡯⣿⢽⣿⡟⣿⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠈⢺⡟⣽⡟⠍⡓⡅⠙⠒⢌⠉⣯⠄⣰⠃⠀⠈⢆⠠⣼⡉⣩⠴⠉⢪⠖⣉⢛⣯⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠜⠷⠢⠪⢈⠈⠌⠀⠊⣁⠜⠁⠀⠀⠀⠈⠣⣈⠁⠀⠑⠐⡑⠃⡡⣾⠍⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠒⢑⠈⣠⡡⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠴⢄⣄⠀⠈⠑⢰⡿⠁
⠀⠀⢀⠀⢀⠀⣀⠀⢈⡷⣠⠞⣉⡀⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠉⠓⣄⢰⡁⢀⡀⠀⢀⡀⢀
⠀⠀⠏⠀⠿⠖⡿⠆⢸⠀⢡⠆⠈⠃⠈⠉⠐⡆⠀⠻⣧⣾⠀⣷⣬⠟⠀⠈⣉⠧⡄⢠⠼⣋⠉⠁⣠⠸⣿⣤⣿⠇⣄
⠀⠀⠘⠀⢀⠛⢃⠀⠁⠀⠈⠃⢀⡆⠘⡄⠀⠁⠀⠐⠛⢋⠠⡙⠛⠃⠀⠠⡿⠀⠀⠀⢀⢿⡇⠀⠇⠰⠈⠭⠁⠦⠸
⠀⠀⢠⣠⠏⣿⡈⣄⡀⠀⠀⠀⢿⡇⢈⡿⠂⠀⠀⠰⠉⡜⠀⢱⠈⠐⠀⠀⡆⠡⠀⠀⠄⠂⠄⠀⠾⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠳⣇⠙⢆⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣈⡷⣄⢈⣶⡞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⢤⡟⠋⠙⢣⠀⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⢀⣀⣀⣨⠷⠋⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣉⠁⠂⢉⡰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀⡤⢤⢤⡴⠶⢶⡄⠘⡿⠉⠁⠀⠘⢆⠀⠀⠀⠀⢀
⠀⡀⠄⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠈⠀⠉⠀⠀⠀⠈⢻⡦⣯⡀⠀⠀⠀⠘⡇⠀⠀⠠⢁⡆
⠀⠱⡌⠀⣇⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠛⠀⠀⢀⣀⣠⣠⠦⢶⡆⠀⠀⠁⠈⡹⠆⠀⠀⠀⡧⠀⠀⡀⢠⠁
⠀⠀⢹⡘⢧⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣌⠀⠀⣀⠿⣩⠀⠩⠦⠈⢹⠄⡀⠀⠀⢹⣦⠀⠀⠀⡇⢠⠒⠃⣹⠦⠄
⠀⣰⣻⠟⠲⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡻⠀⠀⣯⢈⠁⠀⣠⠂⠐⡈⠙⣿⠀⠀⣼⡁⠀⠀⢠⢃⠷⣻⡋⠹⣤⡖⢢
⠀⡜⢧⣾⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⡀⢀⣸⠇⡀⠰⢻⠜⢄⡡⠋⠐⠰⠇⠀⣹⠇⠀⠜⣳⠀⠀⡘⢠⠈⠛⠉⠲⠀⢻⡁
⠀⠈⢿⣙⣃⠀⠇⠀⠀⢀⢀⣠⣴⣿⣿⣿⣿⣿⣗⡆⠡⣷⣾⣅⠀⠀⠀⡸⠿⠗⠀⣰⣾⠟⠀⢠⠃⠀⠑⠤⡀⠠⠇⣰⠃
⠀⠀⠀⠀⠙⡀⠈⠄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⢨⠤⠽⢿⣿⠟⠓⠶⠶⣳⣿⣷⣤⣤⣧⣠⣤⣄⣀⠿⠒⠃⠉⠐⠲⢄
⠀⠀⠀⠀⠀⢣⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣒⡠⠄⣀⠉⠀⠀⢰⣏⠛⠋⠭⠩⠁⠐⠚⢡⣇⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⡆⠀⠹⣿⡿⠻⠦⠉⠱⠿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠶⢧⣶⣦⣿⣿⣧⣄⣠⣠⣄⣤⣶⣿⣿⣿⠟⠀⢀⡤⠚
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠤⠄⠀⣴⢤⠄⡁⠈⠽⣿⣿⣿⣿⣿⠟⠙⠲⠾⣿⣿⠟⠛⠯⢬⣉⡙⠛⠛⠛⠛⠉⣁⣤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⡦⠊⠁⣀⣢⣀⢠⣶⣿⣿⣿⣿⣟⠀⠀⠀⠀⠸⣁⠀⠀⠀⢀⣟⠉⠍⠋⡿⠫⢥⣟⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⣼⠋⣍⣹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣷⠄⠀⠙⡷⢀⡤⠚⣄⠀⠀⠉⠉⠙⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣠⣾⣝⣛⣛⣫⣾⣿⣿⣿⣿⡟⠀⠀⢀⡀⠀⠀⣰⠋⠁⠀⠀⢠⠃⠊⡴⠎⠘⣷⡀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⠃⠀⢠⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢧⣀⣠⠞⠳⣴⡞⠁⠀⠀⠀⠀⢿⡀⠟⠀⠀⠀⠙⡙⢦⡀⠄⢀⠀⢀⡟
⠀⠀⠀⠀⠀⢨⡰⠟⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠑⢄⠈⠙⠁⠀⠀⠀⠀⠀⠀⢠⣄⠈⠫⡳⣤⡀⠀⠀⠘⢄⠋⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣩⣉⣥⢠⣤⠴⡌⠀⡆⢶⣶⣲⡔⢲⠀⠀⢀⠀⢀⣴⠟⠀⠑⠒⠊⠀⠙⢦⡀⠀⢀⢣
⠀⠀⠀⠀⠀⠻⠿⢿⢀⣙⣊⣛⣉⡩⠭⠽⠓⠲⠦⣅⢨⠴⠒⠒⠛⠛⠛⢶⠚⠋⢁⡴⠒⢲⡀⢠⠤⠶⣦⢙⠦⠞⠋
⠀⠀⠀⠀⠀⠀⡟⠉⠉⣩⣄⣠⣿⡄⢰⡏⠉⣳⠀⢈⣿⠀⠀⠺⠿⡍⠛⠛⠀⣠⠎⡴⢦⠀⢧⢸⠀⣴⢫⠏⠳⣦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠴⣧⠀⡇⢠⣍⠉⢥⡶⠋⣿⠀⠀⢾⡿⠷⠒⢳⡞⠁⣤⣝⡋⠀⠘⡿⠸⣧⢎⣠⡴⠾⠃
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠰⡞⠛⠀⢷⣤⡿⠳⠴⠟⠂⠸⢤⣤⡤⠤⠶⠶⠚⠛⠲⠟⢼⣥⣀⣰⠃⠀⣀⠯⢥⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢹⡄⠀⠀⠠⢲⢉⡶⢨⠀⢱⡤⠐⣦⢠⠼⢷⢘⢵⠛⣂⢢⣤⠏⠙⣦⣸⡏⢧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⡾⠁⠀⠀⠘⠠⠊⠉⡈⠄⠣⢃⠀⢸⠌⡢⣼⠀⣨⢞⡑⢫⣐⠀⡀⠈⠻⠃⠘⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣶⠋⠀⠀⠀⠀⠀⣈⡦⠞⣤⠆⢠⠆⠳⠋⠴⠷⠾⠶⠯⢼⡇⢀⡇⢠⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⢸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⡲⠞⣁⡴⠖⠛⠉⠉⠉⠉⠉⠛⠓⠦⣄⡀⠀⠀⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⣠⠖⠉⠀⠀⢀⣾⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⡶⠒⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⣻⠁⣴⠷⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢿⡄⠀⢀⡞⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡰⠋⠀⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠃⠀⠀⠀⢀⡿⢄⡀⠀⠀⠀⠀⢀⡠⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⢀⡀⠙⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠙⣦⠀⠀⠀⣾⠁⠈⢧⠀⠷
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣉⠭⢍⡀⠀⠀⢿⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠓⠦⡀⢸⠀⠀⠀⠀⠈⠳⣄⠀⣼⠋⠉⢢⣏⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠙⢦⠀⢸⠀⡴⠋⠀⠀⠙⣆⢀⣤⢤⣄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⡇⠘⣇⠀⠀⣠⣀⣤⣈⡄⣏⠀⢀⡏⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣷⡀⠀⠸⣇⠤⣇⢧⣠⣤⣤⣄⠀⢾⠀⠀⠘⠃⠀⠠⠴⠖⠛⠛⠛⠿⢿⡷⠀⠙⢦⡜⢡⡏⠀⠈⠁⢉⡒⠋⠀⠀⢀⡜
⠀⢰⡆⢀⡴⠞⠓⢶⠋⠀⢀⠙⠀⠀⣇⠀⢸⠛⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⢸⠁⠈⢳⡖⠓⡆⠘⠃⠀⢠⡴⠋
⡄⠀⠀⢸⠀⠀⢀⣼⡀⠀⢹⡄⢠⠀⠉⠀⡇⠀⠀⢀⠔⠚⠉⠉⠉⣱⠶⠤⢄⣀⡀⢀⣀⡡⠏⠀⠉⠁⠀⠀⠀⣤⠤⠚⢀⡜⠀⠀⠀⠀⢸⠁
⢱⡀⠲⡀⠳⠖⠁⠀⠙⣶⠚⠀⠘⢦⠀⠀⣇⠀⠀⠻⡀⠀⠀⠰⠋⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠎⠀⠀⠀⠀⠀⡸
⠀⠹⢄⠙⠆⠀⠀⠀⠀⠈⠛⠒⡞⠀⠀⠀⠘⠦⣀⠀⠙⠲⠤⣀⣀⣀⠤⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣀⠜⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠈⠉⠒⠶⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⣰⠖⠒⠙⠋⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⢀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣐⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⠙⠲⠦⢤⣀⣠⠤⠤⠖⠛⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠲⣶⠶⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⢤⣀⡀⠀⠀⠀⣘⢻⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡏⠉⠛⠶⣄⡟⠘⡇⣠⡴⠶⠖⠲⣞⣸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠳⣄⠀⠀⡟⠃⠀⠛⠒⣷⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣤⣤⣬⠦⢤⣄⢸⡇⣸⡳⡄⠀⠀⠀⣷⠶⠚⠛⢛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⣭⡟⠁⠀⣎⠩⠷⠤⣒⡶⠾⠷⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠢⣤⡀⡀⠀⠀⠀⠀⣨⠇⣀⣠⠔⣫⠃⠀⠀⢸⣇⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢌⣉⣑⣒⣒⣚⣉⣉⠭⠔⠊⠁⠀⠀⠀⠈⢯⡛⠧⣄⣀⠀⠀⠀⢀⣠⠼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⠉⠉⠉⠁⠀⣤⠖⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⣀⣤⣄⠀⠀⢠⡀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⣤⠀⠀⢠⣴⣶⠄
⠀⠀⠀⠀⠀⠀⢺⡇⣸⣿⡀⣿⡇⠀⢸⣿⠛⠋⠀⠀⣽⡇⠀⢸⡿⠈⣿⠀⠀⠀⣿⠙⣿⡄⠀⢀⣿⠀⠀⣿⡏⡁
⠀⠀⠀⠀⠀⠀⢸⣧⣿⢿⣧⣿⠀⠀⢸⣿⠶⠀⠀⠀⣾⡇⠀⢸⣷⣶⡏⠀⠀⠀⣿⠀⢸⡇⠀⢠⣿⠀⠀⣿⡟⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡟⠈⢿⣿⠀⠀⢸⣿⣤⡄⠀⠀⢹⡇⠀⠀⣿⠈⢿⡆⠀⠀⣿⣤⡿⠁⠀⠀⣿⠀⠀⢻⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠈⠛⠀⠀⠈⠛⠋⠁⠀⠀⠘⠃⠀⠀⠉⠀⠀⠁⠀⠀⠙⠋⠀⠀⠀⠀⠉⠀⠀⠈⠉







⣤⡶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠈⣧⡀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣴⠂⠀⠀⠀⠀⠀⠀⣸⣷⣄⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡾⠁⠀⠀⠀⠀⠀⠀⣰⣿⠃⣙⠳⢶⣬⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⠶⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠚⠋⢁⣀⡀⠀⠀⠀⠀⣤⣤⣈⠈⢿⡟⠷⣦⡀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣤⠾⠛⠻⠿⣦⡀⢀⣴⠋⠉⠉⠻⢮⣿⣆⠈⢷⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣠⣴⠾⣻⣿⠏⠀⢠⡿⠁⠀⠀⣾⣿⣦⢻⣼⣇⢠⣶⣶⠀⠈⣿⣿⣷⡄⠀⠀⠀⠀⠙⢦⡀
⠠⣤⣤⠄⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠈⠁⣤⣾⣿⡏⠀⠀⠘⣧⡀⠀⠀⠈⢉⣡⠞⠉⢿⣌⡛⠋⠀⢀⣿⢻⡇⠹⣦⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠙⠇⠀⠀⠙⠾⣿⣿⡾⠿⠋⡁⠀⠀⠀⠙⢿⡿⣾⣿⡷⠛⣿⣤⡜⣇⠀⠀⠀⠀⠈⢿⡄⠀⠀⣠⣤
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣀⣀⣀⣼⡇⠀⠀⠀⠀⠙⢫⡿⠙⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢿⣄⠀⢰⣄⠀⠀⠀⠀⢸⡇⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⣛⣋⣁⣀⡀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣙⣷⣤⣀⠀⠀⠻⠦⢉⣻⣿⡶⠂⢀⣴⡾⠿⣿⠛⠛⣿⠛⠋⢻⡟⠙⢻⡟⠻⢷⣄⠈⣷⡀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠟⠟⠛⠛⠛⢻⡟⢀⣴⣿⢿⡄⠀⢻⣄⣀⣿⣄⠀⣸⣇⣀⣿⣧⣤⣾⣿⣀⣿⡇⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠙⢲⣄⣾⠁⠻⢿⣿⡾⢿⡶⠟⣿⠋⣿⠉⣿⡿⢿⣏⠙⢷⣄⠙⣿⠻⣿⣷⣶⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠙⠿⢷⣦⣼⠟⢠⡿⠁⢀⡿⠀⣿⠀⢻⣇⠀⠙⣧⠀⢹⡆⢸⣇⢹⣧⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⠁⠀⠸⠃⠀⠁⠀⢸⡟⢳⣄⠀⠀⠈⠁⠀⠋⢸⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡿⣦⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡄⠀⠀⠀⠀⠀⣸⠏⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⠀⢻⣆⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⢠⣿⣀⣀⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢀⡀⣿⡏⣠⡾⠟⠁⠀⠀⠀⠀⠀⣠⠟⠁⠀⠙⣧⡀⠀⠀⠀⠈⠻⣯⣁⡀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠞⠛⠋⠉⠛⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠈⠻⣿⡛⢶⣌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢀⣠⣴⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠈⠹⡆⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣠⣾⠟⠁⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⠿⣷⣶⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠶⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠄⠀⠀⠀⠀⢀⣐⣮⡉⠐⣀⣀⠀⠀⠐⠀⡞⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⣠⡄⠠⡖⠁⢸⠀⢀⡀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢽⠋⡄⠈⣦⠸⡜⠁⠈⠻⣶⡆⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠃⠰⠳⣘⡀⢃⠀⠀⠺⣇⠀⠀⣀⡔⢁⣀⣀⡤⠄⠒⠂⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠁⠒⡺⠉⠉⢉⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⡓⡲⠿⠁⠀⠀⠈⠀⠀⠀⠀⠈⠓⠲⠤⣄⣀⣀⣀⣀⣀⣀⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣉⡉⣤⠀⠀⠀⡀⠠⡤⠀⠒⠖⠐⡆⠀⢠⡌⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⢀⡔⠈⡇⠀⡇⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠊⡇⠀⡆⠀⡥⠄⠼⡀⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⡄⢃⠀⠘⠉⠀⠀⠀⠘⠶⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⣀⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢊⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣩⣉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠈⠉⠒⠤⠄⣀⠀⠀⠀⣀⠄⣀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠂⠄⠀⠀⠀⠉⠁⡲⢧⠀⡏⠱⠾⣗⠢⢄⣀⠀⠀⠀⠀⢀⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⡖⠀⠀⠈⠐⠢⠼⢀⠀⢛⠳⡀⡇⢞⠦⠦⣝⠓⠢⡤⠠⡄⠼⡡⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡶⠁⢱⡀⠀⠀⠀⠐⢄⣀⡁⠘⠄⠘⢿⠀⡱⠐⢌⠑⢲⣇⢀⠃⣸⠀⠡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡔⠢⡜⢀⠌⠀⠀⢸⠆⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⢾⣿⡇⠀⠀⠑⡙⠨⢎⢡⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠷⡝⠾⡄⠃⠀⢹⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢀⣤⡊⠑⡀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⢚⡠⢤⡵⠠⠊⠀⠀⠀⠸⠤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠀⠚⠉⠁⠀⢈⡦⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠆⠧⠀⠠⠖⠀⡄⠀⠀⠑⢤⡀⠀⠀⠀⠀⠺⠠⠤⠀⣀⠀⣰⠻⠅⠀⣈⣣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠐⠲⠋⠙⡀⠉⠒⢢⡤⣤⢀⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⣨⠖⠂⠹⡖⠛⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⠀⠤⠔⠚⣭⠞⢆⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠲⠂⠂⠀⠶⠖⠒⠚⠒⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠞⠛⠁⠀⠐⠳⣦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢉⡤⣦⣄⢠⢤⣤⡈⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⠀⠀⢸⡀⠿⣿⣇⠘⢛⡇⠀⠀⠀⣿⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡀⠀⠀⠈⠙⠛⠁⠈⢛⣩⡤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠖⠀⠀⢴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⠀⡄⠈⠙⠛⠶⠶⣤⣤⣶⠶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠘⢧⡀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⠉⠳⢤⣀⣠⣉⣉⡉⠀⠀⢀⣀⣤⡴⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⢻⠀⠀⠀⠈⠙⠛⠛⠓⠛⠛⠛⠉⠁⠀⠀⣿⠻⣄
⠀⠀⣀⣀⠀⠀⠀⠀⢀⡴⠋⣡⠶⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣌⠳⣄
⢀⣀⣑⣬⡑⠖⠋⠉⠉⢠⠞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢧⡌⠳⣤⡀⠀⠀⠀⢀⡀
⡜⠒⠒⠒⠂⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠣⣌⠹⠶⠦⠴⠡⠗⡲
⠈⣏⣉⡩⠤⠒⠉⠉⡇⠸⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⠠⠞⣀⡀⠀⠀⣐⣋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠼⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⠤⠞⠁⠉⠲⣄⣉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠳⣄⠀⠀⠀⠀⠉⠙⣿⢲⡶⠶⣤⣤⣤⣤⣶⡶⠶⠟⠋⠁⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠹⣦⣤⣀⣀⣀⣀⣹⠀⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣠⡞⠉⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠉⠉⠉⠉⠉⠃⢷⡄⠀⠀⢠⠞⠐⠛⠛⠛⠛⠉⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⢦⣤⣤⣤⣴⠇⠀⠀⠘⠦⣤⣤⠤⠴⠶⠶⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢁⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠓⠟⠂⠀⠀⠀⠀⠀⢀⠀⠀⠠⠒⠀⠇⠰⠁⠀⠄⠀⠀⠀⠀⠀⠀⣉⣁⣀⡀⣈⠄⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⢰⠀⡠⠀⣀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢁⡿⠀⠀⠀⠀⠀⠀⠀⠂⡤⢄⡀⠀⡀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠉⠊⠙⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠤⠛⠳⠤⡠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠐⠐⠂⠔⠄⣁⡀⠀⠀⠀⠀⠀⠴⠀⠀⡰⠀⠀⠀⠀⢀⡀⣄⡤⢠⣴⠒⢺⠀⣏⠣⣅
⠀⢀⣅⠀⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠂⠚⠒⠒⠚⠠⠎⢥⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⡇⠁⣧⣻⣏⢻⣠⡜⠦⠟⠓⠚
⠀⠀⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠑⠆⠀⠀⠀⠀⠀⠳⠞⠛⠉⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣌⠑
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠟⠀⠘⠷⡀⠀⠀⠀⠀⠀⠀⢰⠀⣠⣦⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⢤⡤⠀⠀⠔⠀⠀⠀⠀⠀⠀⠹⠀⠽⠿⠇⠀⠀⠀⠀⠀⢰⠊⡇⣿⡁⣏⣹⣱⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠛⠛⠻⠏⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⣀⣝⣒⣒⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⣡⣤⣶⣶⣤⣤⣀⡀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣠⠖⢋⣭⣶⣿⡿⠟⣿⣿⣿⣿⣿⣾⣕⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣵⣿⣿⡿⠟⠛⢛⠻⣿⣿⣶⡄⢿⣧⠀⠀⠀⢀⡞⠁⡴⣿⣿⠏⠹⢀⣀⠙⢻⣿⠿⣿⠉⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣩⠕⠒⠊⠉⠈⠑⠮⢻⣽⠿⣾⣿⡆⠀⠀⢸⢃⡞⣼⣿⢣⣶⡟⠻⠃⠀⠈⢩⠟⠱⣄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠀⠀⣿⠈⣾⢋⣴⡿⢻⡿⢀⠀⠀⠀⠁⠀⠀⠈⠙⠢⣀⣤⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⣰⠏⡀⣡⣿⢿⡿⠈⢀⣿⣄⡸⣆⠀⠀⠀⠀⠀⢾⡇⣽⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡎⠛⠋⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣰⡇⢸⡇⢹⣍⡀⠀⠀⠘⣿⣿⢇⣸⡇⠀⠀⠀⣠⣾⣿⡟⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⣾⣷⣦⣤⡀⢠⣤⣴⣶⡬⠹⢟⣿⣿⣾⢣⣾⡗⣴⢻⡿⣶⠦⢒⡋⠥⠾⠀⢤⡀⠀⠸⣿⣿⡿⠃⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⠛⠋⠉⠃⠀⠈⠿⠟⠀⠀⣿⣿⣿⣿⣿⠋⠁⠈⠳⣄⠀⡤⠦⠀⠀⠀⠘⣿⠯⠷⠈⢛⢛⠃⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⡀⠀⢀⠆⠘⠀⠀⢀⣾⣷⣿⣿⣿⡿⢰⣾⣆⠀⣾⣯⠀⢆⠀⠈⠂⠀⠀⠉⠀⠀⣺⣭⣏⠀⠀⢼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿⠆⠠⣄⣠⠄⠆⢸⡿⣿⣿⣿⣿⢇⢸⡏⣻⣧⡙⣧⠀⠀⢄⠀⡱⠀⠀⠀⠀⢠⠀⣿⣿⣷⡀⢈⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣌⠀⢀⣈⣁⡄⠀⠈⡹⠋⣿⣿⣿⢸⢀⣽⣿⡿⠿⣿⠀⠀⠀⠀⠀⠐⠁⠢⠄⢸⣼⣿⠈⠛⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣼⢣⡼⣿⣿⣿⣿⣿⠛⣩⢡⡆⢹⡄⠀⠀⠀⠀⢠⡀⠄⡙⠋⣀⢰⡄⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡟⠿⢻⣿⣷⣦⣤⣴⡿⠃⢳⢰⣿⣿⠋⡹⠴⠋⠀⣽⣇⢨⣿⣆⠀⠀⠀⠤⢻⠷⣧⣘⡋⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡿⠷⣶⣼⠻⣿⣿⠿⣿⡇⠰⢿⣿⣿⢡⣾⣿⣴⣿⠇⣿⣿⡿⣿⣿⣷⣄⡀⢀⠞⠲⣾⣬⡛⠵⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠷⠀⠸⠿⠀⠙⠟⠀⠘⠀⠓⠾⠿⠧⠿⠿⠿⠿⠿⠤⠿⠶⠆⠼⠿⠿⠿⠿⠀⠀⠴⠗⠘⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⢀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⡀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⢰⣦⡀⠀⠀⠀⠀⢺⣿⣷⠀⠀⠀⠀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠻⠿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⣷⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠈⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠸⣿⣿⣇⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢻⣿⣿⡀⢹⣿⣿⡇⢹⣿⣿⡇⠀⠀⢿⣿⣄⣀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⣸⣿⣿⡇⠀⣿⣿⣿⡈⣿⣿⣿⡀⠀⠈⠛⠛⠻⠿⣷⡄⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠃⠀⠈⠻⢿⠇⠘⣿⣿⣷⣶⣤⣤⣤⣀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⢶⡄⠀⢠⣶⣶⣤⡀⠀⠀⢈⣙⠛⠛⠿⠿⢿⣿⣿⣆⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⣿⠀⢸⣿⡇⠈⢿⡆⢸⣿⡟⠻⣦⠀⠀⣠⣄⠈⠙⠃⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣀⣀⣀⢸⣿⡇⠀⠸⣷⢸⣿⣧⠀⢸⣧⢰⣿⡟⢧⠀⢀⡀⠀⠻⠿⠃
⠀⠀⠀⠂⠐⠢⣄⠠⠀⠀⠀⠈⢿⣿⡄⠹⣯⠘⣿⣇⠀⠀⣿⡈⣿⣿⠀⠀⣿⡄⢿⣧⡀⠀⢸⣿⡶⣤⣄⡀
⠒⠤⡤⠄⠈⠀⢹⠿⢷⡆⠀⠀⠈⠻⣷⣼⡿⠀⠘⣿⣄⢀⣿⠇⢹⣿⠀⠀⣻⡇⠀⠻⣷⡀⠘⣿⣇⡀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠒⠀⠤⠄⣀⣀⣀⣀⠀⠈⠛⠛⠋⠀⠀⠻⣧⣀⣿⡇⢀⡀⠹⣷⡀⣿⣿⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠱⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠲⠦⣤⣀⡀⠈⠉⠉⠀⠈⠳⣴⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠖
⠀⠀⠀⠶⠄⡀⠘⢆⠀⠀⠀⠀⠙⠛⠄⡀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠁⠀⡀⠒⠤⣤⣀⣀⡀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡀⠀⠀⠑⠀⠀⠀⠙⣄⠀⠀⠈⠉⠉⠉⠒⠶⠄⣀⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠀⢀⠀⠀⠀⠢⠀⠈⠢⢄⠱⠄⣄⠀⠀⠀⠀⣴⣼⣯⠭⡏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠤⠀⠀⠈⠀⠀⠈⠠⡀⠀⠀⠙⠂⠀⠈⠀⠔⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠎⡠⠀⠉⠓⢤⡀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠤⠴⠀⢀⣤⠓⣡⠀⠀⢀⡀⡤⠚⠀⠀⠀⠀⠑⣶⡄⠀⠀⠀⣸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠛⠣⠤⠀⠠⠤⠤⠄⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠩⠭⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣄
⠀⠀⠀⣀⣾⣟⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣷⣦⡀
⠀⠀⢰⡿⠉⠙⠿⠿⠿⠛⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣬⠀⠀⣴⠎⢻⡄
⠀⠀⢹⡇⠀⠙⠆⠀⠠⠏⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠰⠄⠀⣈⡁⢀⡴⢠⡷⠆
⠀⠐⢀⡀⠀⠶⠀⠂⢠⠖⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠑⠆⠀⠰⠈⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⣬⠤
⠀⠀⠀⠈⡀⠀⠀⠚⠂⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠲⠤⠔⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠈⢲⠄⠀⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠁⠐⠲⠂⠈⠁⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⢀⠂
⠀⠀⠀⠀⢀⡠⡀⢀⡴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡜⢣⠄⣜⡧⠤
⠀⠀⠀⠐⠊⣴⡏⠉⣦⠙⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠯⢁⠹⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢠⡀⠀⠀⠈⣴⠋⠁⠀⠀⣀
⠀⠀⠆⢀⣄⠈⠀⠖⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⣥⠄⣀⣀⡀⠀⠀⡀⢀⡈
⢠⠌⠀⠘⡇⠀⡀⠀⡆⠀⢷⠀⠀⠀⠀⠀⠀⠀⡅⢸⠁⠀⠀⢀⣤⣀⡰⠀⠀⠀⠘⣇⠘⠀⠀⠀⠀⠀⠐⠛⠁⠀⠉⠉⣀⡀⠤⠙⠤⠵
⡄⢀⡆⠀⡇⠀⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠇⢶⠀⠀⢈⡀⢉⠉⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢲⠞⠓⠸⠏⠉⢹⠒⠂⣤⡤⠔
⠀⠀⠡⠀⡇⠀⠀⠀⢘⠀⢠⡄⠁⣿⠀⠀⠀⠀⠐⡀⠐⠖⠉⢻⠃⠀⢀⡉⠰⢂⡠⢾⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⠀⠀⣷
⠀⠈⢄⡘⡷⠘⠣⣀⠺⠐⠻⣧⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⣠⣽⠆⠀⠸⣧⠄⠉⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⣴⠁
⠀⠀⠀⠛⠀⠀⠀⣀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠨⠁⠀⠀⢀⣉⣀⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠣⠔⠋⠘⠃
⠀⠀⠀⠀⠀⠀⡰⠉⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠠⡆
⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡞⠸⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⠀⠠⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠸⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠈⣣⠀⠿⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠃⠀⠀⠀⡇
⠀⡀⠀⠂⠀⢤⠀⠀⠀⣴⠄⠴⢇⣀⡀⠀⠀⠀⠀⠀⢀⡀⠸⠀⠃⠀⣆⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠈⠀⠀⠀⢸⡇
⠓⠤⠖⠚⠓⠊⠀⠀⠀⠉⠐⠀⠠⠔⠋⠀⠀⠀⠀⠘⠃⠀⠀⠙⠁⠀⠈⠃⠰⠾⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⣁⡄⠀⠀⣎⡋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠰⢿⣯⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⠀⢈⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠰⣷⠀⠀⡆⠀⢀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡽⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠻⣿⣿⣿⡇
⠀⠀⠀⢿⣀⣶⣿⡿⣿⢶⣤⠖⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠠⠌⢻⣿⣿⣿⡿⢂⠀⠀⠀⠀⠀⠀⡎⣽⠛⠁⢀⠀⠀⠙⠛⠋⠓
⠀⠀⠀⠀⢻⣇⣀⣾⣇⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠰⣾⠿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠑⠶⡀⠀⠘⢃⡰⠂⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠿⠛⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡽⢿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠈⢉⣟⠁⣠⣿⡀
⠀⠀⠀⠀⢠⡴⢂⠀⢘⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠉⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠰⠿⠿⠏⠙⠋⠁
⠀⠀⠀⠀⡿⠓⣿⠓⢻⡧⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢣⣀⠾⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⢀⠞⠀⠀⠘⢃⡜⠁⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠿⣿⠂⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠙⠊⠉⠐⣄
⠀⠀⠀⠀⢺⠀⠀⢸⠃⠀⠀⣶⠘⣇⠀⠀⠀⠀⠀⠀⡼⠀⣀⠀⢠⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠻⠀⠀⠀⠀⠀⠀⡇⡄⢹⠀⢸⣿⡄⠀⠻⢺⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⣄⠀⠀⠀⠀⠸⣦⣴⡇
⠀⠘⠀⣸⠸⡇⠀⠀⡀⠀⠀⢸⡇⠀⠂⠀⠀⠀⠀⠀⠀⠁⣸⠀⣿⣿⡇⠀⡀⢻⠀⠀⠀⠀⠀⠀⠀⠸⠛⠟⠏⠀⠀⠀⠀⠀⠈⢡⠃
⠀⠀⢆⠈⢳⡿⠀⠀⠇⠀⢷⡌⢁⠖⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⣿⠇⣠⠟⡌⠀⠀⠀⠀⠀⠀⠀⡇⣠⠀⠁⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⢰⣏⠀⠀⠀⡆⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠚⠙⠂⢹⡖⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣶⠀⠀⠀⠀⠀⢀⣹⣮⠃
⠀⠀⠀⠘⣿⠀⠀⠀⡇⠀⠀⢴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⣶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡾⠷⠀⠀⠀⠶⠞⠉⢻
⠀⠀⠀⠀⠘⠛⠓⠈⣉⠀⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠘⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠠⢄⠀⣀⠀⡄⠀⠈⡆
⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠤⡾⠀⡿⠀⢳⣤⠀⡇
⠀⠀⠀⠀⢸⡇⠀⣦⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⢸⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣈⣁⠴⠀⠃⣀⣀⡠⠋
⠀⠀⠀⢠⡏⠀⠀⠉⡆⠠⡟⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠈⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⠀⠀⠀⠀⡙⢃
⠀⠀⣾⡿⢿⠿⣷⣶⡇⠐⣿⣿⢿⣿⣿⡆⠀⠀⠀⠀⣀⡤⠨⡧⠀⠀⠀⣾⣧⣤⣄⠀⠀⠀⠀⠀⠀⡀⠐⠿⠛⢛⣄⠀⠀⢰⠋⠙⠻⢀⡄
⣤⣤⣈⣉⣉⣀⣈⣄⣠⣄⣠⣀⣄⣀⣄⣠⣀⣀⣀⣀⣉⣋⣠⣀⣤⣀⣀⣤⣀⣀⣠⣀⣤⣠⣀⣤⣄⣷⣶⣶⣤⣾⣁⣠⣄⣨⣤⣤⣤⣀⣀
⣿⠿⢿⠿⢿⠿⠿⠿⠏⠿⠿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡟⢿⣻⠟⢻⣿⣻⣿⣿⠻⠿⠿⠿⠿⠛⠿⠿⣿
⣿⣶⣶⣶⣾⣶⣾⣶⣷⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⠇
⠀⠸⡘⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠄⠔⠄⠀⠀⢀⣀⣀⣀⠤⠴⢞⠊
⠀⠀⠙⠲⢦⡈⠉⠉⠓⠢⣄⣠⠊⡰⠧⡇⢠⡠⠚⠉⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⡤⠞⢦⢠⠼⣾⠀⢀⠀⠀⣰⠿⠁
⠀⠀⠀⠀⠀⠰⠛⣦⡀⠀⡼⠡⣶⡟⢸⣶⡽⡀⢸⢦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⠁⠂⢴⠃⠘⡆⠀⢱⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⢄⣰⠃⠀⠀⢧⣀⢼⡏⠧⡀
⠀⠀⠀⠀⠀⠀⠐⠊⠀⡐⠔⠃⠀⠀⣀⣘⣢⣚⣁⡀⠙⣆
⠀⠀⠀⠀⠠⠆⠀⠀⠀⠁⠉⠉⠁⣧⣀⣀⡍⠈⢉⠃⠀⠬⠦
⠀⠀⠀⠀⣈⠀⠀⠀⠀⠈⠃⣼⣿⣿⡿⣿⣷⠈⢸⠀⠀⠀⣤
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⡈⠰⠁⠀⠀⠈⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠀⠀⠤⠠⡀⠀⢣⠀⡃⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠇⠐⠚⠟
⠀⠀⢀⠃⠀⠔⠘⠋⠥⠀⠈⠁⠤⣤⣤⣀⠀⠀⠁⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠆⠀⠀⠂⢐⠢
⠀⡠⠁⠀⡎⠀⣘⠀⠀⢦⠘⠀⠀⠇⠀⠀⠀⢠⢒⣨⠄⠸⢄⡀⢣⠀⠀⠀⠀⠀⠀⠈⣐⣆⡀⠀⠐⠀⠂
⢰⠁⠀⠠⢀⠂⠠⠑⢆⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠐⢁⠀⠀⠱⠀⡆⠀⠀⠀⠀⠀⠈⡎⠨⠰⡆⠲⣄
⢨⠄⠀⠘⡬⠀⢤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠇⢀⢤⠀⡄⠀⢹⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠖⠠⠠⠃⠀⠘⣄⢀⠀⠀⠀⠈⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠢⠂⠀⠀⣀⡏⠀⠈⠀⠉⠉
⢣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁
⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠌
⠀⠀⠀⠈⢢⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⢀⣠⠐⠫
⠀⢄⣀⡊⠤⣴⠁⢒⠆⡰⠄⡀⢀⠀⢸⠀⢖⢣⠎⢠⡢⠓⠜
⣀⣀⣀⣈⣉⣀⣗⣉⣀⣀⣀⣀⣀⣈⣉⣉⣩⣞⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢳⠀⠀⠀⠀⠀⠀⡈⠒⠀⠀⠀⠀⠀⠀⠀⣿⠐⠂⠀⠀⠀⠀⠀⠀⠐⠏⠆
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠆⠐⠐⠀⠐⠠⠀⠀⠁⠀⠁⠁⠀⠀⠡⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠈⠁⠀⠆⠀⠄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⠀⠀⠀⠀⠀⣀⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠆⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠀⠀⠠⠘⠆⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⢢⠄⠤⠂⠲⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⡀⠨⠃
⠀⠀⠓⠀⠀⠀⠋⠀⠀⠀⠀⠃⡀⠀⣀⡀⠀⠀⠀⢠⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⢀⡀⡀⠅⠀⠀⠀⠀⠁⠀⠀⠝⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⡀⠀⠀⠀⠀⠀⠀⡒⢲⡄⠀⠀⡀⣠⡔⠦⡄⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⠀⠠⢤⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠑⠀⣰⠀⠀⠀⠀⡀⢦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠒⢀⠀⡦
⠀⠀⠀⠙⠀⠈⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢠⠄⠀⠀⠀⠀⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀
⠀⠀⠀⡴⠢⠰⠀⠀⠀⠀⠀⠀⠐⠚⠲⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠀⠀⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠒⠂⠐⠒
⠀⢀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢤⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠠⠀⠄⠀⠌⠁⠀⠀⠐⠀⠐⠀⠀⠂⠀⠚⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠑⠖⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠢⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⣠⠶⠛⠉⠉⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⢹⢀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⣀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢗⣺⣿⠀⠀⠀⢸⠻⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠠⠶⢶⢦⡄
⠀⡴⠛⣏⠉⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠀⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠳⣤⣀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣤⣤⡤⠶⠋⠁⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠉⠙⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢀⣴⠞⠋⠉⠀⠀⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⣶⡄⠀⠀⠀⠀⠈⣇⢀⡔⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢲⣿⡿⠀⠀⠀⠀⢀⡟⠸⡗⣺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⣸⠇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⢷⣤⣀⣀⡀⢀⣀⣠⣴⠾⠋
⠀⢰⡿⠟⢿⣏⠉⠙⠛⠦⠀⠀⠀⠙⠳⠶⣤⣤⣤⣤⣤⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣼⠋⠛⠻⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠞⠛⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢰⠇⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣤⣤⣤⠼⠧⠤⢤⣤⠶⠶⠶⠶⠾⠶⢶⡶⠶⠶⠶⠶⠿⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣄⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⠶⠶⠶⠶⠶⠿⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣽⣭⣭⣯⣮⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⢰⠀⣆⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣈⢠⡆⢸⠀⣇⣿⢀⡿⢠⡿⠁⠠⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⢳⣶⣿⣿⣆⣷⠠⠂⣽⣿⣼⣷⣿⣾⣼⡃⢠⡟⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⡼⣿⣿⣿⣿⣿⣿⣳⣞⣿⣿⣿⣿⣧⣿⣿⣡⣟⣮⣞⣡⡶⣡⣤⡞⣁⣄
⠀⠀⠀⠀⠀⠀⢀⠀⢇⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡾⣛⣥⢾⣫
⠀⠀⠀⠀⠀⢀⠺⢧⣌⢶⣄⣜⢿⡿⣿⣿⣿⣟⡿⠿⠟⠙⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣻⣯⣥⣴⣴⣆⣀⡀
⠀⠀⠀⠙⢢⣝⡳⣄⡙⢷⣌⣻⢿⣿⣾⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠄
⠀⠀⠀⠈⡆⢌⡙⣻⣟⣿⣯⡻⣿⣿⠏⠀⣠⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⣿⣿⣿⣿⣿⣿⣵⣾⣿⡗
⠀⠀⠀⠈⠘⣿⣿⣾⣷⣶⣍⢿⣿⠋⠀⠎⠁⢀⡀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠐⠠⠻⣿⣿⣿⣿⣿⣿⡿⠀⠈⢀⡦⠞⣛⠃⢄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡌⠳⠀⢸⣿⣿⣿⣿⣿⣯⣽⣶⣯
⠀⠀⠀⢈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠸⣇⣾⣿⡇⠈⠀⡀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣹⣻⡿⣻⣿⠿⣷⣇⣀
⠀⠐⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠄⠰⢷⡆⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⣿⣿⣟⣻⣯
⠆⠀⢉⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠰⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⢂⣭⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣭⢿
⢀⠘⢿⣥⣦⣾⣟⢫⣽⣿⣿⠿⣏⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡠⠴⠦⠴⠦⣄⡀⠘⠀⠀⠀⠀⢸⣿⣷⣴⣿⣭⣟⣻
⠁⠀⠘⢿⣿⣿⣽⣿⣿⣿⣿⣾⣷⠀⣠⣄⠀⠀⠀⠂⢀⣴⣼⣤⣤⣦⣤⣶⣶⣷⠀⠆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠹⢯⠷⣿⣿⣟⣽⣿⣿⠆⠙⣿⡄⠀⠀⠀⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⠀⠃⠀⠀⢰⣿⢷⣽⣿⣿⣿⣯⣟
⠀⠀⠀⠐⣥⠣⠾⢋⣿⣿⣿⣿⣿⣷⡄⠸⣿⡀⠀⠀⢿⣿⣬⣄⠒⣹⡟⠈⢻⡟⠀⠀⠀⢠⡟⣿⠿⣿⣿⣿⢏⣿⡁
⠀⠀⠀⠀⠊⢀⣔⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⢿⡄⠀⠈⢻⣄⠄⠀⠐⠒⣂⣼⠁⠀⠀⣰⣿⣿⣾⣿⢾⣿⣿⡄⠐
⠀⠀⠀⠀⠠⠚⠛⢻⠿⣼⣿⣿⣿⣿⣿⣿⣄⡘⢿⣄⠀⠀⣍⠻⣷⠶⠖⠪⠀⠀⠀⣰⣮⠟⢿⢿⣿⣷⣯⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠡⣾⣿⣿⣿⣿⣿⣿⣧⣄⡻⣶⡄⠙⠓⢶⠶⠆⡀⠀⠀⠀⢻⡛⣦⢼⡆⠙⠉⠫⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣧⡿⠋⢿⣿⣷⣿⣽⢦⣀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⡿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣷⣷⣿⣶⡦⠀⠀⠀⠀⣿⣿⣺⣤⣄⢁⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠀⠀⠀⣠⣶⣟⣿⣿⣿⡋⠀⠀⠀⠈⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠾⠿⠿⣿⣿⣿⣿⣻⣿⣿⣿⡁⠀⠃⢀⣴⡟⡿⣿⡿⣿⠛⠉⠀⠀⠀⣀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠠⠐⠊⠓⠀⠀⠀⠈⠁⠀⠉⠛⠿⣻⣿⣻⣿⣿⣦⣴⣿⢿⠯⣷⣿⠇⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⠿⠟⠃
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣺⡅⣻⠈
⠠⠤⠄⠤⠤⠤⠄⠄⠀⠀⠀⠀⠐⠂⠈⢉⠀⠀⠀⠀⠀⠀⡄⠞⢻⢿⡇⠶
⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣾⡧⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⠷⣄⠀⣠⠶⣄⠀⠀⣀
⠀⢀⡤⣾⢻⡏⣷⣀⠀⠀⢀⣀⠀⢸⠉⠳⢤⡏⠀⠈⠛⠁⠀⠹⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⣀⣴⠳⣄⣤⡀
⠀⠸⣄⡈⠀⠀⣉⣹⠆⠀⢸⡍⠙⠛⠀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠘⠛⢹⠇⠀⠀⠀⢸⣍⠛⠀⠋⠼⠷⡄
⠀⠀⠈⢧⣀⠀⠻⣅⠀⢶⠤⠇⠀⢠⢎⠏⣯⢱⠀⠀⠀⠀⡏⣼⢹⢹⠀⠀⠛⢲⠆⠀⠀⣀⡾⠁⢀⣠⠤⠞⠁
⠀⠀⠀⠀⠉⢧⡀⠀⠙⠾⠇⠀⠀⠘⢾⣦⡯⠞⠀⠀⠀⠀⠑⠿⠾⠋⠀⠀⠴⠯⠶⠶⠛⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⢠⠴⣞⡿⠉⠉⠉⠹⠽⠋⠳⡀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⡇⠀⠀⣀⣀⡀⣀⣤⣀⠀⠀⡇⠀⠀⢙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠀⠀⠀⠳⣄⣞⡁⠀⠉⠁⢀⣈⡦⠞⠁⠀⢘⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⢶⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣀⡀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡇⠀⢤⣀⡤⣄⣀⡴⢶⠀⠀⣿⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠀⠈⠉⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⢸⠀⠀⠀⠀⠀⢸⠀⢀⠟⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⠤⠭⠥⠤⠾⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⠀⣀⣀⡀⠀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⢤⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⢀⡤⣶⣞⣷⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⢀⠄⠃⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠠⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠸⣿⣿⣯⡇⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣠⣼⣿⠔⢦⠀⠈⠐⠪⠅⣀⠀⠑⠒⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠐⠁⠀⠀⠉⠿⣛⢿⡗⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⣀⣤⣼⣿⣦⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⡼⠋⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⢤⣄⠀⠀⠈⠀⢠⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣠⡔⢾⠖⠠⠤⣀⣀⣤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⢀⡠⠇⠀⠀⠀⢰⠋⠈⠀⢐⢈⠉⡟⠃⠨⠀⠀⠀⡁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣳⠧⠀⠀⠀⠀⠀⡀⠀⠉⠀⠐⠀⠀⡅⠀⠁⠀⠀⣇⠀⠀⠀⣿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠁⠀⠀⠀⠀⡀⠘⠦⡀⠀⡴⠀⠀⠀⠀⠆⠀⠄⢊⠁⠀⢀⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠘⢆⡆⢶⡷⣦⡤⠀⡀⠀⡐⠂⠀⠀⠉⠀⢀⣾⠀⠀⠀⠐⠀⠐⠑⠤⠾⣧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡇
⠀⣼⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣸⡇⢰⣸⢻⣗⠀⠈⠉⠉⠂⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠀⠀⢹⡆⠀⠙⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢺⠁⠸⠙⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠇⢠⢾⠄⠘⡟⡆⢸⣗⠀⠀⢠⡐⠂⠀⠈⠀⠀⠀⠀⢰⡇
⠀⣿⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⠈⠀⠀⠀⠠⠆⠀⠘⠀⠀⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⢸⡇
⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⣀⣦⢄⣄⣀⡆⠀⠈⠛⠉⠉⠀⠀⠈⠁⠀⠀⠀⢸⡇
⠀⢠⠀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠟⠐⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡀⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⣃⣀⣰⣀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸
⠀⠀⠴⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⣚⣛⢛⣃⣀⣉⡉⠛⡀⠀⢀⣠⣀⢀⣀⣀⣀⣠⡶⢿⠿⣤⣤⠤⠤⢤⣤⣤⡤⣤⣤⠤⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⣽⣿⣿⢿⣿⣿⣿⣿⣧⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⠉⠉⠘⠛⠛⠛⠈⠉⠁⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⠁⠀⠀⠠⠦⠀⠀⠀⠀⠠⠦⠀⠀⠀⠈⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠤⢶⣶⣄⠀⠀⠀⠀⠀⠀⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣠⠾⢻⡟⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠠⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡠⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠉⠐⠀⠤⠀⢀⡀⠀⠀⠤⠀⢰⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⣻⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣮⣿⣿⣿⣿⣷⣿⣝⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣷⣶⣶⣿⣽⣿⣿⣭⣭⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⡍⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣂⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⣿⣿⣿⠋⠉⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼⣿⡿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣧⣦⣤⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⢀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⢉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠇

⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⣿⢟⣿⣿⣍⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡿⠿⠟⢻⣿⣿⡟⠛⠛⠋⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠛⠁⠉⠻⣟⣣⡀⠀⠀⠀⠻⠿⠟⠻⣿⣿
⠀⣿⣿⣿⣿⠀⢀⣴⡿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠜⢿⣱⣆⠀⠀⠀⠀⠀⢠⣿⣿
⠀⣿⣿⣿⣿⣵⣾⣍⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⡟⢲⡶⣶⢲⣞⢿⣿
⠀⣿⣿⣿⣿⣷⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠛⠚⠋⣻⣿
⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡄⠀⠀⣀⢀⡀⠶⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿
⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⣿⣿⡏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⣿⣷⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⣿⣿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣧⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤
⠀⣿⣿⣇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣹⣿
⠀⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢻⣯⢙
⠀⣿⣿⣟⡿⡿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⣿⣿⣿⡇⠘⣯⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣯⣬⡀⣼
⠀⣿⣿⡿⠃⠀⢸⣏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡏⠹⣿⡇⣹⣿⡇
⠀⣿⡃⠠⠄⠀⢸⣛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢿⡎⣉⠿⣿⣁⣿⣦
⠀⣿⡇⠠⠄⣤⣛⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⡟⣻⣷⣼⣿⣏⣿
⠀⣿⣿⣧⣾⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⣿
⠀⣿⣿⣿⣮⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⠀⠐⠃⠀⣿
⠀⣿⣿⡾⠟⠆⠀⠀⣴⣦⠄⠒⠲⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⣤⣄⣈⢷⣾⠀⠀⣀⡾⢻
⠀⣿⡿⠃⠀⣦⠰⢶⠛⡇⠘⠿⠀⣴⡆⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡀⠸⡇⣤⢹⡟⢹⣽⠁⢠⣿⡻⣿
⠀⣿⣿⠀⠀⠈⠓⣾⣦⣽⣮⣢⡤⠄⠒⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠘⠃⢀⡠⠋⣰⣿⣾⠃⢠⣿⣛⡇⢸
⠀⣿⣿⡇⠀⠀⢀⠈⣿⣿⣷⣭⣽⣿⣯⣥⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⣻⣘⣷⡼⠶⣤⣤⣔⣸⠏⣿⡁⡄⠘⣿⣻⣇⢸
⠀⣿⣿⡇⠀⣠⣤⡞⢻⣤⣍⣩⣧⣼⠿⣇⢁⣌⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣤⡄⠉⠙⠿⣿⣿⣟⣻⣛⣦⣤⣴⡿⠿⣇⠛⠀⠙⣿⣿⣿
⠀⣿⢋⣵⠾⣿⣈⣿⡾⣿⠛⢉⣷⠿⣷⡿⢯⣽⣅⣠⡋⠹⣦⠖⠛⠓⠤⢞⣀⡛⢯⣼⣥⣄⣰⡳⣦⡴⠛⠛⠻⣿⣋⣉⣠⣴⠿⣦⡀⠀⠈⢻⣼
⠀⣿⣿⣿⣤⡉⣿⢏⣤⠜⠿⣿⡿⣶⠛⡵⢫⡄⢀⡙⢏⣹⣧⡀⠠⡶⠈⡏⠀⣰⢊⣤⠀⠈⢻⠖⢺⡀⠦⠙⠂⣿⢿⣿⠛⣁⣴⣿⠟⣷⣶⣤⣿
⠀⣿⣿⣟⠻⢿⣿⣏⠶⠘⠃⣹⢀⣇⡔⣇⠠⠆⠻⠃⣼⠽⣄⠁⢀⣠⣾⣩⣧⡇⢰⠆⠺⠃⣸⣄⣤⢿⣦⣀⣴⡍⠘⢉⠻⣿⣋⣠⣾⢋⣴⣟⣹
⠀⣿⣿⠿⡷⣿⣧⡙⣶⣾⠾⢃⣋⣤⣤⣛⡶⢊⣿⣻⣱⣶⣌⣿⠿⢞⣣⣥⣜⣳⣦⣶⢤⢞⠛⠸⢋⡈⢁⣛⣛⣥⣾⣿⣷⣾⣿⣿⣷⣾⣏⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⡀⠄⠈⠀⠀⠄⡄⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠄⠀⠀⠀⠉⣁⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡔⠉⠀⠀⠒⢠⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⡌⠀⠀⠊⠁⠈⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠠⠀⡠⠂⠐⠢⢀⠢⡀
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠤⠄⠀⢧⠀⠀⠣⣀⠀⣠⠃⠀⠠⢀⠀⠀⠀⠀⠀⢀⠎⣠⠒⠶⡄⠀⢹⡌⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣉⠠⠤⠒⠁⠀⠀⠀⠀⠀⠀⠸⣄⠣⣀⡴⠃⢀⠌⠁
⠀⠀⠀⠀⣤⢄⠉⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠉⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡐⠁⠀⠀⠀⠱⠀⠀⠀⠀⡠⠀⠒⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠐⠤⠐⠊⠀⠀⠀⠀⠀⢰⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⡠⠌⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠀⠀⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡠⠖⠋⠀⠀⡠⠂⠞⢐⡁⠀⠀⠀⠀⠠⠨⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠞⠈⠠⠊⠀⠀⠈⠀⠀⠀⠈⠠⠀⠀⣠⠀⠀⠈⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢃⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠈⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⢹⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣨⣭⣍⠻⢿⠿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣷⣆⠀⠙⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⡟⣉⣿⣿⣿⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢹⣿⡟⡟⠛⢻⣿⣾⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⡇⠀⢹⣿⣿⣿⣿⣋⣿⣿⠃
⠀⠀⠀⢸⣿⢳⣿⣿⣿⣿⠛⠿⣽⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠟⠋⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⡇⢸⣿⢿⣿⣿⠀⢀⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣷⣌⡉⠈⠙⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣦⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣌⠃⠀⠑⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣦⣀⠀⢀⡶⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣷⣦⣤⣀⣀⣴⣿⣿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⠻⠄⠀⠀⠀⢰⣧⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣟⡋⠀⢻⣿⣿⣿⣿⣿⣿⣤⡉⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢸⡟⣿⠆⠀⠀⠀⠀⠀⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡜⠀⢀⠀⠀⣀⣀⡀⠀⠘⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣸⠀⠘⣦⠀⡀⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢀⡿⠀⠀⠹⣿⣻⣾⣧⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⡜⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣠⣉⠀⠀⠐⢦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⡏
⠉⠛⠻⣶⣤⡈⣷⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠊⡙⠛⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⢀⡀⣀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⡍⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣀⣀⠀⢀⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠁⠀⠀⢄⠀⠀⠀⠀⠀⡔⢈⣰⣀⠎⣭⠑⣼⣥⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⢠⠋⠀⠀⢹⠛
⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠀⠀⠀⣀⡑⠂⠛⠉⠓⠒⠊⠈⠓⠒⠁⠀⠀⠀⠒⢲⡶⠶⢶⡤⣼⣿⠤⣤⣄⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣴⠟⠁⠀⠀⠀⠀⠘⢹⠀⠀⠀⠀⢲⣀⡇⠀⠀⠀⣹⠳⠀⠀⠀⠈⠳⣤⡿⠷⠋⠘⢄⣰⣿⣽⠃
⠀⠀⢀⣠⠄⡌⠁⣀⠄⡨⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠈⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣭⠭⢴⡆
⠀⠀⠀⠀⠈⠉⡨⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠘⠿⠟
⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠲⢲⡆⠰⢪⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⣠⡆⠀⠀⠀⠀⠀⠀⠈⠨⠋⠘⠮⠝⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡖⢁⡄⡆⠌⢀⣤⢨⡄
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⡍⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠑⢬⠿⠃⠘⠚⢿⠛
⠀⠀⠀⣀⣐⠁⠀⠀⠀⠀⠀⡐⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡳⡀
⠀⠀⡆⠀⠁⡈⢦⠀⠀⢀⡴⣥⣿⡇⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⢀⡠⡀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡷⡱⡄
⠀⠀⠑⠤⣄⣀⣼⠤⠖⠉⠀⢹⡁⠀⢠⠀⠀⠀⠀⠀⠀⢳⠀⠀⠃⠒⠀⢀⠇⡱⠁⠀⠀⠀⠀⠀⠀⠀⡄⠿⠀⠛⢸⡷⢃⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡋⡘⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⠤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡠⠯⣀⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠴⡂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠈⣡⠴⠶⢬⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠚⢓⣦⣄
⠀⠀⠀⠀⠀⠀⢠⣘⡸⣜⡣⢄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢿⡇⢶⣷⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢋⣾⣙⡟
⠀⠀⠠⠴⠤⠆⣚⣼⣐⣸⣄⡣⠇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢈⣹⣶⡷⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡌⢿⣿⣿
⠀⠀⠀⡆⠀⠀⠈⣘⠸⢑⣢⠄⠀⠀⠹⠒⠀⠀⠀⠇⠀⢀⣄⠀⣿⣆⣸⣿⢳⣦⡀⠀⠀⠀⠀⠀⣠⠞⠀⠱⣈⠖⠉
⠀⠀⠀⠱⡀⠀⠀⡀⠀⠀⢀⠀⠀⡠⠁⠀⠀⠀⢸⢠⣆⠸⠉⠆⣿⣿⣿⣿⣾⣥⣿⠀⡆⠀⠀⢸⠁⢤⡁⠀⠀⠤⣤⣄
⠀⠀⠀⠀⠙⠢⠌⢀⣀⣠⠤⠔⠋⠀⠀⠀⠀⠀⠘⡈⡿⡌⠀⠸⠋⠊⠍⣛⡛⠋⠁⠐⠁⠀⠀⠈⠣⡀⠀⢀⣀⠔⣹⢀
⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣐⡘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠙⢎⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⢀⡀⠀⠀⠀⠀⢸⣇⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⡀⠙⠦⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢈⡁⠀⠀⠀⠀⠀⠷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⠦⡬⠵⠊⠒⠀⢤⣀⡄⡀⢄⢤⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠆⠀⠻⡀⣆⠀⢀⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠃⠛⠁⡈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠤⢀⡇⠀⢠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⠀⢀⠀⠁⠠
⠀⠀⠀⠀⢠⣀⢀⠄⣽⣢⡈⠉⠀⠀⢨⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣿⡍⠠⢄⠏
⠀⠀⠀⠀⠀⢸⠁⢰⣭⣍⠁⠀⠀⠀⣴⡀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠉⠘
⠀⠀⠀⠀⠀⠀⣅⠺⠀⠄⢁⠀⠀⢠⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡌⠶⠴⠃⠀⠄⠁⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠺⢷⡀⠀⠀⠀⠀⠠⠖
⡀⠀⠀⠀⠀⠙⠦⢷⡀⣀⢀⣴⡆⠀⠀⢤⣯⠏⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠂⣄⣱
⣷⠀⡷⠀⣾⡄⣴⡆⠉⢡⠀⢀⣤⢀⣸⡾⠌⠀⠀⠀⠀⠀⠀⠀⢓⡿⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠉⠉⠉⠀⠀⠐⠁
⠙⠀⠛⠀⠟⠁⠟⠁⢀⠈⠀⠘⡟⠸⠏⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⢸⡆⠀⠠⢶⣾⢀⡄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣮⠃⡇⠀⠀⠀⠃⠀⠉⠀⢀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡙⢚⠒⠂⠀⠀⠀⠀⡀⡄⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⣄⠀⠀⡄⢀⣈⣀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠉⠉⠉⠉⠂

⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡦
⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⡡⠚⠁
⠀⠀⢻⡈⠳⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠸⠋⡄⠀⠀⠀⣀⣀⣤⡤⠤⠴⠒⠿⣫⠎
⠀⠀⠀⠙⠦⠶⣄⠀⠈⠉⠉⠛⠶⣄⠀⢀⡴⠁⣠⣇⣼⠀⢁⣀⠶⠛⠉⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠛⡽⣡⣮⡁⠀⢈⣧⣸⠉⠂⠀⠀⠀⠀⢠⣮⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⢰⡟⢁⡀⣷⢰⣯⡉⣿⠀⠀⣧⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠋⢹⢷⡀⠀⢠⡟⠀⣿⢇⡏⢸⡿⣷⡞⣆⠀⣞⠳⣤⡼⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⡿⠀⠠⠠⡼⠁⠈⣷⠀⠀⢸⡆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣇⠀⠀⣰⠃⠀⠀⢹⡀⠀⢸⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠉⢰⠈⠓⣾⠃⠀⠀⠀⠈⡟⠶⠋⣿⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⡓⠚⠁⠀⠀⢀⣀⣀⣉⣓⣚⣃⣀⡤⠀⢙⡆
⠀⠀⠀⠀⠀⠀⢠⣮⠀⠀⠀⠀⠋⠙⡋⠉⠛⠈⣇⠀⠉⠀⡏⠀⠉⢨⠃⠀⠠⣍⣢⡀
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢸⡠⠇⢰⣦⣶⣿⣶⣿⣿⡇⢀⠃⢺⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠒⠖⠁⠀⠀⠀⠀⠀⡆⢠⠘⣿⡿⠻⠟⠻⣿⡷⠂⠀⡿⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢡⠀⠘⡏⠀⠀⠀⠀⢀⡴⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⠃⠀⠀⣤⡀⣀⠀⠀⠈⣄⡀⢰⠐⠂⠐⠋⠹⠀⡧⢠⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡔⣡⡃⠠⠴⢖⡶
⠀⠀⠀⠀⡇⠀⠀⢀⠄⣶⠬⠇⠀⠀⠵⠖⠞⠒⠓⠚⠂⠲⠦⠧⣤⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⡄⠁⠈⢀⣀⢒⣁
⠀⠀⠀⡤⠂⠀⢠⠋⠀⣀⣈⠑⠒⠒⣄⠀⠒⢺⠟⡷⠶⠤⠀⠀⠀⠀⢀⡤⠒⢲⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠙⠠⣀⡀⠉⠙⠀⠀⠀⠀⠠⠄⠚
⠀⢀⡜⠁⠀⢰⠃⠀⢈⠣⠄⠬⢻⠤⠍⠄⠀⠘⠂⠁⠀⠀⠀⠠⡖⠂⢉⡤⠀⠸⢄⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢝⠚⣠⠀⠀⠀⠒⠀⠃
⠀⡜⠀⠀⠀⡎⢠⣄⢫⡉⢢⡀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡡⠖⠀⠀⠀⢹⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣞⠋⠹⢁⡴⠀⢠⡈
⢰⠁⡀⠀⠀⠁⢠⠈⡀⠉⠢⠙⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⡴⡪⠃⠀⠀⢨⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠁⢸⡠⠁⠙⠂
⠰⡞⠀⠀⠀⠒⠞⠀⠳⢎⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⡶⠋⢀⠖⡲⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠁
⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠤⡾⠂⡠⢀⠎⠀⠀⠈⢣⡀⠠⠀⡀⠀⠀⢀⢢⠃
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣠⢁⡄⠀⠀⠀⠀⢸⠏⠉⠉⠒⠠⠤⠕⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠦⢿⠁
⠸⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤
⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠊
⠀⠀⠀⠀⠀⢙⣶⣤⣄⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠢⡀⣀⣠⣔⠊⠑⠆
⠀⢰⢈⠉⠒⠩⣀⣰⠋⠤⢰⡄⠈⣆⡀⠀⠀⠀⠀⢠⡇⠠⡖⠲⣠⣏⢠⠴⡬⠦⡠⠂
⠀⠀⠊⠉⠓⠖⠒⠛⢆⡠⠚⠒⠉⠀⠈⠁⠒⠒⠲⠖⠳⠶⠶⣀⡇⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⢀⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢀⣾⣧⣴⠶⠿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣾⢿⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠾⠛⠉⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠀⢠⡿⠁⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣦⣴⠟⣠⣶⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣼⠏⣤⣴⠟⠁⢀⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢿⡄⣼⠏⢀⡾⠋⠉⠀⠙⢷⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⢻⡇⠀⠀⠀⠀⠀⣼⠏⠀⠾⢋⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⢸⡇⠀⠟⠀⠀⣸⡇⠀⠀⠀⢠⡞⠉⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣤⣤⡴⠟⣠⡄⠀⠀⢸⣇⠈⠛⠀⠀⣼⠇⠀⠀⠀⠀⠀⠸⣿⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠀⠀⠀⠀⠙⠶⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⣤⣈⣉⣀⣀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡾⣋⣿⣄⣉⠉⠉⠉⠉⢁⣀⣴⡏⠈⢻⡆⠀⠀⠀⠀⠀⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢸⡿⠋⠀⠀⠉⠉⠛⠛⠛⠉⠁⠉⠻⣦⣘⣷⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⡼⠏⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠉⠛⠛⠻⠶⠟⠛⠛⠉⠀⢀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⣤⠤⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠐⠀⠆⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⠁
⠀⠀⡈⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠐⠂⡀⣄⣤⡄⢠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⡄⠀⠀⠀⣲⠂
⠀⠀⠀⠀⡀⠈⠐⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠉⡔⡁
⠀⠀⠀⠀⠁⠖⣀⠶⢄⡀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠆⢸⡆⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⠉⠐⠒
⠀⠀⠀⠀⠀⠂⠀⠙⠉⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠐⠒⠠⢄⠄⠀⠈
⠀⢰⠋⠃⡇⣔⠀⡀⡀⢤⢠⠀⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠆⠀⠙⠂⠀⠉⠁⠀⠻⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡀⠀⠛⠛⣤⣀⠀⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠀⢀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⢆⠀⢀⠈⠉⡀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣦⡀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢿⠀⡀⡞⠀⠉⢠⠸⢃⠗⢠⣀⡴⠀⠀⠀⠀⠀⢸⣷⠒⠋⢸⣧⠈⠒⣲⣿⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⡀⠈⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⠠⠯⠬⠅⠡⠌⠋⢅⣲⠚⣄⡠⡴⠃⠀⠀⠀⠀⠀⢻⠙⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀
⣧⣄⣀⠀⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⠈⠉⠚⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄⠀⠱
⠙⢿⣿⣤⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠁⢂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣌⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⢹⠁⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢈⡀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠚⠹⡀⡴⢻⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣷⡀⣿⣦⣸⢀⣨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣾⣽⡿⠛⠾⢻⣿⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⣰⣄⡀⢙⣟⠱⡏⠐⣄⡏⠉⢇⣉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⠀⣿⣿⡷⣁⠘⣾⣀⠄⢈⣀⡉⠉⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢸⡿⡇⠈⣢⡌⠉⠢⣯⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⣶⡌⢱⠙⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢀⠿⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢯⠀⠀⠈⣂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠄⣿⠀⢨⠀⠄⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⢠⠃⣺⡛⠀⣯⠀⠀⢸⠛⠓⠢⠴⠤⠤⣤⢄⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⢰⣦⡟⠀⠇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⢁⢸⢛⣿⢖⣄⠀⣀⢾⠛⣷⡄⡀⠔⠀⠀⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⣿⠃⣼⠃⠈⣧⠀⠎⢰⡼⠥⠃⡇⢸⣀⠀⠘⠆⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠈⠀⢀⡀⢼⠇⣻⣟⠀⡟⣠⠀⢹⣦⠷⠀⠘⢦⣾⠁⠘⢁⡤⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⠠⠤⠒⠒⠂⠤⠀⠀⠠⠤⠀⠀⠈⠀⠀⠀⠶⡀⡟⢹⢱⡟⣴⡇⣿⡉⢸⠙⠀⢠⡾⡇⠋⠀⠈⠻⠄⠠⢿⡠⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡏⠀⡎⠹⡗⠛⠏⠀⣠⠘⢠⢠⣸⠇⢁⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣰⢃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠐⠒⠀⠀⡈⠹⡀⢰⢁⡼⠁⠐⢠⡇⠀⢰⠜⡛⣿⣀⢵⣠⡙⠚⠀⠐⣴⡖⠀⠀⠀⡿⣸⣏
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠤⠤⡀⠀⠀⠀⡀⠠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠻⠦⠞⢤⣀⣰⠀⢋⠸⣇⡽⠃⠀⢹⣗⠤⣀⠐⠏⠀⠀⠄⠂⠛⠛⢿
⡠⠤⠖⠚⠊⠁⠀⠒⠕⠀⠂⠀⠿⠀⢰⠋⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠓⠊
⠀⠀⠐⠀⠀⠀⠀⠐⠄⠄⠀⠰⠂⠀⠘⢄⠀⠀⠀⠀⠈⠁⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠠⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠍⠀⣧⡄⠐⠚⠛⠛
⢀⢺⣧⣀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠏⣶⡗⠰⣀⣠⠖⢹
⢺⢸⢹⢸⡉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⠀⠀⣠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠀⠀⠀⠀⡰⡄
⠸⣾⡜⠻⠄⣧⠉⣾⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠘⠀⠀⠀⠖⠀⠀⠴⠀⠀⠀⠠⢆⠀⡐⡄⠀⠫⢹⠾⠙⠀⢀⣸⣡⡇
⠀⢻⣴⠀⡆⢸⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣄⡄⡒⠃⡂⣀⡄⠀⢰⡦⠄⣀⠺⢟⣻⡴⣿⣧⠀⠀⠱⠂⠘⠀⠀⣽⣿⠃
⠀⠙⣟⠸⡾⠀⠀⠙⡙⠙⡂⡀⣀⠄⢠⣄⢈⠉⢉⡔⠀⢔⣌⣻⡁⢶⠶⠄⠉⠀⢿⡿⣖⠿⠟⠀⠀⠸⠀⠀⡀⠐⣟⢉⡆
⠀⠓⡃⣿⡗⢤⡦⡛⡜⠄⢊⠉⢿⡛⢺⣿⡄⢀⣸⣶⠚⣈⡹⠉⡅⢸⢘⠀⠶⠆⢨⣀⣼⢠⢄⠀⠤⠀⠀⠀⠀⠀⠚⠅
⠀⠀⠉⠁⢁⠘⣇⡠⣷⡦⠄⢠⠶⠄⢾⣻⠇⠋⣁⡻⠛⢷⣶⠶⢧⡸⢾⡾⠛⠛⠙⠯⠉⠙⠛⠛⡿⠚⣦⣴⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠁⣬⡷⠽⠃⠀⣴⠄⠂⢸⣿⠇⠀⢿⠷⠐⢾⡿⣷⡘⢡⣾⣗⠒⠐⠐⡂⢛⠓⠒⣻⣿⣀⣼⣷⣾⣿⣿⣿⠟
⠀⠲⠖⠀⠀⠠⠉⣦⣾⠀⢀⣿⡏⠁⠀⠇⡸⠀⢸⣤⣤⣌⣀⣛⣡⣛⡁⣻⢹⠀⢩⠃⢘⡉⢭⣻⢿⡿⣿⣿⣿⡿⢿⡟
⠀⣠⣴⠀⠀⡆⢠⢽⣷⠇⢨⠉⠠⡀⠀⠀⠁⠀⡌⣽⣿⡇⢘⣿⣿⣿⠃⣤⢨⠀⢈⠀⠀⠀⠨⣏⢸⣷⣄⣦⣤⡄⢻⡄
⢀⣁⣄⠀⠀⠈⢰⣿⣿⢸⠿⠁⠈⠅⡄⠀⡀⣀⡠⢻⠛⠓⢼⣿⣁⣿⠀⠧⣀⡟⠈⢄⣶⢀⣱⣟⡁⠷⢿⢻⣿⣿⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠈⣷⠄⣤⣬⡴⡶⠶⢮⡕⠶⡤⠤⣴⣶⠾⣵⣾⣦⣈⣿⣯⡁⠛⠋⠉⠁⠁⠉⠀⠐⣂⡤⠔⠒⢂⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠨⡅⣾⢛⣿⠏⣰⠁⢸⡈⠘⠃⣌⡛⠻⠓⠋⢉⣤⠤⠭⠭⣀⣀⡀⠀⠀⠀⠀⢀⣠⠞⢁⣤⣖⣛⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢿⣾⠟⠳⣄⣿⠒⠲⣧⢀⠊⠉⠙⢄⡠⠚⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⣠⣿⣭⣉⣁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠌⡙⠲⣄⡼⠁⣀⡀⠘⢿⠀⠀⣠⠊⠀⠀⢀⣤⠶⠖⠶⠦⣤⣀⡀⠀⠀⣀⣠⣞⣍⡤⢥⣭⡭⠞⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠸⣇⣴⣿⣷⡄⠈⣢⠖⠁⢀⣤⠾⠟⢛⠿⡿⢛⡒⢄⠈⠉⠋⢋⡉⢜⣪⡵⠞⠉⢿⣿⣿⡟⢠⠖
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⢀⢿⡻⢾⣷⠿⠋⠁⣠⡴⢋⣤⠶⠛⠛⢻⠁⡡⢦⡈⠻⡿⠿⣿⢾⠿⡟⢩⣭⣽⣷⣿⣿⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣝⡲⠫⠿⢶⣤⡤⠶⢚⣡⡶⠋⠀⠀⠀⠀⡙⡜⠰⠷⡱⡄⠉⠀⠀⠀⣰⡾⣫⣿⠿⣿⣿⡟⢸⢏
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣬⡿⠛⠛⠿⠿⠿⠿⠿⠛⠃⠐⣚⠛⢲⣶⣾⣿⣮⣵⣓⡱⣝⢄⡠⣤⠾⣩⣞⣵⣮⣿⣿⡟⠁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡙⠛⠃⠿⠷⠶⢤⣤⣤⣤⡶⡿⠖⢋⣩⡾⣟⣛⡳⡶⠿⢭⣽⡿⣭⣔⣶⣿⣯⣼⡿⠟⠁⢀⣼⡷⣽⣢
⠀⠀⠀⠀⠀⠀⠀⢸⣛⢽⡜⠲⠄⠀⠙⠢⣘⡄⣀⣃⣤⣞⢫⣟⠋⣷⢮⡗⢀⣾⠿⣚⡿⠟⣛⣉⣤⣥⣤⡴⣶⣾⢏⣧⡿⣦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢷⠀⠀⣶⣶⣾⠋⠉⠉⠙⠢⣄⡉⢿⠖⡀⠉⠈⢻⠇⣴⣻⠶⠛⠉⠛⠙⠚⣶⣾⡿⣻⠻⠛⣹⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡄⠀⣏⢻⣿⣖⣞⣿⣶⡾⢿⠿⣆⢟⡔⢀⠂⠀⢰⣿⣿⣟⡒⣲⢚⣾⠾⢛⣩⡭⡿⣿⠟⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⡟⠤⠙⠻⠮⣭⣎⢚⡭⢝⣿⡾⢠⠎⡜⣀⣾⢹⣾⣶⡆⣠⡟⢁⡾⠿⣧⣴⣤⣁⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠐⣇⠠⣿⣀⣈⣘⣲⣶⡍⣯⠒⢦⣹⡇⣏⡼⢀⡿⣧⡈⢛⣼⣷⠋⢠⣿⢗⡥⢷⣼⣿⡿⠛⢻⣶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣆⠙⢷⣒⠠⢿⢿⢧⣟⢩⡳⠿⠛⣛⣑⣋⡛⠺⡷⡿⠛⢁⣴⠟⠓⠉⠚⠳⢏⣍⣬⢢⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⠈⠻⣾⡿⠾⠾⣶⣚⣯⡁⣶⣊⡡⢀⡤⢬⠯⣽⣶⡒⣛⢋⣡⠶⠛⡹⣛⣗⠶⣦⣾⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡷⣄⣀⠑⣄⠔⠉⠀⠉⢻⣿⣿⣉⡄⡀⠤⠺⣿⣟⡿⡿⢻⡎⣳⣆⡘⠿⢋⠟⣷⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⠛⣛⣫⡏⠛⣦⢠⡇⠈⢨⠋⡰⠣⢾⠋⡼⢠⠃⡞⣱⠸⣮⡝⠛⣯⠘⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣧⠹⠿⣿⡢⡿⠎⠀⢔⣡⣐⣁⢐⡁⣊⣴⣁⢊⠀⢩⠐⢁⡿⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣙⠋⣁⣿⠛⠔⢦⣬⢭⣤⠨⠍⢭⣤⣤⡬⢳⡞⣻⣤⡀⣈⣿⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣞⣷⠠⣿⡷⡌⠛⠪⢨⢥⣤⡤⢐⡈⠉⠀⣏⡻⣓⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣅⣔⣏⠀⠀⢀⡴⠃⡴⢋⡴⢊⠔⡉⡽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣾⠞⢋⣠⠎⣠⠏⣰⢃⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠃⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⡤⠤⠤⠶⠶⣤⠀⠀⢹⣟⣿⡿⢿⠿⢿⣭⡧⠀⠀⠀⠀⣰⣿⡿⢿⣿⠿⣿⡇⠀⠀⠿⢿⣿⠿⡆⠀⠀⢠⣿
⣿⣿⣿⣿⣿⣿⢹⡿⠀⠀⣿⡆⠆⣀⣀⠰⢿⠀⠀⠀⠛⠿⠦⠾⠷⠞⠋⢁⣀⠀⠀⢀⣿⠿⣄⣸⣟⣀⣼⡇⠀⠀⢳⣾⣽⣾⡇⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣧⣤⣻⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡇⠠⡾⠟⢰⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⡀⠀⠛⠋
⣿⣿⡿⣿⣿⣿⠀⠀⢀⣾⠛⣿⠟⠛⠛⢻⡿⣦⠀⢠⣾⣿⣧⣀⣠⣄⠘⢿⠟⠀⠀⠀⠀⣼⡇⣿⣷⣿⢸⠇⠀⠀⠀⠛⠉⠛⠛⠃
⣿⣿⢃⣿⣿⠇⠀⠀⠘⣧⣀⣿⣇⣠⣤⣼⣧⡟⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢠⠆⠀⠀⠙⠁⣿⠛⠟⠀⢀⣴⣲⣷⡖⣶⣦⣄
⠿⠋⠚⠛⠁⠀⠀⠀⠀⠈⠉⣿⡿⠿⠿⢿⡏⠀⠀⠸⣿⡛⣿⣿⣿⣿⣤⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢭⣽⣿⣿⣛⣿⠇⠀⠀⢠
⠀⠀⠀⠀⠀⠠⡀⠀⢠⡴⠶⠿⠇⠀⣠⣼⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠛⠋⠁⠀⠀⠀⣠⣤⣤⣤⣄⠀⠃⠀⠀⠈⠉⠉⠀⠀⠀⠰⣤
⣿⡶⣦⣄⠀⠐⠋⠀⠘⠀⠀⠀⠀⠸⠯⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢿⣽⣧⣀⠉⣀⣽⣷⣶⠀⠀⠀⠀⣶⠾⠛⢧⣤⠞
⠛⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣣⣿⠀⠀⠀⠸⠷⠶⢶⠀⠀⠀⠀⠉⠛⣿⡟⠋⠉⠉⠀⠀⠀⠀⡇⢠⠖⠋⠙⢦
⢀⡀⠀⠾⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⠀⠀⠀⠀⠀⠀⣴⠞⣿⣿⣿⠲⣦⡀⠀⠀⠚⠿⠛
⣿⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⣴⡇⠀⠹⢦⣿⣿⣿⣀⣼⠇⢀⣀⡀⠀⠀⠀⢰⣳⣾⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠀⠀⠋⢻⣤⣄⡀⠀⣀⣿⠀⠀⢾⢿⣿⠇⠀⠀⠀⢹⣿⣿⡏⠀⠀⣾⣿⣿⠀⠀⢀⣿⣿⣿⣿
⣿⡿⡿⠀⠀⠛⠀⠀⠀⠀⢀⣴⣖⣧⠀⢀⣤⣄⣀⣼⠿⠿⠇⠀⢸⣿⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠻⣿⣿⡀⠀⠹⣿⣿⡃
⣿⡄⠀⠀⠀⣄⠀⠀⠀⠀⠈⠛⠋⢹⣶⣾⠿⠿⠛⣻⠀⠀⠀⠀⠘⠿⠗⠀⠀⢤⡟⣻⣷⠀⠀⠀⣹⣼⡇⠀⠀⠀⠀⠈⠁⠀⠀⠘⠿⣷⣷
⣿⣿⠀⠀⠀⠁⠀⠀⠼⠆⠀⠀⠀⠼⠀⠀⠀⠀⠀⠉⠀⣠⣴⢶⣶⠶⣤⣄⡀⠀⢿⣿⣿⡶⠂⠀⠉⠛⠁⠀⣰⣞⠛⣳⣄⠀⠀⢠⣾⡿⣉
⠁⡏⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⢀⣶⠾⠟⠀⠀⠀⠀⠀⣯⣭⣽⣛⣿⣟⣻⡿⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢰⣇⡹⠿⣅⣿⠀⠀⢸⡈⣿⢿
⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣸⡇⠀⠀⢠⣤⠀⠀⠀⠸⠃⠀⠉⠉⢉⠉⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⠀⢸⣿⠓⠂⣾⡏⠀⠀⠀⠳⣿⢸
⠀⠀⠀⠀⣦⢰⡆⠀⣵⠆⠀⠉⢙⣁⣀⣀⣼⣹⣄⣀⣀⡀⠀⠀⡀⠀⢾⣦⣀⠀⠀⠘⡏⣴⠰⣮⢳⠆⠀⢸⣿⣦⢾⣿⡇⠀⠀⠀⠀⣿⠹
⠀⠀⡆⠀⢹⣎⡟⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠛⠲⣴⣿⣿⣦⠀⠛⣧⡻⠆⣂⡼⠃⠀⠈⠛⠓⠘⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣿⠃⠀⢠⣿⣿⡟⣫⣍⡻⣿⣿⣿⣿⣿⣿⣿⢿⣷⠀⠀⢹⣿⣿⣿⣧⠀⠀⠈⠋⠉⠁⠀⠀⣠⣶⣖⠲⡄⠀⠀⠀⠀⠘⠃⠰
⣦⡄⠀⠈⢳⣿⠀⠀⢸⣿⣿⣎⣁⣼⡧⣿⣿⣿⣿⡏⠉⢋⣼⣿⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⢀⣶⡿⠿⢶⣿⡟⣹⢇⡇⠀⠀⠀⠀⠴⢦⣄
⠀⠀⠀⠀⣼⣿⡇⠀⠘⢿⣿⣿⣿⡿⠃⣾⣿⣿⣿⠿⢿⠿⠟⠋⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠸⣾⣼⣽⠎⣿⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠙⠛⠛⠀⠀⣀⣈⠉⠁⠀⢀⣿⣿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠇⠀⣠⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⢠⣤⣤⣴⠄⠀⠀⠀⣼⣿⠿⠇⠀⠀⣼⡿⠁⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⠀⠀⢿⣿⣿⣟⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣵⣀⣶⣶⣤⣀⠀⠁⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠐⠛⠉⠀⠿⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉
⡿⣿⣿⣿⣿⣿⣿⣷⠀⠐⠂⠄⠈⠀⠀⠀⠠⠀⢠⣤⣤⣶⣿⣿⣿⣷⣾⡗⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣄⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⢀
⣿⠿⠿⠿⣿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⡿⢴⣿⡇⢿⣆⡀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠶⡿⠟⠛⠁⠈⠛⠁⠈⠛⠁⠈⠛⠁⠈⠻⠿⣷⣤⣤⣚⠈⠀⠀⠀⠀⠁⢀⡄
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣀⣼⠴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢉⣸⣦⣀⠀⠀⠀⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣛⣭⠖⣫⠵⣫⢞⠗⡴⣠⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⡜⢿⢯⡻⣿⢿⡷⣶⣦⣄⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⣀⠴⠥⠎⣛⣩⡴⢛⡵⢚⡵⢫⡾⢡⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡌⢷⡙⢮⡳⣽⡺⣝⠿⣗⡦⣄⡀⠀⠀⢸⠄
⠀⠀⠀⠀⣠⡞⢱⣤⢐⣤⠘⣯⡔⢋⣴⠏⣰⡟⢠⣿⢁⣿⠃⢀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⡀⢸⣷⠘⣿⡌⢻⣎⠻⣦⡙⣮⣙⢦⣝⠲⣍⡳⢤⣄
⠀⠀⠀⠰⣿⣦⡎⢉⢈⠉⣀⢸⣷⡎⢹⣦⠋⣷⡎⢱⣴⠉⣶⡎⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⢱⣾⠉⣦⡏⢱⣼⠉⣶⠉⣷⣾⡽⠀⠉⠁⠈⠹⣼⣽
⠀⠀⠀⠀⠉⠙⠀⢀⣭⣭⠋⣸⣏⠁⢸⣿⠀⠿⠇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇⠸⠿⠀⣿⠀⠈⠉⠁⠀⢀⡀⠀⠂⠈⠉
⠀⠀⠀⠀⣦⠀⠐⢰⡦⣜⠀⢻⢃⡆⠸⣿⢠⠖⣀⠰⣄⠀⣿⠇⡞⡞⡆⠆⠀⣀⠀⠂⢰⣶⣳⢸⣿⠀⢡⠖⢤⠐⣄⣿⠀⣿⡄⠀⠄⠀⠀⠀⠌⠀⣴
⠀⠀⠀⠈⠟⠆⠄⠈⠉⡁⢠⣤⠘⠇⠀⠉⢻⡄⢿⠀⡿⠀⠀⠀⠇⠰⠇⡦⠀⠐⠀⡇⠸⡟⡿⠈⠉⠀⢹⠀⣾⠀⡿⠈⠀⠙⠃⠀⠄⠀⠀⠀⠀⠀⠛⡀⡀
⠀⠀⠻⡯⠀⣾⣶⣦⠗⠳⠷⠼⠁⡴⠀⢀⣶⠀⠘⠀⢱⡆⣶⢂⠂⠀⢲⣷⠀⠐⠰⣾⣖⠀⣰⠐⣶⠀⡞⠈⠛⠀⣰⡶⠀⡤⣠⡄⠀⡄⠀⠀⠀⠀⢺⠿⠁
⠀⣽⡤⣥⠟⣹⠟⠃⣄⠀⠀⢠⣾⡒⢲⣤⠃⢀⠀⠀⢸⢳⠃⠀⢄⠀⣿⣿⠀⠀⠀⣿⣼⠈⠥⠀⠈⠓⠀⠄⠀⣀⡝⠀⠚⣀⠠⠁⠀⡁⢐⡿⣷⠀⢸⠀⢤
⠀⠁⠔⠁⠐⣁⢀⠄⣓⣀⠈⠉⠈⠁⠀⠉⠅⢐⡄⣀⢼⠻⠄⠀⠆⠀⠿⠟⠀⠀⠀⠛⠿⠀⠀⢠⣄⡀⠀⣀⠐⢫⣥⠀⠉⠉⠀⠈⢁⡇⣰⡇⢸⡄⢸
⠀⠀⢠⣤⡄⡌⢸⣛⣷⡡⠀⠂⣀⡄⢀⣠⠀⢀⣌⠟⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣏⣷⣿⠀⠀⣿⠁⣠⣴⣀⣀⠀⡇⢨⣡⣬⡦⢰
⠀⠀⠘⢲⡿⣷⢛⣿⣟⠁⠈⢩⠉⡏⠀⠆⠀⣶⠃⠀⠍⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣀⣉⣛⠱⡄⠻⠦⠄⣿⠀⠀⠀⡀⠀⡏⠛⠁⢸
⠀⠀⠀⣼⠂⣻⠸⡇⣽⡀⢈⣴⣶⡿⠂⢀⣞⡁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⡉⣿⠀⢈⣛⣠⣶⣿⠶⢈⣁⡅⢀⠁⢠⠀⣸
⠀⠀⠀⢻⣦⣿⡶⠛⣿⡇⠚⠛⠁⠀⠀⠾⠃⠀⠃⣰⡄⠠⣤⠀⢀⠮⠀⠀⠀⠀⠀⠀⣠⣄⠀⢸⠀⠀⠀⣿⢰⣿⡑⠛⠋⠁⠀⠛⠉⢰⣿⣿⡿⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣠⣿⣿⣿⣄⠀⢀⣴⡾⠋⠉⢻⣿⣦⣀⣠⡾⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣠⣤⣼⡿⠁⠀⠹⠿⡿⠟⠁⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠈⣿⡀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠉⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⠜⠋⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣦⡴⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣴⣶⣾⣿⣷⣶⣦⡤⠶⠿⠿⠿⠿⠿⠿⣷⣿⣿⠿⠿⣿⠃⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠀⠀⠀⠀⠀⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢶⡄⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣤⣤⠀⠀⢰⡆⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⣶⠀⠘⠇⠀⠀⠀⠀⠀⠀⠸⡃⠀⠘⠿⠿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡄⠉⠉⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣶⡆⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣄⡀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡀⢸⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣦⡀⣤⣦⣄⠀⣴⣶⣀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⢿⣿⣿⠿⢿⡿⠟⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⠀⠀⢀⣀⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠉⢻⣿⡿⢳⣶⣿⡟⢱⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠋⠀⠀⠈⠉⠁⠈⠙⠋⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⣴⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⣿⣿⠁⣀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣶⣿⣇⠀⠀⠀⣿⣿⣿⡟⠛⠛⠛⠛⠛⠻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠀⠀⠙⠿⠿⢿⣿⡦⠼⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣾⠟⠛⠁⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⣠⣤⣤⣤⣤⣤⣤⡾⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⢠⣶⣶⣶⣶⣿⡿⠉⠙⢻⣿⣿⠟⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠰⣿⣯⣀⣀⠈⠙⠃⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣠⣤⣾⠿⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⣾⠟⠁
⣶⣟⠉⠉⠉⠀⠀⠀⠀⢰⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡟⠀⠀⣠⣶⣾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋
⣿⣿⠟⠛⠂⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠿⣿⣿⣷⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠛⠷⣶⣶⡄⠀⠀⣀⣀⣶⡶⠛⠛⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠟⠋⢱⣶⣟⠉⠉⠉⠉⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢻⣿
⠀⠀⠘⠻⣧⣤⣾⠿⠃⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⡴⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⠂⠀⠀⠀⠀⠘⢿⣧⣄⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣋⢴⣶⠄⠀⠀⠀⠀⣿⣿⣷⣄⣰⣾⠟⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⡀⠀⠀⣠⣿⠟⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⣿⣁⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⠟⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⠋⠉⠛⠛⠛⢿⣿⣷⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠃⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣄⣶⣶⡶⠛⠛⠛⠻⠿⠿⢿⣿⣿⣷⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣬⣛⣻⡿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⢷⣶⣤⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣤⣴⠛⠛⠛⠛⠛⠛⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡀⠀⠐⢾⣇⡀⢀⣸⣧⣠⠆⠀⡔⠂⣰⢀⣠⢖⡤⠂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡺⢧⣈⡹⣿⡦⣿⠲⣟⣟⣮⣎⣻⣈⣰⣹⢯⢚⣿⣵⡯⣶⡡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠶⠖⠛⠚⠛⠋⠛⠉⠉⠉⠋⠉⠙⠛⠿⠿⠿⢿⣿⣾⣿⣾⠭⢅
⠀⠀⠀⠀⢈⣓⣤⠤⢤⣾⣥⣝⣰⣄⣰⣄⢠⡀⠀⠀⠄⠠⠤⠠⠄⠀⠀⠀⠀⠀⠀⠉⢻⣟⣯
⠀⠀⠀⠀⠠⠀⠈⡹⢛⢟⡫⠟⠉⠁⠉⠉⠉⡓⢦⣄⣀⣀⡀⠀⠀⠀⠀⢀⣜⣠⣀⡀⠈⢯⡿
⠀⠀⠀⠀⠀⠀⠞⠀⡾⠋⠀⢡⡖⠀⠁⠈⢁⠀⡂⠈⢂⠀⠙⠉⠉⠉⠙⡛⢋⠍⠻⣷⣤⣘⡁
⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⡁⡏⣶⣿⣟⢣⠀⠁⢡⠀⠀⡆⠰⠀⠀⠀⡅⢌⣹⢇⢀⠃⠀⠀⡅
⠀⠀⠀⠀⠀⢸⡀⢸⢷⠀⢁⠙⠿⣷⠧⠋⠀⠆⡇⣸⣸⠃⢠⡀⠀⠀⠐⠢⠭⠑⠁⠀⠀⠰
⠀⠀⠀⠀⠀⢸⢸⣌⠪⣧⣈⠀⠐⠀⢈⣠⡪⢞⣴⡽⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠎⢇⡜⠑⢤⣉⡉⠛⣛⣉⡭⡶⢻⢻⠀⠂⠀⠘⢿⠓⠒⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡾⢘⡀⢆⠄⢨⠋⢸⠀⡠⠃⡌⠹⡘⠀⠀⠀⢸⠴⠶⠶⠖⠒⠂⠀⢸
⠀⠀⠀⠀⠀⠰⠃⢸⠀⠸⣸⠀⡇⠸⡄⢣⠀⠇⠰⡇⠀⢀⡠⠁⠀⠀⡴⡄⠀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢛⡂⠃⠈⡇⠁⠀⢀⡰⠇⠒⠁⠀⠀⠀⢀⠁⠇⠀⠀⠀⠘⠒⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠒⠂⠉⠁⠀⠀⠀⠀⠀⡀⠀⡞⠀⡇⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢛⣳⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢸⠁⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠥⠤⠤⠤⠄⠀⠒⠒⠒⠠⠤⠤⠄⠒⠊⠀⠀⠉⠉⠉⠉⠉⠛⠛⠒⠒⠒⠲⠶⠚⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠃⠸⠇⠰⠀⠚⠆⠛⠄⠀⠈⠄⠃⠆⠺⠀⠖⠂⠀⠀⠘
⠀⠐⠶⠒⠲⠾⠟⠒⠀⠛⠒⠀⠒⠒⠊⠙⠋⠉⠁⠈⠉⠃⠉⠙⠋⠉⠛⠉⠙⠈⠉⠛⠚⠛⠒⠚⠒⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠒⠀⠀⠐⠂⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠰⠤⡀⠄⠑⢤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠀⠀⠀⠀⠈⡁⠀⠁⠀⠀⠀⠀⠐⠀⢠⣤⡂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠻⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠂⠛⠋⠛⠉⠀⠀⠈⠙⠻⢿⣿⣿⣮⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡽⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣴⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣅⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⢻⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠝⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⠂⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣃
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣵⠀⠀⠠⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⣛⡀⢀⡄⠀⠀⠻⠁⠘⠀⠈⣾⣅⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⠋⣻⡯⠭⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⢹⡇⠀⠈⠀⠀⠐⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⣿⡟⠁⠀⣴⣿⣿⣆⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠸⠇⠀⠀⠀⠀⠀⢀⡉⠳⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣛⡉⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣝⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠸⡾⠂⠰⣆⠀⠀⠒⠒⢐⣢⠔⠂⠁⠀⠀⠀⠀⠘⣟⣿⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠁⢆⣀⠙⠣⢠⣴⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⣻⡟⠉⣻⡟⠋⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⣿⣿⣴⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣾⠃⠀⠘⣷⠧⠀⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣬⠹⣯⣩⡴⠎⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠈⢰⡏⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⠉⢁⣴⣦⠀⠀⠀⠀⠑⠀⢄⣀⠘⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡈⠛⠁⠀⠀⠀⠀⠀⠔⠋⠁⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣤⣀⣀⡠⢴⡶⠋⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠋⠻⡿⣿⣆⠀⠀⠀⠀⣀⡿⠁

⠀⠀⢀⣐⣠⣀⡒⣶⡦⠄⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣄⡀⠀⠠⠤⣤⣰⣶⡔⣶⣾⣿
⠀⠀⢸⢟⡓⠾⣛⣩⣑⣦⡤⠤⢤⣤⣃⣸⣿⣭⣽⣥⡤⠤⠈⠀⣀⠈⢿⢀⢅⣛⣡⣾⣿⣿⡛⠿⣿⣿⣿⣿⡟
⠀⠀⢠⠙⠉⠂⢀⣄⠉⣽⠀⠴⠶⣤⣉⣉⢁⡉⠤⠉⢙⡷⠶⠦⠙⢛⡓⢊⣉⣉⢉⣹⠙⢀⣈⣖⣩⣧⣭⣿⣿
⠀⠀⢀⠀⠀⠈⠉⣛⣃⠁⠐⠀⠃⠀⠀⠀⢀⠘⠿⠿⣿⡷⣶⢶⠶⠄⠐⠶⠤⠧⠀⠀⢆⡈⠀⣊⣐⡟⡛⣿⣟
⠀⠀⠸⣦⠀⡠⡶⠈⠁⠀⠠⠄⠀⠀⠀⠀⡈⠀⠀⡀⣀⠀⠁⠈⠴⠊⠒⠀⡀⠀⡀⢀⣴⣿⣷⡀⡿⠁⡤⠨⡍
⠀⠀⠀⠄⠉⠉⠡⢤⣤⣤⠀⠙⣤⠄⠴⣾⠓⠒⠶⠄⠠⠀⣄⠀⠠⠄⠀⠀⠁⠀⣼⣿⡿⢟⡟⢿⣿⣾⠿⠓⣅
⠀⠀⠀⠀⡂⠀⠄⠤⠀⢀⣀⣄⣀⣀⣄⡟⣠⣆⣀⣐⣒⡀⠀⠀⠀⠆⢐⣶⢦⢰⣿⣿⣷⣭⠛⣿⣽⣷⣾⣶⡆
⠀⠀⢀⣀⣧⣀⣀⠙⠒⠒⠋⠭⠭⠿⠭⠤⣶⣿⠼⠿⠻⣇⡠⣉⠉⠫⠀⣹⡻⣿⢿⣿⣿⣿⣽⡟⣏⢩⣟⠛⠟
⠀⠀⢀⣿⣽⡯⢙⢿⣆⠐⠒⣲⠄⠀⠀⠀⠀⢷⢰⡴⢶⣶⣮⣤⠀⣐⠒⣿⣿⣟⣾⣿⣿⣿⣭⣻⣿⣿⣿⣷⣏
⠀⠀⠘⣭⣝⣿⣿⣟⣗⠚⠳⢭⡉⠁⠀⠀⡃⣿⢹⣿⡀⠓⣮⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣦
⠀⠀⢈⣿⣿⡏⠍⠅⣙⢀⣭⢘⡷⠗⢺⣥⡷⣿⣦⣿⣥⠜⣿⡾⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣯
⠀⠀⢸⣿⣿⣧⣽⣿⣻⣭⡹⢏⣿⣧⡾⣿⣿⣿⡏⢹⣽⣾⣿⣿⢯⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢘⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣌⣹⣿⣿⠷⡋⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣽⣴⣓⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣏⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁