Sail Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠈⠀⠀⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢀⡞⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⣠⣯⠾⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢃⡞⢀⠾⠋⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⡶⠉⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⡟⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⡟⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⡜⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⡞⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⡞⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢀⡞⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⣡⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⢰⣤⠏⠀⠀⠀⠀⠐⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣾⣿⣟⣭⡤⠴⢖⣾⣯⣁⣤⣀⣀⣀⣀⡤⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟
⠀⠐⠛⠛⠉⠭⠤⢤⣤⣤⣮⣤⣤⣤⠄⢠⡐⢲⠽⣟⠒⠂⠀⠀⠀⠀⠤⠤⠤⣤⣴⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠝⠉⣙⣿⣶⣦⣤⣅⣈⣓⣒⠢⠤⠤⠤⠤⠤⠤⣙⣀⡤⠇⢀⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠄⠀⠙⠻⣿⣍⠉⠉⠿⠛⠻⢿⣿⣿⣭⣤⡴⠶⠋⢀⣨⣥⠶⠚⠋⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠄⠀⠀⠈⠻⢿⣶⠶⠂⠀⣀⡿⠙⠻⠷⠶⠶⠞⠋⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢷⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣦⠀⢿⠏⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠄⠐⠒⠀⠈⠉⠀⠀⠀⢸⣃⣾⠟⢳⡄⠀⠐⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⠄⠀⠀⢀⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢿⢸⡆⠀⢫⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⠾⠿⠯⣿⢿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠃⠒⠒⠶⠤⠤⣤⣴⡇⠀⣠⣾⣦⣴⣯⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠉⠙⠿⠿⢯⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⡿⠰⢿⢰⣶⠤⣤⣤⣤⣀⣀⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣠⣰⣶⣿⣿⡏⠉⠉⠀⠀⠀⢀⡀⣀⣒⣒⡐⠺⣥⡿⣷⠶⠞⠛⠫⠉⠁⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⡤⣦⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠷⠶⣾⣧⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡴⠛⠛⠁⢠⣤⠤⠤⠄
⠀⠀⠀⠀⠈⠙⠛⠚⠛⠋⢉⠉⠀⣀⣀⣀⣀⣉⣉⣉⣀⠀⠀⠀⠀⠀⣉⣡⣤⣄⡓⠛⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠀⠈⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢁⣿⣅⠒⠒⢻⠀⠀⠀⣀⣠⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⣀⣀⣼⣿⣽⣶⣴⣾⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠠⠁⠠⠤⠄⣀⣀⣙⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣉⣭⣿⣽⣿⣀⠀⠄⠀⢀⡈⠐⠀⠀⠐
⠀⠀⠠⠖⠒⠚⢿⣿⣛⣛⣻⣿⣿⣿⣿⣏⣿⣯⠏⠿⣷⣶⣿⣿⣿⣿⣻⣿⣿⣿⣾⡓⠂⠀⠀⠈⡉
⠐⠒⠛⠻⠷⠶⠶⢿⡛⠛⠋⠝⠛⢿⣶⣾⠟⢻⣿⣿⣿⠿⠯⠝⠛⠛⣳⣞⣿⢿⣿⡏⠉⠉
⠀⠀⠉⠛⠛⠃⠀⠤⠬⣍⣉⣉⡀⠀⠙⠛⠋⠬⢽⣿⡇⠀⠐⠒⠒⠒⠒⢒⣺⡏⠉⠁⠀⠒⠒⠒⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠓⠒⠀⠀⠛⠃⠀⠀⠙⠓⠒⠈⠉⠀⠀⠈⠛⠒⠒⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⢃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠒⠈⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⣤⠤⠞⠓⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠉⠁⠀⠀⠀⠀⠀⠀⠘⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠠⢀⡀⣀⣠⣵⠶⠗⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠒⢻⡏⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡏⠁⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠃⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡇⠀⠀⠀⠀⠀⣸⡇⣤⣀⣠⠤⠒⣺⠈⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣀⣤⠤⠴⠒⣿⠉⠁⠀⠀⠀⠀⡈⠠⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠉⠁⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣧⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠃⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡄⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠄⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠐⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢡⡶⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⡴⢀⣴⡟⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣬⣴⢖⣋⢁⠀⣱⠋⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⣸⡂⣤⣤⠤⠒⣿⡩⠐⠉⢀⣀⣸⢠⡋⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⢠⡴⠓⠋⣿⠀⠀⠠⠒⢁⣤⣶⣾⣿⣿⣿⣾⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⢀⣁⠼⠒⢈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣩⣤⣴⣾⣿⣿⣿⣿⠿⠋⠻⡿⣿⣿⣿⣟⡁⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠠⠝⠻⠿⠿⠛⠛⣛⣋⡉⠷⢛⣿⣶⡓⣿⣻⣯⡍⠻⣿⣿⣿⣿⠿⠷⣶⣤⣿⣿⣶⣛⠋
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠐⠶⠚⠃⣲⣖⢚⣳⣶⣿⣢⣾⣿⣟⣃⣠⣾⣿⠿⢿⣿⡿⠿⢋⣤⠘⠃⡙⠳⠾⠿⠿⣿⣿⣾⣿⣦⣭⣿⣟⣻⠿⠿⠋⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣄⠸⠧⠤⠾⠟⠛⠉⢻⣿⣿⠿⠷⠖⠒⢻⣿⣿⠿⠿⠿⢃⡄⠐⣺⣿⡿⠂⠀⣝⡛⠷⣲⠂⠀⠈⣛⣋⣩⣍⣉⠛⠛⠛⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠓⢀⠲⠿⠦⠤⠄⠈⠛⠋⣀⣤⣤⣤⣉⠁⠀⠰⢿⣫⠄⠀⠀⠘⠿⣷⣯⣭⠖⠛⠙⠻⠿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡆⠀⠀⠀⠀⠀⠀⠑⢄⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠀⢀⡠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣀⠀⠀⣿⢣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢂⠀⠀⠀⢀⡇⠀⠀⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣭⣏⢛⣖⣦⣀⣀⣀⡀⡀⠀⡟⠀⠀⠘⣀⠀⢤⣈⣱⣤⠖
⠀⠀⠀⢀⣐⣒⡲⣶⣶⣤⣤⣶⣤⣸⣿⣿⣿⣿⢶⡾⠷⠾⠧⠼⠷⠶⠗⣶⣿⣏⣉⣙⡫⢋⣀
⠀⠐⠂⠀⠿⡛⠻⣿⡷⠿⠛⣽⣯⣍⠛⠡⠿⣿⣯⣽⣻⠿⠆⣹⣿⣓⠛⢛⡓⠐⠛⢻⠁⠶⠂⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠒⠀⠤⠀⠻⠿⢦⡀⠀⠨⠀⠠⣆⠙⠛⢃⡀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣇⠠⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠀⠀⠀⠀⠀⡀⢀⣀⣧⠜⠖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠦⠶⠒⠚⡟⠉⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢬⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡢⠐⠀⠀⠈⠉⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠚⠋⠉⠁⢸⠇⠀⠀⠀⠀⠰⡆⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢻⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡃⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡇⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣴⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠷⣾⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⢤⠖⠻⠟⠁⠂⡂⡾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣀⠦⡜⡾⠉⠉⡠⠘⢁⣤⣴⣾⣿⣯⣷⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⡀⠿⠐⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣬⣥⣶⣿⣿⣿⣿⣿⣋⠻⢿⣿⣿⣿⣁⣀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⢀⡖⢮⣙⣛⡛⠉⢩⣽⣭⣙⣋⣿⣶⠳⣿⢿⣯⢙⣿⣿⣿⠝⣛⠷⣴⣶⣶⣾⣍⣀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠠⠌⠋⣁⣬⣿⣭⣿⣿⣿⡞⢛⣯⣥⣶⣿⣿⣛⠛⠛⣉⣵⣆⠁⢨⣉⣛⣛⠙⠻⣿⣿⠿⠶⣿⣯⣟⡛⠁⠠⠄
⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠉⣭⣥⠀⢼⡿⠛⠋⠁⠤⠼⠛⠋⠉⠩⠕⠂⢀⣿⠿⡋⠀⠰⣮⣍⣻⠥⠤⠤⢴⣶⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠁⠁⠀⠀⠀⠘⠻⢿⣷⣶⡄⠀⠒⠉⠀⠀⠀⠈⠙⠛⠉⠁⠀⠢⠭⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣽⠳⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠛⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆⠃⠀⠀⠀⠈⠁⠀⠀⠀⠂⠐⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣀⣤⡶⣿⠻⠷⠀⠀⠙⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣚⣹⡭⠧⠟⢳⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣀⡤⠰⡞⢛⠟⢀⠀⠀⠻⡉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⣀⣴⡾⠷⢻⡉⣿⣗⣶⡾⠒⠛⠚⢻⡿⠓⠒⠒⠒⠒⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⣟⣭⠴⠖⠚⠉⠉⠁⠃⠀⠀⠀⠀⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣴⣶⣿⣤⣽⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⢀⣀⠤⣶⢹⣤⡼⢛⣽⣿⣿⣿⡷⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡰⠛⠁⢀⣿⡼⣫⢿⣬⣷⣿⣿⣿⣿⣷⣦⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣼⣶⣾⣯⡾⠷⣿⣿⣿⣿⣿⣿⡿⠿⡇
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⡇⠈⠉⢀⠀⠙⢿⠟⠀⠂⠀⠂
⠀⠇⠀⠀⠀⠀⠰⠶⣤⡶⠖⠒⠓⠲⢶⣦⡯⠿⠿⠋⢤⣦⣀⣀⣀⣲⣾⡀⠠⣄⣛⡐⠶⠖⠀⠀⠀⠐⠒⠓⠒⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠶⠿⢏⣽⠿⢶⣶⣶⡆⠀⠚⣯⣾⠇⠠⣿⣿⣿⢡⣀⡛⠿⣿⣥⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢿⡶⠌⢩⣤⡞⠫⠟⠁⠀⢉⣻⢵⡎⠻⣿⣦⢀⡀⠀⠀⠀⠀⠈⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⢄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣵⠠⠂⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡂⢀⠄⣽⡯⠟⠁⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢉⣾⢻⣗⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡰⠛⢡⣿⡏⢸⢹⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠡⠂⠀⢀⣿⣷⠇⢸⠈⡧⠀⠀⠀⠡⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⣀⠐⠁⠀⢠⣾⣿⣿⠀⢸⠀⣿⡄⠀⠀⠀⠈⢄⠀⡇⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣏⡚⠋⠉⠀⢸⠀⠸⣗⠀⠀⠀⠀⠀⠀⣯⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣦⣶⣿⣦⣤⣴⣆⣀⣀⡗⠀⣀⣉⣀⣀⢀
⠀⠀⠀⠀⠀⡤⠶⠾⠯⠤⣤⡖⠛⣛⠛⠛⠿⠟⠛⠛⠿⠿⠿⣿⡿⠻⠿⠿⠿⣿⣿⠏⣠⠤
⠀⠀⠀⠀⠀⠐⠋⠉⠁⠠⠖⠋⠉⠙⠒⠚⣙⣶⣀⣤⠾⠒⠛⠫⠴⠒⠶⣶⣶⠄⠳⢶⣀⠀⠀⠠⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠊⠘⠒⠀⠂⠶⠖⠚⠋⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠈⠉

⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠉⠉⠓⠲⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠙⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡟⠀⣯⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡏⠀⢸⡌⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡷⠀⠀⣇⠈⡎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡇⠀⠀⣿⠀⢸⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⠀⢹⠀⠀⣇⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡇⠀⠀⣸⠀⠀⢻⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠸⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡧⢀⡏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢺⣇⡼⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡤⢤⣒⣢⣤⣤⣞⣿⣛⣭⣽⠷⠶⠦⢤⣼⣿⣿⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡯⠭⠟⠛⠋⠉⠉⢀⣉⣀⣀⣤⣤⣴⣶⣾⡿⠿⠛⢋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣶⣶⣶⣶⣿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⣉⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⣉⣀⣀⣀⣠⣤⠤⠤⠤⠶⠶⠒⠒⠚⠋⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣤⡐⣦⣠⣆⠐⠿⣿⠷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠊⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⢠⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣴⡥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣟⣿⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣶⣶⣶⣤⣄⡀⠀⣻⠀⠀⠂⠀⠄⠀⠀⠀⣷⣾⣿⡇⠀⠀⠀⠒⠀⠀⠀⡀⣤⢦⣴⣾⡿⢈⣩⡅⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠤⠤⠤⠀⠀⢒⣀⡁⠙⢯⣟⣻⠿⣿⣿⣿⣿⣿⣧⣤⣦⣁⣄⠀⠀⢰⣾⣿⣿⣿⣁⣈⠤⠄⠖⠚⠁⠉⠁⠀⢸⣿⣿⠃⢸⡟⠁⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠚⠛⠂⠈⠭⠅⠀⠜⢾⣭⣑⣒⠶⠝⢯⢿⡧⣿⣿⣿⣿⣿⣿⣿⣿⢏⣩⣤⠤⠤⠄⠲⣒⠺⡋⠙⣁⣐⡀⣿⠇⠀⠈⡄⠀⡀⠀⠉⠀⠐⠶⠆
⠀⠀⠀⠀⠀⠀⠠⠭⠤⠤⠴⠦⢤⣤⣤⣻⣿⣿⣿⣿⣳⣲⡦⠴⠤⣤⢤⢤⣤⣤⣄⣐⣀⡀⠀⠤⠌⢚⣓⣲⣋⢯⣝⣶⣴⡿⢄⡁⠀⢳⠀⣀⠘⠛⠋⣀⣀⣀
⠀⠐⠒⠀⠒⣒⣒⣒⣒⣂⣀⣀⡀⠀⠒⠚⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣾⣴⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇⡔⠂⠀⣼⠤⠤⠄⠀⠀⢀
⠀⠀⠀⠀⣉⣉⣩⣭⣉⠉⠙⠛⣻⣿⣿⣧⣤⣀⣀⡀⢀⣉⡉⣛⠛⢛⠙⠛⠿⠿⠟⠛⠿⠿⢿⡿⢛⣻⠟⠛⢋⠀⠀⣀⢀⡤⣄⠙⠀⠙⠛⠓⠒⠒⠀⠀⠙⠀⠒
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠈⠉⠀⠶⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠙⠉⠀⠀⠉⠃⠀⠰⣆⣀⣉⡳⠦⠔⢛⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁
⣶⣶⣴⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣤⣦⣶⣦⣴⣴⣶⣤
⣿⣿⣏⣀⣁⣨⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣽⠃⠀⠀⠸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⣼⠏⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣸⡿⠀⠀⠀⠰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣿⡇⠀⠀⠀⡆⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢘⣿⠀⠀⠀⠀⠁⠀⠸⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢸⣿⠀⠀⠀⢘⠀⠀⠀⢸⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠀⠀⠈⣿⡇⠀⠀⡀⠀⠀⢀⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠚⣿⣿⠀⢀⠃⠀⠀⠀⣸⣇⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠈⢹⣿⣇⡸⠀⢀⠃⠀⣸⣿⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⠋⣿⣿⡇⠀⠘⠀⠀⠹⣿⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠄⠀⠂⠀⠈⢿⣿⡄⡆⠀⠀⠀⣿⡄⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠰⣿⡇⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠐⢮⡦⠘⢻⣷⡀⠀⣿⣽⣷⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠈⠛⠿⣿⣷⡀⠀⢸⣿⣧⡀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⢰⣿⣿⣿⡷⡀⢸⣿⣿⣵⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⣠⣴⣿⣿⠿⠋⠁⠀⠘⠟⠉⠉⠛⣩⣍⣉⠉⠉⣡⣾⡟⠀⠐⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⣶⣴⣖⣴⣾⣷⣄⣀⣚⠛⣻⣿⣷⣾⣿⣿⢷⣤⣾⣶⣷⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣒⣀⣈⣙⣉⡉⠍⠙⠒⢾⠟⠻⠿⢴⡿⠾⠶⣶⠶⢾⡷⠛⡛⠛⠿⣾⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣤⣿⣿⣿⣯⣿⠿⠷⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠉⠉


⡿⢿⠿⠟⠟⠛⠿⠿⠟⠿⣿⣿⣛⣿⣿⣛⣿⣻⣿⣿⣿⣿⣟⣻⣿⣿⣿⣻⣛⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿⣻⣿⣿⣿⣻⣿⣿⣛⣿⣟⣿⣛⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠤⠴⠂⠀⠀⠀⠀⠀⠿⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢸⠇⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⢠⡄⠀⢳⠀⢀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢰⡀⠀⠀⢀⠉⠉⠉⠉⠁⠀⠁⠀⠈⠁⠁⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣄⡄⠀⢠⠀⠀⠀⠀⠨⠿⠟⠠⠄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⡗⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⢘⡗⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢤⡤⡄⡤⠀⠊⠀⠀⠀⠀⣀⣀⠀⠀⠀⢴⠂⠈⠁
⠀⠒⠼⢢⣀⣀⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠰⣏⠀⣠⠀⠀⠀⠀⠀⠀⢀⢠⣀⢀⢸⡇⣪⡏⠀⠀⣀⣀⣠⡤⣠⡴⡦⠀⠀⠀⠣
⠀⠀⠀⠸⣯⣽⣏⣙⣛⠛⠓⠛⢸⢻⢿⢻⠿⠾⠿⠼⠷⠤⠤⠾⠿⠷⠶⠾⢿⡿⣞⣛⣟⣛⡿⠿⠿⠯⠿⡟⠻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣶⣶⣶⣶⣾⣼⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣯⣧⣦⣾⣶⣶⣶⣶⡶⠖⠛⢉⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣭⠉⠩⣩⣉⠛⣛⣋⡉⣟⣛⡛⣛⡛⠛⢻⣛⣛⣛⡛⠋⣛⠓⢛⣚⠀⠚⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠈⠙⠛⠉⠀⠉⠛⠁⠐⠀⠀⠉⠈⠁⠈⠁⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡇⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢸⣧⠗⡠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⡏⠀⠀⢸⠁⠀⠚⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⣿⠁⠀⠀⢸⡆⢀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢰⣿⠀⠀⠀⢼⠟⡿⠗⣟⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢸⣿⠀⠀⠀⢸⡅⢚⠊⠁⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⣿⠀⠀⠀⢸⡃⠄⢲⣾⣏⡀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⣿⡀⠀⠀⢸⡗⠀⠈⢛⣉⠀⠄⣀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡇⠀⠀⠈⣷⣲⠉⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣉⠀⠀⠀⠀⠀⠈⣷⢹⣆⠀⢸⣷⢟⡋⠀⠉⡕⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡁⠀⠀⠂⠀⠀⢀⠻⣾⣿⣆⢸⣿⣿⡗⠀⠀⠰⣀⣀⣤⡴⠶⠻⠗⠒⠛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠀⠀⠀⢀⣀⣈⣀⣹⣿⣿⣿⣿⣿⣯⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠠⠤⠄⢀⡀⣰⣿⣃⣤⣴⣶⡶⠿⠿⠿⠿⠛⠛⠛⠛⢿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣷⣿⣿⣿⣽⡧⠤⣤⡤⠤⢤⡶⣶⡶⢷⣶⡿⣿⡶⢶⢶⡶⢿⡾⠿⢿⡷⣤⣤⣼⣴⣶⣦⣤⠴⠂
⠀⠀⠀⠀⠀⣀⣀⣀⣙⣯⡉⠙⠛⠃⠈⠓⠒⠚⠓⠒⠒⠒⠛⠒⠒⠒⠀⠐⠒⠟⠂⠀⠐⠚⠓⠀⠚⠒⠒⣶⣾⠶⢛⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣷⣤⣄⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⢤⣼⡯⠭⠭⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠯⠍⠙⠛⠷⢛⣻⣿⣟⡛⠛⢿⣯⣭⣉⡉⠻⢟⣯⣍⣩⣻⣭⡵⠾⠿⠒⠒⠒⠒⠶⠶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⡉⣛⣋⣉⣉⣭⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣀⣛⠛⠿⢿⣿⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⣉⠁⠀⠀⠖⠒⠚⠛⠓⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣤⣶⣶⣶⣾⣿⣦⣤⣤⣤⠠⠄⠀⠀⠤⠄⠀⠤⠤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠒⠂⠲⠶⣤⠀⠉⠉⠙⠿⠿⢿⣷⣿⣿⣶⣶⣦⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠔⢶⡚⠩⡥⠆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠁⠀⠠⠄⣀⣀⡈⠁⣄⣀⠀⠀⠀⢈⡉⠉⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣇⠠⡁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢸⡗⠒⠂⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⣩⣿⣁⠛⠒⠒⠺⡇⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣧⣀⣀⣀⣿⣤⣤⣴⣶⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠰⠤⠈⢁⡤⢤⣀⣀⠂⢀⠘⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⣁⣠⣶⣶⣭⣿⣟⣂⠀⠀⠀⠀⠀⠤⠀⠐⠒⠂⠈⠉⠉⠁
⠀⠀⠉⢉⣠⣤⣤⣤⣶⣶⣿⡿⠿⠿⢿⣷⣶⣾⣿⣿⣿⡿⠋⣿⣿⣛⣟⣻⣯⣭⣯⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣀⠀⠀⡈⠉⠉⠉⠉⠉
⠀⠀⠤⣬⣥⣤⣄⣀⣈⣛⢿⣶⣾⣷⣶⣾⢿⣿⣿⣿⣟⠳⠾⢿⡿⣯⣤⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣗⠛⣻⣿⣿⣿⣿⠤⣤⠤⠤⠤⠈⠉⠛⠂
⠉⠙⠋⠛⠛⠻⠿⠛⠛⠛⠿⣆⣉⠉⠁⠐⠛⠋⠙⣛⣭⣿⣟⠛⠉⣽⣿⣿⣾⣷⠛⠶⠶⠂⠀⠉⠩⣭⣭⣭⣭⠯⣽⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠙⠛⠒⠖⠓⠀⠠⣤⣤⣉⡛⠛⠛⠛⠁⠀⠀⠚⠓⠛⠛⣃⣙⣻⣿⣿⡃⠀⠀⠴⠶⠶⠦⠴⠶⠖⠒⠒⠚⠚⠛⠛⠀⢠⡴⠶⠶⠶⠦⣤⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠀⢠⣤⣤⣄⣀⡉⠉⠉⣿⣿⡅⠀⠀⠀⢤⣤⣀⣀⠒⠒⠚⠛⠉⠩⠭⣤⣄⣀⣀⣠⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣷⢄⡀⠀⣿⠀⠀⢸⣷⡦⢀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⠸⡆⠙⢢⣿⠀⠀⢘⡟⡈⠂⠱⣤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⢚⣿⠃⠀⢸⠅⠀⠀⠀⠉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠚⣿⡄⠀⢨⠀⠀⠀⠀⠐⢉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⢹⠀⡆⠀⠀⠀⠀⠀⠀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡯⡶⣴⠁⠀⠀⠀⠀⠀⠀⣿⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⢸⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠃⣸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⡀⠀⠀⠀⠀⠀⠂⠰⠒⠎⠢⠒⠘
⠀⠀⠀⠀⠀⠀⠀⣈⡥⠀⠀⠀⠀⢀⣈⣺⣿⣇⡤⠄⠂⡀⠀⠀⠀⠀⢺⣿⡀⡇⠀⠀⠀⠀⠀⣫⡤
⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⠄⠀⠀⠉⠉⢻⣯⣥⣞⣟⣛⡛⣀⡀⠀⠀⢘⣿⣧⡓⠀⠈⣁⣤⣾⡟⠁
⠀⠀⠀⠀⠀⢸⠝⢻⣟⡐⠒⠒⠒⠒⠛⠾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣩⠽⣉⣥⣶⣿⣿⣿⡿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠟⠉⢻⣶⠂⢀⣀⣤⣿⣿⣿⣿⡿⠿⠙⠙⠿⠃⡠⠇
⣀⣀⣀⡠⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣶⠗⠂⣰⡆⢘⣿⡆⢀⣄
⣿⣿⣿⠿⢷⠂⠑⠳⠄⢰⣦⣁⡘⠻⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣍⠉⣈⣱⣶⣿⣿⣿⡇⢸⡿⠛⢸⣿⣦⣤⣀⣀⣠⠔⡂
⠉⠉⠉⠑⠀⠀⠀⠘⠷⣦⣤⡉⠙⠻⣿⣾⣶⣿⡿⣿⣷⣾⣿⠿⠿⠿⠿⣿⡛⠿⠿⠿⠿⠃⠘⠇⡀⢸⣿⣿⣏⠛⠿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠢⠄⠉⠉⠈⠛⠳⠶⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠘⢷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢀⡼⣃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇⢀⠀⠀⣠⣾⠿⠋⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡟⣠⡾⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⡟⢀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⠜⡿⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⡼⠁⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡾⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡾⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⢉⣴⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠉⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡀⣰⠏⠀⠀⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⣀⣴⣋⣡⡤⠀⠀⠀⢀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠘⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⣠⣾⣿⣿⣿⡷⠶⠶⠚⠛⣹⠿⣷⣶⣶⣦⣤⣤⣤⣤⣴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠁
⠀⠀⠚⠛⠋⠡⠤⠤⢤⣤⣤⣤⣾⣥⣤⣤⡤⠤⠤⢶⠒⠒⡆⢿⣋⠛⠒⠒⠒⠀⠀⠠⠤⠤⠤⢤⣤⣴⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡹⠯⠛⠻⣷⣦⣤⣅⣈⣁⠘⠳⠦⠬⢥⣤⣀⣀⣀⣀⣀⣐⠺⠗⠀⣀⡵⠀⠀⡈⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡟⠋⡉⠙⠛⠿⣿⣟⡛⠛⠲⠶⢶⣶⣶⣶⣶⣶⣶⣖⣚⣋⣉⣠⣤⠞⠀⠤⢒⣋⣭⣤⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠈⠻⣿⣦⣄⡀⠋⢀⡀⠈⠹⣿⢿⣿⣿⣛⣛⣉⣁⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠀⠀⠀⠀⠈⠙⠿⣯⣍⣉⣀⣠⣴⠟⠀⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢾⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢸⡏⢹⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢸⡇⣰⠌⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⡏⠁⠨⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣾⠇⠀⠀⢸⡇⠀⠀⢐⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢠⣿⠀⠀⠀⢸⣿⢶⣞⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⣿⠀⠀⠀⢸⡇⢻⡅⠞⠃⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢸⣿⡀⠀⠀⢸⡇⠀⢂⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⣿⠁⠀⠀⢸⣗⡀⠈⣿⡯⠖⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠈⡶⣒⣈⠄⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠸⣟⢿⡄⠀⢸⣯⣀⣤⠀⠺⡿⠂⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠄⠀⠀⠀⠀⠀⠀⣿⢘⣷⡀⢸⣿⣫⣥⡀⠀⠸⠀⠀⠀⠀⠀⢀⣱⡀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠀⠀⠀⠀⠀⠀⠠⠈⢿⣿⣷⣸⣿⣿⣿⡁⠀⠀⣡⣤⣶⠾⠛⠋⠙⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⡷⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠐⢒⣖⠠⣤⢠⣿⣿⣤⣶⠾⠿⠟⠛⠛⠛⠛⠉⠉⡉⠉⢻⣿⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⢿⠾⠶⠶⠶⠶⠶⠶⠷⠾⣿⠿⠿⠿⠿⠿⠶⠶⠷⠟⠿⠿⡟⠿⠷⠶⢶⡶⠶⣿⠷⣦⠔⠃
⠀⠀⠀⠀⠀⠠⠤⢤⣤⣵⣤⠈⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠛⠒⠒⠀⠀⠙⠛⠋⠀⠀⠛⠛⠓⠈⠛⠛⠛⠾⠟⠛⢋⣷⣤⣤⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣷⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠶⡾⠿⠯⠥⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠬⠉⠙⠛⠖⢛⣻⣿⣟⣙⠛⢻⣯⣭⣄⣉⡙⢛⣯⣭⣥⣭⣭⣥⠶⠾⠟⠛⠛⠛⠛⠒⠶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢉⡙⣛⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣄⣘⡛⠻⠿⢿⣿⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⢉⣉⠁⠀⠤⠖⠒⠒⠛⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣤⣤⣶⣶⣶⣾⣻⣂⣀⣀⣀⠀⠄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠉⠉⠉⠉⠉⠉⠥⠤⣤⡀⠉⠑⠓⠲⣶⢶⣶⣶⡶⣶⣶⣶⣦⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠖⢛⣩⠤⠀⠀⠐⠀⠐⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⣀⡀⢀⣀⠈⠃⣀⡉⠀⠀⠀⠀⠒⠒⠂⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⢋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⣀⡴⣫⡾⣫⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢃⠀⠀⠀⠀⣠⠞⣫⢾⣫⡾⣻⠿⣋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⣠⡴⢛⡵⢚⡵⢟⡥⢞⡡⢞⡥⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⣟⣡⠖⣫⠞⣩⢖⡫⢖⡩⢖⣡⠞⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢻⡴⢚⢵⡪⢒⣩⠖⣋⡴⢛⡵⣫⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠀⠀⠀⠀⠀⠀⠀⢺⡕⣫⠖⢋⣴⠞⡩⢖⣭⢞⣩⢞
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣻⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⠾⣋⠔⣫⠖⣫⠶⣫⣴⣿
⡶⠮⣍⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⣃⡭⠾⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⢞⡵⣊⡵⣻⡵⠞⣉⣤⣴
⣉⡛⢦⣍⡳⢬⡙⠷⣦⣀⠤⠤⠤⠤⠶⣶⡿⠿⠛⣉⡡⠔⢚⣉⣥⣶⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣾⡿⣺⣵⠾⠛⢋⣩⣴
⡯⣝⡓⠮⣝⡶⣌⡳⢬⡙⢷⣦⡤⠭⠵⠖⣚⣋⣭⣴⣶⡿⠿⣛⣩⠵⣒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣠⣤⠶⠟⣛⣽⣿
⣭⡓⠭⡓⢮⣙⠦⣝⠶⣝⠦⣈⠻⣿⡛⠉⠝⣓⡫⠭⠔⣒⣩⠥⠒⠋⣉⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣡⣴⡶⢟⣿⣟⡵
⢌⡙⠷⣦⡑⠈⢑⠈⠳⢾⣿⡮⣓⠌⠳⣌⠡⠶⠒⠓⢉⣁⣤⠴⠞⣛⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⣠⣴⡿⣫⠞⣡
⠶⢦⣅⡈⠙⠷⣦⣈⠢⣄⠙⢿⣮⣷⢤⡈⢣⡐⢒⠈⠡⣁⣢⡼⢟⣋⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢛⡿⢋⡼⢡⡞⢁
⣿⣷⣮⣝⡳⢦⣄⠉⠻⣮⣓⢷⡿⣛⣷⣹⣤⡴⠶⢾⣛⣉⠙⠲⠦⣬⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⠔⢉⡴⣫⡶⠋
⠭⢍⣛⣻⣿⣷⣬⣽⡤⠶⠿⠛⠛⠯⠭⣉⡓⠲⠶⣤⣍⡙⠛⠶⣤⣀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣥⠞⣫⡾⢋⣤⠾
⢷⣶⣤⣐⠲⢤⣤⣀⣀⡀⠐⠛⠛⠛⠒⠶⠦⣭⣝⣒⠾⠿⣷⣦⣤⣉⡛⣿⣀⣀⠠⠤⠄⠀⠀⢴⣢⡼⠛⣁⡴⣛⣡⠆⠙⠴⣋⡵⢛
⢦⣌⠓⠮⣽⣒⠶⠬⣭⣝⣛⣛⡛⠛⠛⠒⠲⠶⠾⠽⠿⠷⣶⣾⣿⣿⣿⣿⡋⡏⠉⠉⠉⠀⠀⠒⠒⠒⣈⠡⠬⠭⠭⠥⠵⡯⠵⠚⢋
⡳⢭⣓⠦⣬⣉⡛⠶⢤⣮⣭⣭⡭⠿⢿⣿⣶⣖⣒⣒⣲⣶⣶⣶⣯⣽⠟⠉⣇⢱⠀⠀⠀⠀⠀⠀⠭⠀⠀⣒⣠⣤⣶⡶⠶⠷⠖⣛⣋
⣍⣓⠮⣭⣛⣿⡿⣿⣦⣶⣤⣍⣩⣭⣭⣧⣤⠤⠶⢿⣿⡿⣟⣫⣭⠿⢷⣾⣿⡘⣤⣶⣤⣀⣠⣴⣶⡾⠟⠛⢉⣁⣤⣴⡶⠿⢛⣛⣉
⣚⢿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣶⣶⣶⡶⠿⢛⡩⠥⠴⠶⣖⣒⣛⣉⣽⠿⣱⣭⣿⡿⠟⠋⢉⣠⣤⣶⣿⣿⡿⠿⠿⠛⠛⢉⣉⣉
⠿⣿⣶⣾⠭⢭⣭⣭⣽⣿⠿⠿⣛⣋⣩⣴⣶⣬⣥⣬⠄⠩⠭⠭⠭⢖⣪⣶⠿⠛⢉⣁⣤⣶⡿⠿⠛⣋⣉⣡⠤⠴⠒⠚⠛⠛⠉⠉⠉
⣽⣶⣶⣿⣿⣿⣯⣭⠶⢂⣉⣡⠤⠤⠒⣒⣒⣒⣒⣒⣒⢒⣿⡿⠿⠋⣁⣤⡴⠾⢛⣉⣥⠴⠖⠚⠉⣉⣀⣀⣤⣤⣤⡶⠶⠶⠶⠟⠛
⣭⡭⠭⣷⢒⣪⡭⠖⣛⣩⠥⢖⣛⣋⣉⣭⣭⣭⣉⡭⠟⢋⣡⡴⠾⠛⣉⡥⢖⣚⣩⣥⡶⠶⠟⠛⣛⣉⣭⣭⣴⠶⠶⠶⠶⢞⣛⣛⣛
⢖⡶⠟⣋⡭⢄⣒⣭⣶⡶⠿⠿⠿⢯⣤⡤⢞⣫⣭⠶⣛⣩⣥⣶⣾⡿⠶⢛⣋⣩⣥⣴⠶⠾⢛⣛⣉⣭⠤⠴⠖⢒⣛⣉⣉⣉⣉⣉⣉
⠛⠚⠛⠑⠚⠛⠻⠟⠛⠻⠿⠷⣶⣿⣿⣛⣫⠵⣖⡿⠿⢛⣋⣭⡴⠶⠛⣛⣭⡭⠶⠖⢚⣛⣭⡭⠶⠶⠚⣛⣛⣉⣭⣭⣭⣥⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠹⣵⠋⠳⢴⣖⢦⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⠦⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠁⣟⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣠⢤⠀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⢆⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠳⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠤⡄⠀⠀⠀⠝⠃
⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⡇⠉⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠼⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⡋⠁⣀⢀⢀⠀⠀⢠⠇⠀⡇⠀⢀⣀⡠⠔⠐⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⢧⡀⠀⠠
⠀⠐⠀⠀⠀⠀⠉⠛⢹⣸⣠⠃⢀⠞⠀⠀⠉⠉⣉⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡞⡳⠋⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⢠⠒⢢⣀⡄⢠⠤⠼⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡙⠁⠀⠀⢰⢇⡠⣢⠖⢠⠤⠤⣄⠀⠀⣇⠀⠀⠈⠁⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣣⠞⣵⡵⢦⡀⣳⠀⠈⠓⠂⠀⠀⠤⠦⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡡⠚⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣱⣄⣀⡀

⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉
⣛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣤⣤⣴⣾⣿⣷⣤⣤⣤⣤⣤⡶⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣤⣤⣾⣿⠇⣀⠀⣀⠀⠀⠀⡀⢶⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠟⠛⠉⣛⠉⠉⠉⠉⠙⠛⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣀⢴⣀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣁⣀⣀⢀⡘⣿⣿⣿⣦⣀⣀⣀⣀⣴⣥⣤⣈⣤⣴⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠉⠉⢉⣻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⡤⠖⠉⠉⣿⡿
⠀⠀⠀⠀⠀⠀⠸⡇⢻⣿⠳⣄⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠄⠸⠀⢻⣆⠈⢃⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣼⣯⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠘⠀⢠⠀⠈⢿⡄⠈⠀⣱⣮⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣴⣶⣶⣀⣤⣀⢀⣀⣠⣌⣹⣿⣿⣆
⠀⠀⠀⠀⠀⠆⠀⢾⠀⠀⠘⢷⡀⢠⡿⢏⠳⣄⠀⠀⠀⢠⣿⡿⢿⡿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⠀⠀⣼⠁⣼⣿⣿⣿⣿⣦⡪⣷⢈⠳⣄⠀⢆⡰⠀⣼⣿⣿⣿⠀⣿⣷⠀⠈⠛⠻⣿⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣀⣼⡟⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⣦⣌⠷⣤⠁⣂⣿⣿⣿⣿⣿⣿⣿⣀⣄⣀⡀⠀⠀⠺⢍⣿⣿⣿⣿⣿⣷⡀⠀⢁
⠀⢠⣿⠏⠀⠀⢀⣾⡿⠿⣿⡿⠛⢿⣿⣿⣿⣿⣿⡿⣴⣿⣿⣿⣿⣿⣿⣿⢿⠟⠛⠋⠉⢻⡏⠉⠛⠓⠦⣬⣭⣟⣿⣿⣿⣧⡀⢸⣷⡀
⠀⣼⡏⠀⠀⠀⠈⢻⠀⡾⠋⠁⠀⠀⢻⡟⢻⣿⡿⢠⣿⣿⣿⣿⠿⠟⢹⣿⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣷⣼⣿⣿⡀
⠀⠻⠀⠘⠃⠰⡶⢾⣶⣶⣶⣶⣶⣶⣶⣿⣾⣶⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣶⡶⣾⣿⡿⡷⡶⠶⠶⠶⠶⠶⠶⠶⡶⠾⠿⠿⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠛⠉⠉⠉⢸⣿⣿⠀⠀⠀⠀⢀⡝⠂⢠⠏⠉⢶⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠻⠦
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠆⠀⠀⠙⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣀⣀⣀⣀⣀⣀
⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡏⠉⠉⠈⣭⠉
⣧⣴⣧⣤⣤⣤⣤⣿⣦⣤⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠿⡿⠿⣿⡿⠶⠶⠶⠶⠶⠶⠶⠶⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣌⣿⣧
⢥⠀⠀⠀⠀⠀⠀⣶⠐⢦⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⣠⠶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢘⣿⣿⡏
⠀⠃⠀⠀⠀⠀⠀⠹⣷⣦⣾⢶⣤⠀⣀⣶⣄⣴⣟⣿⣿⣋⣴⣯⣤⣤⣤⣤⣴⣶⣶⣶⣶⣷⣶⣤⣤⣄⣀⣄⣿⣶⣤⣤⢻⣤⣿⣿⣿⠁
⠀⠀⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⢿⣿⣿⠿⠿⠋⣿⣿⠿⣿⣿⡿⠋⢻⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⣿⣿⠀⠀⠐⠀⠀⢸⡏⢫⢧⡀⠀⠀⠉⠁⠀⠀⠈⠉⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣶⠀⣿⣿⣀⠀⠀⠀⠀⢸⡇⠸⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⣠⣔⠶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣤⡀⣿⣀⣾⣇⠀⡇⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⣶⣶⣤⣄⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡈⠉⠉⠙⠋⠉⠉⠉⠑⠶⠦⣤⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠉⠙⡻⠟⠛⠛⣉⣀⣀⣀⡈⠉⠉⠛⠋⠛⠛⠛⠛⣻⣿⣿⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠛⢿⣿⡉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢡⣴⣾⡿⠟⠛⠙⠛⠻⠷⣶⣶⣤⣴⣶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣸⣿⣿⣟⣛⣉⣭⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠾⠟⠛⠛⣛⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠀⠒⠒⠒⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣍⣤⣬⣽⣏⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠂⠀⠀⠂⣿⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠇⠀⠀⠀⡏⢠⠀⠰⣤⣤⣍⣮⣤⡆⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⠀⠀⢀⣄⣥⣸⠀⠀⠘⢻⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠲⣾⠗⠉⠙⠻⠻⠛⠻⢗⣿⣿⡗⠒⠲⠲⠖⣢⣴⡦⢠⣌⣩⣽⣋⠋⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣏⠀⠀⠰⣺⠷⠀⠀⠀⣿⢻⡇⠀⠀⠀⠀⠰⠀⠀⠀⠀⢸⢿⠋⠉⠁⠑⠚⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠸⡄⢿⠀⠀⠀⠨⠉⢓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠘⣿⠃⠀⠀⠀⠄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠃⠉⠆⠰⢹⣇⠀⠀⠀⠀⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠤⣶⣿⡿⣷⠀⠁⠀⠀⠀⢸⢻⣆⠀⠀⠀⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⠀⠐⠆⠀⠁⢸⠀⠘⢧⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣿⣿⡆⠀⠀⠀⠀⣼⠀⡀⠈⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡼⢃⠐⠂⡀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⣀⡀⠀⠈⢀⣁⠀⠀⠀⠀⠀⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠸⠀⠰⣿⣿⡟⢠⡫⢁⠀⡞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⣯⣿⣿⣿⣿⣯⠙⢻⡀⠀⠀⠘⠀⠀⠀⠂⡀⠀⠂⠀⠀⠀⢠⠀⢠⡀⠀⠀⠀⣶⡷⣾⣶⣿⣷⣿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣾⣤⣵⣮⣤⣧⣽⣗⣰⣧⣄⣄⣀⡀⠀⠀⢀⣵⣆⣔⠐⠒⠐⠁⠀⠘⠧⣄⡀⣸⡯⣛⣿⣿⡿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣌⣉⢙⠚⢳⣦⠈⠘⠉⠃⠟⠚⠓⠒⠐⠓⠀⠀⡉⠉⠠⠁⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⢻⣿⡎⢹⣿⠈⡽⠉⠉⠙⠛⠻⢿⠿⣏⡉⠁⠁⠘⠀⠂⠀⠖⠆⠀⠚⠂⠀⢉⠁⠄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⣛⢷⡄⠾⠄⢰⠄⢠⣄⠻⢿⣿⣷⠶⠒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠉⢽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣬⣽⣷⣶⣶⣯⣿⣷⡀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠄⠀⠀⢠⣤⡀⠀⠀⠀⢾⠆⢀⠀⠉⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⣿⣧⣿⣿⡉⣿⣿⣿⣶⣶⣶⣦⣤⣤⣟⣡⣤⣤⣤⣼⣿⠥⠄⠐⠀⠀⠀⠀⠀⢀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢴⣾⣷⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢯⡇⠒⠂⠀⠀⠀⢀⠀⠄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣶⢀⣿⢩⣽⣟⢿⡆⣹⡿⣿⡿⣿⡩⢽⣯⠭⠭⠄⠀⠖⠀⠒⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠓⠚⠛⢻⣿⠁⠻⣯⡍⠀⠉⠰⣲⣿⡋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠋⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠆⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢃⢠⠜⢀⡀⠠⡄⠀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢠⠀⣸⣼⡟⢛⣻⣿⣿⣾⣿⡿⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠛⠉⣰⣿⣿⣿⠛⣦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣠⣦⡀⠀⢸⣿⠟⢯⢃⡴⠞⢋⣴⣿⣏⣴⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣻⣿⣥⠶⠋⣷⣶⠖⣩⣴⣶⣿⠿⢿⣿⣿⡿
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⢽⣿⣿⣾⣯⣥⣤⠟⠋⠩⣥⡾⣿⣿⣿⣧⣄⠐⠂
⠀⠀⠀⠛⠾⠷⣶⣶⣿⣿⣦⣭⣿⣿⣾⣧⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠈⣻⣷⠶⠄
⠀⠀⠀⠀⠀⠀⠩⠷⠋⠉⠉⠉⠀⠉⠉⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠢⠤


⣿⡏⠉⢻⠟⡿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡟⢉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣴⣬⣤⣴⣧⣤⣄⣀⣤⣤⣄⣀⣀⡀⣀⠔⢸⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣓⣲⣶⣶⣶⣶⣶⡶⠶⠦⠤⠴⢦⡥⠤⠤⠼⢧⣤⣬⣴⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣾⣛⣛⣛⣛⣻⣟⣛⣛⣟⣡⣾⣿⣿⣛⣛⡋⠉⣿⠛⠛⠻⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠲⠶⢶⣶⣶⣶⣶⣶⢾⢧⣭⡵⣿⣭⣍⣍⣉⣉⣉⣁⣀⢿⣀⣀⣀⣀⣈⣻⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⣿⣿⣿⣷⣶⣷⠀⠀⠶⣶⣶⣾⣮⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠛⠿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⠁⠀⠀⠀⢀⣤⢖⡋⣡⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠇⠀⢀⣠⢞⣭⣾⢏⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡟⠀⠀⢀⣭⣭⣿⣿⣿⣿⣿⢸⣿⣿⠏⢀⡴⣫⣵⣿⡿⢣⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢽⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⢃⡴⣫⣾⣿⣿⡿⢡⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⡝⠀⠀⠾⢿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣚⣩⠥⠴⠖⠀⠈⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣽⠅⢀⣾⣿⣿⣿⣿⣿⣿⡟⣡⠿⣟⣯⣿⡶⠿⡛⡿⠀⣆⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡴⠟⠛⠉⠁⣿⣇⡠⢖⣯⣷⣿⣟⣫⣵⣾⡿⢻⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣾⠇⠀⠀⠀⠀⠠⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡏⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⣒⣋⣛⣛⣛⣛⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡾⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣰⣶⣀⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡼⠁⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣯⣯⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡼⠁⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠎⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠁⠴⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣿⣿⣿⣿⣛⣛⣛⠛⠛⢛⣋⣉⣀⣀⣀⣀⡀⠀⠈⠙⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠻⠿⠿⠿⠿⠿⠿⠿⠭⠍⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠑⠢⣼⣆⣀⣀⣀⠀⠀⠀⠀⠀⣄⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠁⠲⠢⠬⣦⡀⠘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠀⠑⠐⠒⠓⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⣉⣉⣉⣉⣙⣛⡲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣈⣦⠖⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠔⠚⠉⣉⡠⠤⠤⠤⠬⠭⣽⣓⡻⠽⣟⣓⡶⠶⠶⠶⠶⠶⠛⣋⣩⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠪⠭⢭⣙⣛⣛⣋⣉⣡⠤⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣄⡀⢈⣛⡛⠿⣿⡿⠛⠛⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣽⣿⢿⣿⣿⣏⢻⣷⣶⣿⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠘⠿⠲⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣠⣿⢷⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣧⢀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠿⠃⣸⡿⠿⠟⠹⠗
⢸⣿⣿⣿⣿⣾⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣬⣿⣿⣟⠿⡿⠛⢻⣿⣿⠀⠀⠁
⢸⣿⣿⣿⣯⣽⣀⡈⠏⠉⠀⣤⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⡦⠤⢴⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠤⢤⣿⠇⢀⠀⠀⠀⠄⣀⣀⣀⣴⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⡿⠽⡿⠿⠛⠛⠉⠉⠉⠉⢸⡿⠀⡈⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣾⣇⣶⡇⠠⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣉⣀⣀⣸⣃⣼⣿⣿⣿⣹⣽⣯⣽⣿⣿⣿⣿⣯⣽⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠉⠉⠉⢩⣉⣉⣭⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡀⡀⢠⣐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣧⡀⠀⠀⠲⣟⣻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣂⣿⣿⣽⣟⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠐⠢⢄⡀
⠀⠀⠀⠀⡀⠀⠁⠁⠢⠄⡈⠑⠢⠄⡀
⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠈⠐⠠⢄⠉⠐⠠⡀
⠀⠀⠀⠀⢆⠀⢀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠒⠠⣀⠙⠂
⠀⠀⠀⠀⠨⡀⠸⠧⡌⠃⠒⠂⠀⠓⠀⠀⢘⠀⠀⣘⡖⠠⡀⠐⠠
⠀⠀⠀⠀⠀⢿⠀⢁⠀⠂⠀⠢⡈⠉⠢⣀⠀⠁⠠⢀⠀⠀⠀⠁⠢⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡝⠂⠄⢂⡀⠀⠈⠢⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⠙⠠⢀⠁⠢⣀⠉
⠀⠀⠀⠀⠀⠀⠈⢌⠓⣀⡥⢀⡰⠄⠘⠂⠀⠀⠉⠁⠀⠀⠀⠋⠀⠀⠀⣀⡄⠑⠢⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠙⠻⣥⣠⠀⠄⠐⠂⠀⠀⠀⠂⢁⠀⡀⠀⠀⠉⠢⣀⠀⠀⡉⠢⠄⠀⠰⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠛⠲⢦⣀⡀⠠⡀⠒⠀⠑⠄⠑⠀⡀⠀⠘⠁⠀⡇⠀⠀⠀⢠⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠙⠓⠾⢦⣀⣀⠀⠀⠀⠀⠀⠐⠀⢀⠁⠀⠀⠀⡀⢀⡇⢀⣤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣄⣀⠀⠀⠀⠀⣂⣴⠶⠿⠟⠛⠛⠓⠛⠋⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢈⡑⢄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠳⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣄⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠐⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣠⣶⣶⣶⣼⣭⣥⣀⡠⠀⠀⠁⠀⠒⠛⠋⠉⠈⠈⠉⠉⠀⠀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⣶⣦⠈⠀⠉⠉⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠀⣠⣿⣿⡁⠀⠀⢀⣀⣰⠾⠟⠉⠀⠀⠀⠀⢀⣼⡤⡄⠰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠿⠦⣴⣷⣿⣿⣿⣿⣿⣦⣿⣿⡟⣠⣤⡤⠾⠛⠩⠰⠤⠴⠤⠤⠶⣤⣤⣴⣉⣉⣀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣴⣿⣠⡟⣿⠏⠛⠉⠉⠉⠉⠉⠉⢭⡥⠿⠉⠀⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠻⠋⠉⠉⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⣳⣿⠞⠁⠀⠀⠀⠀⠀⣀⣰⣶⣿⠀⠀⠀⠀⢲⡴⢶⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠿⠟⣋⣸⣿⢿⣿⣶⣤⡤⣶⡾⣾⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠦⠴⠷⠏⠀⠀⠄⠀⢰⡄⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠚⠟⠭⢄⠚⠉⣀⠀⢀⣏⣰⣚⣡⣶⣴⣦⣦⣤⣴⣤⣀⢀⡀⣼⣀⣀⣀⣀⡀⢀⣤⣤⣤⣬⣼⣏⣩⣩⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣿⣻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⣉⣛⣛⣫⣭⣿⠿⠿⢿⠿⣟⣻⣯⣭⠭⢭⠿⠛⠉⠉⠋⣉⣉⠉⠡⠋⠉⠙⢡⢶⡮⣿⣿⣿⣿⣳⣶⣶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⣠⡄⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣾⣿⣶⣿⣿⣿⣿⣿⣆⣴⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠿⢟⣫⣶⣿⣿⣿⢭⣽⣛⡝⠂⠩⣭⣛⡛⠻⠿⠿⠿⢟⣛⣫⣵⣿⠟⣿⡿⢟⡪⣔⣮⠳⢍⡙⠛⢫⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣑⣃⣀⣠⠄⣐⣬⣽⢟⣋⡭⠷⠦⢶⣢⣤⡉⢔⣲⣭⣿⣿⣷⣮⣭⣿⣿⣿⣯⣙⣶⠾⠫⢾⣿⣿⣭⣖⣿⡶⠿⢛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠙⠋⣉⣉⣭⣭⣅⣛⠻⢭⣛⣫⡴⢮⣽⠷⠾⠿⠿⠶⡾⢿⣿⠿⣭⣷⣶⣮⠭⠽⠿⠛⠛⣒⣙⣫⡵⠾⠛⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⠻⠷⠶⢖⡫⠽⣛⣴⣬⠭⣿⣛⠤⣶⠾⢝⣟⣭⣝⣿⣿⣾⣷⣽⣍⣋⣙⣀⣈⣊⣛⠽⣵⣒⠴⢞⣋⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣭⣭⣭⡭⢽⣓⠶⢿⣻⣶⣬⡵⢞⣛⣯⡥⢶⣿⣥⠯⠽⣟⣿⣶⠭⢭⣭⡉⣉⣩⣭⠭⢴⣞⡯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢭⣿⣶⣢⢌⣉⣑⡒⠒⠂⢘⣫⢭⠒⣉⣽⣖⣛⣩⣭⣽⣭⣭⣽⣛⠤⠮⣌⡕⠀⠚⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣭⣛⡻⠿⣙⣻⣤⣤⠈⠶⣯⣝⡻⣛⠭⣑⣻⣧⣶⣾⣿⡿⣛⡿⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⢿⣤⣞⠿⠯⠉⣭⣯⣿⡶⠾⠧⣬⣽⣿⢥⣼⣟⡻⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠟⠛⢭⡴⣬⠴⠾⠿⠟⠟⠋⠛⠈⠁




⣶⣶⣶⣶⣶⡾⢶⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣾⢿⡾⠶⡶⠶⢾⣶⡾⠾⠾⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠘⠀⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣤⣤⠐⣲⡿⠁⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⠀⢸⣷⣮⣿⡯⣹⠋⠀⢀⢠⣤⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣾⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠢⡤⡘⣿⣿⣷⣌⢻⣿⣿⢽⢿⣤⣌⠁⣈⡿⠆⠀⠀⠀⢨⠁⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣫⡿⣿⣿⣷⣿⣿⡅⠁⠉⠉⠉⠉⠉⠉⠁⠀⠈⢸⠳⢴⢃⡘⠈⠻⠋⠑⠉⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣿⠟⠉⠓⠒⣲⠖⠒⠒⠒⠒⠲⡖⠒⠚⠢⡀⠙⠷⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⢀⡴⠞⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⢄⣀⢝⢶⣫⣤⣾⣿⣶⣤⣤⢀⠀⢀⠀⢀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⡇⠄⣩⣵⣶⢻⣻⣷⣾⢣⣿⣿⣿⣷⣷⣷⣷⣾⡏⣽⣿⣶⡆⣴⠀⡦⢤⡄⢠⡶⠀⠀⠀⡀⠀⠘⢿
⣿⣿⣿⣿⣿⣿⣿⣛⠿⠟⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠾⠁⠺⠦⠻⣅⠠⠔⠈⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣣⣿⣧⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡀⠀⠀⡀⠀⠀⠀⣀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠞⢁⠤⡈⡆
⠀⠀⠀⠀⠀⠈⠀⠀⢈⢀⠎⠀⢸⠀⡀⠂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠅⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⢆⡸⠔⡁⠀⢀⠥⠄⢠
⠀⠀⠂⠐⠀⠀⠀⠀⢀⠠⠊⠀⠂⠀⠀⡀⠁⡆⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠰⠈⠀⠀⠈⠑⠒⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⡀⠤⠊⠀⠀⠀⡩⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⢂⠐⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠊⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠀⠀⠒⠒⠠⣄⠀⠀⠤⠯⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣀⠈⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠁⠁⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠬⠄⡂⠨⠄⠀⢅⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠂⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢎⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡡⠝⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⠉⠉⠉⠀⠀⠀⠈⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠶⠾⠛⡆⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⢀⠀⠀⠀⠀⣀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠖⠒⠁⢠⣾⡤⠽⠖⠀⠉⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⣿⡁⠀⠀⠀⢠⣾⡍⠀⠀⠀⢰⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠘⢀⣄⣼⣶⣿⣿⡷⠀⠀⠀⠀⣇⠠⠀⢀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡿⠟⠊⣿⣍⣅⣤⣴⡶⠎⣿⠀⠈⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢤⡯⠔⠂⠁⠝⡿⣯⠉⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢿⠂⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⡟⡀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⡆⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣸⣧⢧⠀⠀⠀⠈⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢺⣿⡇⣠⣄⣬⢰⣸⣿⣿⣷⡿⠟⠛⣿⢸⣤⣤⣀⣠⣀⡈⢠⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣿⣯⣿⣾⣿⣿⣯⠀⠀⠀⠀⢿⡀⢛⣛⣋⣛⠛⢿⡛⣿⡖⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢼⡿⠻⣿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠼⣿⠿⣿⣏⣿⣛⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡍⢀⣇⣷⣿⣿⣿⢿⣿⣿⣿⣧⢬⠃⣈⣿⣷⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣮⡟
⠀⠀⠀⠀⠀⣠⡄⠀⢰⣿⣦⡾⠾⠿⠿⠿⠿⡾⠿⢿⣿⣿⣦⣴⠋⠋⣿⣤⣶⣿⣿⡯⠋⠀⠀⠀⠀⠀⠠⠴⢮⣝⣿⠁
⠀⠀⠀⠀⢸⣟⣥⣖⣿⡆⠈⠀⠀⠀⠀⣀⣁⢃⣀⠒⢿⡝⢉⣭⣴⣾⣿⣿⣿⣿⣿⡠⠀⠀⠀⠠⠀⠀⢤⠲⠀⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣡⣶⡀⠀⢰⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⡿⣛⣉⡙⠻⠿⢿⣿⣄⠀⠀⠀⠀⠀⣄⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠓⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠸⠛⢋⣿⣇⣤⣄⣿⣿⣱⣤⣐⣆⣄⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣶⣿⣶⣭⣍⣭⣽⣤⣬⠉⣤⣤⣤⣴⣿⢿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣙⣿⣿⣿⣿⣥⣀⣀⣀⣐⣶⣾⠋⠿⣿⣿⠟⠱⢿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⢈⣥⣷⣶⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠻⠿⠿⢿⣿⣶⣶⡾⠿⠿⠛⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⡴⠚⠛⠛⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⢧⠀⣀⡀⣠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠙⠁⠀⠈⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠸⠅⠘⠀⠀⠁⠀⢀⠀⠀⠉⠆⠀⠀⠀⢤⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⣤⣃⡀⠀⠤⠊⠀⠀⠀⢠
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠉⠉⠙⠛⠿⠿⢯⣭⣍⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠲⠶⠶⠦⠤⣤⣥⣤⣦⣤⣀⣀⣀⣀⠀⣀⠀⠀⠀⠐⢶⢺⡇⠀⠀⠐⠠⣄⠀⠀⠀⠁⠈⣂⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣏⡙⠂⠂⠀⣸⠘⣧⡄⢀⠀⠀⠀⡄⠀⠀⠀⠀⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠶⠃⠀⢸⡀⠈⠀⠉⠈⠀⠀⠁⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣤⣄⣤⡼⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢅⠀⠐⠀⠀⠀⡀⠀⠔⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠢⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⠀⠐⠁⠠⡀⠀⠄⠐⠠⢄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠄⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠑⣀⠌⠀⠀⠀⠈⢀⣤⣴⣶⣶⣿⡿⣷⣶⡶⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠴⠶⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⡿⠷⠿⠿⠿⠷⠤⠄⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⡐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⢀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡏⠙⣷⢸⣷⠀⡖⣿⠁⢠⣿⡇⠀⡦⠉⣿⠁⣾⡇⠀⣸⣼⣿⠉⢷⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⠀⣿⠈⣿⣴⠇⣿⠀⢸⣿⣿⢸⠇⠀⣿⠀⣿⡇⢀⡇⣿⣿⣠⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣿⢸⣿⢀⡟⠀⢻⣿⠀⣿⠖⢸⡏⢻⣿⠀⠀⣿⠀⢻⣧⡾⠁⢿⣿⠻⡄⢸⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠋⠉⠀⠀⠈⠉⠀⠙⠃⠈⠁⠈⠁⠀⠀⠉⠀⠈⠉⠁⠀⠈⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣀⣀⣀⠀⠔⢢⠀⠀⣀⠔⠂⠀⡃⠀⠠⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠈⠑⠑⠀⠁⠊⠁⠀⠊⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣲⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⣟⣵⣋⣁⣀⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢬⣙⠲⢤⣈⡩⠤⠶⠒⠒⠒⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⣹⢂⡇⣀⣀⣠⠤⢤⡄⡖⠒⢮⣳⡄⠀⠀⠀⠀⠀⣿⠤⠶⣈⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠾⣍⣓⠦⣤⠇⣼⢰⠁⢸⣿⠀⠀⢷⠹⣄⣤⠷⠙⢦⠀⠀⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢲⣟⣉⣀⣀⣀⣤⡬⠽⠶⣤⣇⡘⠒⠚⠉⣉⣉⣉⣠⣤⠤⠤⠤⠘⠳⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⣹⣋⠔⠊⠁⣸⠉⠉⡏⠀⠀⢱⡀⠀⠘⢦⢻⡉⠉⣉⣉⠉⣀⣀⠀⢠⠤⡀⢰⠒⡌⠳⣤⢼⣧⣤⡀
⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠻⣄⡀⠀⠀⠇⠀⠀⢃⠀⠀⢀⣇⣀⡠⠜⠊⢧⣀⣓⣚⠀⠓⠚⠀⠘⠒⠁⢈⣉⣀⣀⣬⠧⠔⠋⢻⡄
⠀⠀⢠⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣻⡿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⡉⣉⣭⣍⢩⠵⢦⠀⠀⠀⠀⢀⡹⣤⣀
⠀⠀⠈⠙⢿⣟⡛⠛⠛⠛⠛⠛⠋⠉⣩⣭⠤⠴⠖⠒⣦⠀⣀⣀⣀⢀⣤⣤⡄⣴⠶⠖⠒⠛⠛⡇⣏⣉⡇⢧⣤⠼⠘⠒⡚⡁⠀⣠⣤⣼⣷⠟⣸
⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⣀⣀⠀⠀⠀⠙⠲⣄⠀⠀⢘⡆⢳⣀⣸⡏⣧⡤⠇⠸⠶⠒⠒⠛⠋⠁⠈⠁⠀⢀⣰⣲⣿⣾⡶⠛⣟⢉⡤⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠉⠙⠛⠿⢶⣤⡀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣠⣶⡶⠶⢶⢿⣿⣿⠿⠧⠨⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡃⠀⠀⠀⠀⠀⠉⠓⠶⣦⣴⣿⠶⠶⠶⠶⠶⣿⡟⢿⣭⣽⣿⣾⡷⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⠞⠛⠿⠟⠂⠸⠦⠔⠶⠖⠲⠿⠝⠛⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣇⣦⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡷⣄⡘⣦⠻⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢉⢻⢿⣿⣿⣿⣡⡱⣾⡝⢦⡈⠛⠛⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠈⠫⠳⣄⠻⢿⣿⣝⣾⣿⣷⠙⠲⣴⣷⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⣛⠛⠛⠉⣠⣴⣾⣰⣤⡴⠶⣤⣀⠀⠉⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⡿⡋⣠⣦⡈⡛⠿⣷⣾⣴⣴⣴⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣀⡀⠀⢀⣠⠴⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⣿⣷⠋⠀⠀⠀⠀⠀⠀⠉⣧⣀⣤⠤⣤⣀
⠀⠀⠀⢀⣤⣶⣿⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢙⣦⡤⠖⠒⠲⢤⣀
⠀⠀⡴⠁⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠈⢧
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡴⠟⠋⠉⠉⠙⠲⢤⡀
⠀⠸⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠙⠂⠀⢠⠴⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀
⢀⣴⠿⠦⠀⠀⠀⠀⠈⠛⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠰⠟⠀⢠⣤⠀⠁⠀⠈⠙⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠉⠳⣄
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠬⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠈⢳⠀⣰⠂⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠹⣄⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⢀⠀⠀⠀⠀⠈⡇⠙⠶⢶⣄⣀⣀⣀⡠⠇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢾⡁
⠀⠈⣳⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣠⣼⠃⠀⠀⠀⠘⣯⠉⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠘⠧⠀⠀⠀⠙⡆
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⡽⠀⢀⣀⣀⡀⠀⠀⠉⠳⠶⠤⠶⠀⢀⣀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⣄
⠀⢻⡀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠸⠋⠀⠀⠀⠈⠙⠻⣟⡛⢶⣶⣤⣤⣤⣄⣀⣉⣛⠛⠛⠁⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⣸
⠀⠀⠓⢄⣀⣀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠉⠲⣍⡛⠿⣿⣿⣿⣾⣭⣭⣭⣉⣛⠛⠛⠛⠿⢿⠧⠤⣤⣄⠤⠴⠋
⠀⠀⠀⠀⡾⠛⠉⠙⢿⡀⠀⠀⠀⠰⣄⣀⣀⣠⠋⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠙⡄⠀⠀⠙⢦⡈⠙⠲⣍⡻⢿⣶⣤⣍⣙⣿⡶⠶⠯⠄⣀⣀
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠙⢦⡀⢠⡝⠳⢮⣙⠳⢯⣿⠲⠤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠙⣾⠷⠤⠤⠽⠻⢯⡀⠉⠒⠦⣍⡃
⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀⠀⠀⠀⠘⢶⣤⣤⣀⣠⣤⠶⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⢶⣦⠤⠤⠖⠋⠉⠳⡄⠀⠀⠀⠀⠉⠲⣄⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠻⣇⠈⠁⠀⠀⠀⠀⣠⠟⠉⠉⠁⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⠦⢄⣀⣀⣀⣠⡤⠞⠁⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠈⠈⠙⣧⠴⠶⡄⠀⠀⠸⠷⠞⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠱⠀⠀⢠⣤⣤⣤⣿⣭⣭⣿⣭⣭⣭⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣭⣭⣭⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⣿⣿⣿⡉⣻⣿⣿⡏⢙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠿⣶⣶⣶⣾⣿⣭⣿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣶⣾⣯⣭⣭⣭⣿⣿⣿⡭⠽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠞⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⣼⠃⡇⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡋⠙⠷⠠⠀⠀⠐⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⡀⢀⢸⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠤⢬⣿⡀⠀⡧⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣸⠂⠈⠉⠛⢶⠿⢤⣑⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⢀⡤⠀⠀⢀⢿⡏⠀⠀⠀⠀⠈⣇⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣘⡂⠘⠀⠦⣽⠇⢀⣴⣾⣿⣁⣀⣀⢀⡀⠀⢹⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠉⠙⠛⠻⠿⣿⣿⣿⣾⣿⣿⡏⠉⠙⠛⠿⠤⣀⣿⠚⡇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⢻⠃⠀⠀⠀⠀⠀⠀⠈⠙⡟⠲⠦⢄⣱⡀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⠏⡐⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠇⠀⠀⠄⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣣⠠⢂⢀⣄⣠⣀⣤⢄⣴⣿⠋⠀⣥⢀⡄⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⢐⣱⠷⣿⢿⣿⣿⣿⣿⣿⣿⣡⣤⣤⣤⣤⣀⡀⠀⠀⠀⣠⣿⠇⣀⡽⠛⠦⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡛⠛⠿⣿⣿⣿⣧⣽⣿⠟⠽⢻⡿⣿⣿⣿⣿⣯⣅⣀⣈⣷⣿⣿⣴⣯⣤⣀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⡿⠤⣌⡙⠛⠛⠀⠀⢹⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⣴⣞⣛⢿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠏⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡜⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⢸⡛⠹⢿⣾⣿⣿⣿⣿⣿⣿⢋⣥⠶⠦⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠰⣻⣭⣴⣶⣷⣾⣿⣿⣿⣿⣿⣿⣩⣾⣿⠄⠀⠀⠛⠿⡻⡛⠛⢛⣇⠘⠛⠀⠀⢀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠈⠳⢦⣬⣿⡿⠛⠟⢛⣻⡿⠿⠿⢿⡿⢭⣍⣛⠧⣀⠀⠀⢀⣐⣛⢦⣘⠿⣿⣦⣴⣦⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠉⠴⣶⣿⣿⣿⣷⣦⡌⠛⢷⣬⡀⠐⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⡶⠶⢟⣉⣙⣿⣿⣿⣿⣿⡧⢀⣌⠻⣄⡀⠢⢭⣯⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠋⠙⠿⣿⣿⣟⠹⠇⢀⣿⣦⣀⣙⠛⠲⠶⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠹⠿⠷⣤⡐⠈⠙⠻⠿⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠃⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠉⠙⢻⣿⣿⣿⡏⠙⠟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣷⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠿⠽⠿⠛⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣸⠟⣿⡿⠋⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠉⠉⢀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣧⣬⣜⣓⠤⡦⠶⠶⠤⣄⣚⣶⠈⠹⠹
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠈⠙⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠁⢀⡸⠀⠀⠀⠀⣠⠚⠉⠉⠈⠲⡀⠀⠀⣀⡀⣀
⠀⠀⢀⡀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠠⠄⠐⠉⠀⠀⢀⡸⠀⢠⠊⠉⣁⠠⢤⡀⠀⠈⠁⠉⣷⠀⠀⠀⠗⠠⡀
⠀⠈⠉⠀⠀⠀⠒⠠⠍⡑⠂⠄⣀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠂⢮⣀⣀⠜⠁⠀⢸⠀⠀⠓⠀⠀⡱⠀⠀⠀⠴⠋⢀⣀⠀⠀⠤⠘⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⢀⠉⠒⠤⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢀⠀⢀⡀⠀⢣⡀⡀⠠⠐⠁⠀⠤⠄⠀⠀⠉⡸⠀⠀⠐⠁⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡐⠀⠀⢀⢀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠈⠉⠁⠠⢀⣤⡤⠴⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠢⠤⢠⣀⣁⡀⠒⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣒⡒⠢⠭⡂⢄⡀⠀⠀⢠⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠢⡀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠂⠤⠤⠤⠤⣀⠀⢁⣹⡀⡀⠈⡆⠈⠉⠉⠁⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠲⢄⡀⢀⠤⠤⠤⣀⡀⠀⠀⢠⡄⠀⠀⠀⡹⠉⠈⠦⠤⠔⠁⢠⣀⠀⠀⠈⡧⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⢬⠉⠒⠀⠀⠠⢿⠀⠀⠈⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡦⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣆⠀⢀⡤⠤⠚⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠲⠄⠠⠴⠆⢀⠠⠔⠁⠈⠁







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

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠠⠒⠶⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠀⠀⢀⣵⡼⣶⣿⠀⠀⠀⣤⣤⣀⣀⣀⣄⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠒⠲⠶⣾⢟⢭⣴⣶⣿⣏⠁⠒⢶⡟⠀⡫⠅⠉⠛⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢉⠃⣾⣿⠟⠙⠓⠈⠐⢄⢻⡄⠋⠀⠀⠘⠞⣦⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠂⢀⣒⣤⣄⡌⢠⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡷⡀⢰⠚⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⣉⡠⠤⠐⠛⡭⡙⢀⢎⠎⠀⣦⡀⠀⠀⠀⣼⡟⠛⢻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⣻⣿⠀⠀⠀⠀⢠⠁⣌⠃⠀⠀⠈⢿⣄⠀⠀⠙⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠁⡀⣿⡏⣀⠄⠀⠀⣼⡞⣤⡴⠈⡲⢟⣛⠿⠄⠐⢲⣾⢎
⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⢸⣿⠏⢁⣀⢠⣤⡿⣿⠩⠔⠁⠀⠉⠉⠀⢢⣄⠀⢧⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣼⣿⠟⠉⠀⠀⠹⢷⡝⣥⠀⠀⠀⢠⣦⠚⠀⠹⣄⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⢠⢊⠄⢀⠀⠀⡅⡟⠂⠀⠀⢹⠏⠀⠀⠀⢻⣷⠀⠀⠀⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣿⣧⣤⠁⣠⠎⠀⠀⠀⠀⣤⣼⠿⣦⣀⡀⠀⠈⠓⠤⣀⠀⠀⠉⠻⢶⣖⡈⠒⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢝⠻⣧⣾⣧⣄⣀⣀⢀⣀⠿⢏⡀⠈⠓⢮⣳⠶⢤⣀⠀⠈⠁⠒⠢⢤⣈⠻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⠀⠀⢘⣶⣦⣴⡞⠁⠀⠀⢀⡞⠀⠀⠀⣽⢧⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠐⣹⢿⡇⣿⣷⡄⣀⣴⣿⣡⣤⣴⠤⠉⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⡘⠻⢿⣿⢀⠏⠈⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠃⠇⠀⠀⠈⠁⠀⠀⢶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⢿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣠⡾⢧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡿⢄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⣷⣤⣤⣤⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠩⠁⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⣀⣡⢴⠶⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠒⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿⣷⢸⠀⠀⠈⠹⠷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⢠⠆⢀⣀⣀⣴⢨⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠘⠿⠷⢖⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣂⣈⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠁

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⣧⠀⢻⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠠⠐⠒⠁⠈⠉⠀⠀⠀⢸⣧⣼⠟⢻⡄⠀⠰⡀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⣄⠀⠀⢀⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⢿⢸⡆⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣶⡾⠿⠯⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡀⠀⢠⡾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠛⠃⠒⠶⠶⠦⢤⣤⣴⡇⠀⣠⣾⣦⣴⣯⣀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠉⠙⠻⠿⢯⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⠸⢿⢰⣶⡤⣬⣤⣤⣀⣄⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣬⣵⣾⣿⣿⡏⠉⠉⠀⠀⠀⣀⡀⣀⣒⣒⣴⠺⣥⡿⣿⠶⠞⠛⠫⠉⠑⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⣦⣦⣤⣤⡤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠬⠽⠷⠶⣿⣿⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡼⠛⠛⠁⢠⣤⡤⠤⠄
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⣉⡉⠀⢀⣀⣀⣀⣉⣉⣉⣀⠀⠄⠀⠀⠀⣉⡥⣤⣄⣓⡙⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠍⠀⢈⡻⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⣸⣿⠖⠋⠉⢛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡏⠉⠙⠛⠛⠉⠙⠛⠛⠋⠉⢉⢍⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⡇⢀⣼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡎⠀⠀⠀⠀⢀⣀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠙⠚⠚⠛⠉⠉⠁⠀⢀⢢⠃⠀⠀⠀⠀⠀⠀⣈⡟⠉⠉⠉⠉⠉⠉⢉⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⡀⢤⠄⠀⢀⢣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⠤⠤⠤⠴⢶⣦⠀⠒⠂⣊⠜⢀⡐⠒⠒⠒⠂⠉⠉⠁⠀⠉⢸⠋⠀⠀⣌⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢀⠀⠠⢄⠀⠈⡀⠀⡸⠀⠀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠌⢫⠉⠉⠙⠛⠋⠀⠀⠀⠐⠒⠈⠂⠀⡨⢻⠊⠀⠉⠀⢠⣷⣿⣿⣟⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⠀⠀⠀⡸⠀⣀⡠⠔⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣀⣀⠤⠒⠋⠉⠀⢀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⢶⡒⠛⠒⠦⡉⠓⢤⣀⡴⠃⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠒⠒⠉⠉⠁⠀⠀⣀⠠⠔⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠘⢏⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠒⠈⠵⢦⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠈⢷⠀⠀⠊⠀⠀⠀⠀⢀⣄⠈⠁⠉⠀⢠⣞⡆⠀⠀⠀⣀⠑⠉⠂⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠣⣀⠀⠀⠀⠀⠀⠑⠘⢀⣀⡀⠤⠤⠀⠐⠒⠈⠉⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡒⠀⠀⠀⠒⠒⠒⠒⠂⠈⠉⠉⠉⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠽⡜⠙⢛⠒⠤⣄⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠠⠔⠐⠚⠛⠋⠉⠉⠉⠋⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⣠⡴⠉⠀⠄⠀⠒⠂⢀⣀⣀⣤⣤⣤⣤⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⠠
⠀⠀⠀⠀⠈⠙⠛⠛⣷⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠰⠞⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠓⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠐⠂⠈⢥⢈⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣶⣶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠁⠉⠉⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⠟⠋
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣠⣠⣤⣽⣭⣭⣿⣿⣿⠟⠁
⠀⠙⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣦⣄⡄⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣳⡤⡀⠀⣿⠀⠀⢻⣦⢀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡆⢃⠈⠲⣿⡀⠀⢸⠋⠀⠑⣦⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢢⣿⡀⠀⡔⠀⠀⠀⠈⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠃⠀⠀⠀⠈⠸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⣿⣸⡰⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠿⠇⠀⠀⠀⠀⠀⠀⣿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣬⠆⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⠀⠀⠀⠀⠀⠀⡤⠌⠀⠄⠙
⠀⠀⠀⠀⠀⠀⣘⡁⠀⠀⠀⠀⣈⣺⣿⣇⡠⠀⠀⠀⠀⠀⠀⣿⣇⢸⠀⠀⠀⠀⠐⣢⡄
⠀⠀⠀⠀⠀⣰⡿⠁⡠⠀⠀⠀⠈⠹⣿⣥⣬⡟⢋⣠⡁⣀⠀⣹⣿⣄⠀⢀⣡⣴⣾⠏
⠀⠀⠀⠀⠠⠟⠹⠋⠀⠈⠀⠉⠉⠛⣿⣿⣿⡇⠀⠀⠀⠀⣀⣽⣫⣤⣾⣿⣿⣿⡏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⣀⣿⣇⣠⣴⣶⣿⣿⢿⠟⣛⡈⢀⠙⢠⣞
⣶⣶⣶⣤⡐⠄⣀⠀⢀⡀⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⣟⣻⣤⣼⡇⣼⣿⡄⣾⣄⠀⠀⠀⠀⢀
⠿⠿⠿⠍⠁⠀⠠⣤⣈⡛⠲⢦⣽⣿⣿⣯⣿⣿⣭⣾⣿⣿⣷⣤⣶⣿⣿⣿⣿⡆⢻⠁⠀⣿⣿⡿⢿⣾⣿⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⡈⠙⠛⠛⠿⣭⡭⠉⠁⠀⠀⠈⠙⠂⠈⠉⠀⠀⠀⠀⠀⢻⣿⡟⠀⠸⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⣤⠦⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠒⠢⣤⡄⠀⢰⡆⢰⣄⣠⣾⠟⢀⣄⡀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠈⣷⠀⠀⠀⠀⠀⣬⠀⠀⢻⣿⢞⣫⣿⣿⣶⣿⣿⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⣠⡤⠞⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⢐⣿⠶⠖⠬⢒⣿⣿⣿⣿⣿⣿⣯⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⣅⣀⣀⣀⣀⣀⣀⣤⣬⡷⠶⠚⠛⠉⠀⢀⣄⠔⠉⣿⡿⣿⣿⣿⣿⣿⢿⣾⣿⡇⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣤⣭⣍⣉⣈⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣤⣀⡄⠀⠈⠙⠛⠋⠛⠈⠉⣛⣧⣄⣼⣿⣶⡆
⠀⠀⠀⠀⠀⠈⠀⠘⠒⡚⠉⣛⢿⠛⠛⣿⣿⣿⡿⠿⢭⣭⣷⣦⣶⣶⣶⣶⣦⣀⣈⣭⣵⣼⣿⣿⣶⣼⣧⣶⣿⣇⣴⣿⣶⣌⢻⣿⣿⡿⠟
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⣿⣿⠿⠿⢟⣛⣉⣤⣶⣶⣿⣿⣿⠟⢁⣀⠀⠶⣿⣯⡉⠉⠀⣉⣟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢦⣷⣶⡾⡿⠿⠿⠿⠷⠿⠿⠟⠉⠀⠀⠛⠿⣿⣿⠆⠀⠈⠀⠀⢹⣿⣿⣤⣄⠈⠉⠙⣻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣀⣤⣄⣐⠾⣿⢿⣿⣿⣿⠿⠖⠤⠄⠀⠀⠄⠘⠻⠟⠛⠛⠛⠀⠀⠀⠀⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠻⠿⠿⠿⠟⠛⠂⠘⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠛⠓⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠏⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈
⠀⢀⡟⠁⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠒⠈⠉⠉⠉⠙⠒⠢⣄
⢠⠟⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠈⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠈⢳⡆⠀⠀⠐⠒⠉⠉⠉⠀⠀⠀⠀⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠄⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏
⠀⠀⠉⠒⢲⡶⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⡀⠀⠀⠉⠉⠒⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠐⠻⣍⠉
⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠄⠀⠀⢀⡴⠒⠲⣌⠢⠀⠀⠑⠲⢄⡀⠀⠀⠀⠉⠁⠀⠀⠀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⣀⡼⠀⠀⠀⠀⠀⢀⣈⠑⠄⠀⠀⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢯⠈⣹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⠇
⠀⣀⣀⣠⣤⣤⣤⡀⢀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁⠀⠀⠀⠀⠸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣽⣿⣶⣶⣶⣶⣦⣴
⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣠⣤⣶⣿⣿⡇⠀⠀⠘⡇⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿
⠀⣄⡀⠀⡀⠀⠀⠀⣤⣤⣀⣀⣀⠀⠀⠀⠈⠤⠖⠲⠾⡇⠛⠋⠉⠉⠙⠋⠉⠉⠀⠀⠀⠉⣭⣿⣿⡿⠿⣿⣿⣿⡇⠀⠀⠈⠉⡉⢭⡉⣿
⠀⣿⣶⣾⣿⣿⣿⣶⣾⡇⡼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣿⣿⣷⣶⣶⣶⣶⠆⢤⠆⠰⣿
⠀⣿⠛⣿⣿⣿⣿⣿⣿⠟⢀⣀⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤⠀⠀⠤⢶⣿
⠀⣿⣶⣿⣿⣿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⠿⢿⡿⠿⢿⠿⠿⠿⠿⣿⠿⠿⣿⠿⣿⣿⣿⡿⠆⠀⠀⠀⢾
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⠉⠙⠁⠀⠠⠭⠄⠈⣽⡋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⠀⠀⠀⡄⠀⠠⠄⠀⢀⡀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠂⣆⣠⣤⣼⣅⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠀⠀⠀⠈⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣷⣶⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⢂⣾⣧⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣻⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠂⠀⠰⡶⠾⣿⣦⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠓⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⣭⣁⡄⠀⢶⣿⣶⣼⣧⣤⣤⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣿⣿⣿⣿⣿⡟⠛⠿⠿⠿⢿⣶⢦⣤⣤⣤⣀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣟⠻⢿⠷⣶⣦⣤⣤⣤⣤⣄⠁⠈⢉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠭⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠤⠤⠾⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢼⣤⣤⣮⣤⠾⠤⠴⠒⠐⠒⣀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠿⠇⠀⠀⠉⠉⠉⠈⠁⠀⠀⣤⣴⣶⠶⠟⠛⠉⠑⠦⠄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣷⣶⣒⣂⣀⣤⣀⣀⣤⣄⠄⠠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣄⡀⢐⠲⠶⢶⣶⡶⣶⣶⣶⡶⢶⠖⠁
⠀⠀⠀⠀⠀⣠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⢛⣓⡛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⠞⣛⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠞⠋
⠀⠀⠀⠀⠻⡏⢹⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣴⣿⣿⣶⢿⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣰⣟⢩⣭⢿⣷⠋⣶⣾⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠞⣁⣴⣿⡿⠛⡯⢚⣿⣿⣿⣟⣻⣻⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣰⣿⣿⣿⣿⣿⣶⣿⡿⣬⡟⠛⠛⠟⠛⠦⠐
⠀⠀⠀⡀⠀⠻⢱⠜⣿⣿⣿⣿⣿⡿⢿⠋⣽⠛⠓⠃
⠀⠀⠀⠸⠷⢸⣿⣼⣿⣿⣿⣿⡿⢶⠖⢛⡇
⠀⠀⠀⠀⠀⠹⠿⣿⡿⢿⡛⢿⡷⣬⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠈⠁⠀⠀⠈⢻⣷⡠⢀⣀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠤⠄⠀⠀⠚⠛⠚⠻⠿⠶⠿⠿⠿⠷⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡄⠀⠀⡇⠀⠀⡄⠹⢿⣿⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⣿⣴⠀⣿⣇⠀⢰⣧⠀⣦⣧⠀⢀⠀⠀⠀⢀⣷⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⢀⣿⣿⣿⡴⣋⣠⡕⠺⢿⢀⣿⣿⣤⣿⡄⠀⠀⢸⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⢀⠜⣉⣙⣿⣿⣿⣿⣵⣶⣆⡬⣿⣿⣿⣿⡇⠀⢄⠹⣿⡅⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⡙⡟⢁⣚⡟⢫⣾⣿⣿⣿⣿⣿⣿⡇⠙⣿⡟⢻⣿⡤⠀⠀⢹⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⢀⣿⡿⠿⠀⡾⣿⡷⢮⡽⠿⣿⡿⡷⢾⣭⠀⣤⡬⣏⣀⣀⠖⢿⣀⣙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃⣚⠋⢘⠃⣾⠆⣼⠇⣿⢁⡚⢀⣡⡿⢀⡖⢸⡟⢰⡿⢠⡟⢠⡟⣰⠆⣿⠁⡣⡮⢶
⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⠁⠇⢾⣦⣵⣤⡵⣦⡴⠃⠑⠬⠭⠦⠞⠣⠬⠲⠬⣴⣬⢥⣬⣴⣿⠬⠕⠤⠭⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠒⠁⠀⠧⠚⠙⢿⣿⣷⣼⢻⣷⡀⠀⠰⡢⢔⠆⠀⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣬⣻⣦⡈⠠⠄⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣍⢻⣶⣾⡿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁

⣿⣭⣭⣍⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⠿
⣿⣿⣿⣿⣿⣿⠷⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣶⣶⣶
⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿
⣯⣭⣭⣭⣭⣭⣍⣉⣉⣉⣉⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠛⠛⠛⠛⠉⠙⠉⠙⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐
⣟⡛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣒⣲⣶⣾
⣿⣿⣿⣛⣛⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣻⣿⣿⣿
⣛⣛⣉⣉⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠿⠿⣿
⣿⣿⣿⣿⣿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶
⣿⣿⣿⣛⣛⣛⣛⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣚⣛⣿⣿⣿⣿
⣷⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛
⣿⣟⣋⣉⣉⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣩⣽⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩
⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶
⣿⣿⣿⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿
⣿⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⠿⠿⠿
⣿⣯⣭⣭⣭⣭⣭⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣭
⣓⣒⣒⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠾⠿
⣿⣿⣿⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⠙⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⠖⠛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⣝⣴⠶⠀⠐⠻⢷⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠺⠀⠀⠀⢈⡔⢢⠀⡴⠋⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡤⠤⠼⣦⢸⠀⣧⡀⣨⠷⠒⠒⠀⠐⠂⠄⠀⠀⠠⠒⠋⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠊⠉⢹⠀⠀⠀⠈⠧⠤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⡳⢤⣀⡐⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠐⠒⠀⠐⠀⠀⠸⠤⠄⠀⠠⢏⠉⠉⠉⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⢶⣚⣠⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠉⠀⠀⠀⠂⠈⠁⢈⢧⡀⠀⠀⠉⠀⠈⠙⠒⠦⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡶⠃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠓⡤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⣠⠞⠳⢄⣀⠀⣀⣠⠴⠋⠳⣄⠀⠀⠀⣀⡴⠋