Musically Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣄⡜⢧⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠄⠂⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣀
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠘⣿⡶⠏⢙⠿⢿⠿⢷⠲⢿⡇⠀⠀⠀⠀⠀⠀⣶⡿⠋⠀⠀⠀⠀⠈⠙⣿⣤⠀⠀⠀⠀⠀⣼⣧⠀⠀⡆
⠀⠀⠀⠀⠀⢼⣿⣻⣇⡇⠀⠀⠀⠀⢻⡇⠀⣎⣵⣾⣶⡞⠗⢸⡇⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⣾⣿⠀⠐⡇
⠀⠀⠀⠀⠀⠈⢻⣿⡟⠁⠀⠀⠀⠀⢘⡇⢠⣏⡷⣯⢽⢓⡇⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢾⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢴⣶⣿⣿⣿⣿⣵⡶
⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠰⣿⣷⡀⠀⠀⠀⢠⣴⣶⣬⠀⠀⠀⠀⠀⡇⢿⣿⣿⡏⠃⡇
⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⠀⠀⢸⣿⡗⢛⢩⣿⣼⠏⠉⠖⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠳⣜⣻⡿⢀⣠⠇
⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⢸⢨⣿⣧⡈⢠⡇⠀⡰⢎⣠⣿⠀⠀⢀⣠⠀⠀⠘⢿⣿⡇⠀⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠘⣶⣿⣿⣿⣦⣤⣦⣴⣼⣿⣏⠀⠀⠈⠀⣠⠀⠀⠀⢀⠇⢠⠟⠀⠀⠀⠀⠀⢀⡤⠖⠂⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⠂⠀⠀⠀⡴⢃⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡌⡴⠁⠀⠀⡠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣼⡟⠀⠀⣟⢸⡇⠀⣤⣤⡀⠀⠀⠐⠀⠀⣀⣀⣠⣤⣤⡶⠾⡆⠀⠀⠀⠀⢸⠜⠀⠀⠀⡰⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣷⣤⣴⡯⢼⡇⠀⣸⠈⣧⠀⠀⢸⢸⡏⢉⣿⣇⣸⡿⠷⠔⠒⠀⠀⠀⠀⠸⡇⠀⠀⢠⠃⠊⠀⠀⠀⠀⠀⠀⠈⠳⠦⠀⠀⢸⣿⡇
⠀⠸⣿⣿⠛⢻⣿⣿⣷⠦⠋⢠⡏⠀⠀⠘⡆⠛⠉⢉⣉⠠⠄⠐⠀⣀⢠⠀⠀⠀⠀⠙⠲⠒⣡⡡⠄⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡆⠘⠻⣿⠟⠀⠀⣼⠀⠀⠀⠀⢷⡄⠀⠽⠷⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⡰⠛⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⢸⣿⡇
⠀⠰⣿⡿⠀⠺⣿⣿⡿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠒⠈⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢰⢃⣿⣇⠀⣠⣬⣿⣄⡀⡀⠀⢳⡀⠀⢀⡤⠖⡆⠀⠀⠀⢦⠀⠀⠟⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠐⠂⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⢤⡀
⡇⠀⣿⣿⣷⣿⡿⠿⣿⣧⡑⠆⢸⡇⠀⢀⠀⠀⠖⠀⠀⢠⡆⠀⠀⠀⠀⠀⠙⠀⠀⢈⣓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⢻⣦⣿⣿⣿⣿⣼⣷⣿⣿⣇⣉⣾⠃⠀⠀⠀⠀⠀⠀⠀⡎⢞⠓⡄⠀⠀⠀⠘⠀⠀⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡟
⠀⠙⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⡎⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠴⠶⠖⠒⠲⡀⠀⠈⠀⠀⠀⢀⣠⡤⠖⠂⠀⠀⣤⣶⣂⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡚⢣⠄
⠀⠀⢸⠉⠉⠉⠁⢀⢀⣀⢀⡀⣄⠠⠤⠤⢴⣔⢀⣤⣷⣄⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⡤⠶⣾⣿⣶⢶⣶⣶⣿⣿⣶⣶⣶⣿⣿⣿⢷⣺⠂
⢠⣤⣾⣷⣯⣥⣟⣦⣀⡬⠄⠠⠀⠠⢼⣿⣿⣿⣏⠹⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠁⢀⣾⣿⣿⣿⣿⠿⠛⠛⠿⣿⣝⢿⣿⣿⡻⠁⠘
⢸⢹⣿⣿⢿⣿⣿⣿⣿⠛⢿⣿⣿⣿⡿⠋⣽⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣷⣾⣿⣇⠀⠐⠀⠀⣸⣿⣿⣿⣿⡀⢐
⠈⣿⣟⣴⣿⣿⣿⣿⣿⣿⣦⣴⣭⣼⣷⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡿⠛⢻⠇⡈⡀
⠀⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣭⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆⠙⢁⡀⢛⡻⢿⣾⣿⣿⣿⣿⣿⣿⡷⠟⠋⣀⠁⠁⢸
⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣍⣋⣃⣠⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⠄⠀⢠⣟⣡⣄⣀⣈⣁⣀⣀⣈⣩⣭⣭⣁⣤⣠⣄⣤⣬⣴⣾⣽
⠀⠺⢿⣟⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢻⣿⢿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠯⠩⡙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢧⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⢁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡻⣿⣿⣷⣈⢙⣛⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢠⣾⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⡇⠈⠉⠉⢿⣿⣿⣷⣶⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⠀⣴⣿⢟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⡙⠉⠙⠻⠛⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣏⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣤⣇⠀⠀⠀⣴⣦⣤⣄⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠙⠛⠀⠀⠀⠙⡟⠛⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠅⠀⢈⣿⣿⡿⠿⠿⠿⣷⣦⣝⣿⣆⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⢀⣾⢁⡶
⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⣿⣿⣿⣷⣤⣄⠀⠈⠻⣿⣿⣿⡆⠀⢀⡬⠟⠉⠀⠀⠀⠀⠀⢰⣾⣧⣾⠃
⠀⠀⠀⠀⠀⠀⢻⣿⡧⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⡇⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⢏⡿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⣟⠋⠁⠀⠀⠀⠀⠀⠀⣤⣧⣾⡗⠀⠙⣿⣷⣀
⠀⠀⠀⠀⠀⠀⢸⢻⣿⣧⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣆⠀⠀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠈⢆⣿⣿⣆⠀⢸⣿⡇⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⢰⣾⣿⠟⠁⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⢻⣷⣦⣿⡇⠀⢀⣤⣾⣿⣿⡿⠁⠀⠀⠀⣀⣠⡶⠛⠻⣿⣆⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣄⣤⣶⣿⠟⠋⠀⠀⠀⠙⣿⣧⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⣿⣵⣼⣯⣴⣦⢾⢿⣿⡅⠀⠀⠀⠀⠀⠀⠘⢿⣷
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⣿⣿⡀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⣠⡶⠛⣿⣿⠋⢉⣡⣿⣿⠉⠉⠉⠀⠀⠀⠀⠸⣿⣇⠀⠀⠈⢿⣧
⠀⣰⡿⠁⠀⢹⣿⡀⣿⣿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠘⣿⡄
⠀⣟⣧⣶⠀⠸⣿⡇⠟⠁⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢼⡟
⠀⠘⠛⠙⠀⠀⣿⣧⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠠⢿⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡻⠝⠙⣄⠀⠀⠀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⠭⠊⠙⣆⠀⢠⣼⡆⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢰⣾⠂⠈⠉⠀⠀⠈⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⡆⠀⠀⠀⠀⢾⠿⠀⠀⢀⣀⡀⠀⠀⢠⣒⠇⠀⠀⠀⢀⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡆⠀⢨⠗⠀⠀⢀⡀⠀⠀⠀⠀⠚⠒⠁⠀⠀⠀⠀⠀⠀⢀⣤⣼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣴⠃⠀⠀⠑⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣃⠀⠀⠀⠀⠀⠀⣠⣯⠀⠀⣖⠙⣦⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣴⢿⡟⠳⡆⠀⠀⠈⠉⠁⠀⠀⠙⠁⣽⠐⠂⠀⠀⠀⣶⣒⡒⢲⠀⠀⠈⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⠀⡧⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⣷⠀⠀⢸⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢺⡋⠀⠀⠐⠯⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠏⠀⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢤⣾⡦⠀⣀⣀⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣹⣾⣅⢀
⠀⠀⠀⠚⠉⣡⣾⣿⣿⣷⣤⠶⠞⠋⠁⠀⠀⢀⠀⠂⠀⠂⢠⠀⠁⠀⠈⠀⠀⠀⠈⠀⠀⢠⣦⢺⡷⢷⡿⠏⠋
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⡀⠹⡟⢿⡿⠟⠈⠁⠀⡸
⠀⢻⣿⣿⣿⡿⠛⠛⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣤⠔⠁⠀⠀⠠⠙⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠙⠛⠋⠀⠀⠘⠋⠀⠀⠀⠀⢀⡀⠀⡀⣒⠀⠁⠀⣀⢈⡴⢶⣄⣛⠃⠀⠀⠀⠀⡰⠀⠀⠀⠀⢰⠁⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠁⢰⡟⠻⠟⢹⠋⣿⢻⡟⣉⢻⡟⢻⠏⣉⣻⡄⢀⣤⡞⠀⠀⠀⠀⠀⠠⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⢸⡇⣤⡄⢸⠠⣿⢀⣜⠳⢾⡇⢸⢸⡏⡉⠀⠈⠉⠀⠀⡴⠀⢀⡤⠀⢩⣉⡙
⠀⢶⣾⠇⢀⠰⣄⠀⠀⠀⠁⠀⣼⠀⣿⣇⢸⣇⠀⣸⣟⠗⣨⡇⣼⣌⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠝
⠀⠀⠉⠰⢍⠓⠀⠀⠀⠀⣀⢀⡈⠛⠁⣙⢋⠙⣛⠉⠙⣛⢋⡛⠋⢙⣛⢋⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢚⡓⠲
⠀⠰⡅⠁⠀⠐⠂⠀⠀⠀⣿⢸⡇⠀⠀⣿⢸⠀⣿⠀⠀⣿⢸⡇⠀⠸⡟⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⡙
⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠁⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⣁⣀⣘
⠈⠵⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠐⠒⠮⣡⡔⢮⢂⢀⣀⡠⣿⣿⣿⣿⣿⠃⢼⣶⣀⠐⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠍
⠀⠀⠀⠀⠀⠲⠯⠄⠀⠀⠙⢿⣿⣿⠃⠐⠙⢿⡃⠀⠀⠀⠶⠀⣿⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠐⠀⠤⠤⠘

⠆⠀⠀⠀⠀⣀⣠⠀⠀⠀⡴⠆⡸⠀⠀⠀⠀⠀⠰⠛⠃⢩⠀⠆⠀⠀⠀⠀⢀⣠⠀⠀⠀⡴⠆⡸⠀⠀⠀⠀⠀⠰⠙⠋⢩
⠀⠀⠀⠀⠀⣫⠼⠂⠀⠀⠈⠈⠁⠀⠀⢀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⣫⠼⡂⠀⠀⠈⠈⠀⠀⠀⢀⠀⠀⠘⠁⠚
⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡄⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡄⠀⠀⢸⠀⠀⠀⠀⠀⢀
⠀⠈⠙⠀⠀⠀⠀⣾⠀⠀⢸⣀⠀⠀⠀⠀⡇⡼⠀⠀⢸⠽⠂⠀⠈⠙⠀⠀⠀⠀⣞⠀⠀⢸⣀⠀⠀⠀⠀⡏⡼⠀⠀⢸⠽
⠀⠶⠂⠀⢠⢖⣽⢅⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠶⠃⠀⢠⢖⣽⢅⠀⠀⠀⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠸⣌⣎⡼⠃⠀⢀⣠⠀⠀⢀⣀⠀⠀⠀⠀⠞⠛⠀⠀⠀⠀⠀⠸⣌⣎⡼⠁⠀⢀⣤⠀⠀⢀⣀⠀⠀⠀⠀⡞⠋
⢀⠀⠀⢰⠶⢠⠃⠀⠀⠚⡉⠤⠀⠐⠗⢸⡆⡀⠀⠸⠀⠀⠀⢀⠀⠀⢰⠶⣠⠃⠀⠀⠘⡉⠤⠀⠐⠗⢸⡆⠀⠀⠸
⢮⠀⠀⠀⠀⠁⠀⠀⠐⠃⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⠉⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠞
⠁⠀⠀⠀⠀⠀⠀⠀⠥⡄⠀⠀⢠⡄⠀⠀⠀⢠⣄⠀⠐⠶⠁⠁⠀⠀⠀⠀⠀⠀⠀⢅⡄⠀⠀⢠⡄⠀⠀⠀⢠⣄⠀⠐⠶
⠀⠘⠁⠋⠀⠀⠀⠠⡤⠀⠀⣠⢼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠁⠋⠀⠀⠀⠠⡤⠀⠀⣠⢼⠁⠀⠀⠀⠀⡀
⠘⠀⠈⠁⠀⣀⠀⠀⠀⠀⢸⠱⢻⢹⠆⠀⠀⢰⠃⠀⠀⠀⠀⠘⠁⠈⠁⠀⣀⠀⠀⠀⠀⢸⠱⢻⢹⠆⠀⠀⢰⠃
⡂⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡑⢺⠋⠀⠀⠀⠀⢀⢤⡄⠾⠉⡂⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⡑⢺⠉⠀⠀⠀⠀⢀⢤⡆⠾
⠁⠀⠀⠀⠀⡶⢺⡀⠀⠀⠫⠡⠎⠀⠀⠀⠀⠀⢨⠄⡠⠘⠀⠁⠀⠀⠀⠀⡴⢺⡀⠀⠀⠫⠡⠊⠀⠀⠀⠀⠀⢨⠄⣠⠘
⠀⠀⠀⠀⠀⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠸⢤⠀⠀⠀⠀⣠⠀⠀⢠⠚⠃⠀⠀⠘⡄⢠⠀⠀⢠⢴⡂⠀⠸⢤⠀⠀⠀⠀⣠⠀⠀⢠⠚⠃⠀⠀⠘⡤⢠⠀⠀⢠⢴
⠀⣀⡀⠀⠀⣀⢤⠟⠀⠀⠘⠶⠀⠀⠀⠀⠧⠋⠀⠀⠘⠉⠀⠀⣀⡄⠀⠀⣀⢤⠗⠀⠀⠘⠶⠀⠀⠀⠀⠧⠋⠀⠀⠘⠉
⠀⠉⠀⠀⢸⠱⡻⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠉⠀⠀⢸⠱⣻⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⢀⣈⠑⡗⠋⠀⣠⠴⠚⠀⢀⣔⢢⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⠑⡗⠋⠀⢠⠴⠚⠀⢀⣔⢢⡀⠀⠀⢠⠃
⡰⠀⠀⠘⠍⠜⠀⠀⠀⡄⠂⠉⠀⠀⠁⣸⠃⠂⠀⠈⠀⠀⠀⡰⠀⠀⠘⠭⠞⠀⠀⠀⡄⠂⠉⠀⠀⠁⣸⠃
⠝⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠌⠟⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀
⠀⢠⠤⡠⠀⠀⠀⠀⠉⠃⠀⠀⢸⠇⠀⠀⠀⠘⠓⠀⠀⠁⠀⠀⢠⠤⠤⠀⠀⠀⠀⠑⠃⠀⠀⢸⠅⠀⠀⠀⠘⠓⠀⠀⠉
⢠⠀⠠⠄⠀⠀⠀⠘⠋⠀⢠⢞⣽⢦⡀⠀⠀⢀⡆⠀⠀⠀⠀⢠⡄⠠⠄⠀⠀⠀⠈⠋⠀⢠⢞⣿⢦⡀⠀⠀⢀⡆
⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠘⠌⣸⠼⠁⠀⠀⠘⠀⠀⡀⣠⡤⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠸⠌⣸⠼⠁⠀⠀⠘⠀⠀⡀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠡⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠁⠒⠒⠒⠈⠀⠀⢇⣠⡟⠀⠀⣀⣠⢿⠶⡛⠉⠀⢀⣀⡤⠶⠒⠉⠉⠀⠀⠀⠀⠀⠀⣰⡝⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠓⠉⠉⠁⠀⠘⣓⣮⡶⠚⠉⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⠐⠀⢳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢱⣀⣀⡤⠴⣚⣋⡁⠀⠀⠀⣀⣤⠴⣚⣭⡅⠀⠀⠀⢾⣿⡿⠁⠀⠘
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⢸⡟⢫⠙⢷⠀⠙⢿⣯⠴⠖⠋⠁⠀⠘⢿⣛⡡⠤⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⣣⠬⣦⡼⠒⠋⠉⠀⠀⢀⣠⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠹⡀⢀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡷⠒⡟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⢿⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⠉⠀⣀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⠶⠟⠋⠉⢹⡇⢠⣴⡶⠶⠾⠿⠶⢶⣦⣤⡀⠀⠀⠀⠻⢿⣶⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⡶⠿⣿⡇⢀⣤⡶⠾⠟⠓⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡆⠀⠀⢿⣤⣴⠿⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⣠⣿⠃⢸⡇⠀⠀⣠⡔⠀⠀⢠⣴⠿⠛⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⣿⠃⠀⢀⣿⠁⠀⢰⡟⠁⢠⣶⢶⣄⠈⢻⡆⠀⠀⠀⠀⠀⠈⠿⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠀⠀⢸⣇⠀⢻⣧⣼⡟⠀⣸⡇⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠶⠀⠀⠀⠻⣦⣀⠈⠁⣀⣴⠟⠁⠀⢰⡿⠛⠉⠁⠀⠀⢀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⢸⡇⠀⣀⣤⣴⠾⠛⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣿⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠛⠋⠀⠀⠀⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⣀⣀⣀⣠⣤⡶⠇⢸⡇⠀⠀⠀⢠⣶⠟⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⣀⣀⣸⡇⠀⠀⠀⢻⣇⣀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠛⠀⠀⠀⠀⣴⠟⠛⠙⢻⡇⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣀⡠⠶⠆⠀⠠⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠳⠘⢁⣤⣶⠀⠀⠐⠀⠀⠐⠀⠨⠄⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⠀⠀⠀⠀⠀⣉⠑⠻⣿⡦⡄⢀
⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠆⠀⠀⣠⠄⠀⡠⠒⢢⡀⠀⠀⠀⠀⠀⠈⠒⠀⡌⡙⡠⠊⣀⠴
⠀⠀⠀⠀⠀⣈⠀⠠⠀⠀⠀⠀⠚⠀⠀⠀⢧⡀⠈⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠤
⠀⠀⠀⠀⠈⠀⠀⠀⣠⡼⠳⡤⠀⢬⣤⠀⢟⡃⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠄⠰⠐
⠀⠀⠀⠀⠑⠀⣤⠞⠋⠀⠉⠀⠘⣉⡉⢁⡀⣀⢀⣀⣉⠉⣉⡙⣉⣁⢀⣀⠀⢀⠀⠰⠒⠛⠓⠂⠤⠄
⠀⠀⢐⠐⠖⠆⠄⠀⠀⠀⠀⠠⠀⢹⣇⢻⣿⣿⢻⣏⢹⣷⣿⡿⣿⡏⠀⠠⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠓⠂⠤⠤⠂⠀⠀⠀⠀⠀⠀⠸⠿⠸⠏⠿⠸⠿⠿⠋⠿⠇⠿⠿⠀⠀⢠⠐⠂⠀⠀⠀⠐
⠀⠀⠔⢶⡴⠲⢤⠀⠁⢀⠀⠀⣴⣦⣴⣷⢰⣇⢰⣷⢾⡿⠆⣿⢠⣾⠿⠆⠈⠘⠓⠚⠧⠠⡀⠀⡀⠁
⠀⠀⠀⠀⠀⡤⠜⢰⣀⠀⠁⠀⣿⡟⠟⣿⡼⣿⣼⣏⣬⣿⣦⣿⠸⣷⣤⠀⠤⠀⣔⠒⠒⠒⠊⠘⠦⠃
⠀⠀⠀⠐⠊⠰⠐⠑⠩⣄⠖⠀⠉⢀⠤⠈⢀⡈⠡⢄⡈⢩⠠⣀⡀⠀⡀⠀⠃⢀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠠⣐⠐⢠⠪⡂⢂⡀⠀⠀⠈⠀⠠⡍⠀⣀⠀⠑⠐⠉⠒⠁⠀⠄⠘⠀⠟
⠀⠀⠀⠀⠀⠪⡖⠋⠀⠘⠂⠀⠀⠀⠀⠀⠀⡎⠀⡄⢠⠀⠀⠘⡧⠀⢘⠀⠀⠀⠀⠎⠀⠐⡾
⠀⠀⠀⠀⠀⠀⠀⠠⢂⠠⠀⠒⠀⣄⠦⢀⠀⢲⡔⠀⠀⠂⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⣀⢸⡿⢂⡼⠇⢠⠀⠀⠎⢀⠠⣡⠀⠂⠀⢤⣖⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠾⠦⠀⠀⠀⠀⠀⠀⠈⠙⢠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠂⠀⠀⠄⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠤⠤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⢀⠀⠄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠐⠂⠐⠒⠀⢀⠐⠀⠀⠈⠐⠠⡀⠀⢀⣀⣤⣤⠀⢈⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠂⠐⠠
⠀⠀⠀⠀⠀⠒⡄⠄⡰⡋⠄⠂⠀⠀⡀⠀⠀⢠⡾⣶⣄⡀⠀⠘⡈⠛⠻⣿⣿⣷⣶⡀⠀⠀⡀⠀⠀⠀⠀⢐⠁⠀⠈⠈⠀⠀⡀
⠀⠀⠠⠀⠀⡈⠀⠀⢁⠀⠄⠀⢡⠀⠡⠄⠠⣈⠀⠀⠀⡄⠀⢁⠑⠀⠀⠈⠉⠁⢻⣿⡀⠊⠀⠂⢀⠀⠀⡜⢂⠀⠀⢀⡂⠀⢄
⠀⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠒⠀⢎⠙⣧⣀⠀⢀⠌⢣⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠀⠂⠈⠞⠀⠄⠠⠀⠒⠉⠀⠄⠢
⠀⠀⠀⠀⠈⢃⠁⠀⠀⢡⠀⢠⠀⠐⠀⠈⡈⠀⢠⡈⠻⣿⠀⡀⢀⠋⠀⠡⡈⠛⠉⠉⠀⠈⠀⣘⣁⠜⢀⠀⠀⠀⡇⢄⠐⠀⡀
⠀⠀⠀⠀⠂⢈⢂⠱⣿⣿⣆⠀⠄⠀⠁⡀⠀⠠⠀⠕⣄⠙⡆⠁⡀⠇⢀⠀⠈⠀⠀⡠⠋⠙⣠⢠⡀⠀⠄⢀⠠⢀⣇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠈⣷⡙⠿⠿⡦⠐⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⠂⠀⠀⡌⠐⠠⠀⢀⡂⠀⠀⢄⠶⢀⣀⡴⠁⠀⠀⠀⡀⠀⢀⠐⠰
⠀⠀⢀⠀⢀⠀⢤⡜⢹⣄⠲⣤⠠⣄⣀⣠⡥⠶⠟⣡⡔⠙⢿⣤⣤⡌⠀⠑⠠⠀⠐⡆⠀⠀⡘⠐⣼⠁⢺⡈⠀⠀⠴⠁
⠀⠀⠂⠀⣴⠰⣿⣇⠻⢿⣧⣘⢂⣿⣟⣿⠟⠒⠓⠈⡙⢢⡰⠟⢅⣠⢶⡷⢠⠀⣰⠁⢁⢀⠃⠼⠳⠞⠉⣿⡶⡃⠀⡀⢠⡠⠀⠀⡀
⠀⠀⠐⠀⠘⣧⡙⠿⣿⣿⠿⢫⡜⣿⠟⠁⠀⠀⠀⠀⠀⠂⠔⠁⣠⠁⡸⠃⡢⠔⠁⠀⠀⠒⠌⠀⠀⠀⣠⡿⠁⢊⠕⠄⠅⠀⠠⠁
⠀⡄⠀⠐⠄⠹⢿⣶⣶⡶⠾⢿⡷⠀⠀⠀⠀⠀⠠⠊⠄⠀⠀⠀⠌⠉⠀⠊⠐⠀⠀⢀⠀⠈⠀⠀⡠⣾⡟⠁⠀⠈⠐⢂⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣴⡆⣧⡈⠃⠀⠀⠀⠀⠀⠀⠈⠁⢂⠀⠱⠀⠀⠂⠀⠀⢀⠁⠀⠰⢰⣷⡌⡄⠀⡔⡀⠞⠁⠀⠘⣷⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠓⠄⠙⠛⠋⣡⣄⠀⠀⠀⠀⠨⠀⠀⠀⠐⠀⣰⡀⠀⠀⢄⠀⠀⠁⠀⠀⠈⣭⡇⢸⡰⢧⣴⣉⢂⠀⣰⡿⠃⠁
⠀⠀⠀⠀⢀⠀⡛⠁⠀⠖⢢⢻⡿⠛⠋⠁⢄⠈⠂⢀⠒⠄⠀⠀⠦⠐⠈⢤⡀⠀⠀⠀⠀⠁⠀⣸⡿⢁⠇⠘⡦⡙⣇⣠⢸⠉
⠀⠀⠀⠀⠀⡐⢤⣴⡊⠀⢀⡌⠀⠀⢀⡀⠀⠐⢢⡤⠓⠀⠁⡅⡈⠉⠂⢔⠙⠒⠦⡄⠀⠦⢮⠩⠔⢍⠃⠠⣌⣷⢨⠄⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠸⢋⡵⠋⡰⠀⠀⠄⡛⠋⠀⠀⢁⠀⠁⠀⠀⠛⠀⠀⡷⠄⠰⡀⢀⡁⠀⠀⢀⢀⡀⠐⢄⡔⠛⢶⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠆⠈⠣⣀⠀⠀⠀⢀⠁⣶⡞⠀⠀⡀⠀⠀⠀⠠⠄⠂⡱⠁⠀⣘⠼⠀⠙⣄⠔⠃⠂⠁⡜⡀⢀⠀⠂⡄
⠀⠀⠀⠆⠀⡉⢐⠂⡆⠀⠠⢁⣾⣿⣶⡾⣿⠸⣿⠁⠀⠀⠘⠿⢀⠀⠀⠀⢨⡀⠀⠠⢉⠇⠀⠙⠏⣤⣤⡤⢊⣼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠂⡑⠘⡃⡇⠀⢀⠸⢻⡿⠛⠛⣿⣆⢻⣷⠄⠀⠈⠠⠀⠀⣤⣄⠀⣼⣦⣄⠁⠀⠠⠒⠦⢩⣭⣶⡿⠁⠠⠁⠀⠀⠈
⠀⠀⠈⠈⠀⢀⠜⠡⠈⡀⠀⠂⠹⠵⠾⠃⣾⣿⣦⡀⠨⣁⣶⣶⣶⡿⠟⢁⣼⣿⣿⣿⣷⣦⡈⠀⠀⠊⠈⢉⠀⢠
⠀⠀⠈⠂⠠⠄⠀⠁⠀⠐⠦⣄⢀⠀⠀⠄⠙⠋⣹⣿⡶⠤⠍⠩⠄⠀⠀⠀⠈⡣⠊⠉⠻⢿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠈⠀⠒⠀⠠⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠗⠀⠀⢰⠋⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⠀⢀⣠⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠞⢺⣶⠉⠒⠣⢴⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠁⠀⠀⠀⢀⣴⣟⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣿⠀⠀⠀⠀⠈⠳⢮⡢⠀⠀⠀⠀⢠⡅⠀⠀⠀⠰⠆⠀⠀⠀⢸⠱⡏⢽⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠖⠎⠀⡿⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡀⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠈⠣⣄⣿⣤⠟⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⠃⠀⠐⢂⣀⡇⠰⣖⡀⠀⠀⢀⡼⠁⠀⠀⠀⢀⠀⠀⠛⠐⠂⠀⠀⠀⢰⡿⠿⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢢⢃⣨⣿⣿⡏⠙⠛⠃⣠⣴⡟⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠅⠀⠀⠈⠷⣤⡴⠋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡞⣿⣿⣿⣿⣁⡀⡀⢀⣼⠛⠀⠀⠀⠀⠀⠀⠀⢹⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⣿⣿⣿⣿⣽⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣡⠀⠀⠘⠆⠀⠀⠀⠘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠠⠞⠁⣀⣠⡤⠶⠶⠛⠛⠀⠀⠀⠀⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠸⣿⣿⣿⣿⣏⣿⣿⣯⡤⠒⢛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡄⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⠟⠙⢛⣿⣿⡈⡟⢻⣿⣿⣿⣿⣿⣿⣦⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⠀⡠⠶⣻⣿⣿⣷⡘⠀⢻⣿⣿⢿⣿⣿⡟⠋⠉⠀⠀⠀⠄⣀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣃⠐⠀⢀⣠⣿⣿⣿⣿⣷⡄⠀⠙⢿⣾⣟⣿⠿⠄⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⣿⢋⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢦⡀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢰⡎⠁⢙⣿⣂⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⡀⠀⠀⠀⠁⠀⢔⡀⢀⡴⡟⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣈⣿⠛⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠺⠿⠟⠋
⣀⠀⠀⣀⣠⣼⣶⠈⣿⣧⣼⣥⣮⣀⣀⣠⣵⣏⢙⡄
⠙⠛⠋⠁⠀⠀⠱⣈⣿⠀⠀⠈⠉⠙⠛⠻⣿⣿⣿⡿⠛⠀⠰⢶⡶⠴⡶⠶⠒⠀⢀⣀⣀⡀⠀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⢰⣾⣯⣀⣻⣿⣶⣤⣤⣶⣿⣿⣿⡟⠛⠀⠰⠆⠀⢶⣿⣿⣿⣿⣿⣿⡿⢯⣶⢧⠀⡀⠠⢤⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣶⣌⣉⡙⠛⠻⠿⠿⣿⣿⣿⣛⣿⣿⣿⣿⣟⣿⠻⠿⠟⣛⡩⠥⠀⠁⠘⠃⠈⣀⣀⣠⡟
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣴⣦⣍⣉⣩⣻⡿⠁⠈⢀⣁⣀⠠⠤⠤⠒⠒⠚⠉⠉⠁⣼⠋⢹
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⡷⠷⠶⠮⢽⣻⣿⣿⣿⠙⣿⣿⣹⠛⠋⡡⠄⣄⣀⣀⡠⠤⠤⣔⣶⠂⣿⠀⢸⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣜⢷⣆⡄⠀⠠⡝⢯⣿⣴⠀⣿⣮⣿⠀⠀⠅⡿⣿⢉⣄⣀⣀⣤⣼⣇⡀⣿⠀⠸⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣵⣿⣛⣶⡴⡶⣯⣟⣿⠀⣿⣿⢿⠀⠀⠀⣤⡿⠿⠿⠟⠛⠋⢉⣋⣤⡿⠀⢈
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣤⣾⣩⣍⠙⡛⠿⣾⣯⣿⣯⢰⣿⡿⢿⠀⠀⣆⣤⠄⠀⠒⠂⠀⠈⠁⠀⠠⠠⠝⠒⠶⣄
⠀⠀⠀⠀⠉⠙⠻⠿⠿⠾⢯⣿⣿⡻⠿⣿⣿⣿⣿⣿⣾⣿⣿⢻⠀⢀⣷⣤⣀⠠⠤⠐⠂⠀⣀⣀⣠⣄⠒⢒⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣿⣟⡿⣿⣿⡿⠛⠋⢉⣉⣤⣤⠤⠖⠚⠋⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⢿⡿⠋⠋⠉
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣦⠔⠑⠜⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠁⣶⡆⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢧⢡
⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⣠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣵⠆⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠫⠖⠨
⡀⠀⠀⠀⡡⡄⠀⠀⠘⣿⠀⠀⠀⠀⠀⣀⢧⠀⢠⣦⠀⠀⠀⠛⠃⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑
⣴⡄⠀⠀⠓⠁⠀⠀⠀⣿⡶⠛⢳⠀⠀⠀⠊⠑⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡄⠀⠀⠀⣠⠞⠡
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⢅⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠀⠀⠀⠀⠀⣬⠞⠀⢀⡴⠚⠡⡀⠀⣼⡦
⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⣞⠁⠀⠀⠈⠂⠀⠀⣼⡆⠈
⠀⠀⠀⠀⠀⠀⠬⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠁⠀⠘⠿⠂⠘⠛⠀⠀⢾⠉⠉⢹⠀⠀⠀⠀⣴⡄⠈
⣤⡀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠓⠋⠀⠀⠀⠀⠉⠀⠀⠀⡀
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠃⠀⠀⢀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠛⠃
⣠⡤⠆⠀⠀⠀⠀⠀⡖⠀⠂⢠⡄⠀⠀⣀⠀⢀⢠⢤⠀⠀⠀⠘⠀⠀⠈⠛⠀⣄⡀⠀⠀⠀⢀⠄⠈⠻⠇⠀⠀⠀⢸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⡀⠙⠃⠀⠀⠛⠃⠀⠀⢈⡧⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢘⡴⣶⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠺⠷⠐⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡐⡄
⡀⠀⠀⠀⡷⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⡶⠚⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠌⠀⠀⢀⡤⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠰⠿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡖⠉⠳⡀⢤⠕⠀⠘⠿⠃⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⡋⡇
⠘⠿⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣛⣦⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡳⣖⡠⠀⠀⠀⠘⠛⠱
⠀⢀⠄⠛⠇⠀⠀⠀⠰⡿⠀⠀⠀⠀⠊⠐⠊⠁⠀⠉⠉⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⠈⠙⠻⠍⠃⠀⠀⠀⠀⠀⠀⠀⠠⣮
⠛⠁⠀⠀⠀⠀⢸⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⣀⡈⠣⡀⠀⠀⠀⠀⢀⢀⠔⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢼⡷⠀⠐⠄⣠⠞⠁⠀⡎⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠈⠻⡁⠀⢨⠇⠀⠀⠀⠠⢸⠀⣤⡀⠀⠀⠀⠛
⠀⠀⢾⠗⠀⠑⢄⣠⠞⠁⠀⠀⣸⢴⡦⠀⠀⠀⠓⠀⠀⠐⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⡔⠃⠈⣳
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠁⠀⠀⢰⠂⠠⢀⣀⠀⠀⠈⢦⠀⠀⠠⣶⠆⠠⣶⠆⢠⣶⠆⠀⠀⠀⣀⠀⠙⠶⠞⣽
⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠘⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⣿⡀

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣿⠁⢤⡼⠂⡾⢤⣥⠬⣯⠜⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣀⣤⣤⣥⡞⣸⣯⣀⠼⠁⠸⣶⣸⠟⠋⠀⣄⢀⠀⠠⠄⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣱⠾⢹⡿⠁⠀⢀⡀⢀⡩⣆⣤⡾⣫⢋⡇⣀⠙⠋⠇⠀⠴⠀⣠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡖⠋⠘⢂⣸⣆⢠⣀⣤⠀⣠⣠⠿⣅⣿⣭⡅⢿⡮⠷⣤⣈⠉⡀⢸⠃⣵⢄⣠⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡷⢀⠀⠀⢠⡛⣿⠁⠏⢙⠾⣿⣷⢻⣿⣿⣿⣷⠸⠲⣾⢟⠓⠄⠿⠀⡿⣻⠞⠃⠈⠁⡄
⠀⠀⠀⠀⠀⠀⢀⣈⣿⡿⠟⠁⠀⣐⡛⢩⠀⠈⠛⠻⠿⠇⢹⡿⠘⠃⠙⠋⠙⣴⣶⡇⣿⠿⠖⡠⠀⣿⡁⠀⢀⣶⣝⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣿⠛⠀⠃⠀⢂⣽⠹⡀⠀⠀⠀⠀⠀⠀⣩⣧⣜⢘⣾⡟⢨⡿⠛⡆⡇⠀⣰⠁⠀⢹⢻⣉⡹⠋⣾⣼⣷⣧
⠀⠀⠀⠀⠀⠀⠄⣸⣧⡄⢀⣤⠔⠙⢷⣬⣀⣀⣀⣥⣁⠾⣋⣴⣾⣟⣟⡏⠽⠆⢠⢱⢳⡀⠃⠀⢠⣴⡆⢫⢹⡎⠘⣿⣿⣯⡆
⠀⠀⠀⠀⠀⠰⠠⣯⡿⠁⡿⠣⡖⠓⣦⣶⣮⡭⠉⣈⠶⣢⣿⣾⣏⡿⣿⠃⠊⡻⣈⢷⠃⢣⡠⢰⣿⠿⠁⢸⣿⣷⠀⢿⡟⠇⠙
⠀⠀⠀⠀⠀⡿⣿⠁⣰⣀⣷⣠⣧⣺⣿⣿⣫⠰⢆⣉⢘⡏⠀⢿⢿⣉⣤⠿⠶⣤⣥⠊⠀⣀⣷⣀⡅⣶⠀⠀⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣿⢸⣿⠙⠛⣿⠛⢋⡁⠀⢟⣤⡈⣥⣌⠁⠀⣼⣿⣿⣿⠀⠀⣿⢸⡇⣸⣿⣿⣿⡇⣿⡄⠀⠻⠙⠀⠀⠀⡄⢄
⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⣿⠀⣴⠳⣤⣁⣀⣠⠿⣿⠀⠀⣿⣿⣿⣿⡀⠀⢿⠟⠇⠿⠟⠻⠛⠇⠙⠁⠀⠀⠀⠀⢀⢠⠔⠛
⠀⠀⠀⠀⠀⠈⠋⠸⠇⠀⢰⡿⢰⡿⣾⡎⠉⠉⣷⠒⣿⠀⠀⢿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢬⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢿⠀⠀⢀⣿⢀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠐⢠⢲⠦⣄⡀⠀⠀⠀⠀⠉⢀⣴⣄⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⢃⣼⠈⡕⣴⡜⠋⣠⣾⣯⣿⠛⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⢟⡇⠴⠎⢴⠄⠀⣿⣿⡉⠉⢻⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⢤⢤⠤⠄⢀⡠⠞⠁⢀⠤⠀⠱⠔⢤⢀⡺⡇⠘⠀⡨⠈⢨⡟⢿⠁⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⣥⡤⠅⠀⠠⠀⡤⠬⢴⠖⢶⡀⣴⡦⣦⠠⠀⣽⢦⡿⢆⠄⠠⠾⠶⠄⠚⠐⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢞⣩⣾⡗⢤⡳⣤⣭⣶⡨⢍⢾⡗⠈⠐⢈⠰⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠒⡚⣁⢉⣰⡾⢷⣤⡛⡾⠛⢀⡀⠰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠠⠿⠿⠿⠿⠶⠿⠿⠏⠓⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣦⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠛⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⣀⣤⣤⣤⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢁⣴⣦⠀⢀⣴⣾⡿⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢀⣾⡿⠁⣰⣿⠟⠁⠀⠀⣀⣤⣤⣤⣤⣄⡀⢸⣿⣿⠀⠀⠀⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣾⣿⠁⣰⣿⠏⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⣿⡏⢀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢸⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⡆⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠸⣿⣇⠀⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢰⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢿⣿⡀⠹⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⣿⡿⠀⣼⣿⠁⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡈⢿⣷⡀⠙⣿⣧⣀⠀⠉⠛⠿⢿⣿⠿⠿⠛⠁⢀⣴⣿⠟⠁⣼⣿⠃⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡈⠻⣿⣦⡈⠛⢿⣷⣦⣄⣀⣀⣀⣀⣠⣤⣶⣿⠟⠁⣠⣾⡿⠃⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡈⠻⣿⣦⣄⡈⠙⠛⠿⠿⠿⠿⠟⠛⠉⣀⣴⣾⡿⠋⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣌⠙⠻⢿⣿⣶⣶⣶⣶⣶⣶⣾⡿⠿⠛⢁⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣈⣉⣉⣉⣉⣁⣤⣤⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡆⢀
⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⢸⡗⢦⡄
⠀⠀⠀⠀⠀⢀⡷⠂⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⡴⠊⢁⠀⠀⠀⣶⣾⡏⠀⠁⠀⠀⢠⣿⣿⣿⣷⣶⣶⣤⡄⠀⠀⠀⢸⣿⠿⠿⠿⣛⠻⠃
⠀⠀⠀⠘⠀⡴⢻⠛⢦⡄⠀⠉⠀⠀⣀⡀⢀⠉⠀⠀⣨⠉⠙⠛⡿⠁⠀⣶⣦⠀⠀⣀⠰⠺⢿⣿
⠀⠀⠀⢀⠀⢄⣘⡇⠀⡀⠄⠀⠀⠈⠿⠿⠂⣴⣶⡄⠁⣀⡀⠐⠃⠀⠈⠙⠋⢀⠹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠓⠦⣭⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⠗⠀⣶⣦⠐⠶⠦⠀⠀⠀⣤⣤
⠀⠀⠀⠀⡴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⣀⢠⠄⠀⠀⠀⠀⠾⠇⠀⢠
⠀⠀⠀⠀⠳⣭⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠛⠂⠀⠀⣤⣀⠋⠉⠀⠀⢱⣶⣎⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⢈⡞⠀⠈⠻⡿⠀⢴⣶⡖⠀⠛⠃⠀⢿⠢⠄⠀⣀⣀⠀⡄⠀⠀⢀⠂⠀⠉⠙⠃
⠀⠀⠀⢠⠞⢙⣀⣄⡁⠀⠀⠈⠉⠁⠀⠀⠀⠀⢟⠳⠆⠀⠹⠿⠃⠰⣿⣶⠂⣀⣀
⠀⠀⠀⠈⠀⠏⢩⡄⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠃
⠀⠀⠀⠀⢀⡐⠶⣿⣰⠏⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣶⡦⠀⣀⣀⠀⣄⠛⠀⢀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢉⣭⣍⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⠋⠀⠈⠀⠀⠉⠉⠁⠀⠛⠛
⠀⠀⠀⠀⠀⢸⡙⢟⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢷⡶⠁
⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣦
⠀⠀⠀⠀⠀⢛⠀⠀⠀⣀⣀⠠⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⢰⡞⠀⠀⠙⠛⠀⠸⠷⠂⢀⠀⠀⠀⠀⣠⣄⣀⠁⠀⠀⠈⠈⠉⠛
⠀⠀⠀⠀⡰⠋⣡⣤⡄⠀⠀⠀⠀⠀⠀⣸⢂⡀⠈⠀⠙⠛⠁⢿⣷⠂⣤⣄⡀
⠀⠀⠀⠀⠀⠘⠁⣄⠀⠄⠀⠀⠀⠀⠀⣘⢃⣈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠤⣛⣫⣴⠡⠄⠀⠀⠀⠀⡏⠁⠁⠀⠀⠀⠀⠀⠐⢠⣤⡀⠐⠤⣀⣀⣀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⣶⣾⠁⠀⠀⠀⠐⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠿⠿⢿⢿⣿⡟⠀⠀⠀⠀⠂⠸⠀⢀⣀⡄⠀⠀⣴⣶⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⠀⢀⡀⠈⠻⣿⠇⠀⠀⠀⠠⣴⡄⠀⠈⠃⠁⠀⠀⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠄⢴⣧⣤⣤⠄⠀⠈⢓⣒⣶⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠐⠀⠀⣵⣟⡈⠀⠀⠀⣀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠄⠀⠀⠀⡘⠀⣰⣿⠟⠀⠀⠀⠀⡅⠀⢸⣶⣖⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁⠀⠙⢻⠗⠁⠀⠀⢰⡄⣤⣿⣿⠂⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⢹⡟⠒⠀⠀⠀⠐⠉⢹⢃⠀⠀⣸⣶⢲⣶⠶⠆⠀⠀⠀⣹⣿⣃⡀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠀⢀⣼⠴⠒⠟⠿⢿⡞⠃⠀⠀⠀⠀⣤⢿⠉⠛⠒⣶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡦⡄⡀⠴⠂⠀⠈⢄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠻⢧⠄⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠀⢸⣆⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣯⠈⢂⣦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⣾⣿⢉⡤⡾⡿⠲⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠣⠏⢱⢳⣇⡴⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡋⠀⠀⠀⠀⠾⠀⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠛⠉⠁⠀⠙⠦⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠝⠋⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠵⠊⠁⠀⠀⠀⠀⠀⣠⠔⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡽⠚⠁⠀⠀⠀⠀⠀⢀⡤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⢖⣩⢾⣥⣀⣀⣀⣀⣀⢀⡠⢖⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢉⠔⠛⠓⣯⠔⠋⠀⠀⠀⠀⣀⣠⣠⠝⢋⠴⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠸⣄⠀⣀⡜⠀⠀⠀⠀⠀⣼⠁⠀⠈⡟⠁⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⡇
⠀⠀⠀⠀⢸⡷⣌⠙⠲⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠚⠉⢀⣠⠔⡇
⠀⠀⠀⠀⠀⡇⠈⢑⡦⢄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⡠⠤⡖⠊⠉⠀⢠⡇
⠀⠀⠀⠀⠀⡇⠀⡆⢡⠀⠀⠉⠉⢉⠗⠒⠒⠒⠒⠒⠒⡚⢏⠉⠉⠉⠉⠀⠀⠂⢱⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠘⡆⠀⠀⢀⠎⠀⡇⠀⠀⠀⠀⡰⠁⠘⡄⠀⠀⠀⠀⢸⠀⠀⢣⠀⠀⣇⡇
⠀⠀⠀⠀⠀⣧⠇⠀⠀⢣⠀⠀⡌⠀⠀⢃⠀⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⢀⠇⠀⠀⠈⡇⢰⢹⡇
⠀⠀⠀⠀⠀⢻⢦⡀⠀⠘⡄⡸⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⠀⢳⠀⠀⡘⠀⠀⠀⠀⠸⣸⢸⠁
⠀⠀⠀⠀⠀⠘⢦⡙⠢⣄⣁⠁⠀⠀⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⣀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣉⠑⠲⠤⢄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⠤⠤⠒⣉⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⢰⣿⣿⣦⣿⣿⢰⣿⣿⣾⣿⣿⢹⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⣿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣶⣶⡆
⠀⠀⠀⢳⣶⣿⣎⣁⡄⣤⣤⣥⣤⢿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣸⣿⡟⠿⠿⢿⡿⣿⠿⠿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠻⠿⠿⢿⡿⡿⢿⣿⣿⡛⠿⠛⠋⠉⠉⠀⠐⣾⢾⠿⢶⠶
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠁⣰⣤⣿⠿⠿⣧⣴⣀⠀⡀⠀⠀⠀⡇⣸⡀⢸
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡃⢰⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠳⣶⠀⠀⠀⣿⣛⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⡿⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣴⡿⣷⠛⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡷⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡯⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠫⢀⣀⣁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢜⡇⢀⡙⠆⢀⠀⠁
⠀⠀⠀⠀⣀⠴⢊⠁⠀⠀⠀⠀⠁⢸⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡏⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⡄⠉⠛⠶⠶⠶⠶⠶⠾⠋⢁⡄⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠐⣧⡈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠈⠛⢦⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠙⢦⣤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⢲⣶⠋⣠⣾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠑⠦⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠞⠁⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠑⠦⣙⠲⣄⡀⠀⠀⠀⢰⠟⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡙⠓⣆⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠉⠛⠓⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠙⠳⢦⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠶⠟⢩⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡄⠀⢀⠌⠉⢟⠛⠒⠶⠶⠶⣶⠶⢶⡶⠶⠶⠶⠶⠶⢻⠟⠛⢍⠀⠀⢀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⡴⠁⠀⠀⠈⢦⠀⠀⢀⡔⠁⠀⠀⠑⢄⠀⠀⠀⡰⠃⠀⠀⠈⠳⣄⡼⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠊⢣⠀⠀⠀⠀⠈⢧⡠⠋⠀⠀⠀⠀⠀⠀⠱⣄⡜⠁⠀⠀⠀⠀⠀⣸⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣀⠀⢧⠀⠀⠀⣠⠞⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⣰⠃⠀⢙⡶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⢦⣧⣴⠊⠀⠀⠀⠱⣄⠀⠀⠀⢀⡴⠁⠀⠀⠙⢆⠀⠀⣰⣁⣤⠶⠋⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠉⠙⠳⠶⢦⣤⣬⣦⣀⣠⣎⣀⣀⣀⣤⣤⡤⠷⠞⠛⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠠⢈⠉⠉⢓⣶⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢾⠷⠐⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡾⠁⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣈⡑⠒⠒⢺⠒⠒⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⣾⡹⠀⠤⠄⢻⢈⡆⠀⢸⣀⠠⠤⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣧⣤⣤⣾⣼⣧⠀⡚⡂⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣤⢤⡮⢤⡤⣤⣤⣄⠀⠀⠀⠀⠀⣸⠀⠹⡁⠀⠀⠒⣿⣿⠟⠈⣯⡇
⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⡟⣸⡗⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣾⣏⢙⡃⠀⠰⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣄⣿⡇⠀⢘⠿⠫⠉⠀⠉⠹⢆⠀⠿⠿⢿⣿⣿⡟⣿⠅⠀⠙⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⡾⠉⠀⠀⠀⡀⠀⠀⠀⠈⢲⠀⠀⢉⢿⠀⢹⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⢹⣟⠋⢺⠋⠋⠛⡿⠁⠀⠀⠐⢧⡀⠠⡂⠀⠀⠀⢃⠀⢀⣏⠀⣿⢆⡔⣾⠂⠱⡀
⠀⠀⠀⠀⣠⠴⢻⢻⢧⣸⠀⠀⠀⣇⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⣸⠀⠀⠏⡸⢹⠞⢳⡟⠂⠒⠙⡄
⠀⢀⡤⠚⠣⣠⠎⢸⡸⣿⣄⠀⠰⢅⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⡦⠤⣶⣤⣸⣀⡼⣿⠁⠀⠀⠈⢄
⡴⠉⠀⠀⠘⠀⠉⠙⠀⠈⠾⢢⠃⠀⠙⠈⠂⢄⣀⣀⣀⣤⡄⠘⠋⠀⢱⠈⣹⠛⠼⠛⠒⠛⡆⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠀⠤⠴⠒⠒⠋⢀⣀⣀⣠⡄⣾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡍⠉⠋⠋⠉⠑⠤⢄⣀⠠⡄⠀⠰⠤⠤⢤⡤⠖⠒⠀⣀⠀⠀⠈⢻⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⠀⢀⣤⣇⡀⠀⠀⠀⠀⠈⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠛⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣤⣀⣀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣿⣄⠀⠀⠈⠸⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠀⢹⠀⠙⠢⢄⠀⢂⡤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣤⢒⡟⣉⣧⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠃⠀⠀⠙⠁⠀⠀⠀⠉⠺⠇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⡨⢠⣃⣤⠟⠫⠄⡇⠝⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠳⢦⣀⣀⣀⣸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⣆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⡿⠯⠽⠿⠛⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣬⡙⠛⠿⣿⡷⣶⢶⣦⣤⣤⣤⣤⣤⣤⠴⠶⠚⠛⠉⠁⠀⣡⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣄⣒⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⢹⠛⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣗⡋⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣟⡺⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢻⡿⣿⡏⠀⠀⠄⠸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⠃⠀⠀⠀⠸⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⡇⠊⠀⣿⣍⠐⠂⠠⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠘⣿⣷⢶⣤⠄⠈⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣀⡴⠛⡅⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠸⣤⣉⠛⠿⣾⣤⣽⣧⣄⣀⣀⣀⣀⣀⣀⣹⣧⡴⠶⠛⠉⢀⣴⠇⠀⠀⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠷⣧⣀⡑⠈⠉⠙⠩⠍⢉⡉⠉⠉⠉⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⣠⢄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⠤⠤⠶⠴⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⣠⣾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠃⢹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢧⠠⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⣰⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠈⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠄
⠀⠀⠀⠀⠈⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠀⠀⠈⠵⠿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠴⠢⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣠⠄⠀⠀⠄⢁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠁⡀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠀⡀⠀⠀⠙⠔⢫⠂
⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠆⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠎⠀⢠⣀⠀⢀⡀⠄⠊⠀⠸⠀⠀⠀⠁⠀⠀⠂⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⡈⠁⠠⡀⠀⠀⠁⠀⠀⡠⠊⠀⠘⠅⢀⡀⠚⠀⠠⠂⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⢀⡬⠃
⠀⠀⠀⠀⠈⠃⠄⠀⠀⡀⡁⠰⣀⠀⠉⠀⠀⠀⠀⢀⠁⠚⠀⠀⡀⠀⠀⠀⠀⠁⡦⢀⣀⡀⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠔⠈⠈⠀⠑⠆⠀⠀⠀⠀⠸⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠤⣀⢺⣀⡀⠀⠀⠀⠀⡄⠀⡐⣂⠀⡠⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⠀⠀⠠⠀⠀⠀⠀⠈⠈⠀⠨⠉⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀⠌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠂⠘⠂⠈⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡠⠀⠀⠘⠀⠰⠀⠀⠀⠀⠀⠘⠀⠂⠀⠂


⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣟⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣤⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⣿⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⣿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⢣⡞⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣿⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣾⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠡⡷⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⡾⠗⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⣤⠀⠀⡘⣦⣠⣯⣷⣿⣶⣶⣦⡀⡜⠒⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣯⣻⣿⣿⣿⣿⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢻⣿⣿⣿⣿⡿⠛⠉⠹⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠸⣿⣿⠁⠈⠁⠀⠀⣠⣏⣀⣿⣇⠀⠀⣀⣤⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠏⠀⢀⣠⣴⣾⣟⣿⣯⡝⠿⠿⠿⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣷⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣃⣭⣿⣿⣿⣿⣿⣻⠟⠛⠃⠀⠀⠀⢰⣿⡿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⡶⣿⡟⢿⣿⣿⣿⣟⣿⢿⢦⡄⠀⠀⠀⠀⣿⣿⣄⣀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⣶⣿⣏⣺⠿⣿⠇⣄⡐⠿⣢⣄⠀⣿⡟⠿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⠿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣮⣡⣤⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣻⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠙⠻⣿⣭⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠉⠙⠻⢿⣿⣿⠏⠀⠎⢁⡈⢑⡀⢹⣿⢿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⣿⡍⢱⣶⣦⣶⣥⣴⣦⣾⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣦⠀⠀⠀⢸⣿⡅⠀⠙⠿⣿⣿⣿⠋⢸⡇⣾⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⢸⡇⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠋⡇⠀⢸⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⣸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⡇⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠰⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⢀⣤⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣠⡟⠉⠳⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠈⠉⠉⠀⠀⣇
⣴⣶⣦⡿⠀⠀⠀⠀⠀⠉⠑⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⣯
⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⢀⣴⣶⣴⣾⣿⣿⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⠟⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡿⠃⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣈⠄⠀⠀⠀⠀⠘⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠚⠲⠄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠆⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢀⠤⣀⠴⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠟⠉⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡄⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡼⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⠁⠀⠀⠀⠀⣼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠁⠀⠻⠇⠀⢠⡿⠀⠀⠀⢀⡼⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⢤⡴⠞⠁⢀⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⡀⠀⠈⢷⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠙⠛⠶⣦⣤⣤⣤⣤⣴⣶⠶⠞⠛⠁⠀⠘⣷⡀⣨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣵⣤⣤⣴⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⠋⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢹⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⡀⢸⡀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⠶⠿⢿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⡼⠏⠉⠛⠀⠀⠀⠀⠀⠈⣿⡉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣭⣬⣯⣬⣥⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠈⡏⠀⠀⢹⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⢾⡇⠀⠀⢸⡷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣾⡇⠀⠀⢸⣷⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠄⠀⠀⢤
⠀⠀⠈⠀⠢⡀⠀⠀⠒⠀⠀⠩⢂⠐
⠀⠀⠀⠀⠀⠈⠙⠋⣩⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠒⢲⣾⣿⣿⡟⢻⡆⠠⢤⣶⣶⣾⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣌⠀⠀⢸⣿⣿⠛⡇⠘⢓⣢⠄⠠⠻⣿⡇⡆⣿⣿⠀⠀⠀⠀⢠⣶⣒⣚⠡⠤⢤⣼⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⠀⠀⠈⣿⣿⠀⡿⢸⣾⠁⣾⢠⣿⣿⣇⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢀⣀⣸⡿⣧⠿⠼⢿⡇⣿⣠⣿⠿⢟⣚⣾⡿⢃⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⣀⣈⣃⣈⡑⠛⠒⠚⠓⠚⠉⠀⣿⣈⣯⡉⠀⠀⢸⠷⣷⡶⠶⠶⠶⢾⢿⣿⣾⡇
⠀⠀⠀⢸⣿⣿⡟⣿⢉⡏⠩⢭⣭⣿⡇⠀⠀⣠⠤⠒⠒⢚⡷⠞⠛⠛⠒⠲⢤⡀⣿⡇⠀⠀⣇⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⣿⢸⣇⣸⠘⠻⣿⣇⣴⣯⣄⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⣻⣻⠟⠛⠋⢉⡧
⠀⠀⠀⢸⣿⣿⡇⢿⢸⣿⣿⣰⠀⠏⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣷⠀⢸⣿⣷⣼⡆⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⢀⠴⡷⣄
⠀⠀⠀⠈⣿⣿⣿⣀⣸⣿⣿⣿⣇⣿⣿⣿⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⢠⠏⠀⡇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣰⠯⢤⣀⣇⣠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣷⠃⠀⠀⠈⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⣴⢃⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⣀⣀⡠⠤⠖⠋⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⢶⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣀⣸⠁⢈⣀⣉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⢁⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⢱⡞⠁⠂⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡾⡯⠀⠻⠛⡳⠆⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠤⢼⡇⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡴⣿⣅⣀⣀⣠⢤⡶⠞⣍⠉⠉⠛⠻⢲⡧⣭⠤⠀⣀⣐⣒⣒⣤⣤⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣸⠀⠀⢠⠀⠀⡇⠀⣿⠀⣀⠄⠀⠨⡇⠀⠀⢨⠀⠀⠀⠀⠉⢹⠇⢠⣼⣃⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⢸⠀⢠⡅⠀⢉⣀⣻⣀⣟⢸⠋⠉⠀⣸⠀⠀⠀⠀⡇⢸⡧⠤⠼⡧⠀⠀⢒⣛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⠀⠀⢠⡆⠀⠙⠿⣷⠿⢿⣿⣂⣀⣀⣤⣤⣀⣠⣤⣼⠁⠀⠀⣇⠀⠀⠁⢘⡂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣄⠀⢸⣇⡀⠀⠀⣇⢀⣀⣘⣀⣠⡇⠀⠀⠉⠉⠙⠁⠉⠀⠈⡬⠒⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡄⢀⡾⡿⡄⠀⠉⠙⠛⣸⢿⡍⠉⠁⠀⡨⠙⠓⣀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⢀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣼⣁⣧⣨⣆⣀⡀⠀⠑⠚⠷⣂⠀⠀⠀⠀⠀⠙⢿⣆⠰⡄⣇⠀⢀⣟⠀⠀⠀⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⢉⣿⡷⠾⠟⠯⠭⣤⣀⠈⠁⠀⠀⠀⠀⠀⣠⠆⢻⣇⠢⣿⡀⢈⣉⣀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⡿⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⡑⣄⢀⡤⠀⠀⠀⠈⣿⡄⠬⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠹⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠎⠀⠀⠀⠀⠀⡿⠃⠀⠀⠴⡆⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⢐⠃⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢃⠀⣀⠴⢁⡟⠀⠂⣠⣘⣻⠈⠉⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡄⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⠉⠀⠈⠃⢠⠚⠁⡹⣷⠤⠧⠂⢀⡀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠏⠀⢠⠿⠛⠒⠶⠫⠾⠒⠀⠀⠉⠓⠤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⢀⠴⠁⠀⠸⠆⠀⠀⢠⣦⠀⠀⠀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠱⢿⣶⠀⠀⠀⡀⡀⢀⣴⠟⠉⢀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠛⠊⠈⠀⠀⠀⠘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡙⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣌⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠁⠀⠀⢀⣴⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣷⡈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⣰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠘⣿⣿⣷⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣟⠁⢀⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⢿⣿⣷⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣅⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠙⢿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣦⣴⣿⣿⡿⠋
⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⢙⣿⣿⣿⡿⠋
⠀⠀⣾⣿⣿⠟⠉⠉⠉⠉⠙⠻⣿⣿⣿⠀⠀⠀⢀⣠⣴⣶⣾⣿⣶⣦⣄⣠⣾⣿⡿⢿⣿⣿⣶⣿⣿⡿⠋
⠀⠀⢿⣿⣿⣦⣄⣀⣀⣀⣠⣴⣿⣿⡟⠀⠀⣰⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣏⠀⠀⣹⣿⣿⡿⠋
⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣼⣿⣿⠋⠀⠀⠀⣠⣾⣿⡿⠻⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⢀⣀⣠⣾⣿⡿⠁⠀⠀⣠⣾⣿⡿⠋⠀⠀⣨⣿⣿⣿⣏⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⣿⣿⣷⣄⣠⣾⣿⡿⠛⣿⣿⡇⠀⠀⢸⣿⣿⡿⠿⠟⠛⠛⠛⠉⠉⠁⠀⣀⣀⣠⣤⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⢰⣿⣿⠏⠉⠉⢻⣿⣿⣿⡿⠋⠀⠀⣿⣿⡿⠀⠀⢸⣿⣿⣇⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⢀⣸⣿⣿⠋⠀⠀⠀⣠⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠉⠉⠁⠀⣿⣿⣿
⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣾⣿⣿⠏⠀⠀⠀⢸⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣠⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠟⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠈⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣄⣀⠀⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⣾⣿⣿⠛⠉⠉⠉⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣤⣀⣀⣀⣠⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⢻⣿⣿⣷⣦⣤⣤⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠶⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠠⣽⡉⠀⠀⣠⢀⣀⠤⢾⣥⡤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣬⡳⠤⣀⡀⣄⠀⠀⢉⣿
⠀⠀⠀⠀⠈⢉⡟⠁⠀⠀⠀⡌⠀⠀⠀⠸⠉⠙⠓⢦⡀⢠⡴⠚⠋⠉⡅⠀⠀⠀⢡⠀⠀⠀⠈⢻⡉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠤⠄⠀⣀⡇⠀⠀⣾⣆⠃⠘⣸⡇⠀⠀⢰⣀⠀⠠⠬⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡸⡄⠀⡘⠛⠛⠒⢦⣤⣀⠈⠁⠒⣯⡟⠀⠀⢯⡵⠒⠉⠀⣀⣤⡴⠚⠛⠛⢂⠀⢠⢇
⠀⠀⠀⠀⠀⡇⡇⠐⣃⠐⠂⠠⢎⡀⠈⠙⢯⡀⢻⡁⠀⠀⢸⡇⢀⡽⠋⠁⢀⡡⠄⠐⠂⡸⠂⢸⢸
⠀⠀⠀⠀⠀⢹⠁⠀⠀⠑⠂⢴⣆⡈⠑⠢⡜⠀⢸⡃⠀⠀⢸⡇⠀⢢⠔⠊⢁⣰⡦⠐⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⣠⠞⠛⠉⠉⠙⠳⣤⡀⠉⠉⠁⠘⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠰⢣
⠀⠀⠀⢀⡴⢳⠷⣄⠀⢀⡴⢿⠷⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⢶⡿⣆⠀⠀⣠⠾⡞⢦⡀
⠀⢀⡴⠋⠀⠘⠀⠈⣷⡞⠁⠘⠃⠈⢳⣄⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⣠⠞⠁⠘⠃⠈⢳⣾⠁⠀⠃⠀⠙⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣁⠀⢹⣧⣿⣿⣿⣿⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠼⣿⢷⣿⣸⡟⣿⣧⣻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢠⣀⢲⣶⣶⡓⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣎⢿⣆⠿⠛⠁⠁⣤⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣏⢻⣿⣿⣷⡄
⠀⠀⢀⡀⠀⠄⠀⠀⠀⠠⠷⠿⠛⠛⠋⠉⠟⠛⠛⢦⠀⠀⠀⠀⠀⢻⣿⣻⢹⣿⣾⣿⣿⣻⢿⣿⣿⣆⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠀⢲⠀⠙⠢⡀⠀⠀⠀⠀⠀⠈⢣⠀⢻⣥⡄⠀⠀⠀⣼⣿⣿⣾⠿⠛⠉⠁⠙⡏⢢⣿⡻⣿⣼⣿⡀
⠀⠀⠀⠀⠀⢠⡊⠀⢀⡀⠑⠆⠀⠀⠀⠀⠀⠐⡗⠀⢸⡇⠀⠀⣨⣵⣿⣿⠋⠀⠀⠀⠀⠀⠃⠀⠙⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠄⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⡇⢠⣿⣿⢟⡿⠉⠀⢠⡤⠴⠛⠀⠀⠀⠀⢻⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣏⡵⠋⠀⢠⣀⠀⠀⠀⠀⢀⣜⣥⠠⣯⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⣷⠘⠋⠀⠀⠀⠀⠙⣿⣶⣤⡀⣼⢿⠁⣼⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⢤⣾⣿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠏⠙⠈⡡⢂⣘⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⣀⣠⣶⡛⣿⠿⠛⡛⠋⠀⣀⡴⢡⣾⣿⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠛⠉⠙⣿⣿⣿⣷⣷⣻⣆⣀⠙⣶⣿⣟⣹⣿⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⣿⣿⣿⡙⠛⡋⠁⠛⠿⢾⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠋⠉⠉⢱⣿⣿⣳⣶⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣠⣤⣠⡄⠘⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⠠⢄⣀⣀⣨⣿⣿⡏⠉⠉⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣶⡛⠋⠙⢛⢿⡛⠛⠓⠒⠒⠒⠒⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣾⣿⣶⣻⡶⠀⠀⠀⢠⣤⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢉⣉⣭⡌⣇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠛⠉⠀⣀⣾⠂⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠈⠓⠀⠀⠀⠀⠲⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠳⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⣿⠈⠙⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠀⣿⠀⠀⣴⠶⡶⠿⢿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⢠⢟⣴⣿⣿⣓⠮⣝⢯⣉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⠀⣿⢸⣿⣿⣿⣿⣷⣌⢧⣩⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⣿⢸⣿⣿⠟⠻⣿⣿⣇⣧⢻⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⢹⡞⣿⣿⣄⣀⣿⣿⣿⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⣿⣈⢿⣿⣿⣿⣿⢏⡾⣸⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠈⣩⣷⣭⣛⡿⢿⣋⢵⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⢠⡿⣨⡽⠿⢭⣛⢿⣏⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⡿⣼⣿⣿⣿⣷⣌⠳⡝⢶⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠸⡇⣿⣿⣿⠿⣿⣿⣷⡹⡜⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⢿⢹⣿⣧⡀⢈⣿⣿⡇⣷⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⢘⡧⠻⣿⣿⣿⣿⣿⡇⡿⣼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠈⡇⠀⠀⠛⠳⣍⡻⣿⣿⣯⠞⣱⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠈⠙⠳⠶⢖⣛⣛⣁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣤⣤⣷⣤⠶⠶⠶⠞⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠁⣀⠈⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡠⢄⡄
⠀⠀⠄⠀⠀⠀⠑⢷⢈⠀⠀⢠⠖⠉⠉⠛⢦⡀⢀⠀⠀⠀⠀⠁⠀⠐⠛⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠁⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⢀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣀⣨⢚⣿⣶⠀⠀⠀⠀⠀⠀⢄⠀⠰⡄
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡟⠀⠠⣄⣠⠤⠙⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠂
⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣗⡞⣞⣾⣸⣵⣇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⢀⡉⢉⡉⠈⠁⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡡⠔⠂
⠀⠀⠀⢠⣄⣀⠀⠀⠆⠀⠀⠀⠀⢤⣤⣤⠀⠀⠏⢠⣞⡵⡾⡋⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠉⠐⠀⠀⠀⠀⠀⠀⠈⢣⠞⠀⣀⣴⠻⡟⣸⢱⣷⣧⣷⣿⣴⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠘⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢖⠒
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⡠⡄⠀⠠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⢠⠀⢽⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠍⢡⣨⠆⠄⠀⠀⠘⠂⢰⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢠⡇⠘⠄⠀⠀⠀⠀⠀⣝⠃⡞⢉⢹⣦⣐⣫⠵⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⢸⡇⠀⠀⠀⠀⠐⠀⣾⠏⠀⠘⠊⠹⡿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠄⠉⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣀⠀⠀⡲⣄⠀⠀⠀⢀⡀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠁⡤⠀⠀⠄⠀⠀⠰⠂⠀⠀⣴⠄⢀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠸⢊⠆⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⣐⠀⠨⠥⠤⣀⣈⣓⣀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠈⢇⣒⠀⠀⠀⠋⢐⡖⠂⠬⠭⠭⣝⣚⣒⣒⣒⣶⣒⠒⢲⡖⠒⠒⠒⡾⠉⠉⠀⣀⠄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠒⠒⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢀⣠⣧⣤⣬⣤⣴⣶⠮⠍⣉⣽⡋⠉⠉⠒⠒⠢⠤⠾⣀⣀⠐⠀⠀⠀⠀⠐⣒⠚⠉⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠉⠑⠒⠢⠭⢍⣓⡒⠤⢄⣐⡀⠀⠀⣨⠉⠁⠀⠰⠋⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠛⠛⠓⠒⠒⠂⠀⢠⠀⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡴⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⣠⠞⠓⠤⣉⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⠀⢠⠾⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠱⠤⠞⢁⡤⢄⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠑⠴⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠜
⠀⠰⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡰⠁⡜⠉⢢⡀
⠑⢮⠀⠀⣎⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣴⣶⣶⣤⡀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣱⢦⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠙⢿⣿⣿⣻⠗⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⣤⠖⠛⢤⡙⢦
⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠁⠸⣄⣠⣶⣇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠤⡄⢀⣹⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠁⠀⠀⠀⠀⠀⠒⠀⠀⠈⢻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣹⠤⠖⠙⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡄⠀⠀⠀⠼⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⢠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠄⣠⠁⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢧⡀⠀⠀⠀⠀⠀⠀⢹⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⡀⣀⠀⢸⣷⣄⠀⠈⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠠⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠷⠄⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⣿⠿⠟⠋⠉⠂⠈⠓⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣰⠟⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢷⡴⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠉⠙⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣠⣤⠶⠶⠶⢦⣤⣀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣼⠋⠀⣠⡾⠋⠁⠀⣀⣀⣀⡀⠀⠉⠻⣦⡀⠈⢻⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠃⠀⣴⠏⠀⢠⡶⠛⠉⣉⡉⠙⠳⣦⠀⠈⢷⡄⠀⢻⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⡏⠀⢰⡟⢀⣴⠛⠉⠙⢷⡄⠘⣷⠀⠈⣷⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⢾⠀⢸⡇⠀⠀⠀⠀⣿⠀⣹⠄⠀⣿⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠸⣇⠀⠸⣧⠈⠻⣤⣄⣤⡾⠃⢠⡿⠀⢀⡿⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡄⠀⠻⣆⠀⠘⠷⣤⣀⣈⣀⣠⡴⠛⠀⢀⡾⠃⠀⣼⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⣄⠀⠙⠷⣄⡀⠀⠉⠉⠉⠁⠀⣀⡴⠟⠁⢀⣼⠋⠀⠀⠀⢸
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠙⠛⠒⠲⠖⠚⠛⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⢼⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣼⣿⣿⡍⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣠⣤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⢺⠉⠉⠉⠉⠉⠉⠉⢩⣿⣿
⣿⠉⠋⢀⣤⠶⢞⣛⠲⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⠶⠖⣛⡳⠶⣤⡀⠙⠉
⣿⠀⣰⢟⣵⢟⣫⣭⣝⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣵⠞⣫⣭⣝⡻⣮⡻⣆
⣿⢰⡏⣾⢡⡟⠁⠀⠉⢳⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣡⣤⣤⣤⣌⣉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢺⢣⡟⠁⠀⠈⢻⡌⣷⢹⡆
⡿⠸⡇⢿⡸⣧⡀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⢀⡾⠃⣴⠟⢉⣤⡤⣤⣌⠙⢷⡄⠻⣆⠀⠀⠀⠀⠀⠀⢺⡜⣧⡀⠀⢀⣼⢇⡿⣸⠇
⣷⠀⠹⣌⠳⣬⣛⣛⣋⣵⣿⠀⠀⠀⠀⠀⠀⣾⠃⣼⠃⣴⢋⡴⠶⢶⣌⢷⡄⢻⡄⢻⡄⠀⠀⠀⠀⠀⢸⡿⣮⣙⣛⣛⣥⠞⣱⠟
⣿⠀⠀⠈⠻⠶⣭⣭⣥⠶⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⢸⣇⠀⢀⣿⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⢽⠛⠶⣬⣭⣭⠶⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣆⠹⣆⠘⢷⣍⣛⣋⣵⠟⢀⡿⢁⣾⠁⠀⠀⠀⠀⠀⣽
⣿⠀⠀⠀⢀⡶⢛⣙⡛⢦⣿⠀⠀⠀⠀⠀⠀⠀⠻⣦⡙⠳⣤⣌⣉⣉⣤⡶⠛⣠⡾⠁⠀⠀⠀⠀⠀⠀⢼⢀⡴⢛⣉⡛⢶⡀
⣿⠀⠀⠀⢸⡅⢿⣭⡷⢸⣿⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⠶⣤⣌⣉⣉⣤⡴⠞⠋⠀⠀⠀⠀⢀⣀⠀⠀⢼⢺⡇⢾⣭⡿⢸⡇
⣿⠀⠀⠀⠈⠻⢦⣤⡴⠟⢿⡀⣴⠏⠙⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⣷⠀⣼⠀⠻⢦⣤⡴⠟⠁
⣿⣾⣳⠄⠀⠀⠀⠀⠀⠀⠘⢧⡘⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠟⣰⡟⠀⠀⠀⠀⠀⠀⠀⠠⣞⣷
⣈⣻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣟
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣍⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢣⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⣤⠀⠀⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣬⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢐⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⣿⣿⣿⣿⣿⣿⡟⠳⠤⣀⠴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⣿⣿⡋⢠⣅⣠⣨⡆⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⣿⣿⡿⢿⡇⠀⠀⡀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠛⠛⠛⠛⠲⠶⠷⠾⠾⠃⠀⠀⢠⢄⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⡤⠀⠀⠀⢷⡾⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⣀⡀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠰⣤⣶⣶⣶⣧⡴⠀⠛⢵⣦⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣬⣦⣴⣦⣶⣷⣤⣾⣯⣼⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⡴⠋⣡⣴⣿⣅⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⢋⣭⣭⣤⣄⡴⠛⢁⣤⠞⠉⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⣻⡶⠛⠁⣠⣾⣉⡀
⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣷⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⠞⠁⣀⣴⠟⠁⠀⠉⠉⠛⠶⢦⣄⡀
⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣿
⣿⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⣿
⣿⣆⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠁⠀⠀⢀⣴⣿
⣿⠈⠳⣄⠀⠈⠉⠛⠳⠶⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉⠀⠀⠀⠀⢀⣠⠴⠋⢸⣿
⢸⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⡾⣿
⢸⠀⠀⠀⢀⡏⣇⠉⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠚⠋⠍⢦⠀⠀⠀⠀⠀⠀⢠⡇⡿
⢸⠀⠀⠀⣼⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠉⣹⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⢯⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣆⠀⠀⠀⠀⠀⢸⢰⡇
⢸⡄⠀⢰⠇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠀⠀⠀⠀⡟⢸⡇
⢸⡇⠀⡟⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⢸⡇
⠸⡇⢸⠃⠀⠀⠀⠈⣇⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢳⠀⠀⢸⡇⢸⠇
⠀⣇⡏⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⠁⣿
⠀⣿⣦⠀⠀⠀⠀⠀⠘⡇⠀⢀⡞⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⣿
⠀⢻⡌⠳⢤⡀⠀⠀⠀⢹⠀⡼⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡿⢀⡿
⠀⠈⢳⣄⠀⠙⠲⣄⡀⠘⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣻⠇
⠀⠀⠀⠈⠳⢤⣀⠀⠙⠓⠾⣤⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡏⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠙⠓⠶⠦⣤⣀⣀⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣼⣁⣤⡤⠴⠶⠚⠉⠁⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠒⠒⠓⠒⠒⠒⠚⠛⠛⠛⠉⠉⠁⠀⢀⣀⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⡿⠉⠉⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⢠⣶⣶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⠁⡠⠀⡀⢸⣿⣿⣿⣦⣴⣿⣿⣿⡇⢀⠐⣄⠈⣧⠀⠀⢸⣿⣿⢣⣿⢟⣿⣿⡇⠀⠀⠀⠙⠋⠀⠀⠀⢸⣿⣿⢻⣷⠹⣿⣿
⠀⠀⡇⠀⡇⢸⡀⢸⣿⡟⣋⣭⣭⣙⢿⣿⡇⢈⡇⢸⠀⣸⠀⠀⢸⣿⣿⢸⣿⢸⡇⣿⡇⠀⣠⠶⠒⠒⢦⡀⠀⢸⡷⣹⡇⣿⡇⣿⣿
⠀⠀⢻⡀⠙⠀⠁⢸⡟⣸⣿⠋⠙⣿⡇⢻⡇⠈⠠⠋⢀⠟⠀⠀⢸⣿⣿⣌⣿⣮⣿⣿⡇⢰⠇⢀⣴⣦⠀⢹⡄⢸⣿⣿⣼⡿⣰⣿⣿
⠀⠀⠀⠁⠀⠀⠀⢸⣷⡸⣿⣦⣴⣿⢃⣿⡇⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠘⣇⠀⠙⠋⢀⡼⠁⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣭⣭⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠒⠒⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢴⣿⠄⠀⠀⡀⢀⣢⣽⣿⣭⣔⡠⠀⠀⠀⠰⣿⡆⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡐⣴⡿⠋⠁⠀⠈⠙⢿⡆⡂⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⡈⣿⠀⠀⣶⣿⣦⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⣷⠀⠀⠰⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⡠⣿⡀⠀⠙⠛⠋⠀⢠⡿⡙⠀⠀⠀⠀⠀⠀⢹⡀⠀⡀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⣶⠟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢝⢿⣦⣤⣀⣤⣴⢿⠫⠂⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣶⡀⠈⢻⣆
⠀⠀⠀⠀⠀⣰⡟⠀⣰⡟⠁⠀⠀⠀⢸⠀⠀⠀⣠⣦⡀⠀⠀⠀⠈⠚⠥⠤⠌⠓⠉⠀⠀⠀⢠⣶⡄⠀⠀⠸⡇⠀⠀⠈⢻⣆⠀⢻⣆
⠀⠀⠀⠀⢀⣿⠁⢰⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣇⠀⠀⠀⠀⢻⡆⠈⣿⡀
⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠡⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⠀⣸⡇
⠀⠀⠀⠀⠈⣿⡀⠸⣧⠀⠀⠀⠀⠀⢿⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡇⠀⠀⠀⠀⣼⠇⢀⣿⠁
⠀⠀⠀⠀⠀⠸⣧⠀⠙⣷⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⠀⡄⠀⠐⠀⠀⢰⡇⠀⠀⢀⣾⠋⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⠻⣦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠟⠁⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⣇⠀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠄⠀⠀⡀⠀⣼⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⣿⠀⠀⢀⠢⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⡿⠀⠀⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠐⠀⡀⠁⠂⠀⠀⠒⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣴⡟
⠀⠀⠀⠀⠛⣿⣿⡟⠁⣠⣤⣤⣄⠀⠀⠀⠀⢀⣠⣤⠾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡦⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡶⣦⢻⣿⢷⣀⣤
⠀⠀⠀⠀⠘⠁⠈⣡⣾⣿⣿⣿⣿⣷⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⢀⠀⠘⡄⠀⠑⠀⣤⠀⠈⠀⢀⠀⣀⠀⡀⠀⠀⠈⠀⠀⠀⣤⣶⡜⢿⡷⠶⢶⡿⠧⠉⠉
⠀⢀⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠅⠀⠀⠸⣷⣿⣿⡻⣾⣿⠀⠛⠁⠀⠀⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠢⢄⠙⣧⠈⠻⠊⠁⠀⠀⠀⠀⢰⠃
⠀⠈⢿⣿⣿⣿⣿⡿⠋⠀⠈⢉⡉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⣴⣶⠎⠁⠀⠀⠀⠀⢣⠈⠁⠀⠀⠀⠀⠀⠀⠔⠃
⠀⡀⠀⡙⠻⠿⠋⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠀⠆⠀⠚⠀⠀⠀⠀⢘⣤⣶⣤⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡾⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠀⢀⣶⠿⣦⣴⠶⣶⠿⣦⣴⣤⣶⠶⢶⣾⣯⢴⣿⡿⠿⢷⣄⠀⠀⠀⢀⣤⠊⠀⠀⡄⠀⠀⠀⠀⢁⠠⠀⠼
⠈⠀⠀⠀⠀⠴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠙⠃⠀⡇⠀⣿⠁⢹⢁⣴⠀⣹⡇⢸⡏⢀⣶⣶⡿⠂⠀⣴⣿⠏⠀⠀⠀⠇⠀⠀⠀⠀⢉⣤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠤⠀⠀⣿⡇⢰⣤⡄⠀⣇⢸⣿⠀⢰⡈⠛⠿⣿⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⢀⣤⠄⠈⣆⣀⣀⠘
⠀⢠⣤⣴⡦⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠁⠀⣿⡇⢸⣿⡇⠀⣿⠀⠉⠀⣾⡿⣷⠄⢸⡇⢸⣇⠘⠿⠿⣷⠀⠀⠀⠀⠠⠀⠀⠀⠆⠀⠀⠀⠀⠘⢋⠉⢉⢳
⠀⠈⠻⠿⠁⣠⠰⣄⠜⠂⠀⠀⠀⠀⠀⠀⠀⢿⣦⣾⠛⢷⣴⡿⣷⣤⣾⠿⣷⣤⣴⡾⣧⣼⠿⣷⣤⣼⠟⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠄⠀⠀⢀⣈⣉⣉⣀
⠀⠀⠀⡀⡊⠓⠆⠈⠀⠀⠀⠀⠀⢠⣤⠀⣤⡄⠀⠀⢠⣤⢠⣤⠀⣤⡄⠀⠀⢠⡄⢠⡤⠀⠀⢠⣤⠀⣤⡄⣤⡆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⠄⠀⠙
⠀⠀⢾⣀⠙⠀⠀⠀⠞⠁⡆⠀⠀⢸⣿⠀⣿⡇⠀⠀⢸⣿⢸⣿⠀⣿⡄⠀⠀⢸⡇⢸⡇⠀⠀⢸⣿⠀⣸⡇⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠉⠹
⠀⠄⠄⢀⡀⠀⠀⠀⠀⣀⠆⠀⠀⠈⠛⠀⠛⠃⠀⠀⠘⠋⠈⠋⠀⠛⠁⠀⠀⠘⠃⠘⠃⠀⠀⠈⠉⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠐⣋⠉⠉⢳
⢐⡶⠀⠠⢈⠑⠢⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠠⠥⠤⠤⢤
⠀⠠⠞⡑⠤⣌⠢⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⠸
⠀⢒⠤⣬⠑⢌⡱⣜⠧⠀⠠⠀⠀⢠⣾⣿⣿⣾⣯⣿⣿⠃⢀⣷⣤⠀⠀⠶⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢘⠛⠛⢛⠻
⠀⠀⠀⠀⠙⢏⠱⣈⣦⠳⠊⠋⠛⠘⣿⣿⣿⣿⣿⣹⠇⢀⣼⣿⣿⡯⠁⠀⠀⠀⢀⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄⢀⡢⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠈⣆⣹⠚⡚⡂⠀⠀⠀⠈⠛⠧⣿⣿⠃⠀⠀⠀⠈⠹⡄⠀⠀⠀⠀⠘⠃⠀⣿⠇⠀⠀⢈⠀⡀⢈⠂⠄⠘⠤⠀⠐⠐⠉⠀⠉⠀⠐⠒⠒⠃⢸

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠿⣿⣿⣿⡿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠴⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⡷⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⡷⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡷⢶⣶⣶⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠿⣯⠀⡇⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠿⣧⠀⡇⠀⠀⠀⣽⠋⠁
⠀⢠⣶⣿⣁⣀⣀⣿⣿⣿⣷⣾⣿⣶⣶⣿⣤⣷⣶⣶⣴⣿⣿⣿⣷⣾⣿⣶⣶⣷⣤⡷⠀⠀⢠⠿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⠿⠟⠛⠉⠈⠁⠉⠉⠛⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠺
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⢟⠝⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠪⣻⣿⣿⣿⠿⠿⠿⠿⢇⡀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⠀⠀⠀⢀⣴⣿⣶⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣶⣾⣧⡼
⠀⣿⣿⢿⠛⠛⠛⠛⢻⣿⠃⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣿⡉⠉⢹⡟⠉⠉⣿
⠠⣿⣿⣿⣷⣷⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠰⣿⡇⠀⢾⡷⠀⡰⡿⣆
⠀⠙⠿⣿⣿⣿⣿⡿⢻⣿⡄⠀⠀⠀⠀⠀⠘⠠⣟⣿⠠⠇⠀⠀⠀⠀⠀⢠⣿⡧⡀⣸⣧⠞⠁⠿⠈⠳
⠀⠀⠀⠀⠀⠈⠉⡄⠀⢻⣶⡀⠀⠀⠀⠀⠀⠐⠿⠿⠀⠀⠀⠀⠀⠀⢀⣷⡟⢈⣮⠛⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⠟⣴⠟⢉⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠈⠻⣷⣢⠄⣀⠀⠀⠀⠀⢀⢠⣰⣽⠟⠁⠚⠋⠀⣰⣦⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⡔⠂⠄⣀
⢿⡷⠁⠀⠀⠀⠉⣲
⠀⠀⠀⠀⠀⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠒⠛⠋⠉⢿⣠⣄⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢉⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠿⠇⠀⠈
⠀⠀⠀⠀⢀⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠓⢦⣄⣧
⠀⠀⠸⢧⡄⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠉⠉⠛⠒⠦⣄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣾⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⡏⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠈⡇
⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⢸
⠀⠀⠀⠀⢻⣄⣤⣄⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣾⠁⠄⢀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠓⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⠀⠀⡠⠖⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠟⢿⡆⠀⠀⠀⡼⠃⠰⣯⠛⠄⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢀⡽⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡯⢴⢶⣤⠶⠋⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠉⠉⠛⠶⠶⠶⠖⠛⠛⠛⠉⠉⠀⠀⠈⢷⠀⠸⡇⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⡐⢄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢹⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣦⡼⠚⢷⡶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠃⠀⠀⠈⠳⣦⠀⠀⠀⣀⣀⣀⡀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⡀⢀⣈⣹⡅⣀⡀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡖⠒⠒⢻⠗⠒⠒⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⣿⣶⣾⣇⣠⣤⡶⠶⠶⠶⠶⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⢋⣠⣴⣾⣿⣿⣿⡏⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⠟⢀⣤⣶⡿⠟⠛⠛⠛⠻⠿⣷⣦⣄⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⡿⠿⠛⢁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣤⠀⠀⢰⡟⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⢹⣆
⠀⠀⠀⠀⣠⣶⠇⠀⠀⣿⠁⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢿⡀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⢸⡇⠀⠀⠀⣿⣇⣠⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣆⣀⣿⡇⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⣴⣦⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⣾⠏⢸⡇⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⣿⡟⡟⣿⣦⡄⠀⣿⣿⣿⣿⣿⣿⣷⢸⡏⠀⣸⠀⠀⠀⡰⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⣿⣿⣿⣿⣿⣿⡇⠀⢸⣷⣷⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⡟⢸⣧⡼⠁⠀⠀⠀⢌⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⠻⠿⢿⣿⣿⡇⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⢋⣴⠿⡏⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⢰⡟⠁⢀⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠀⢠⣿⠛⡏⠉⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣄⣀⣠⣤⣤⡶⠾⠛⠉⠀⣿⠀⠀⠻⠀⢿⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡟⠀⠀⠘⢷⣄⣀⡀⢸⣆⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣥⡀⠀⠀⠀⠈⠙⣛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣥⠋⠀⠀⠀⢠⣿⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠿⢤⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢒⣒⣈⠁⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⣒⣊⡭⠤⠖⠒⠋⠉⠁⠀⠀⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠠⠒⣊⣭⠥⠖⠒⠋⠉⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⡴⣻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⡜⣼⣿⢟⣹⣻⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⢗⢻⢱⡘⠽⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠘⢎⡻⠿⠾⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⡾⢋⣥⣶⣶⣦⣍⠻⣿⢯⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⢠⡟⣰⣿⣿⣿⣿⣿⣿⣷⡜⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⣿⢰⣿⣿⣿⣿⢛⡭⣭⡙⣿⢸⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⡇⢸⣿⡿⠟⠃⣾⣿⣿⣿⢼⡇⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⣧⠸⠡⠐⡠⡀⠹⣤⣹⠟⣾⢣⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⢹⡄⢀⠣⡐⢄⠠⠀⢤⣾⠏⣼⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠙⣄⠑⠌⣂⠠⡀⠔⢃⣼⡻⡃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠈⠳⢦⣤⣤⣴⣾⡿⠟⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠒⠛⠉⠩⠿⠿⠿⠟⠛⠓⣒⠒⢒⣒⡒⠲⣶⣶⡀⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋

⣿⣿⠟⠿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠻⠛⠓⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣶⣾⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣦⣤⣤⣴⣶⣶⣴⣶⣿⡷⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣷⣶⣿⣷⣶⣦⣿⣿⣤⣤⣤⣴⣿⡷⣤⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣰⣿⡏⠀⠻⣿⠏⣩⠝⠉⢹⣿⠉⠙⢻⣿⣙⣿⣿⣿⣿⣿⣬⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⢹⣿⡇⠀⠘⣿⠇⠁⠀⠀⢸⣿⠀⠀⠽⢿⣩⡿⠟⢹⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠸⣿⡇⠀⠈⣿⠀⠀⠀⠀⢸⣿⠀⠀⢼⣿⣤⠴⠚⢹⣿⣿⢋⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟⠀⠶⡶⢼⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡤⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⢹⣿⠒⢶⡦⢸⣿⡿⣦⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣧⠀⣿⡀⠀⠀⠿⡇⠀⠀⠀⢹⡿⠀⠀⢸⣿⢰⠞⠀⣼⣿⡇⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢉⣿⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⢸⡟⢀⣿⣴⣿⣿⣤⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠸⣿⡄⠀⢸⣧⠀⠀⠀⣾⡏⠀⢀⣿⣿⣋⣿⣿⣿⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢹⣷⠀⢹⣷⠀⠀⠀⣿⡇⣤⣸⣿⠏⣛⣩⣿⡟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠸⣿⡄⢨⣿⡅⠀⠀⣿⣿⡂⣾⣿⣰⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣹⣷⣤⣽⣧⣤⣤⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠻⢶⣿⡏⠹⢿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠤⠀⠉⡿⠟⢿⣿⣿⣟⡵⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠉⠀⣛⣛⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠛⠶⠶⠶⠦⠾⣿⣿⣿⣿⣿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡤⠰⠿⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⣃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⡿⢂⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠺⣟⣿⣏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣧⣶⣿⣿⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠙⠯⣹⣿⣿⣶⡁⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢲⣤⣿⣯⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠈⠋⠐⠲⠛⠋⠹⠟⣻⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⢦⣤⣬⡥⠾⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



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

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⣰⢟⣿⣿⠿⢛⣉⣩⣭⣭⣉⣙⠻⢿⣿⣟⢷⡄⠀⠀⢠⡾⣻⣿⣿⠿⠛⣋⣉⣉⣉⡛⠻⢿⣿⣿⣻⣦
⠀⠀⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⢻⣿⡇⠀⠀⢸⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿
⠀⢠⣿⢃⣾⣿⣿⣿⢟⣯⣭⣭⣟⢿⣿⣿⣿⣆⠹⣧⠀⠀⣼⡟⢁⣼⢛⣭⡛⢋⡛⣫⡛⢫⣝⣻⣝⢂⠙⣿⡄
⠀⣼⡇⣼⣿⣿⡿⣵⡟⠋⠁⠈⠙⣷⡽⣿⣿⣿⡄⣿⡄⢠⣿⠁⣌⢋⣬⠋⣤⠉⠀⠀⠀⢙⡉⣍⢋⣌⠀⢹⣧
⠀⣿⡄⡿⠛⠛⡇⣿⣿⣦⣄⣤⣼⣿⡇⡟⠛⠻⡇⢸⡇⢸⣿⠰⡟⠛⡿⠛⣿⣿⣶⣾⣷⣿⣿⡟⠻⡿⡟⢸⣿
⠀⢹⣇⢹⠀⠀⠐⠙⣿⣿⣿⣿⣿⡿⠑⠁⠀⢀⠀⣿⠁⠘⣿⡜⢻⡞⠛⡿⠛⣿⣿⣿⣿⣿⡟⠳⡞⢻⡗⣼⡏
⠀⠈⣿⣄⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣼⡏⠀⠀⢻⣷⡘⠓⡖⠙⠒⠋⠛⠋⠓⠚⠑⠚⠑⠛⣱⣿⠁
⠀⠀⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⠀⠀⢸⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣾⣿⣿
⠀⠀⠙⠶⠿⠿⠷⣦⣤⣄⣀⣀⣤⣤⣶⠿⠿⠿⠞⠁⠀⠀⠈⠳⠾⠿⠿⢷⣶⣤⣤⣤⣤⣴⣶⡿⠿⠿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣶⣶⣤⣶⣶⡿⠿⠿⣷⣶⣦⣴⣶⣦⣄⠀⠀⠀⠀⡤⢶⣶⣦⣶⣾⣿⣿⣿⣿⣿⣶⣦⣶⣶⠦⡄
⠀⠀⢸⣷⣿⡿⣛⣵⣶⣿⣿⣿⣷⣶⣍⡻⣿⣷⣿⡄⠀⠀⢸⣷⣾⣿⡿⢋⣭⣴⣶⣶⣶⣬⣍⠻⣿⣿⣾⣿
⠀⠀⢸⣿⢏⣾⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣎⢿⣿⡀⠀⠀⢠⣿⣿⢋⣾⣿⣿⣿⡿⡿⣿⣿⣿⣷⣎⢻⣿⣧
⠀⠀⣿⡏⣾⣿⣿⣟⡵⠋⠉⠉⠳⣝⣿⣿⣿⡆⣿⣇⠀⠀⣾⣿⢃⣿⣿⣿⣫⡟⠋⠉⠛⣯⣻⣿⣿⣇⢻⣿⡆
⠀⢸⣿⠀⡿⠿⢻⢸⣷⣄⣀⣀⣴⣿⠸⠻⠿⡷⢹⣿⠀⠀⣿⣿⢸⡿⠿⡇⣿⣷⣄⣀⣴⣿⡏⡿⠿⣿⢸⣿⡇
⠀⠘⣿⡆⢇⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⢀⠇⣿⡟⠀⠀⣿⣿⡘⡇⠀⢳⡹⣿⣿⣿⣿⡿⣹⠃⢀⡟⣸⣿⠇
⠀⠀⢻⣿⡌⠂⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠊⣼⣿⠁⠀⠀⠸⣿⣷⡙⣆⠀⠙⠲⠭⠭⠵⠚⠁⢀⡞⣱⣿⡟
⠀⠀⢸⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠇⠀⠀⢰⣿⣿⣿⣮⣙⠦⢤⣄⣀⣤⠤⢞⣩⣾⣿⣿⣿
⠀⠀⠈⠳⠿⠿⠿⢿⣦⣤⣀⣠⣤⣾⠿⠿⠿⠷⠋⠀⠀⠀⠈⠳⣼⣿⡿⢿⣿⣶⣶⣶⣶⣾⣿⡿⣿⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢀⣶⡄
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣯⠟⢸⣿⣿⡟⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢸⣡⡇⠀⠀⢰⣦
⣿⣿⣿⣿⣿⣇⣤⣶⢸⣿⡟⠁⣠⣿⣿⡿⠇⣿⣿⣿⡟⢻⣿⡇⠀⠀⠀⠀⠀⠀⡟⠛⢹⡄⠀⣠⣾⡟⠁⠀⢀⣸⡏⠀⠀⣴⡦
⣿⣿⣿⣿⡿⠛⣿⠉⣸⣿⢠⢋⣨⡍⢿⣷⣶⣿⣿⡿⢷⣿⣿⡇⠀⠀⠀⠀⢀⣤⡇⢰⣾⠇⢰⣿⡵⡟⢷⡆⠈⠉⠁⠀⢤⣯
⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣦⣻⡟⣃⣾⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠻⢧⣵⡼⠃⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢇⣿⣿⡟⠛⣿⡋⢉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢾⣿⣸⠀⠀⣴⣶⠄⣶⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠈⠁⠀⠀⡏⠉⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⣤⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣄⣐⣲⣶⣶⣶⣶⣶⣶⣖⣂⣀⣤⣄⠀⣀⡀
⠀⠀⢀⠔⢀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣢⣬⣷⣾⣿⠿⠿⠿⠛⠛⢛⣛⣛⣛⣛⡛⠛⠛⠻⠿⠿⢿⣿⣾⣯⣴⣢⢄⡀
⠀⠀⠈⡗⠜⠀⢻⠐⠀⠀⠀⠀⠀⡠⢒⣵⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣮⣕⢄
⠀⠀⠀⡇⠀⠀⣼⢀⠀⠀⠀⠀⡜⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠙⠻⠿⠬⠿⠛⠋⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⢱
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⢠⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠦⣤⣀⣤⡴⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢆
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⡇⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡶⠶⠛⠋⠉⠈⠉⠛⠳⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⢘
⠀⠀⠀⣇⠀⠀⣿⢸⠀⠀⠀⠸⡉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠋⢉⡏
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⢰⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡆⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢸⢸⣿⣿⣷⣦⣤⣀⡀⠉⢹⣯⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢵⡏⠉⠀⣀⣠⣴⣾⣿⣿⡇⡇
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⠘⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣦⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⡇
⠀⠀⠀⣿⢀⠀⣿⢸⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣗⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠱⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡜
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠀⠑⢄⡀⠙⠻⢿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠋⠁⡠⠊
⠀⠀⠀⣿⠸⠀⣿⢘⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠈⠉⠛⣻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠛⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢭⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡭⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠀⠀⠀⠀⠉⢳⣄⡠⠒⠉⠉⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⢤⣀⣿⠁⠀⢰⡛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⡿⠧⢀⡤⢤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠰⠿⠛⠰⠀⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠑⠆
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⢀⣌⣀⡐⢰⣄⣰⡀⣴⣧⣾⣿⣿⣷⣴⣖⠀⠀⢀⣀⣀⡀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠀⢀⡀⠀⠀⠀⢀⣿⣿⣧⣈⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣆⡀⢶⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠻⢷⣷⣤⣼⣷⣌⢺⣿⣿⣿⣿⣿⣽⣏⠸⣿⣿⣷⣾⣿⣿⠃⣸⣷⣽⣿⣾⣿⠘⢳⣬⡐⣶⠟⡤
⠀⠀⠀⠀⠀⠀⠀⡴⠉⠠⡤⡀⠀⠀⠐⢿⢿⣿⣿⠿⠿⠇⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣼⣿⣷⣟⣃⣹⡇⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⢀⠈⡘⣉⡠⢀⣀⣤⣥⣀⣀⢄⣉⠻⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⣤⡉⢻⣿⣿⠏⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⡿⣫⣾⣿⣿⣿⣿⣿⣿⣷⣮⢻⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣆
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣷⣿⡌⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⠁⣠⠴⠂
⠀⠀⠀⢿⠀⠐⢠⣐⡄⠀⠀⠀⠀⠀⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⠟⡛⠋⢀
⠀⠀⠀⠈⠳⣄⠈⠉⠀⠀⠀⣄⡀⣚⠙⠘⣿⣿⣿⣿⣿⣯⣀⣿⣿⣿⣿⣿⠇⠛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣥⡴⠃⠀⠉
⠀⠀⠀⠀⠀⠈⠑⠒⣶⡿⢛⠟⠋⠩⠆⠀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣄⢀⠈⠉⠻⣿⣿⣿⠛⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠉⠁⢀⣤⠚⢿⣮⣝⡿⢿⣿⣿⣿⠿⣛⣵⣿⡋⠈⠀⠀⢐⠻⡿⣿⡇
⠀⠀⢄⡴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⢴⣾⡏⠛⠿⢿⣶⣶⣿⠿⠟⡿⠏⠉⢧⠠⠀⠈⠀⣰⡏⢠⠄⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠀⡀⡀⠀⢼⣏⠉⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠃⢀⠌⠉⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠴⠚⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠉⠈⠀⡀⠀⠀⢹⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⢀⠤⠀⠀⢠⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠶⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠙⢤⠀⠰⠒⡀⠆⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡠⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⡙⠀⠑⣄⣣⣥⠘⣂⣀⣐⣂⣀⣐⣀⣐⣀⡛⠦⠂⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠄⣿⣿⣿⢸⣿⣿⣭⡽⣋⢫⣭⣽⣿⡇⠀⠀⠀⡀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠠⠄⠀⠀⠀⠁⡶⠠⣿⣿⣿⢸⣿⣿⣿⢼⡋⣻⢸⣿⣿⡇⠒⠛⠁⡀⢀⡼⠴⠤⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⢸⣿⣟⣿⣾⣽⣵⣿⣻⣿⡇⠀⠈⠈⣠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡿⣋⣭⣍⡻⣿⣿⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣿⣿⣿⢸⣿⣿⢸⣿⠉⢹⣷⢸⣿⡇⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡀⠐⣿⣿⣿⢸⣿⣿⣮⡻⠿⠟⣣⣾⣿⡇⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⡠⣿⣿⣿⢸⣿⣿⣿⠿⢿⠿⣿⣿⣿⡇⢀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⡠⠊⠀⣿⣿⣿⢸⣿⣿⢡⣾⠟⢿⣎⢻⣿⡇⠀⠑⠄⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡘⢿⣤⣾⢏⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢸⣿⣿⣿⣶⣬⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⡻⣫⢦⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣮⡾⡃⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢢⡨⠯⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠠⢄⣠⠞⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣠⠞⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠐⠁⠀⠀⡠⠚⠁⢀⡴⠃⠐⡀
⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⢀⢖⠅⠲⢷⣄⡠⠊⠀⠀⠀⡁
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢨⢠⠀⠀⢀⢃⠁⠀⠀⢀⠐⠁
⠀⠀⠀⠘⠀⠀⠀⠀⡔⣄⡀⠀⠀⠓⠭⠀⠥⠊⠀⡠⠊⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⡻⣦⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠻⣚⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢀
⠀⠀⠀⠀⢰⣶⣶⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠓⠼⣀⡄
⠀⠀⠀⠀⢘⡉⠉⣁⣸⠀⠀⠀⠀⠀⠀⠀⢠⢊⡔⠛⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠸⣿⠇⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠚⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⣠⠿⢄⠀⠀⠀⣀⣠⣔⣀⡀⠀⠀⠀⢠⠒⠩⠿⠔⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⢻⣱⠊⠀⡄⠀⠈⠈⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡎⠁⣀⡼⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡎⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡠⠃⠀⠀⠈⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢸⠀⠀⡘⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠙⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠋⠀⠀⣧⠄⣀⠠⠤⠤⠒⠒⠂⠉⠉⠹⡀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠑⠒⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣭⣿⣧⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣾⣿⣾⣿⣾⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢿⣤⠤⠤⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠶⠟⠉⠻⣏⠉⠉⠉⠉⠍⠛⢛⠚⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠿⢦⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣾⡭⢷⡏⣿⡴⢛⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡛⠋⣉⣉⣉⣤⠧⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠩⠭⢹⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢞⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⢂⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣽⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣔⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⠭⠷⢿⣀⡷⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠥⣭⣩⣿⣿⣇⣸⢟⣛⣋⣉⣉⣤⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠛⢡⠜
⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣆⣸
⢻⣿⣿⣿⣿⣿⡿⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠀⣠⠿⠛
⠈⣿⣿⣿⣿⠏⠀⠐⠉⢀⣀⡀⠀⠀⠀⠀⢶⣯⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⢀⠎
⠀⠸⣟⠙⠿⠆⠀⠀⠈⠻⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⡴⠁
⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡠⠋
⠀⠀⢨⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡄⠀⢠⠔⠂⠀⠀⠀⠀⢀⠔⠊
⠀⠀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢲⣶⣶⣾⣁⣠⣠⣴⠗⠈⠀⠀⠀⠀⠀⢀⠖⠁
⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡶⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⡰⠋
⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡔⢀⣐⣦⣾⠀⠀⠀⠀⢠⠊
⢀⣼⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⠄⠀⠀⡰⠂
⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣦
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶
⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⡄
⠀⠀⠈⠉⢻⣿⡀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠈⠻⠿⠀⠃
⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠈⠻⠿⢧⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣴⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⢀⣤⣥⣀⣠⣠⡆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣠⣥⣄
⠀⠀⠀⢀⣴⣟⣋⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡛⢻⣷⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣛⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⡻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⠛⠛⠛⠉⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠛⠛⠛⢻⠇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣡⣤⠐⠒⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⢀⣀⣀⠠⠦⠶⠖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠓⠢⠤⣌⣉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠬⣉⣓⠲⠤⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠛⠉⠑⠒⠮⢭⣛⡾⠛⢯⡉⠉⠉⠓⠒⠲⠤⢤⣀⠀⣀⣠⠤⠴⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣀⣀⣠⠤⠤⠴⠖⠒⠒⠒⠛⠒⠚⠳⠦⠤⣤⣤⠤⠴⠒⣋⣉⡥⣤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢰⡇⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢔⣒⣪⠭⠴⠒⠒⠉⠉⠓⠦⣄⢸⡆
⠀⠀⠀⠀⠀⠀⢠⡟⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢻⡇
⠀⠀⠀⠀⠀⠀⢨⡗⢄⡈⠙⢒⣦⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⣴⡒⠋⢁⡠⢺⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠑⢻⣬⢧⣀⣀⣀⠈⠉⠉⠉⡽⢯⠉⠉⠉⠁⢀⣀⣀⡼⣥⡟⠊⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠧⡀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢀⢼⡇
⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠢⣼⢼⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡧⣧⠴⠚⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣽⣴⡋⠉⠉⠑⠒⠒⠒⠒⡟⢻⠒⠒⠒⠒⠚⠉⠉⢉⣦⡯⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠲⠶⠿⠿⠶⠖⠒⠒⠒⠋⠉⠉

⠀⠀⠀⣠⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⡀⡠⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⡀
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣉⡤⢤⢤⣨⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣡⢤⠤⣌⣹⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⡇⢸⣳⠟⠛⢻⣞⡆⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡾⠛⠛⣷⣻⠀⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⡇⠰⡲⣦⣀⣴⢖⠎⢰⣟⣽⣶⣾⣝⢿⣿⡇⠀⡇⡇⠀⣿⣿⢟⣽⣶⣾⣝⣷⠀⢖⢶⣄⣠⡶⡲⠀⣾⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣾⡉⠚⠿⠓⢻⣶⣿⣽⣿⣿⣿⣿⢸⣿⡇⠀⡇⡇⠀⣿⣿⢸⣿⣿⣿⣿⣽⣷⣞⠓⠻⠟⠚⣹⣾⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣷⣶⠾⠿⠿⠿⠷⣽⣟⣿⣵⣿⣿⡇⠀⡇⡇⠀⣿⣿⣷⣽⣛⣿⡵⠿⠿⠿⠿⢶⣾⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⢿⣅⣴⣶⡿⠿⣷⣶⣤⣹⢿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⢿⣡⣴⣶⡿⠿⣷⣶⣤⣽⢿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⡿⢁⣾⡟⠉⠀⠀⠀⠀⠈⠙⢿⣆⠹⣿⣿⡇⠀⡇⡇⠀⣿⣿⡿⢁⣾⡟⠉⠀⠀⠀⠀⠈⠙⣿⣆⠹⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⠃⣾⡏⠀⠀⣠⠖⠛⢦⡀⠀⠈⣿⡆⢹⣿⡇⠀⡇⡇⠀⣿⣿⠁⣾⡏⠀⠀⣠⠖⠓⢦⡀⠀⠘⣿⡆⢻⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⠀⣿⡇⠀⠀⢿⡀⠀⣠⠇⠀⠀⣿⡇⢸⣿⡇⠀⡇⡇⠀⣿⣿⠀⣿⡇⠀⠀⢧⡀⠀⣸⠇⠀⠀⣿⡇⢸⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣇⠹⣷⡄⠀⠀⠙⠛⠉⠀⠀⣰⡿⢁⣾⣿⡇⠀⡇⡇⠀⣿⣿⣆⢹⣷⡀⠀⠈⠙⠛⠉⠀⠀⣴⡿⢁⣾⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣦⡿⢿⣶⣤⣄⣀⣠⣴⣾⠿⣧⣾⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣦⡿⢿⣶⣤⣀⣀⣤⣴⣾⠿⣧⣾⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣦⣌⣉⣛⣛⣋⣉⣤⣾⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣦⣌⣉⣛⣛⣋⣉⣤⣾⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠧⠤⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠥⠤⠇⠧⠤⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠓⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠄⠀⠀⠀⠀⠈⠁⠊⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠹⣷⣵⣊⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠌⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⡀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⢟⣻⡀⠀⠀⠛⠁⠀⠀⠀⠀⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣳⡊⠀⠀⠀⠴⠂⠀⠀⠀⠀⢷⠚⠉⢘⣇⠀⠀⠀⠀⠀⠀⢠⠎⣠⠂⣀⡜⠁
⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿⣿⣽⣿⣽⣿⣷⠂⠀⠀⠀⠀⠀⠀⣸⡀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠉⠉⠉⢉⠉⠀⠒⠲⠏⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⡀⢆⠀⠀⠀⢚⡃⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠐⢰⠀⢰⡠⢄⠤⣀⢠⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠑⠢⡀⠀⠁⠀⠀⠠⠀⠂⠈⠠⠖⠀⠀⠀⠀⠈⠈⡀⠁⠀⡀⠀⠀⢀⡀⣼⠃⡀⠀⡀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠋⡟⢽⢰⡓⢹⡡⣞⠏⢾⠰⡇⠾⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠛⢿⣿⣄⠀⠀⠀⢠⣤⣴⣶⣾⣿⡿⠿⠟⠛⠋⠉⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣿⡄⠀⠀⢸⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⡇⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠖⠲⠾⠤⠄⣀⡀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠁⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⢃⣀⠠⠴⠒⠄⠀⡁⠸⠆⠀⠀⠀⢠⡀⠀⡇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠉⠀⠀⢀⢿⠟⠁⠀⡀⠀⠀⠀⠀⠈⠂⠀⠁⢉⠓⠃⢀⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠴⡋⡨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠀⠠⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡐⠠⠀⠀⠀⠘⠁⠀⠀⢠⠜⠀⢠⠁⠀⠘⡁⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠛⠒⠓⠀⠤⡀⠀⢀⠠⠀⠔⠒⣁⠉⠀⢄⣑⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⡀⠀⢀⢐⠎⠤⠉⠀⠀⠀⢱⠟⠲⠞⢋⡵⣲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡦⠀⠀⠀⠀⠀⠀⢀⣦⡀⠈⠓⠒⠒⠋⠀⠁⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⠀⠤⠠⢤⣀⣘⠀⠑⠄⡄⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⢴⠀⠁⠀⠀⠐⢢⡠⠶⠦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⡶⡄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠄⢣⠈⡦⢀⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣉⣀⣈⣨⠤⠃⢹⠀⠄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⡿⢛⡇
⠀⠀⠀⠀⠀⠀⣧⣾⠇
⠀⠀⠀⠀⣠⣾⢿⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⡟⣡⣾⡷⣶⡄⠀⠀⠀⢸⠉⠁⢡
⠀⠀⠀⠘⢧⡹⠄⢷⣸⡇⠀⠀⠀⢸⠀⢠⣼⠀⠀⠠⣄
⠀⠀⠀⠀⠀⢉⠙⢻⡍⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠒⡵
⠀⠀⠀⠀⠸⣿⣗⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⡀⠀⠀⠀⠀⢰⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⡧⠀⠀⠀⢸⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠐⠛⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠂⠀⠐⠒⠒⠒⠀⠠⠬⠭⠭⠭⠉⠉⠛⢉⣙⣛⠛⠛⠒⠺⠶⢶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⠀⠀⠀⠀⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠉⣐⣒⡒⠠⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠰⢠⠀⠖⢀⠄⠉⡁⠁⣒⠒⠠⠭⠅⢁⣒⠒⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠤⠐⡠⠄⡀⠐⢂⠀⠙⠁⡄⠈⣠⠒⢁⡠⢄⠒⠤⠀⢉⡉⠁⣒⠂⠤⠭⠭⢁⣒⠂⠴⠃⡇⡀
⠀⠀⠀⠀⢀⡠⠒⠉⡀⠀⢁⠀⠈⠠⠄⠀⠡⠀⠈⠀⠒⠈⠔⠂⠄⠈⢡⠐⠈⠀⠠⠆⠀⠤⠈⢀⠀⠀⠆⠀⠤⠀⣀⢥⣟⣉⠇
⠀⠀⠀⠀⢨⣉⠉⠒⠒⠂⠥⠴⢅⣀⡐⠀⠄⠁⠀⠀⠀⠈⠁⠀⠀⠆⢀⡄⠀⣁⠀⠊⠀⠔⠂⡡⠀⠈⣁⠤⠒⣉⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⣀⣉⠉⠑⠒⠂⠤⠤⢄⣀⡠⠂⠐⠆⢀⠀⠈⡀⠀⠃⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣉⡉⠐⠒⠒⠤⡤⠒⢉⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣆⠀⠀⠀⠀⠀⠛⠛
⠀⠀⣴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢻⣦⡀⠀⠀⢰⣶⣶⡆⢰⣶⡆⢰⣶⣶⡆⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠉⠻⠇⠀⠸⠿⠿⠇⠸⠿⠇⠸⠿⠿⠇⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠐⠰⢤⣐⢀⠀⠀⠀⣀⢀⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠓⠲⠄⠆⠀⠀⠀⠀⠀⠐⢒⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥
⠀⠀⠰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣥⢄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠠⡜⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈
⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⠀⡀⠰⠀⠀⠀⠛⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⢄⡀⠀⠀⢈⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⢠⣄⠐⠁⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠃⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠠⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠤⠀⠀⠠⠀⡀⢒⠀⡄⢠⠀⠄⠨⠀⠆⠸⠀⠂
⠀⠀⢀⠁⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⣄⠁⠄⠀⠀⠀⠀⠀⠀⠄⠀⣡
⠀⠀⢰⠀⠀⠀⢀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⣀⠀⡀⢠⠠⡌⠀⠀⡄⠰⠀⡂⠠⠀⡁⠐⠈⣥⠘⠀⠇⠈
⠀⠀⠇⠀⠉⠈⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠂⠀⠘⠁⠀⠀⠠⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢴⣢⠽⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡋⣽⡿⠿⠏⠤⠿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠀⠐⠀⠀⠀⠀⠀⠈⠻⡦
⠀⠀⠀⠀⠀⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣼⡖⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⠠⠀⠀⠀⡇⠀⠀⠾⠛⠃⠀⠀⠰⣦
⠀⠀⠀⠀⠄⡀⠋⠀⠀⢋⠉⢹⣿⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⡀⢀⡀⠰⠆⣀⡀⠘⠷
⠀⠀⠀⠀⠂⠀⠀⢄⠀⠘⢿⣿⡏⠑⠀⠐⠁⠀⠀⠀⠀⠀⠐⠂⠐⠀⠙⠁⠀⠀⠉⠁⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠙⠧⢀⡀⠀⠀⠀⠀⠀⡠⠂⣴⡿⢁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠂⠤⠀⣀⣀⣀⣤⣄⠀⠀⠀⢿⠀⣾⡿⠁⠐⢀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⢲⠠⢀⡄⠀⠀⣂⢡⣀⠠⠄⠀⠑⠀⠀⠀⠓⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠉⠁⠀⠉⠀⠐⢄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠂⠀⢀⠀⡀⠐⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠀⡐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠈⠃⠊⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠐⠤⢀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠀⠀⠀⠀⠳⣶⣶⡄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢨⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⡅⠀⠀⠁⠀⡊⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢨⠃⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠘⠇⠀⠀⠀⠂⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣻⣿⣿⡇⣿⣿⠢⡀⠀⠄⠀⠉⡀⠀⢰⣿⣿⢸⣿⡿⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠈⢸⣿⣿⣧⣿⣿⣿⡇⣿⣿⢀⠈⠢⡈⣄⣠⢀⠔⢸⣿⣿⢸⣿⣿⣧⣿⣿⣿⠐⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣻⣟⢿⣿⣧⣿⠿⠤⠤⠶⠾⠿⠿⠷⠦⠼⠿⣿⣸⣿⢟⣿⣟⢿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡸⣧⣼⢇⣿⣿⡇⢀⣴⣾⣿⣿⣶⣄⠀⣔⣕⡆⣿⣿⣿⠺⣥⡿⣸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠾⠶⣿⣿⣿⡇⣿⣿⣿⢛⢻⣿⣿⡇⠈⢩⠀⣿⣿⣿⡿⢶⠾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢱⡟⢹⡎⣿⣿⡇⢿⣿⣿⣦⣼⣿⣿⡇⠀⡘⠁⣿⣿⣿⢼⡋⣻⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣽⣯⣾⣿⣿⡇⠈⠛⠿⣿⡿⠿⠋⠀⠸⠃⠀⣿⣿⣿⣮⣽⣭⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣴⠋
⠰⣟⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡾⠁
⠀⠈⠳⣄⡈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⠁
⠀⠀⠀⠈⠛⢦⣄⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣳⣦⣙⠷⣄⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡾⣡⣞⣁⡀
⠀⠀⠀⢀⣀⣤⠤⠴⠒⣚⣛⣻⣯⣭⠭⠭⠿⠷⠿⠷⣿⠉⠙⠳⣶⠲⠶⠶⢶⡾⠛⢻⡿⠿⠭⢭⣭⣽⣛⣛⣒⠲⠤⢤⣀⣀
⣠⡶⠛⠉⣡⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⣷⠀⢰⠏⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢦⡈⠙⠳⣦
⣿⣄⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢻⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠀⢀⣼
⣿⠈⠛⠶⢦⣭⣝⣛⣲⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠶⢶⣚⣛⣭⣥⠶⠞⠋⢸
⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠿⠿⠿⠯⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠿⠿⠿⠶⠶⠒⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⣿⠀⢿⡙⠛⠲⠶⢤⣤⣤⣀⣀⣀⣀⢠⡟⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢷⡀⢀⣀⣀⣀⣠⣤⣤⠴⠶⠚⠛⢉⡿⢸
⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡟⠙⠛⠛⢿⡛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⣿⠛⠛⠛⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⣷⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸
⣿⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸
⣿⠶⣄⣀⠀⠹⡄⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⠃⢀⣀⡴⢾
⣿⠀⠀⠉⠙⠛⢿⡶⠤⣤⣾⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣧⣤⣤⠴⣶⠟⠛⠉⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢷⣀⡾⠁⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⢻⡟⠛⢻⡟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠹⣆⠀⣰⠏⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏⠀⠀⠀⠀⠀⠀⣸
⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⡤⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠒⠒⠐⢛⣀⣉⣉⡈⠉⠉⠉⠉⠉⠁⠀⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠖⠞⠚⠒⠒⠒⠒⠚⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠒⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠠⢁⠠⠡⡡⡁⡐⢀⢣⣧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⢁⢁⠂⡄⡑⠡⠀⢀⢂⡴⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠂⠂⠌⠘⠡⠡⢁⠂⣼⠂⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠌⠈⢌⠄⠀⠁⠁⠆⣼⠃⠅⢿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠆⠜⡌⢀⠂⠆⢀⠀⠌⣰⢃⡮⠠⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡐⡀⠘⢠⠋⠄⡄⠀⠌⣰⠇⡾⢁⠐⢻⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡐⡰⡐⠠⡇⠀⡐⠌⡘⢠⠏⡸⢁⢂⠂⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠐⡐⠁⡝⡐⠐⠀⡐⢠⠏⡰⢃⢂⠂⠲⢻⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⡐⢀⠛⠀⠁⠐⠔⢠⡎⣰⢃⢊⢎⢮⠡⢹⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠡⠁⠠⠣⠡⠠⠁⠀⢠⡟⣰⢣⢂⠂⢢⢁⠃⢩⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢡⠣⠁⠡⢠⠁⠠⡰⠁⡙⡠⠃⠃⢌⢢⢁⢂⢒⠉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠃⠃⠴⠡⢃⠶⠠⠁⠰⠴⠕⠌⢎⠊⠆⠀⠂⠞⠄⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡂⣰⣆⣀⢀⣉⣉⣉⣉⣀⢀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣞⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣾⣾⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⢈⠉⠪⣟⠝⢹⣬⡝⣭⣫⣝⣭⣋⢫⣝⣭⣪⡝⣭⢫⣭⣝⣫⣠⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠂⠀⠀⠀⢀⣀⣀⣀⣀⣜⣀⣠⣁⣀⣈⣀⣀⣉⣀⣒⣃⣰⣄⣱⡐⣒⣘⣆⣐⣒⣐⣄⣀⣀⣀⠐⢒⡢⠵⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠃⠀⣠⡀⣀⣀⣀⣀⣀⣇⠀⠀⣢⢐⠒⡀⢣⡐⣤⠚⢢⡀⠀⠀⠀⠀⠈⢆⠲⠈⢧⡙⡄
⠀⠀⠀⠀⠀⠀⢀⠎⡔⠀⠀⠀⠀⠀⠈⣹⠋⣉⣉⣃⡈⣨⡁⠈⠀⣉⠀⠀⢈⡀⠀⠈⡁⠀⠀⠈⣊⠢⡙⠀⡙⣬⣥⠛⣍⠀⠜⢏⠨⡠⡂⠀⢳⣜⡄
⠀⠀⠀⠀⠀⢀⠎⣈⠀⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣬⡍⠋⠁⠿⡹⢀⡀⢉⣈⡛⣁⣁⣹⡻⡀
⠀⠀⠀⠀⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠥⠍⢭⠁⠉⠈⠀⠀⠙⠬⡑⠭⣋⠉⡋⠧⣉⡋⠙⠛⠦⣳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⣉⠢⢕⠨⣙⠣⡢⢅⠂⢳
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⡢⢅⠂⢅⠨⡀⢃
⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠀⠚⠢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⣀⣌⡂⢸
⠀⠀⠘⠯⣀⣒⣖⣶⢖⣖⡰⣤⢤⢤⠤⠤⠤⢤⠤⠤⠤⠤⠤⢤⠤⠤⢤⢤⠤⡤⡤⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⢤⠤⡤⠤⠤⢤⠤⢤⢤⠤⡤⣤⣤⣤⡼⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠘⠁⠙⠋⠊⠘⠉⠛⠋⠋⠚⠋⠛⠋⠚⠙⠛⠛⠉⠛⠛⠛⠛⠛⠘⠑⠃⠛⠉⠚⠑⠛⠛⠋⠋⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣀⢀⠠⢀⠄⠠⠀⠒⠀⢐⠀⠀⠀⠠⠄⠀⠀⠀⡰⠄⡐⠀⠀⠐⢂⡒⠀⡐⠠⢀⠂⠠⣐⠀⠀⠂⠀⠐⠀⠀⠂⠀⠂⠠⢂⠁⢐⡀⠀⡢⠔⡀⢀
⠀⠀⠀⠀⠀⠁⠀⠁⠐⠈⠀⠀⠉⠐⠀⠀⠊⠀⠂⠀⠒⠁⠒⠁⠐⠊⠀⠊⠁⠂⠁⠐⠈⠀⠂⠁⠐⠈⠀⠂⠀⠔⠊⠀⠊⠁⠐⠀⠔⠊⠁⠂⠁⠐⠁⠀⠁