Cardiac Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢾⣰⣃⣿⣖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠠⣶⣾⣦⣿⣿⣿⣦⣤⡆⠀⡀⠀⡀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣐⢎⢘⠒⠴⣘⢶⣦⣀⠀⢢⣿⣵⡿⣿⡿⣯⣿⣿⣿⣿⣧⡆⢀⠁⢀⠘⠂⠼⣖⡡
⠀⠀⠀⠀⠀⠈⠑⡺⠛⠋⣹⣻⣝⣿⣦⣞⠛⢿⣷⣿⠇⣿⣿⢩⡛⣿⣶⣿⡎⣆⢈⣴⠞⣩⠅⠐
⠀⠀⠀⠀⠐⠒⡈⣇⣀⡌⣽⡌⣷⣿⣿⣿⡔⢀⣹⣿⣶⣾⣿⣿⣿⣿⣏⠍⠁⢇⡼⣠⠚⠁⠰⠂
⠀⠀⠀⠀⠀⠀⣭⣬⢹⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣠⣾⠟⣏⠁⢰⡆⠀⠠⠘
⠀⠀⠀⠀⠰⠒⠁⠟⢈⠠⠾⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⠤⠉⠒⠉⠠⠐⠂
⠀⠀⠀⠀⠀⠐⢀⡤⠶⣵⡴⠺⠛⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠳⡭⡔⡊⠁⠰
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠁⠚⠁⠀⠆⠠⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠈⠀⠀⠀⠛⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⠏⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣄⠀⠀⣟⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢻⣿⠀⠀⣿⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⢶⣶⡀⠀⢸⠀⢸⣿⡇⢰⠇⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⣿⣇⠀⢸⡆⢸⣿⣇⣼⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠄⢻⣿⡄⢸⠀⢠⣿⣿⠇⢰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠀⢻⣿⡞⠛⠠⢿⡯⠀⠘⠋⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣤⠀⠀⢱⠦⠘⠋⠀⠀⢀⣀⢤⣀⣴⣅⢶⣶⣤⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣭⢹⣿⣿⡄⢀⠞⠀⠀⢀⣠⣴⣿⣿⠿⠋⢻⠁⠜⢋⠻⣷⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⣿⣧⡎⠀⠀⠀⣿⣿⢧⣾⣿⣾⣷⡬⣿⣿⣷⣧⣬⠛⣇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⠀⣿⣿⡟⠀⠀⠀⢠⣿⢏⣿⣽⣿⣛⣿⠇⠹⠿⠿⠿⠻⠿⠷⠴⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠠⣿⣿⢱⠁⠀⠠⣿⠋⢰⣿⠿⣿⡟⠋⠀⠀⠀⠀⣠⣤⣤⣤⠄⣀⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢧⣈⢻⡟⡿⠀⠀⣽⣟⣷⣿⣇⣰⡟⠀⢠⢐⣼⣾⠛⠈⣩⠹⣝⣓⡈⢫⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠄⠀⠈⠙⢿⣇⠀⢸⣿⣿⣿⣿⣧⣻⠇⠀⣠⣿⣿⡍⣠⣶⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⡉⠀⠀⠀⠀⢸⣿⣄⣿⣿⣛⣿⣿⣼⣟⠁⢐⣷⣿⠟⢰⣿⠟⣧⣽⡿⢷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣺⣦⣀⣠⣄⣀⣠⣿⣿⣿⣿⣿⡿⢿⣧⡾⠁⠀⢸⣿⣿⡆⢸⣿⣀⡼⠋⣴⣟⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⣿⣿⡿⡟⠹⢛⣿⣿⣿⣿⡿⠋⠀⠘⠛⠁⣤⣴⣿⣿⣿⣷⢼⣿⠿⣷⡄⠛⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⣿⡿⠋⠀⠱⢁⡼⣿⣿⢏⣤⠀⢠⣄⣀⣄⣤⣤⣬⣍⣙⠻⠟⠻⣿⣧⡈⠛⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣃⠠⠀⣦⣸⣿⣿⠃⡼⣷⣄⡘⠋⣿⠿⢿⢿⣽⣿⣽⣶⣿⡄⢿⣿⣿⣟⡛⠛⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⢏⡲⣾⣿⠋⠐⣦⠄⠀⠉⠀⠀⠀⠳⢻⡟⠫⡝⢼⢏⣧⡉⠢⠙⠛⠉⡛⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠁⣭⣼⣿⣿⠟⠛⠂⠰⡥⠂⣀⣀⡀⢑⠄⢠⢄⡀⠀⠺⠁⠀⣽⣧⡀⣠⣶⣧⣴⣦⣄⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣿⡿⢛⠁⠀⢀⣠⡟⢰⣾⡿⣿⣿⣿⣿⡛⢦⡜⢀⢦⠀⢰⢾⡇⣰⡇⣾⣿⣿⣿⣿⣇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⣾⠟⣅⠛⣥⠛⣔⡿⣿⢻⡟⣦⠱⡉⣤⠒⠀⣴⠿⠟⢠⡏⠳⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⡞⠀⠐⠇⠘⢠⣤⠙⢿⣤⣿⣯⢥⣟⣷⠖⠁⣆⡠⠂⣀⡴⢃⢾⣵⣛⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⢰⡏⠀⠈⠃⠰⢸⣿⣦⣹⡡⢍⣆⠢⠺⣅⠶⠚⣩⡴⣻⡏⠰⡿⢻⣿⡛⣹⢾⣿⣏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣺⣿⣄⡀⢡⣽⣬⣿⣿⡟⠀⠈⡁⠔⢠⢞⠐⠁⠈⢳⠉⣰⡇⢀⠎⣹⣶⢫⣟⡿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⢻⣿⣟⣸⢣⣿⣿⢿⣧⠈⢀⠀⠀⡄⡘⠃⢀⠐⠏⢰⠏⡼⢛⣴⠣⣯⡛⢼⠱⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡻⣿⣢⡱⠆⡉⠘⠹⡀⢀⣴⢞⣷⣦⠎⣀⠁⠠⠏⠾⣁⡾⣯⡾⢃⡔⠠⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡟⣴⣸⣟⡇⢸⣇⢻⣿⣿⣯⡥⠀⠌⠀⡞⢻⡿⣿⣁⡿⢧⣸⠄⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡛⣿⣷⣭⣿⣿⡟⠻⠋⠀⠀⠔⠁⢨⡏⣱⠾⣏⢘⢻⣅⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣽⣻⡅⣹⢿⢀⣠⢂⣏⣠⡑⢲⣤⣉⣾⣥⠟⣲⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣟⣧⣶⡜⢁⡎⢡⣜⣁⡛⣯⣼⠧⢾⡻⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣧⣾⣟⡻⢿⣿⠁⣿⢃⣶⠜⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⢿⣍⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⠿⠿⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢀⠀⠀⠀⠀⣰⠋⡍⣻⣿⣿⣷⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠋⠀⣠⣿⣿⣿⣿⠟⠡⠉⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠛⠛⠉⢠⣤⣿⣿⣿⡟⠁⣠⣀⣼⣿⠛⠩⢿⣿⣦⠠
⠀⠀⠀⠀⠀⠀⠀⣰⣾⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣧⣀⣴⣿⡿⠁⠀⠠⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⣴⣿⣿⡿⠋⠀⠀⠀⢠⣾⣿⠟⢿⣉
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣿⣿⠿⠳⠀⠀⠘⠀⢸⣿⡇⠂⠀⣹⣿⡄
⠀⠀⠀⠀⠀⢼⡿⣿⣔⢀⠀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣽⡛⢥⡂⠰⠀⠀⣿⣿⠃⠀⣴⣿⣿⡗
⠀⠀⠀⠀⠀⣿⣿⣮⣻⣋⣾⣿⣿⣿⠷⠟⢋⣶⡿⠙⣦⡿⣄⠛⠍⠓⠀⠻⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣯⣍⠉⠉⠁⣀⣤⡶⠿⡏⠁⠀⠹⣗⠻⠂⠠⠀⢸⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⠟⢻⢿⠟⠦⣤⣿⣐⠒⠒⢹⣄⡀⠀⠈⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢀⣾⣋⡳⠦⢼⣇⣠⠌⠉⠀⡀⢀⢀⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣾⣏⣤⠀⠀⢸⡧⠂⠀⠀⠀⢱⣼⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⣿⠄⠀⠀⠠⠞⠇⠀⠀⠠⣸⡿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⠀⠀⠀⠘⠁⠀⢀⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢲⠆⢄⠀⠀⠀⠀⡦⠀⡠⠂⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⡃⠀⢣⡀⠀⢠⣇⡜⣡⠔⣋⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣷⡄⠀⣷⠀⣾⣼⡟⣰⠎⠁⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⢷⣬⣷⣤⣼⣷⠿⠿⠛⠳⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡀⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣾⠿⣷⣿⣿⠋⠀⠀⠀⣀⣴⣶⠶⠶⣷⣴⣦⡼⠞⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠁⢰⣿⣿⠃⠀⠀⠀⣴⣿⠉⠛⠛⠻⠛⠉⠁⠀⠀⠀⠀⠀⢛⣽⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠘⢿⣿⡆⠀⠀⢸⣿⣿⡀⠀⠀⠀⡠⠊⣠⡴⢾⠿⠛⠛⠿⠿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠸⣿⢿⠇⠀⠀⠐⢀⣾⠏⠀⠀⢀⣴⠶⣶⣄⣈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣸⣿⣿⠃⣀⠔⠒⠒⣥⣴⣤⠀⢠⣿⣿⣿⠶⣄⣿⣧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣰⣴⡿⠛⠻⠿⠛⠻⣤⣠⡟⠁⢀⠈⢠⣿⣿⠛⠁⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⣀⣹⠟⠋⣷⢄⠀⠀⠀⠀⠀⢨⣧⣰⡏⠀⠘⣿⡁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⡼⠁⠀⣰⣧⠀⠀⡠⠶⠤⠞⣏⠀⢹⡛⢿⡆⢸⡀⢀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⡿⢹⡀⠀⠀⠀⠀⠀⠘⢳⡄⠉⢀⡟⢸⠷⣾⣷⣤⣠⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠘⠁⠀⠙⠀⠀⠀⣐⣀⣀⠀⢁⣠⠞⣱⡿⠄⢀⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⣁⣠⠄⠈⣩⡝⣡⠞⠉⢿⣏⡛⠀⢈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠊⠉⠀⣀⠼⡽⢱⡏⠀⡠⠚⢿⠙⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣀⠀⠀⠚⠁⡜⠀⣸⡇⠀⠀⠀⠈⠃⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠈⠁⠀⠃⢹⡒⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠂⠀⠀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣤⣀⣀⣀⣦⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣻⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣛⣛⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢹⣷⡀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠤⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢻⣧⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢠⣿⢀⠀⠀⢀⣼⣅⡀⠀⠀⠠⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠘⢿⡘⠀⠀⣸⣿⡿⠏⠒⠒⠁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠄⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣠⣀⠀⠀⢀⣹⣿⣇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⢼⠿⠓⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣟⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠿⣿⣾⣷⣶⣾⡟⠛⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠄⠉⠀⠀⠨⠉⠁⠉⠉⢩⡿⠉⠉⢰⣶⣾⣻⣿⣿⠿⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⢉⣷⣶⠀⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣿⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⠟⠁⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠁⢀⢀⠀⠀⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠃⠀⠀⠀⠀⠀⠐⠀⠠⡀⠁⠀⠀⠀⣠⣼⡿⣣⡤⣴⣶⣶⣿⣿⣦⡀⠀⠀⠀⢽⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠇⠈⣀⣤⣄⣤⠀⠀⣀⠀⠘⠂⠀⠀⢸⣿⡟⣼⣯⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠀⢫⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠶⣆⣀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢀⣶⢤⣾⢿⡿⣾⣷⡆⠀⡀⠀⠀⠈⢻⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣽⠇⠀⢀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⠛⠇⠺⠀⠀⠑⠘⠛⠁⣾⣷⡀⠀⠀⠀⠹⣿⣧⠀⠉⠻⠿⣿⣿⡿⠿⠟⠀⠈⠈⠀⠠⢤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠋⠀⠀⠁⡀⠀⡀⠀⢀⡼⠿⣿⣿⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⢀⠆⠀⡠⠔⠈⠀⠀⠀⣠⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠂⣠⠸⠁⠀⠯⢶⣾⣧⠀⠀⣸⣿⣿⣷⡀⠀⠈⠀⠛⠤⠤⣔⠀⠀⠀⡀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠀⠀⠈⣿⣿⡆⠀⢿⣿⣿⣿⣷⣤⣀⠀⠀⠀⢀⣈⣱⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠀⡀⠀⣴⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣺⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⡠⢆⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣷⡿⠁⣰⡏⣿⠟⠋⠉⢀⣠⣼⠟⠛⠛⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⣓⣠⡄⢠⣾⡟⠹⠃⣀⣠⡶⠋⠁⣽⠀⠆⠀⠀⢻⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠸⣿⠁⠀⠠⠂⠀⠀⠀⠄⠀⠀⠀⢰⡄⣽⣿⣿⣿⠁⣾⣿⣧⠴⠊⢁⠞⠁⠀⠀⡿⠁⠀⠀⠦⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⣾⣿⡿⢃⣶⠙⠛⠁⢀⠔⠍⠀⠀⠀⠀⠁⠀⠀⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢁⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⡞⠿⢡⠀⡄⠀⠀⠀⡀⠆⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣇⣾⣿⢀⠂⠈⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⢀⣴⣾⣿⣿⣿⣻⣿⠁⠈⠁⠀⢀⠀⣈⢀⣤⣤⣤⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⢠⣾⣿⣿⣿⣿⣿⣿⣶⣴⣾⣶⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠀⠀⠖⠀⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⡀⢰⢀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⡎⠃⠀⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣨⣿⡇⠀⢀⣤⣤⣀⡷⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠸⡏⠀⢀⣿⡇⠀⣠⣴⠚⠻⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢰⣿⣄⣼⡟⠁⣠⣿⢿⣀⣞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡶⠟⡿⠏⠛⠓⠀⣿⣿⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡛⠋⠉⠀⠀⠀⠀⢀⣴⡿⠻⣦⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠄⢳⠤⠔⣾⠁⢱⢤⡈⢿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠔⢣⠀⢀⠇⣀⢸⠀⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢉⢰⠃⠀⠑⠌⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠢⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣔⠆⢀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⠟⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⢄⡁⠀⠀⢠⠁⢀⠔⠀⠚⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣤⠀⢘⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⣷⡤⠟⠀⠀⣀⣀⣦⡀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣼⡿⠁⠀⣼⡯⠽⠿⠛⠋⠉⠉⣉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠿⣯⡀⠀⣾⣿⡆⠀⠀⡤⠖⠺⢿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣖⣽⣗⣀⡽⠛⠀⠀⡸⠀⢀⣀⢸⡏⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠙⢿⣴⠏⠀⢰⣷⡾⠋⠈⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡇⠀⠀⠀⠜⠐⠀⠘⣿⣅⣆⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⠀⠀⠀⡠⠔⣾⠉⢢⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠈⠀⣰⡇⠀⠈⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⣰⠃⠰⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⡇⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⣶⡄⠀⠀⢀⡴⢶⡄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡾⠛⠛⠛⠛⠛⠛⠋⠀⢿⠷⡏⡴⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⣦⣄⢴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⣿⠿⠋⠀⠀⠀⠀⢀⣀⣼⣶⣴⣶⣶⣧⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠏⠀⠀⠀⢠⠤⢼⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⡤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⢂⣼⣿⡀⠀⠀⢀⠆⣰⣿⣿⠋⣠⣿⣿⠿⠿⠛⠛⠻⠿⠛⠿⠟⠿⠛⠛⠛⠉⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠛⠉⠀⠀⢹⣄⣠⠟⠺⡩⣿⣧⠾⠇⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣠⣄⣤⣤⣤⣴⣶⣿⡿⠃
⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⡽⠉⠙⢻⣁⣿⢏⡄⠀⠀⡀⢲⣴⣾⣿⣿⣿⣿⡿⢿⡿⠿⠿⠛⠟⠋⠁
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢨⣿⡏⠹⣿⣿⣿⣿⣿⣿⣿⠟⡉⣿⣾⣿⣾⣇⠀⠀⠀⠼⣶⣄
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠁⣾⡿⠁⠀⡿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣞⣿⡄⠀⠀⠀⢻⣿⣦
⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠙⢦⣄⣼⡿⠁⠀⠈⠀⢨⡿⠀⠀⠀⠈⠻⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⣾⣿⡿⡆
⠀⢠⠃⠀⠀⠀⠀⠀⢀⠀⠀⢻⣦⣨⣿⣿⠀⠀⠀⠀⡄⠀⠹⠀⠀⠀⠀⣰⣿⣿⣿⡿⣽⣿⣻⡄⠀⢴⣟⣋⣿⡇
⢠⣷⠀⠀⠀⠀⠀⠀⠸⣆⠀⢸⣿⣿⠟⠀⠑⢐⣤⡀⠈⠀⠀⠀⠀⡀⢀⣾⣿⣿⡱⡇⣿⣿⣿⡿⢷⣿⣴⣿⣿⡇
⢸⣿⣄⡀⠀⢣⡀⠐⣾⣽⣿⡿⣿⠋⠀⠀⠀⠀⣇⠈⠲⣶⠉⠉⠀⢐⣿⣿⠋⢹⡗⣧⡿⣏⠳⣵⣿⣿⣿⣿⣿⣇
⠀⣿⣿⣿⣷⣾⣿⡿⠛⠋⠸⠃⠈⡛⠀⠀⠀⠀⠈⠀⠀⠘⡃⠀⢀⣾⣷⡏⠀⣼⢰⡿⢿⡞⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠘⣿⣷⣮⣽⡛⠿⢏⠁⠀⠄⠀⠠⠀⠀⠀⠇⠀⠀⠀⢠⠉⠁⢘⢡⠁⠀⠀⣷⣿⠇⠈⢣⠀⠀⠙⣿⡻⣿⣿⣿⣿⡀
⠀⠀⢸⣿⣿⡟⢻⣿⠳⣄⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⡀⠺⠀⠀⢀⡟⡹⣧⡀⠈⠉⡆⠦⣝⢑⢻⣾⣿⣿⡇
⠀⠀⢸⣿⣿⠁⣾⠟⡀⢛⠉⠢⢰⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠁⠀⠘⣾⠁⢹⣿⡄⠀⠀⠄⠈⠀⠰⣽⣿⣿⡇
⠀⠀⢸⣿⣿⣄⠁⣷⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⡏⠀⢸⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⢿⣽⣽⡄⠼⡄⠀⠀⠀⠀⠀⢠⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⡇⠀⣿⠑⠻⣄⣠⢴⡈⢏⡹⣿⣿⠃
⠀⠀⠀⠸⣿⣿⣡⣆⢹⣦⠀⠀⠀⠀⠀⠃⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⡍⠃⠀⠈⠙⢦⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠙⣿⣿⡿⢴⡏⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠠⠈⢳⡀⠐⠀⠘⡄⣘⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣿⡀⡟⢲⠄⠈⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⡇⢀⠀⠈⣧⡀⠀⠀⡿⣞⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠁⢧⠀⢢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠈⠂⠀⢺⠻⡖⢤⠃⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢸⣧⠀⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡊⠀⠃⠀⠀⢠⣿⡀⡇⣼⢶⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢾⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠁⠀⠀⡎⢺⡄⣛⡝⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢣⠐⡃⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠃⠀⠀⠠⡀⢸⣧⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣮⠀⠄⠀⢁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠞⠀⠀⠀⢛⣦⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⡶⣤⣸⠄⠀⠀⠀⠀⠀⡆⠀⠀⠇⠀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣽⣵⡣⣀⠀⠛⠰⠳⠀⣀⠆⢐⣽⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠗⣃⣼⣏⣁⣀⣨⣤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣤⣤⣄⣀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⡴⠋⢁⣈⣹⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠙⢶⣶⡀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣴⠋⡠⠈⠁⠀⠈⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⢀⠀⠀⠀⡟⠀⠀⣞⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣴⡟⠀⠀⢰⣇⠀⠸⠟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⠀⢠⣿⣼⠀⣠⣾⠏⠉⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⢸⡏⣿⣿⠛⠉⠀⠀⢀⣀⣠⣴⣷⣦⡖⣼⣿⣆⣀⣀⣠⡴⣾⠿⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣄⢠⣾⣿⣿⠗⠀⠀⠀⣰⣿⣯⣉⣉⣻⢏⡼⠟⠛⠉⠉⠈⠀⠀⠀⠀⣸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣿⡿⠿⠆⠀⠀⠀⣠⣿⣿⡋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⣿⣿⣿⣿⣿⡿⢿⣾⢇⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠻⢛⣿⣏⣉⡙⣾⠿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣉⡰⣿⣧⣐⠀⠀⢰⣿⣿⣿⠟⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⠟⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢟⡵⢾⣿⣏⣁⣀⣨⣿⠟⠋⠀⠀⠀⠀⢠⡇⠀⠀⢀⣀⡀⠀⠀⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣾⣻⡛⣶⣿⣿⣿⣿⣿⡏⠀⣸⡴⠒⠒⠀⢀⡾⠀⣠⣴⡾⠋⠛⢆⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣷⠏⠁⠀⠀⠻⣭⣋⣾⠟⠀⠀⠀⠀⣿⣧⣾⠿⠂⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⡿⢻⡧⠀⠀⠀⠀⠀⠀⠈⡿⢄⠐⠀⠀⠀⢿⣿⣿⠁⢰⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⣴⣿⡏⠀⠀⠀⠀⠀⠀⠘⠁⠀⠙⠈⠀⠀⠈⣿⣿⣆⢸⣆⣀⢠⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡅⠀⠀⠉⠛⠛⠛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠁⠙⣦⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣯⡇⠀⠀⠘⣆⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢼⠁⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢻⠀⠀⠀⠀⠀⠃⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⠏⠀⠀⠇⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣷⣶⢿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⣷⣄⠀⠀⠀⢀⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣭⣤⣶⣶⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⠀⠂⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣶⡀⠀⠀⣶⡀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢀⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢧⠈⢿⣿⣆⠀⢸⡇⠀⠀⠀⠀⢠⠇⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡿⠏⠀⠀⠈⠻⡿⠀⢸⡇⠀⠀⠀⠀⣾⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣀⠰⣿⡦⠄⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠻⠃⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢤⣀⣭⣭⣍⠳⣾⣆⠀⣨⠀⠀⠀⡀⠀⠀⢤⣰⣤⣶⡓⠀⠦⣤⣤⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠀⠀⣀
⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣷⣘⠿⠶⢃⠀⠀⠀⢀⣴⢶⣿⣿⣿⣿⡿⠷⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠈⠀⣦⣠⡞⠙⠃
⠀⠀⠀⢀⣾⣿⣿⢻⠿⠛⢻⣿⣿⣿⠆⠘⣧⠀⠈⣿⣿⣼⣿⣿⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠛⠀⢆⣠
⠀⠀⢠⣿⣿⣟⡁⢁⠀⠀⠀⠀⠀⣠⡦⢠⣳⣤⡿⣿⡿⣿⣿⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠀⠀⠀⠀⠀⢀⡠⠶⣬⠈
⠀⠀⣸⣿⣿⣿⣻⣿⢶⡆⠀⠀⠀⣿⣷⠈⣿⣜⣷⣿⣿⣷⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣁⡴⢆⣤⣶⠂⠙⣿⡟⠁⠀⠁
⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢀⣄⣀⣨⠏⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣿⣿⣿⣴⣿⠀⠀⠈⠃
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠘⣯⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢃⣠⣼⣿⡿⣧⣉⢷⣟⣷⠀⡀⠀⠀⠀⠀⠈⣽⣿⡝⣿⣿⡟⣿⣿⣿⣟⠛⠃⢄⣀⣀⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣼⠋⢺⣳⣾⣎⢻⣷⣿⣇⣀⣀⣠⠀⠀⣀⣴⣿⣿⡟⢸⣿⣹⡿⣿⣿⣿⣷⣆⣤⣿⣿⡿
⠀⠀⢹⣿⣿⣻⣏⣿⣿⣿⣿⢻⣿⡗⠀⠻⣟⢿⡯⣸⡟⢿⠉⠀⠉⢁⠠⣺⣿⣿⡿⣫⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣱⣿⣿⣿⣿⣿⣿⠛⣶⣿⡇⠀⠠⠀⠀⢿⡙⢷⣾⣿⣦⣐⣌⣿⣦⣾⣿⣤⡿⢈⠨⡿⢿⣿⣿⣿⣿⣟⣻⣿⣿⡟⠁
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣣⢿⣿⠄⠀⠀⠀⠀⠘⠘⣷⣿⣽⠚⢿⣧⡛⢻⣿⣿⣿⣿⠸⡅⠀⠙⠿⠟⠛⠃⠙⠟⠋⠁⠰
⠀⠀⢻⣿⣿⣷⣿⡏⢱⣟⡟⢣⢿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠘⠃⠙⠀⣱⣿⣿⣿⢠⡁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣿⣿⣧⣧⣨⡿⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠛⣽⣿⣿⠀⢃⠙
⠀⠀⠸⣿⣿⣿⣿⣿⣨⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡇⡀
⠀⠀⠀⠻⣿⣿⡿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⠁⠠
⠀⠀⠀⠀⠛⣿⣿⣿⣞⠁⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣡⠇⠀⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⠔⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⠈⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣶⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣤⠿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡈⢲⣁⠄⠀⠀⠀⠀⠀⣠⠅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⢉⠁⠔⠂⠀⠀⠀⠹⠀⠀⢷⡄⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣟⣾⢾⡛⢃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣾⡉⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣧⠘⠆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⡌⣰⣅⣤⣠⣶⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⢿⠿⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⠴⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⣿⠂⠀⠀⠀⢀⣼⠇⠸⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠹⠒⢤⣄⠈⠇⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⡴⠋⠁⠀⠐⠓⢾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⣾⠃⠀⢠⡿⢡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣦⡀⠀⠀⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣰⣟⡁⠀⢠⣿⣤⣾⠃⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣿⣭⣃⣤⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡹⣦⡀⠀⠀⠀⠀⢠⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡻⠃⠀⠀⠸⢟⣾⣿⠟⠋⠀⠀⢠⣤⣴⣷⣷⣶⣿⣿⣿⣷⣤⠶⠶⠾⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡅⠀⣴⣿⠋⠉⠀⠀⢀⣾⣿⣯⣤⣬⡽⠿⠾⠟⠉⠀⠀⠀⠀⠀⠀⣀⣼⣇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡷⢸⣟⣿⣶⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⣷⠟⣿⣵⣄⠀⠀⢰⣿⣿⣿⣟⠀⠀⠀⠀⠀⡰⠛⠛⠛⠛⠻⣿⣿⣿⣭⣹⣿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⢿⠃⣀⣄⢻⡿⠟⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣿⠟⠻⠻⣿⠧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡓⢻⣧⠘⠛⢥⢾⣯⣷⣬⣅⣨⡿⢿⡇⢀⡀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣝⣻⣿⢻⣿⠿⢿⣿⣿⣥⢰⣸⡷⠚⠒⠂⢀⣾⣀⣤⣴⠞⠀⠈⠣⡀⢾
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣼⣟⡿⠛⠁⠀⠀⠉⠙⣋⣾⠋⠀⠀⠀⢠⣿⣿⣿⠿⠆⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⡾⠛⠙⣇⠀⠀⠃⠀⠀⠀⠀⡛⢄⠀⠀⠀⠈⣿⣿⣿⠈⢐⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠄⣠⠒⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⠀⠀⠀⠘⣷⣿⣆⠰⣖⣀⣴⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⣂⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡟⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣯⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⠃⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣼⣿⣦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠁⠀⠀⠆⠀⠀⠀⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠄⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣤⣆⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠗⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣤⡴⡞⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣤⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⠇⡈⠀⠀⠀⠀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣶⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢂⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠘⢧⠀⡠⠆⠀⠈⠉⠘⢲⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢸⣿⣿⣟⣯⣿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡴⠋⠀⠀⠀⡔⠁⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣷⡝⣿⣿⣿⣿⣟⣻⠿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠅⠀⣴⠋⠀⠀⠀⠀⠀⠐⣄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠾⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣿⠋⣾⣿⣿⣿⣿⣿⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⠛⠛⠙⣶⡄⠀⠀⠀⠀⠀⣠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠂⠀⠀⠀⠸⠻⣿⣀⣀⣠⣤⣼⣿⡏⠀⢈⡇⠀⣴⣶⡶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠰⠿⠟⠛⣩⠤⣿⣿⣧⣀⣾⡇⠀⣿⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢻⣷⣼⣧⣘⠁⢸⡇⠀⣰⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢿⣿⡇⠀⠉⠻⢧⠝⠀⠈⣷⣾⣯⡔⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣤⣿⡿⣿⣓⣤⠶⠖⠋⠀⠀⠀⠉⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢈⡙⣿⣿⣿⡿⠃⠰⣯⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⣹⣯⣿⣿⠟⣡⠒⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⠀⠙⢦⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⣶⡆⠀⡄⢰⡗⢈⣿⡏⢠⣷⣀⣀⠀⠀⠀⠀⢲⣤⣤⣾⣿⣿⣿⣿⣦⡄⠀⠱⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣴⣾⣿⡇⣀⠀⣤⣹⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠋⠹⣯⡿⢿⣿⣿⣿⣿⣟⠿⣦⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⠃⢀⠀⠀⠀⠈⠻⡏⠛⠀⠀⠀⠀⠖⡀⠘⣿⣿⣿⡀⠙⣿⣿⣿⣷⠞⣷⠀⣄⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠾⠛⠉⠉⠁⠀⠁⠀⠀⠀⠀⢶⢃⡀⠀⠀⠀⠀⠀⠉⣤⣹⣿⣿⣿⣦⣸⣿⣿⣿⣤⣇⣼⠿⠋⠀⢀⣀⣤⢾⡿⣦⣤⣴⣤⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠯⠡⠄⠐⠀⠐⢤⡀⠀⠀⠀⠀⠴⠶⠊⠛⣆⠀⠀⠀⣤⠀⣸⣿⠿⠟⢉⠉⠉⠻⣿⠿⠿⠿⠿⠿⠟⠛⠛⠉⢴⡟⢀⣾⣿⡏⢸⣿
⠀⠀⠀⠀⢀⣾⢋⡶⠀⠀⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠠⠄⠀⠀⠉⠀⠘⢾⣇⣋⣿⡿⠄⠚⢋⣁⣀⣈⣡⣄⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢈⡿⠿⠋⠈⣿
⠀⠀⠀⠀⣼⠇⣼⠐⠀⠂⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⣰⡿⠟⠁⠴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣤⣤
⠀⠀⠀⢰⡟⢰⣇⣠⠾⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣦⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠉⠀⠀⢸
⠀⠀⢀⣿⢁⣾⠏⡁⠐⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣤⡰⢬⣶⠟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣤⣤⣀⡶⠒⠀⠀⠀⠀⢠⣿⣋⣤⣤⣶⣴⣤⣿
⠀⠀⣼⠃⠸⠋⠀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣯⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⢋⡴⣿⠃⠀⠀⠀⠻⣤⣿⣿⡟⠛⡛⠙⠋⠉⢃
⠀⣰⠇⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣸⡿⣧⠘⢁⣿⣆⠀⣀⠈⠤⢙⣿⣿⣿⠿⠛⠛⠻⣤⡄
⢀⣿⠁⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢁⠂⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣾⣿⣧⣟⠻⠿⠇⠙⣧⡬⠶⠾⠟⠉⢻⡃
⢸⡟⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣛⣿⣧⡟⢀⠀⠃⣀⢾⣇⠀⠁⠀⠀⢸⡇
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢻⡟⢸⣿⡈⠀⠈⣿⡆⢿⡀⠀⠀⠀⢸⣧
⣿⡇⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠁⠀⣿⣧⡀⠈⠻⣷⠘⡇⠀⠀⠀⠃⣸⣆
⣿⡇⢿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⢸⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⠀⠀⣿⡽⣷⡄⠀⠀⠒⠀⠀⠀⠀⠀⢡⣿
⣿⡇⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⣴⣿⠀⣷⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣛⡛⠿⣿⣿⣿⣰⡆⣿⣧⡝⢿⣦⠀⠀⠀⠀⠀⠐⣦⠀⢹⡇
⢸⣷⠼⣿⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣏⣸⣿⠀⡿⠀⠀⠀⢤⣤⢤⣦⣤⠬⠤⠬⣿⣶⣿⣿⣉⣽⣇⣿⡛⢿⣆⡿⢷⣄⡀⠄⢀⣴⣼⣧⠸⣧
⠘⣿⡄⢻⡄⠡⣀⣠⠄⠀⢷⠀⠀⠀⣠⣾⣧⣾⣿⢃⣼⠃⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⢀⠉⢻⣿⣿⣿⡏⠿⣿⠀⠹⣧⡾⡉⠻⣿⡟⠉⠙⣿⠿⠃
⠀⠸⣷⡸⣿⡄⠈⠁⠀⠀⢨⣷⣤⣽⣿⣿⠿⣿⡿⠁⠀⠘⢠⣤⣶⠶⠿⠏⢭⣶⠲⢤⣤⣤⣾⠛⢛⡛⢿⡇⣾⢃⡆⠀⠈⠻⢧⣤⡄⠀⣀⣾⡇
⠀⠀⢻⣷⣿⣿⣿⢮⣐⣠⣾⣿⣿⠏⠀⣠⡜⢋⣼⣧⢆⡀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡴⠶⠟⢻⡿⠛⣿⣿⡇⢹⡏⣉⣀⠀⠀⠀⠈⠁⠀⢩⢿⡇
⠀⠀⠀⢻⣿⡟⢿⣾⣭⣿⠟⢉⡽⠃⣼⣏⣼⢏⠐⠀⠀⡁⠜⠋⠀⠀⠀⠀⣼⣱⠟⠀⠀⣸⢇⣴⠞⠛⣿⣇⠘⣷⡊⠘⠆⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⢿⣇⣠⣼⣯⡴⢆⡻⢶⡄⢸⡟⣧⡜⠁⡈⢦⡉⠀⢠⡁⠀⠀⡜⣷⠏⠀⠀⢠⣻⡞⠁⠀⠀⣿⢿⠀⠘⠻⠷⢦⣄⠀⠀⠀⠂⠀⠻⣿
⠀⠀⠀⠀⠸⣿⡋⢸⣏⡴⢿⣄⠀⣷⠸⣿⠨⠀⠆⠘⠆⠘⠉⠃⠀⠀⣼⢻⡟⠀⠀⠀⣎⠟⠀⠀⠀⠀⡇⣿⠀⣾⣿⣶⣶⣬⣛⠠⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣿⠾⠋⡠⠘⢻⣆⠘⣇⢻⡄⠀⠀⠀⣠⣀⣀⣀⣀⠴⠟⢸⠃⠀⠀⢀⣿⠀⠀⠀⠀⠀⡇⢻⢰⣿⣿⣿⣟⣯⠙⣷⠀⡴⢶⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠻⣷⣶⠀⠀⠸⠿⢷⣼⣾⢃⠀⢸⡄⠙⠛⠻⠭⠤⠐⣶⡇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⡇⢸⢸⣯⣀⣴⣿⡈⠀⢸⡇⡆⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠰⢣⡀⠀⠙⠻⣏⠲⡈⢣⡄⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⣸⠀⣿⠟⠛⣁⠀⠀⠘⣇⡇⢀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠮⠀⠠⠇⠀⢸⡆⢱⠄⠉⠈⠛⢷⣆⢀⣹⢻⡀⠀⠀⠸⢇⠀⠀⠀⠀⢀⠃⢻⠀⣿⠄⠙⠀⢐⠀⠀⣽⡇⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⡄⢸⡇⠀⣶⠁⠜⢳⡎⠽⣿⣿⢸⡇⠀⢀⣠⣜⣄⠀⠀⠀⣼⢀⡍⠀⢿⡄⠀⠠⡌⠀⠀⠉⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⡀⢠⣿⡷⢻⡿⣧⠀⠀⠐⠦⣹⡟⢠⣷⢠⡾⠃⠈⣧⠀⠀⠀⠟⡸⠇⢠⡌⢻⣆⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡙⣦⣿⠀⠀⢻⡘⢦⡀⠉⠀⣿⣇⣹⣿⠈⠁⠀⠀⠿⠀⠀⢸⣇⢷⠀⡿⢳⠈⣿⡀⠀⣠⠁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⢠⢀⠀⠹⢦⣿⡄⠀⠛⠋⢹⣿⡦⢀⣤⠀⠀⠘⣰⣿⡿⢃⡾⠁⠀⠇⣸⡇⡰⢇⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡉⠀⢶⡀⠙⣿⠀⠀⠀⠀⢿⡇⠰⣿⠀⣠⢄⣿⠻⢁⡞⠁⠀⠀⡸⠟⣿⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣎⡙⣦⡈⠀⠀⠀⠀⠀⢈⣁⠳⣞⠷⠋⠀⣦⣿⠀⣤⠀⣠⡿⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣏⣠⣀⠀⠀⠀⢁⣴⡿⢿⣿⡟⢶⣸⠇⢠⡌⢠⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⣤⣿⣉⣰⣿⣿⡟⣼⣇⣸⣿⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣾⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠈⣿⣇⠀⠀⣿⠋⠀⡠⣛⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠳⣿⣟⡄⠀⡿⢠⣎⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⣟⢄⣏⡿⠶⠤⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⣿⣿⡿⠉⣴⣾⣿⣿⣿⣧⣠⠤⡾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣶⢦⣾⡟⠁⣼⣿⢧⠼⠛⠋⠉⢀⣁⣠⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⠀⣿⣿⡷⠀⢀⣴⢚⣿⣿⣿⡿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢪⡂⣟⡆⣀⣹⠟⡃⡀⣹⣇⢉⣾⡗⢸⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣷⣳⠿⠛⢟⣡⡞⠍⣹⣿⣿⣟⠁⠙⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣿⣿⣷⢿⢁⡄⠀⠀⢸⠳⠀⠿⢿⣯⢹⣥⣾⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⣿⣿⣳⡌⢀⠗⠀⠀⠀⠀⠀⠀⢸⡏⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣾⢻⣿⢿⡇⠈⢀⠀⠀⠀⠈⠁⢁⠞⠀⡀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣰⣿⠇⠀⠻⣷⣿⣆⡿⠀⠀⠀⠀⠁⢠⡇⣀⠈⡀⢺⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠙⢿⣿⣿⣶⣍⠀⠀⢀⠞⠁⢿⣷⣤⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣤⡏⢀⠄⢈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⢔⠛⠁⠋⡑⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣀⣿⢦⠼⢧⣀
⠀⠀⠀⠀⠀⠀⠸⠷⠟⠛⠷⢿⣵⣦⠐⣟⠂⠀⡀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣠⠞⠋⢀⣀⡤⣤⣬⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣽⣳⠀⠡⠀⠀⠀⣸⡃⢀⠀⣠⣾⠁⢀⠐⠉⠁⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢷⡄⠁⠀⠀⣿⠀⠀⣰⡿⠃⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⢠⣾⠿⠀⠀⢰⣟⠀⢠⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⠀⠀⠀⣿⠧⣿⡄⢀⣼⣿⠷⠛⠟⠁⠀⠈⠉⠱⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⢸⣿⢼⣭⡿⣿⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣠⣶⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀⠀⠀⠀⠙⣧⣿⣿⠗⠀⠀⠀⠀⠀⣠⣦⣼⣾⣿⣯⣿⣻⣿⣿⣧⣤⣤⣶⣶⣾⣿⣿⡶⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣥⣾⣿⣆⠀⣰⣿⣿⣿⠛⠀⠀⠀⢀⣄⣶⣿⣁⣈⣉⣿⣿⢿⡿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣾⡿⢠⣾⣿⣿⣿⡅⠀⠀⠀⣼⣿⣇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣧⣞⡦
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⠲⠀⠀⠀⣸⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⢀⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⡆
⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠉⠉⠋⢀⣼⣿⣿⠟⠁⠀⠀⢹⣿⣿⣯⢿⠀⠀⠀⠀⠀⢀⡶⠛⠉⠙⠛⠛⠟⡻⣿⣿⣦⢤⡙⣿⠛⠿
⠀⠀⢠⣿⣿⣿⣿⠉⠀⠀⠀⠠⠖⠀⣠⣿⣷⣦⠄⠀⠀⢸⣿⣿⠛⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡑⢻⡟⠙⢧⢹⣜⣧
⠀⠀⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠺⠋⣸⣿⣿⣍⣒⣠⣴⡾⠟⠻⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⢄⠀⠀⢀⢺⡁⠀⠀⠀⠉
⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⣤⣶⠋⣹⣿⡿⠿⢿⡿⣋⣩⡤⢠⣴⠖⠀⠂⠀⠀⣰⠏⢀⣠⣴⡾⠋⠈⠻⣀⠸⢸⠃
⠀⢸⣿⣿⣿⣿⣏⠻⢿⣆⣋⣽⠟⠋⠀⠀⠀⠈⠣⣝⢋⣰⣿⠃⠀⠀⠀⠀⣰⣿⣷⡿⢟⣭⡄⠀⠀⠀⠘⣷⣻⠁
⠀⠈⣿⣿⣿⣙⣿⣶⣾⡟⠋⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠀⠀⠀⠀⠀⣿⢿⣿⣝⠖⡡⠄⠀⠀⠀⠀⢉⡇
⠀⠀⢻⣿⣽⡿⣯⠉⢸⣀⡶⡟⠁⠀⢼⠂⠀⠀⠀⠀⢠⠇⠈⠳⠟⠤⠀⠀⠻⡌⣿⣇⠀⣿⠆⠀⠀⢀⣀⢸⡧
⠀⠀⢸⣟⣿⡷⣇⠝⢠⡿⢸⡇⠀⠐⡻⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢱⡌⣿⣷⠸⣀⣀⣦⣼⣿⣿⠇
⠀⠀⠀⢻⣿⣿⡟⣻⠘⣷⠊⠳⠀⢰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠁⠙⢷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢻⡟⣧⠘⡩⣿⣀⠗⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣸⡋⠀⠀⠀⠀⠀⠀⠀⢩⣿
⠀⠀⠀⠘⣿⣷⣸⡅⢽⣿⡤⠴⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣰⡵⠁⠙⣆⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠘⣿⣧⣾⣞⣿⣱⡜⠏⡡⠀⠀⠀⠀⠀⠀⠉⢀⣠⡴⠋⠉⢉⢏⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣼⣿⣤⡄⠀⠀⠀⠀⠀⠒⠊⢰⠏⠁⠀⠀⠀⣼⡄⠀⠀⠀⠀⢘⠆⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠜⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⡆⠀⠀⠀⠀⠀⠘⠀⢠⢶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠠⣶⡄⠀⠀⠀⠀⢀⠮⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠠⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⢹⡄⢠⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⣿⣷⠶⡾⢻⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⠧⣦⡿⣋⠀⠀⠀⠀⠀⠀⠀⣀⢲⢤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡉⣼⢢⠿⠛⠦⢤⡀⠀⠀⣄⣬⣿⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣯⣿⣗⣶⣶⣶⣶⣾⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢿⠀⠀⠀⠀⢠⣶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡀⢈⣘⠛⢀⡀⢃⠀⠀⣰⡿⠏⠀⡿⠁⠀⢀⣄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⠛⠛⠛⢿⣷⣿⡃⠈⠀⢠⡿⠃⠀⠠⠁⠀⢀⣼⣿⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⠀⠁⣼⠃⠀⠀⠀⢀⣴⣿⠛⠇⠉⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢠⣾⣿⠀⢸⣃⡀⡀⣀⢠⡿⣷⡄⠀⡠⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡠⠀⠁⠀⢠⣿⣏⣿⡶⠛⠛⠛⠻⠟⠻⠳⠶⢤⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠂⠀⠀⠀⠀⣠⣿⠟⠁⠐⠀⠀⠀⠀⠀⠀⠀⠂⠚⣿⠿⣦⡀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣵⡞⠉⣶⣾⡟⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⠀⠀⢹⡀⢀⠠⠊⡄⡈⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠋⢀⣠⣿⠁⠀⠀⠀⠀⠀⣰⣾⡿⣛⣛⠭⠭⠿⠧⠤⠐⠂⠒⠉⠀⠀⠀⠀⠀⣾⡿⣗
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠫⠟⣿⣥⣄⣸⣷⠀⠀⠀⠀⣶⣿⠻⠓⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡟⢡⠂
⠀⠀⠀⠀⠀⢀⡊⠁⠀⠀⠀⠉⠋⠈⣹⡇⠀⠀⠀⢠⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠓⠚⠿⠺⠟⣉⠄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠅⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⣀⠀⠐⠄⠠⡀⢈
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢠⠂⣀⡀⠀⠀⢇⠀⢁⠀⢸⠀⠹⡄
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣦⡞⠋⠀⠀⠀⠀⡿⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⣤⠃⣸⠀⠘⢄⠀⠈⡆⠀⠉⠀⠂
⠀⠀⠀⠀⢻⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠑⡄⡇
⠀⠀⠀⠀⠈⠀⢀⣴⣾⡟⠹⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢆⠁⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣴⣿⡟⠙⠣⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣄⠀⢀⡹⠋⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⡟⠁⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠆⠀⠀⡀⠀⢳⡄⠒⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠘⢿⣶⣶⣤⣴⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⠰⠂⠂⠚⠉⠁⠀⠼⠀⠀⠀⠀⠀⠀⢫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⣠⢟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⡍⠀⠀⢈⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢠⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⠂⢰⠀⠀⠀⠁⠄⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡠⠂⢀⠂⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⡀⠀⣀⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣤⣀⣀⡛⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⠁⠉⠛⣿⣿⣇
⠀⠀⠀⠀⢿⣿⣿⠿⠿⠛⠋⠉⠁⠀⠙⠻⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣉⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠉⠉⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣴⣿⢁⣴⠚⣵⣋⠴⢒⡶⠓⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢱⣧⠞⢁⡾⠋⣡⡾⠿⢚⣫⠽⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠀⢸⠁⡼⠋⠀⠈⠀⢊⣡⣶⣛⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠇⠀⠁⠀⠀⠀⠀⢉⣩⠽⠟⠛⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⣠⠊⠁⠀⣠⠆⠀⠀⠂⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣹⡄⠀⠀⠀⢠⠀⠈⢢⡀⢀⡤⠦⢄⡀⠀⠀⢀⠤⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣶⣜⣓⠄⠁⠀⠀⠀⠋⠐⠒⢄⠑⣾⠀⠀⠀⠈⠢⡔⠁⠀⠀⠘⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠟⣛⡃⠀⠀⠁⠀⠀⠀⠙⢅⠈⢧⠀⠀⠀⠀⠤⠘⠟⠀⢸⠇⠀⠁⠀⠀⠁⣀⣀⣠⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⣉⡭⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠠⢄⠀⠀⡠⢄⡀⣠⣿⡤⠤⠴⠒⠊⠉⠉⠀⣠⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣊⣥⡴⢊⣠⠄⠀⢀⡀⠀⠀⢀⡤⠄⣶⠃⠀⠀⠀⠀⠁⠀⠀⢤⣄⣀⣀⣀⣀⣀⣤⠴⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢡⣿⠖⡿⣃⡴⢞⡿⠁⠀⠠⡏⠰⠒⢋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋⠀⣼⠟⠉⠀⣼⢡⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⡀⡴⠶⠦⣤⣤⡄⠀⣰⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠁⡠⠄⠀⣇⣿⡀⣦⠀⠀⡀⠀⠀⠀⠀⠀⡜⢰⠦⠎⢠⠃⠀⢀⡞⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⣇⡏⢣⡀⠈⠢⠀⠀⠀⠀⠥⠶⠴⠠⡇⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠂⠀⠤⢀⠃⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢨⠃⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢠⡏⠀⠐⠒⠤⢼⡆⠀⣶⢾⡥⠤⢤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣀⡀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⣸⠸⡉⣧⣠⡤⠤⠤⣤⠉⠀⢹⡦⣄⡀⠾⣭⠭⠭⠭⠭⠭⠭⠭⣉⢑⠮⣷⠶⠿⠇⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢺⠶⢋⡍⣐⡣⢄⠀⠀⠀⢰⠃⠀⢀⣸⠇⠀⠉⠛⠁⠄⢀⡄⠀⡀⠀⠀⠍⠉⠱⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢋⡼⠀⠘⠚⠀⠀⠀⡏⠀⠀⠀⢛⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣶⠂⠈⠁⠀⢘⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡅⠸⣯⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣹⠆⣘⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣰⡿⠀⠠⣞⠀⢽⡃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⣤⣼⠷⠶⢾⣋⣉⡷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠛⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠆⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠰⠖⠖⠒⠋⠉⠉⠉⣉⢹⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠤⠤⠤⠤⠄⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠘⠛⠀⠀⠀⠀⠸⣿⢸⠀⠀⢘⣿⣿⣶⣶⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣬⣭⣥⣤⡦⡴⠶⠷⠖⠀⠀⠀⠀⢸⠀⠀⠘⠛⠉⠉⠉⠉⢠⡀⠟⣵⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠠⣤⣤⣬⠁⢀⣀⣀⣀⣀⠈⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⢹⣿⠇⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣿⣿⠋⣰⣿⣿⣿⣿⣿⡇⢸⡀⢸⠂⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢻⣷⣄⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⡿⠁⢸⣿⠉⠉⠿⢿⣿⡇⠈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⣧⡟⠙⠃⠀⠀⣼⣿⢏⢿⣷⡄⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⢀⣀⡤⠤⢾⣯⣤⣀⣀⡀⠀⢀⣠⠞⢇⠀⠘⣿⠀⠀⢀⣾⣿⣿⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣀⣀⣀⣼⣿⡏⠀⠀⢻⣿⣦⣿⠿⠟⢻
⠀⢀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠁⠀⢸⠀⠀⠉⣁⣀⣈⣉⣅⣤⣤⣶⣼⣧⣤⣤⣶⣶⣶⣶⣿⣿⣿⡿⠿⠿⢿⠟⠛⠿⠔⠀⠀⠀⠹⣿⡟⠈⠀⢸
⢠⣿⣿⣷⣦⣤⣀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣿⡿⠿⢿⣿⣛⣛⡻⠯⠭⠭⢿⣽⡿⠿⠯⠉⠙⠛⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡙⠀⠀⠀⢸
⢸⡏⠉⠛⠻⢿⣿⣧⣀⣚⣛⠿⠭⠭⠽⠛⠒⠒⠓⠘⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣉⠛⣷⠀⢈⣀⣠⠤⠞
⣿⣇⡲⣦⣤⣇⡀⠀⠀⠇⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⠊⢭⠁⢹⣏⣽⣷⣶
⠸⣮⣉⠒⠏⢷⡍⠉⠈⠀⠀⣴⠟⠋⠉⠉⠉⠙⠻⡷⣄⣧⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡇⠀⢀⣋⣀⣤⠤⠴⠖⠒⠛⠉⠹⢦⣾⣿⣿⣰⣋⡾⠛⠛⠋⠉
⠀⠙⢿⣟⡶⢶⣿⣀⣀⣀⡾⠁⠀⡰⠚⢽⣽⡢⠀⠀⠈⠯⠤⠔⠒⠒⠛⠋⠉⣙⣻⣿⡟⢋⣉⣩⣤⣶⣶⣿⣿⠿⠿⠛⠛⠁⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠙⢿⣷⣦⣭⣥⡼⠃⣄⠀⣄⠀⢈⡍⠇⣤⡴⠒⢶⣶⣒⣒⣺⣭⣭⣽⣿⣷⡶⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠲⢮⣭⣉⣉⡀⠠⣿⣿⣿⣷⡮⣀⣼⣿⢃⣾⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠞⠻⣿⣿⣿⡿⠛⠁⠉⠉
⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⠛⠒⠶⣤⣤⡀⠀⠀⠀⠀⠀⣸⣤⣀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⣤⣄⠀⠀⢻⡈⢷⡀⠀⠀⠀⢸⡁⠀⠘⣆
⠀⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢷⠘⢧⠀⠀⢳⣈⡷⠖⠛⢲⡄⢳⡀⠀⠘⣆
⠀⠀⠀⢸⠀⠀⠀⢸⠇⠀⠀⠀⢰⡆⠀⠈⢧⣸⡧⠖⠋⠁⠀⠀⠀⠸⡇⠀⢳⡀⠀⠘⣦⣤⣀⣀
⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⠀⢸⡇⢠⠖⠋⠁⠀⠀⠀⠀⠀⢠⡖⠚⢻⡄⠀⢷⠀⠀⠸⡆⠙⣏⠙⣦
⠀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⢀⣀⣤⠼⠁⠀⠀⣷⠀⢸⡆⠀⠀⢻⡀⠘⢧⠈⢧⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⡤⢤⣄⡿⠀⠀⣧⠀⠀⠀⢧⠀⠈⢧⠈⢷
⠀⠀⠀⢸⠀⠰⡆⠀⠀⠀⠙⠶⣤⣀⣀⠀⡀⣙⣦⡀⠀⠀⠀⣿⠀⠀⠙⣧⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⢸⠀⠀⡏⢦⡀⠀⢀⣴⡆⠈⣉⠉⠉⠉⠀⠙⠓⠶⠶⡞⢧⡀⠀⠈⠳⣄⣈⣧⠄⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⣄⠀⢿⠀⠳⡴⠋⠀⣧⣰⠋⢷⣀⣠⠴⣶⠀⠀⠀⠹⡄⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠉⢻⡆⠀⢀⣄⠀⢸⡃⠀⠀⠉⠀⢀⡟⠛⠉⠉⠉⠙⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠸⡟⠛⠛⣇⡴⠋⠈⢧⡘⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠹⡄⠀⠙⣶⣤⣄⡀⠀⠀⠀⠀⠀⣀⣸⡧⠶⣄
⠀⠀⠀⠀⠹⣆⠀⠉⠀⠀⠀⠀⠳⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡆⠀⢹⣄⡤⠴⠚⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠓⠲⠦⢤⣤⣀⣀⠀⠀⠀⠈⢳⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⢾⡁⠀⠙⢦⣄⣀⣀⣀⣠⠞⠃⠀⢀⡞⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠉⠉⠉⠀⠀⣀⡴⠋⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋


⣇⣸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⣇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⢀⣀⠀⡀⢀⣀⠀⡀⠀⢀⡀⠀⢀⡀⢀⣀⢀⣀⠀⢀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⡅⠀⠀⡇⡰⢡⢸⡗⡇⡇⠀⡗⠂⠀⠑⢤⠀⡇⢸⠤⡁⡁⠀⡇⠶⡁⢰
⣶⣶⠶⠶⠶⢶⣶⣾⣷⣾⣿⣶⣶⣶⣷⣷⣶⣾⣿⣷⣿⣷⣿⣷⣶⣿⣷⣶⣷⣾⣶⣷⣾⣿⣶⣷⣾⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢰⡟⣧⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠀⣾⠀⢹⡄⠀⠀⢀⡴⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⠰⡇⠀⠈⣷⢀⡴⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⡦⣤⣀⣀⠈⢿⣿⣿⣿⣿⣷⠀⠀⠀⠹⠏⠀⠀⢰⡇⠀⢀⣀⣠⣤⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠉⠛⠈⢻⣿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠛⠋⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⢦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠐⠶⠿⣿⣍⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣩⣽⡿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣀⣤⡤⠶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⢦⣤⣀⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠟⠛⠋⠉⠀⠀⢠⡟⠀⠀⠀⣾⡀⠀⠀⣴⠂⡀⠀⠘⣇⠀⠀⠈⠉⠛⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣠⡾⠁⣿⣇⠀⢠⣿⠀⠻⣦⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡾⠋⠀⠀⣿⣿⡀⣼⣿⠀⠀⠈⠛⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⣿⣿⣧⣿⣿⠀⣴⣿⣿⣷⣆⠀⣠⣤⣤⣤⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣷⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⣭⣯⠈⣩⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣬⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣡⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⡿⢋⡴⢆⡀⣶⣶⣶⣶⡆⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣄⠲⣿⣷⣌⣛⣛⣉⣴⣌⠻⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣇⢸⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣧⣤⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠙⡶⠊⣀⡤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣶⣿⣷⡟⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠉⠘⢶⣟⣺⣿⢦⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⢉⣿⠞⣹⠮⠿⠟⠂⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⣷⠞⢉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠖⠻⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠌⣿⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠻⠿⣯⣟⢻⣿⣾⣿⣤⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠘⠛⠽⣿⢿⣽⣗⣖⣠⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣰⣧⣤⣿⣥⣤⣭⣭⣋⣉⣀⣀⣰⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣝⣛⣩⣛⣻⣟⣽⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡾⠻⢿⣤⡄⠀⠀⠀⢀⡼⠛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣅⠀⣠⡯⢤⣀⠀⠠⣞⡀⠺⣥⡀⠀⢀⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣼⠁⠀⠀⠹⡆⠀⢈⣿⡴⠋⠀⣴⠋⢠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡼⠃⠀⠚⠋⠀⠀⠀⣨⣗⣤⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠬⠥⠤⠤⣤⡀⠀⠀⠀⠘⠛⠉⠀⠀⣾⠛⠛⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⢉⣤⠀⠀⠀⠀⣤⢠⣄⡙⠳⢦⡀⠀⠀⠀⢸⠋⣭⡍⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠠⣞⠋⢸⠀⠀⠀⠀⣶⢸⡏⠙⡶⠀⡿⠀⠀⠀⢸⠀⡃⣰⠋⣹⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣹⡄⢸⠀⠀⠀⠀⠿⢸⡇⣸⣃⣸⠃⠀⠀⠀⢸⠀⢱⡏⠐⠿⣤⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣭⠇⢸⣤⠴⠶⠶⠶⢾⣇⠿⡽⠟⣦⠀⠀⠀⢸⠀⠈⠉⣗⡴⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠁⠀⣠⣤⣀⡀⠉⠙⠳⣆⢸⡇⠀⠀⢸⠀⠀⠸⠟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⢀⡼⠃⠀⠈⠉⢳⠆⠀⣿⠘⡇⠀⠀⠘⢳⡶⢶⠖⢲⡖⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠛⢦⡀⠀⠀⠀⣸⣀⣀⡏⠀⠙⠦⣤⠴⠞⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠷⣄⠀⠙⢶⣄⠀⣿⠉⣹⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡞⢉⡧⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡾⡧⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡷⣥⡀
⠀⠙⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣽⣯⡧⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡷⠷⢚⠆⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠐⠀⠀⠀⣀⠀⠀⠀⣴⣖⣉⡁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⡿⠉⠀⠠⠤⠴⠦⠄⠀⠗⡆⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⢛⣻⣿⠒⠒⠒⠒⠶⠒⠂⠈⢷⡥⢶⠛
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡤⠄⠀⠀⢀⠀⠀⠀⡼⠓⢘⡆⠀⠀⠀⣶⣾⠄⠼⢾⡗⢀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠨⠇⠀⠀⠈⢉⠀⠀⠰⠛⠃⡠⠷⠀⠀⠀⣟⠻⠀⡈⢿⡅⠸⠁⣇
⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘⢶⡀⠀⠀⢀⡤⢾⠭⣧⢀⠀⣂⣀⡀⡰⠛⢠⣤⣼⣉⢛⣿⢻⠄⠀⢀⡇⠀⣧⠁⡞⣇⠐⡟⣿⠒
⠀⠀⠀⡇⠀⣼⣠⣸⣧⣷⣶⣶⣿⢶⠶⠿⠿⢷⣛⣛⣛⡙⠯⠭⢿⣷⢰⣾⠿⠻⣿⡿⣿⡟⣷⣾⣿⣴⣦⣻⣶⣯⣿⣅⣿⣿⣷⡆
⠀⠀⣠⣧⣸⡭⣟⣹⣿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠓⠦⠬⢭⣍⣀⣚⠛⠾⠿⢿⣷⣿⣿⣿⣿⠍⣿⣿⣿⣷⡽⠿⣷⠃
⠀⢠⠤⣗⣨⠶⣫⠽⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢶⣤⣤⣥⣉⡛⠺⠿⠿⣿⣿⣷⣶⣾
⠀⠀⢘⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⣷⣦⣤⣄⡉⠁
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⠴⠉⠎⠃⠦⠍⠦⠤⠔⠉⠀⠐⠈⠑⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣺⣭⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠭⠍⠭⠭⠭⠭⠭⠍⠩⠍⠹⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⡴⠁⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⡐⢠⢡⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢈⢈⠀⠀⠀⣉⢈⠀⠀⠠⠤⠀⠈⠆
⠀⠀⠀⠀⠀⡐⢀⠇⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠠⢀⣀⣀⣀⣀⣀⣀⣠⠀⠐⢅⣓⠔⠀⠰
⠀⠀⠀⢀⠴⠠⠌⠼⠿⡴⠿⠿⠿⠿⠿⠿⠤⠀⠻⠛⠛⠟⠿⠇⠠⠠⠠⠤⠤⡤⢋⡙⡌⠠⠀⠀⠀⠀⠀⠀⡤⠄
⠀⠀⢀⣁⡀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣠⡄⠀⠀⡀⣀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠢⡘⢃⠜⠀⠀⠀⠀⠠⡤⠠⡇
⠀⠀⢸⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠓⠐⡇⠐
⠀⠀⠸⢩⡀⠀⠀⠀⢀⠀⣀⣤⣄⣐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡏⠋⠙⠉⠉⠉⢉⠙⢋⣉⡛⠉⡉⠉⠙⠉⢉⣩
⠀⢀⣰⣒⣃⣀⠀⠀⢠⣾⡿⠿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣡⣾⠿⠿⢿⣷⡌⠀⠀⠂⠈⣚⡀
⠀⠨⠀⠀⠀⢈⡇⢢⢸⣿⢀⡌⣀⢸⣿⡇⡗⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣆⣿⣏⢠⣁⡄⣿⣿⣐⠒⠲⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⣾⣶⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣶⣷⣾⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡥⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⣩⡼⠷⡄⠀⠀⠀⠀⠀⠀⢠⢞⡭⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡞⢡⠘⣿⡆
⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣼⠛⣉⣥⠴⠾⡇⠀⠀⠀⠀⠀⠀⡏⡞⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠹⣌⠆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣋⠉⣤⠶⡟⠁⠀⠀⠀⠀⠀⠀⣧⢣⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡧⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣤⣿⣤⣧⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣯⣼⣤⣿⡇
⠀⠀⠀⠀⠀⠈⣳⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿

⠀⠀⠀⠀⠀⣀⢀⣤⢠⣄⢀⡀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣶⣤⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣞⠛⣶⡀
⠀⠀⠀⢀⣘⠋⢾⡏⠉⡽⠎⢛⣀⠀⠀⠀⠀⠀⠙⠦⣀⣬⣬⡃⣤⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡿⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡙⠿⠾⠏⡁
⠀⠀⠀⠾⠿⠸⡿⠀⠀⢛⡿⠸⠿⠆⠀⠀⠀⠀⠀⢠⣿⣿⡸⢞⣵⢟⣀⠀⠀⠀⠀⠀⣧⣿⣿⡇⠀⠘⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁
⠀⠀⠀⢶⣶⠰⣿⡶⢠⣾⡷⢰⣶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣧⣌⠡⠿⡋⠀⠀⠀⠀⠀⠹⣿⣿⣷⡗⣰⣿⣿⠇⠀⠀⠀⠀⠀⣠⠀⣾⣮⣩⣶⡆⢠⡀
⠀⠀⠀⠈⢩⣄⢶⣷⢻⣶⢆⣬⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢻⡂⢯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠇⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠻⢦⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⣿⣿⣷⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⢸⠟⢉⣉⠛⣿⠀⢠⢀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄
⠀⢀⡄⣿⣿⣿⣿⢰⣀⣠⣶⢾⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣴⣴⠀⠀⠀⠀⠀⣿⣿⡈⠀⢁⣿⣿⠀⠀⠀⠀⠀⡞⠀⢠⡀⢙⣶⣄⠀⠀⠸⡆
⠀⠘⠇⣿⣿⣿⣿⢸⣿⠶⠶⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣾⣷⣿⠀⢿⢿⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⠀⠀⢀⡇
⠀⠀⠀⣿⣿⣿⣿⠸⣿⠛⢻⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠿⠟⠀⢸⢸⠄⠀⠀⠀⠀⣿⣿⣈⣠⣀⣿⣿⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⡇⢀⡼⠁
⠀⠀⠀⠛⠋⠩⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣄⣀⣠⣎⡞⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⢫⣻⠇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⠰⠚⠋⠁⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⠀⣾⣭⣶⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣤⡤⢸⠩⣭⣭⡭⢻⠠⣤⡀
⠀⠀⠀⠉⠛⠗⣶⣶⣤⣀⡀⡶⠒⢦⠀⠀⠀⠀⠀⠀⢨⡿⢿⡅⠀⣴⡟⠉⢻⣆⠀⠀⠀⣶⡆⠀⢬⡥⠀⢠⣥⠀⠀⠀⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⠀⢠⣶⣶⣴⣬⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⣾⣧⠀⢸⣷⠀⠀⠀⠿⠟⣵⡎⠿⠿⠿⠿⠏⣶⠹⠿⠃
⠀⠀⠘⠿⣟⢙⡿⠿⠿⠿⠿⠿⢿⡟⢙⡿⠿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⢸⣿⣮⣴⣿⣿⣿⣿⣷⣭⣾⣿
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠉⠀⠀⠀⠃⠀⠀⠀⠙⠁⠀⠙⠋⠀⠈⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⣤⡄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⠏⠀⠀⠀⣿⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠔⠈⡏⠉⠋⠐⠄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣱⢣⡏⠀⠀⠀⠀⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡆⠀⠁⠀⠁⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣰⢣⣿⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⡁⠀⡆⠀⠒⠀⡘
⠀⠀⠀⠀⠀⢀⠀⠄⠒⠊⠁⢸⠀⠂⠿⠼⠧⠤⠤⠤⠤⠟⠀⠀⡿⠦⠤⠤⠤⠤⠤⠤⠼⢻⠀⠀⠀⠈⠒⠳⠤⠓⠊⠀⠀⠀⠀⠂
⠀⠀⠠⢂⢉⠠⠀⠀⠀⠀⠀⢸⠒⠀⠀⠀⠀⠀⠀⠺⠶⠖⠂⠀⡇⠀⠀⠀⠀⠀⠰⠶⠶⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂
⠀⡔⠛⢶⣤⣤⠀⠀⠀⠀⠀⢸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈
⡄⠟⠛⠉⠉⠉⠁⡀⢐⠈⠤⠍⣑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⢂⠩⠤⠌⣐⠠⡀⠀⡀⠉⠫
⢁⠈⠤⠀⠀⢀⠌⡔⢡⡶⠒⠲⣦⠑⡌⢧⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠌⣠⠶⠒⠶⣄⠱⠐⡄
⠀⠑⠂⠀⠀⢖⢠⠀⣿⠀⠀⠀⢸⡇⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠚⠀⠀⡏⠀⠀⠀⢸⡆⣁⠐⠂⠴⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠣⡙⠶⣤⡴⠞⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠦⣤⡴⠟⡠⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣶⣶⣶⣬⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⠋⠙⠛⠙⠛⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠙⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠋⠉⠀⢻⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⣠⣿⣿⡄
⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⢿⡁⠀⠀⠈⠙⠟⠹⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⡟⠿⢻⡿
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⡿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢠⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠜⣿⣿⠁⠈⠋⠛⠙⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⡻⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⠛⢿⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠉⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠙⠿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⢸⡿⠙⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠇⠀⠀⠰⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡗⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡆
⠀⠀⠀⠀⢿⣿⠇⠈⠙⠿⠿⣿⡟⢿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠋⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠘⠏⠉⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠘⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡷⠀⠀⠠⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣤⣀⣤⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠿⢿⣿⣿⣿⣿⣿⣦⣠⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡵⠚⡛⠲⣯⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⢀⣠⠷⠶⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠠⠄⠀⠤⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣄⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⠁⠠⠞⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠋⢠⣀⡀⠙⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢻⡀⠀⡏⣿⣧⠀⠀⣼⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡄⠀⠓⢰⠇⣿⠸⠀⠚⠁⢀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡗⠀⣠⠉⢀⡏⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⡼⠟⠋⠁⣠⠞⠀⠱⣄⠈⠳⢾⣧⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣠⡴⠛⠁⠀⠀⠀⠈⠳⣦⣀⠈⠙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠶⠛⠉⠰⡀⠀⠀⠀⠀⠀⠀⣴⠙⠳⠶⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠛⠛⠒⠒⠀⠀⠨⣽⣷⣶⣶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠈⠉⠙⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⠛⠻⠶⢶⣤⣤⣄⣀⣀⣀⣤⠤⢴⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⢤⣤⣤⣴⡶⠾⠟⠛⠉⠉⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠸⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⣀⣀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠔⠚⢋⣉⡡⠤⠤⢉⡉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠴⠒⠋⠁⠀⢀⣀⣀⠀⠈⠳⡀⠙⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠞⠁⠀⣀⣀⠀⢰⠁⠀⠀⠉⠳⡄⢹⡀⠀⠈⡇⠀⠀⠀⠀⢀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⡜⠁⠀⢱⠈⠳⣄⡀⠀⢠⠃⢸⠇⠀⢸⠁⠀⠀⠀⠀⢾⠡⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢇⠀⢰⡇⠀⠀⢸⢀⡀⠤⠍⠉⠹⣠⠊⠀⠀⢸⠂⠀⠀⢀⢤⠼⠃⢀⣣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠘⢆⠈⠳⠤⠴⠋⠈⢁⣠⠤⠖⠺⣷⡀⠀⠀⡜⣀⠴⠊⠁⠐⠆⠐⣒⡧⣠⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⢄⠀⠑⠦⣀⣁⡀⠔⢃⢀⠄⢊⡔⡜⢱⡄⣘⠭⠠⠐⡅⠄⢀⡠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠱⡌⢄⠠⣹⣲⢭⠖⡛⣅⠸⠒⢁⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣤⣤⠙⠂⠒⠩⣆⠈⠏⠈⣁⡄⠈⢇
⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⣀⣀⠤⠔⠚⠉⠄⢂⠂⠐⠀⠀⠨⡆⠈⢏⡠⠴⠂⠈⢣⡀
⠀⠀⠀⠀⢸⡁⠠⡹⡓⠴⠋⢅⠀⠒⠤⠠⠀⢉⡠⢄⠀⠊⡁⠀⠰⡀⠈⡦⠒⢉⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠉⠲⢌⡎⡂⠈⠒⠉⢀⡤⠖⠋⠁⠀⠘⡄⠈⠡⠀⠈⠳⣤⡃⠌⠀⠀⠁⡀⠳⢤⣀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡞⢁⠤⠔⠛⠊⠁⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⡂⢒⠄⠍⠒⠠⠖⠡⠀⡹
⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢨⠐⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠐⠀⠢⠴⡆⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀⠄⡀⠁⢀⡤⠒⠉⠉⠀⠀⠀⠀⠈⠉⠉⠚⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠁⠢⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠐⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠃⠘⠠⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢬⡥⣄⡀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣹⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠙⢻⢿⠀⠹⠿⠟⠛⠛⠉⠉⠀⠀⠻⣿⣿⣿⠿⠛⢿⣿⣿⠰
⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⡀⣸⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠠⠀⢻⠒⡄⢴⡁⠀⠀⢨⠆⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠘⠂⢼⠐⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⠄⠈⠀⢸⣿⡜⢄⠘⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⡎⠀⠀⠀⠀⠀⢿⣿⣮⣑⣀⣙⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠙⠻⠛⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⠎⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡜⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢰⠁⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⣿⠿⠛⠁
⠀⠀⠀⡎⠀⠀⢠⠊⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠉
⠀⠀⢰⠀⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠆⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡘⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⣴⢃⡼⣄⢳⡄⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣙⠦⡀⣸⢃⠎⠀⠈⢆⢻⡀⡠⠞⣡⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡇⡏⠈⠁⢸⣇⠀⢠⣄⣤⠀⢀⣿⠀⠉⠈⡇⣿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡲⢦⣸⡀⠀⠀⠀⠸⠀⠀⣿⣧⣿⡇⠀⠘⠀⠀⠀⠀⣽⣤⠶⣚⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠉⠀⠈⣷⢄⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⢀⢴⡏⠀⠈⢹⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⠀⠀⠀⠘⢆⠁⠀⣠⣤⣴⣾⣧⣿⣶⣤⣤⡀⠀⢁⠞⠀⠀⠀⠸⢸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⢏⣀⣀⣀⡀
⠀⠀⠐⢾⡛⢉⣩⡭⠭⠍⠉⠙⢦⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⠞⠉⠉⠭⠭⣭⡉⠙⣻⠖
⠀⠀⠀⠀⠙⢦⡙⢆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠏⢸⣿⣿⣿⣿⣿⣿⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣉⡀⢸⣿⣿⣧⣿⣿⣿⠀⣈⣙⣿⣿⣿⣿⣿⣷⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⣉⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣉⣡⣴⣋
⠀⠀⠀⠀⢀⣴⣞⡉⠴⣄⠀⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠁⢀⡴⠌⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠿⠿⠿⠿⠉⠁⠀⠀⠀⠀⣀⣠⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⣸⣿⣿⡀⠀⢀⣠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢹⣿⣿⣿⣿⣷⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿⡇⠀⠀⢀⣴⣿⣿⣦⠀⠀⠀⠀⠀⣰⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⢸⣿⣇⠛⠿⠛⠁⡀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢀⣴⣿⣿⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⡿⠿⣿⣿⣦⣿⡿⠀⠀⣀⠐⠿⠀⠀⠙⠛⠋⠁⣀⣤⣾⣿⣿⠿⢿⣧⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣤⣶⣿⣇⡉⠻⣧⣄⣼⡟⠀⢠⣶⠀⠀⢠⣶⣿⡿⠛⠛⠛⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠿⣿⣿⠏⠉⠀⠙⠿⣦⠟⢛⠻⢿⣿⣿⡿⠶⣶⣾⣿⣟⣥⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡿⠈⠻⠿⠆⠀⣴⣶⡄⠈⣿⣏⣉⣠⣼⣿⡁⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠰⠶⠆⠀⠠⣤⣄⡀⢰⣿⠛⠛⠉⠉⢻⣿⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣴⡿⢝⣷⡌⠉⣡⣿⡇⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⠗⠲⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⡟⠉⠉⠛⠶⠖⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣅⣠⣼⣷⣤⣤⣤⣤⣤⣄⣀⣠⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠈⢹⣿⠟⠋⠉⠉⠉⠉⠙⠛⠻⣿⣆⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣼⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢸⡇⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣀⣀⡀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⢸⣏⢉⣿⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣟⣛⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡟⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⢩⣉⠛⣷⡀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⢻⡇⣾⠛⠛⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⣶⣾⣿⣶⢸⡧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢷⣄⡘⠛⣠⡾⠃⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣸⡇⣿⣀⣀⣀⣀⣀⣀⣀⣈⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣭⣿⣿⣭⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠿⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣟⣠⣶⣾⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣷⣦⣄⣀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⡏⠉⣿⣿⡿⠋⢉⡉⠛⢿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡿⠋⣉⡉⠻⣿⣿⣏⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⠀⣾⣿⣿⡆⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣾⣿⣿⡆⠘⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⠻⠟⠁⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⠿⠟⢁⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠆⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠾⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡇⠀⠀⠀⠀⠀⢰⠀⠘⠻⢿⣿⠏
⠀⢀⣠⣄⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⠤⠄⠈⠀⢀⣿⡷⢀⣀
⠲⣿⣿⣿⠶⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠄⠀⠀⠀⠀⢠⣾⡟⢠⣿⡿
⠀⠻⡉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⣿⠏⣰⣿⡟
⠀⠀⠈⠑⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠉⠰⣿⡟
⠀⠀⠀⠀⠀⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢠⠂
⠀⠀⠀⠐⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠐⠂⠒⢲⣄⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡄⠀⠀⠆⠀⠀⢾⠋⡁⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢂
⠀⠀⠰⡀⠀⠀⠀⢸⢠⠆⠀⠀⠀⢠⡆⡜⠁⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠑⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢳⣶⠀⠀⣾⢿⠀⠀⠀⢠⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠡⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⡁⠊⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠛⢇⠀⠉⠸⠀⠀⢀⠟⠋⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⢠⢀⠄⢠⠎
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠎⡠⠔⣽⠷⠚⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⡀⠾⠁⢁⠴⠁
⠀⠀⠀⠀⠤⠀⠀⢀⠀⠀⠄⠊⠀⠀⠁⠀⠀⠀⠨⠗⠊⠁⠀⠐⢱⡀⣤⡀⠀⠠⠋⢀⠀⠁⠸⢶⡬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠈⢦⠀⠀⠄⠀⣤⠈⡅⠀⢢⠃⠀⠁⠒⠁⢰⠀⠀⠐⠀⠈⠱⡀
⠀⠀⠀⠀⠀⢀⠠⠽⠒⢃⡄⠡⠤⢄⢄⠄⠀⠐⠒⠒⠘⠀⢣⡘⠤⡵⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢤⣈⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡠
⠀⠀⠀⠀⠈⠉⠈⠈⠉⠐⡀⠀⠀⠤⣀⣀⠀⠀⠘⠛⠓⠲⠤⢤⣀⠀⠆⠀⠈⠑⠀⠀⠀⠀⠀⠀⣠⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⡀⠀⠐⠾⠿⣟⡒⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢂⣠⣤⣤⢶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳⣷⡀⠀⠀⠀⠀⠉⠑⠢⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠉⠉⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⣬⠀⠀⢸⢀⠀⠀⠳⡀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⣼⣿⠛⠁⣀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠉⠐⡲⢤⣿⡟⠀⠀⠀⢀⣼⡿⠗⡊⠅⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡈⠀⠸⡅⠀⠀⠀⣸⡫⠖⢁⢀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠏⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠶⠒⠛⠉⠉⠉⠉⠛⠒⠶⢤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠋
⠀⠀⠀⠀⣰⠋⡠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠟⠀⢴⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠆
⠀⠀⠀⣰⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠉⢠⡀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂⡀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢠⡀⡀⠀⠀⡰⡛⠁⠀⠀⠀⠀⠀⠀⢘⡄⣀⣡
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢆⠀⢁⠜⡝⠁⠀⠀⠀⠀⠀⢀⠤⠚⠛⠛⠋⠀⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠐⠀⠀⠂⠁⠀⠀⠀⡀⠠⠀⠀⠒⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡇⡞⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣤⣴⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⣜⠣⢄⡀⣼⣸⠁⠀⠁⣀⡖⠋⠁⠀⠀⠀⡴⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠂⠈⠁⠀⠁
⠀⠀⠀⠀⠀⣍⠳⠄⢀⠀⠈⠉⠉⠉⢑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⡡⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸
⠀⠀⠀⠀⠈⠆⠉⠒⠢⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⡄⢪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⣀⣀⠀⢀⠔⠒⠀⠀⠀⠀⠀⢐⣉⣶⠆⠀⣀⢬⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡤⣚⡭⠟⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⡠⢚⡥⠋⠁⠀⠀⠀⠀⠐⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠛⣓⡶⠀⠀⠈⠑⣣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠐⢁⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⠞⠁⠀⠀⢀⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠂⠀⠰⠤⠀⠀⠠⠂⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠚⠢⠀⠈⠁⠀⢀⡴⠊⠁⠀⠀⡠⠚⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⡄⠀⠀⢯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠲⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⠋⠀⠀⠀⣀⠬⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡠⠔⠊⠁⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠔⠉⠀⢀⡴⠒⠁⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠐⠋⢹⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣰⣫⠖⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⣀⠀⢨⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡾⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢆⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠉⡄⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢲⡀⠀⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⣤⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡾⠛⢷⡽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣉⠟⣁⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢷⣤⡾⠃⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣄⣀⣄⣿⠟⣿⠻⣿⠛⣿⣀⣀⣠⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⠿⠀⣤⠀⣤⠀⣤⠀⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣛⣿⣿⣛⣿⣛⣿⣛⣿⣿⣛⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠶⠀⠀⠶⠀⠶⠀⠦⠀⠀⠶⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣭⣿⣿⣭⣿⣭⣿⣭⣿⣿⣭⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠒⠀⠀⠒⠀⠒⠀⠒⠀⠀⠒⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⠶⣿⣿⣶⣿⣶⣿⣶⣿⣿⠶⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣛⠀⠀⠛⢀⠛⡀⠛⠀⠀⣙⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠾⠿⠿⠿⠂⠀⠀⠿⠿⠿⠷⠿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣇⣀⣉⣀⣀⣀⣿⣿⣿⣀⣀⣀⣉⣀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⢠⡆⠀⣀⠀⠀⢈⠀⡉⠀⣀
⠀⠀⠀⣸⡇⠀⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⢽⣧⣤⣦⣶⠄⠀⠈⢲⣷⣤⣴⣤⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⢉⡙⢩⡍⢿⣿⣿⣟⠏⠈⠀⠀⣠⣿⣝⠻⢍⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⣲⠄⠀⠀⠀⠀⣀⣶⣶⣦⣤⣤⣄⡴⠼⣯⠿⠄⠰⠓⠀⠛⠛⠛⠑⠐⣿⠟⠻⢿⣿⣿⣿⣟⠛⣀⡈⢙⠻⠟⠛⣿⣿
⣉⣉⣙⣁⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠛⠿⣿⣶⡂⠀⠀⢠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⣀⢀⣀⣀⣐⣻⣿
⠈⠉⠉⠉⠉⠋⠉⠉⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⢿⣿⣿⣿⠆⠀⠀⠁⠀⠀⠈⠀⠐⠂⠚⠛⠛⠋⠉⠉⢉⣉⣉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀⠀⢸⣿⠿⣿⣿⣿⣿⣿⢻⣿⣀⣀⣀⣴⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⣘⣃⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⣠⣶⣿⠿⠿⣿⣶⡄⢀⣠⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣿⠿⠇⣸⡇⠻⠿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠛⢿⡿⢛⢃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣘⣃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⢿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡆⣾⣿⣿⣿⣷⢸⣿⠀⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⡇⣿⡇⢸⣿⣿⣿⡇⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⠀⣿⣿⢸⣿⣿⣷⢸⣿⣿⡇⢸⣿⡇⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⠀⣿⣿⢸⣿⣿⣍⢨⣿⣿⡇⢸⣿⡇⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠇⠿⡿⣿⡿⡿⠸⠿⠀⠿⡿⠸⢿⣿⣿⣿⡿⣿⠇⠸⣿⠇⠿⠇⠸⣿⣿⣿⠇⠿⢿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠂⠀⠸⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢻⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡈⡇⠀⠀⠀⠀⠀⢸⣸⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣆⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⣿⣿⣯⣶⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⣗⣀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠛⠉⢿⣿⡿
⠀⠀⠀⠀⠀⢿⡏⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣼⣇⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠾⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠻⡿
⠀⠀⠀⠀⠀⠀⠈⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⡴⢿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣛⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⢏⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣋⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠉⢁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠻⠟⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠆⣠⣀
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠂⠀⢈⣹⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠛⠉⠀⠀⠮⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⢠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠛⠻⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⣤⠈⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⢿⣄⠈⠢⣄⠈⠻⣿⣟⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣆⠀⡀⠀⠀⠀⠀⠀⢠⣯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⣛⣽⡻⡙⣍⢯⢳⣦⡈⠳⣄⠈⢻⣯⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠹⠞⢻⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⣿⣛⢧⣙⣼⣏⣛⣚⣚⣓⣓⣃⣹⣤⢾⡷⠶⣿⣧⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣠⢤⣘⡇⠀⠀⠀⡤⣤⣤⣤⠶⣚⣻⡭⠽⠷⠒⢛⣛⣉⣽⣿⣏⣏⠏⠀⠉⠀⠀⠀⠀⡜⠁⡞⠀⣼⠟⠹⡿⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠉⠋⢀⣤⣦⣿⣷⡦⢖⣫⡥⠶⠚⠛⠉⠉⠉⠉⠈⠉⠙⠛⠛⠛⠿⢷⣶⣦⣤⣴⣁⡜⢀⣼⡿⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣸⡿⣛⡵⠞⠉⣤⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣫⠵⣛⡵⡋⠁⠀⠀⠀⠘⣯⠛⣁⡘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢞⣫⠴⢛⡥⠞⠁⢰⣿⠀⠀⠀⠀⠀⠘⠛⠉⠻⣼⠇
⠀⠀⢀⣠⡤⠶⢚⣉⡤⠞⣋⣥⠞⠋⠀⠀⢀⣟⣈⠙⢳⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢩⠴⠖⢛⣉⡥⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⣤⠤⠒⠊⠁⠀⠱⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣰⡋⢈⣉⡁⢀⠀⢈⣉⡁⢨⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡇⠀⠀⠀⣼⠀⠀⠀⠀⢸⠜⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣠⠔⠛⢢⣄⣀⣀⣠⡖⠛⠢⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡁⠀⠈⠙⢦⠀⠀⡴⠋⠀⠀⢈⣂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠒⠤⣀⢣⡜⣀⠄⠒⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢠⠏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⢄⡀⠀⠀⠀⣰⣋⣠⠧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⣄⣙⣆⠀⠀⠀⢀⡠⠄⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡯⣄⡀⠀⠙⡆⣠⠞⠉⠁⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠈⠉⠳⣄⢰⠋⠀⢀⣠⢽⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢣⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡜⠁⠀⠀⠙⣆
⠀⠀⠀⠀⢠⡇⡎⡜⢠⡆⢸⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒⠤⣀⠤⠒⠒⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⡇⢰⡄⢣⢱⢸⡄
⠀⠀⠀⠀⠘⢷⠀⠃⡏⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠒⠲⣄⡤⠔⠤⢄⡀⠀⠀⠀⠀⠀⠀⠘⡼⢱⠘⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠛⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠈⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣆⣀⣤⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠈⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢖⠋⠁⠀⠘⢒⡲⠥⢄⠤⠠⢓⡄⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢃⠃⢠⠄⠀⠀⠀⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠠⢀⡄⠤⠄⣀⣀⣀⡀⡈⢸⠀⠘⠀⠀⠀⠂⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠃⠈⠈⠤⠘⠀⠀⠀⠠⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠄⠀⠀⠀⠀⠀⠈⣂⣀⣀⡀⠈⠀⠁⠀⣸⠤⠤⠖⠚⢄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⢄⠀⡀⠀⣀⣀⣀⣀⣀⣸⠀⠀⠈⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠊⠁⠀⠀⠀⠀⠈⠢⢤⣤⠔⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠉⠁⠀⠀⠉⠉⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⢳⠀⢰⠋⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡏⠉⣷⠀⠀⡟⠉⢹⠀⠀⢸⠉⠙⡆⠀⢸⠋⠉⡇⠀⠀⡏⠉⢻⠀⠀⡟⠉⢹⠀⠀⢸⠉⠙⡇⠀⢰⡏⠉⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠓⠒⠋⠀⠀⠙⠒⠚⠀⠀⠘⠒⠚⠁⠀⠈⠓⠒⠃⠀⠀⠓⠒⠋⠀⠀⠙⠒⠚⠀⠀⠘⠒⠚⠁⠀⠈⠓⠒⠃⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡖⠒⣶⠀⠀⡖⠒⢲⠀⠀⢰⠒⠒⡆⠀⢰⠒⠒⡆⠀⠀⡖⠒⢲⠀⠀⡖⠒⢲⠀⠀⢰⠒⠒⡆⠀⢰⡖⠒⡆⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠓⠒⠛⠀⠀⠓⠒⠚⠀⠀⠘⠒⠒⠃⠀⠘⠒⠒⠃⠀⠀⠓⠒⠚⠀⠀⠓⠒⠚⠀⠀⠘⠒⠐⠃⠀⠘⠓⠒⠃⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡄⠄⣤⠀⠀⡤⠠⢤⠀⠀⠀⢠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⢠⠤⠠⡄⠀⢠⡤⠄⡄⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠧⠤⠿⠀⠀⠧⠤⠼⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⠀⠀⠸⠤⠤⠇⠀⠸⠧⠤⠇⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡤⠤⣄⠀⠀⣠⠤⢤⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⢠⠤⢤⡀⠀⢀⡤⠤⡄⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣇⣀⡿⠀⠀⣇⣀⣸⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⣀⣀⡇⠀⠸⣇⣀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣈⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣤⣾⣾⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣩⠆⣩⠙⠛⢷⡶⠦⣤⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⣶⣶⠀⠀⠀⠈⢻⣆⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣿⣿⠋⠁⢀⣐⡲⠀⠠⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⡝⢿⠀⠀⠈⠉⠁⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠘⢷⣄⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠈⠙⢲⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⣫⠗⠙⠉⠛⣿⣿⣿⣿⡆⠀⠀⢤⠀⠈⣿⣿⣿⣿⣉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡟⡀⠀⠀⣤⠸⣿⣟⣿⠷⡀⠀⠀⠀⠀⢹⣿⣿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢮⢷⡵⣄⡆⠀⠙⡇⢻⡟⡞⠀⡽⠆⠀⠀⠀⠈⣯⣿⣄⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢙⢿⣿⣆⠀⣿⡈⠻⣧⣄⡀⠀⠐⠀⠀⠀⢻⠘⣿⡄⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⢰⣿⢻⣿⠘⡟⢻⣷⣶⣴⡄⠀⠸⡄⠈⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠉⠀⠙⠛⠛⠋⠀⠀⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡀⠀⢹⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⢺⡿⠻⢋⣻⡗⠀⠀⠀⠀⠺⣥⣕⡦⡄⠀⣶⡞⠋⠾⠁⢺⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⢿⠘⡶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠀⢙⣧⠀⠈⢣⣾⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⠟⠛⢿⣥⣤⣶⣯⡾⠁⠐⢿⠀⠀⠀⣰⣶⠶⠶⠴⢺⢇⣤⡾⠃⠀⣀⣸⠏⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠏⠀⠀⣴⡟⣩⠦⠤⡈⠙⠶⣾⣿⡧⠀⠀⠘⣧⢺⡋⢡⣾⠋⠀⣶⠓⠁⠀⣿⡸⡇
⣿⣿⣿⣿⣿⡟⠉⠈⢹⣿⣿⣟⠀⠀⢰⣿⣼⢇⣴⠟⠛⠷⣤⡀⠉⠁⠀⠀⢀⡟⠈⣧⠀⠻⣄⠀⠻⣄⠀⠀⠈⢳⣕
⣿⠏⠀⠉⢉⠀⠀⠀⠘⣿⣿⣿⣆⠆⠘⣿⣿⠻⠟⠀⠀⠀⠀⠉⠒⢤⣤⡴⡾⠁⠀⣼⠀⠀⠘⣧⠀⠈⢧⠀⠀⠀⣹
⠋⠀⠀⠀⠀⣀⠀⠀⠀⢿⣿⣿⣿⢷⣆⠘⣿⣧⡴⠾⢷⣤⣀⣀⠀⠼⠃⡠⠁⠀⢠⡟⠀⠀⣼⠇⠀⢠⣾⠀⣾⡾⠫
⠀⠀⠀⣠⠞⠉⠉⢢⣀⣠⣤⣌⡉⠤⢹⢆⣾⡟⡠⣤⠂⠚⠛⠛⠛⠋⢉⠀⢀⣠⠟⣔⣥⠞⠁⠀⠀⢻⡇⠀⢿⣿⡀
⠀⡇⣸⠃⠀⠀⠀⠀⣹⡟⠁⠉⠙⠻⠿⠿⠋⢨⣾⢃⣴⠿⠷⠶⠶⠾⠟⠛⣻⡿⣯⣭⣤⣴⣶⠟⠲⠚⣁⣠⣾⡟⠃
⢀⢹⡇⠀⠀⠀⠈⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠁⣀⢄⣀⣀⣠⠤⠊⠁⠀⠀⠀⠀⠀⣿⠀⢉⣩⣭⡿⠫⣀⣼
⢸⢸⡇⠠⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢸⣿⠿⡩⠁⠸⣿⡁
⠀⣮⣧⠠⠀⠐⠁⠀⠜⢳⣧⠀⠀⠀⣠⢾⡏⣓⠹⣇⢹⠀⠀⠀⠶⢤⠀⠀⠀⠠⠖⠋⠀⠀⠀⢸⣿⣷⢾⣵⣾⣿⠇
⣶⣥⣸⡇⠀⠀⠀⠀⣄⠀⠙⣆⠀⠠⢻⣜⠇⢿⠀⢻⢸⠀⢶⣶⣤⣀⠀⠀⠀⠀⣤⠶⠾⠋⠀⢾⣿⣿⠀⠙⠛⠁
⡏⠙⠻⠄⠄⢀⠀⠀⠌⠂⢀⡿⠁⠀⠈⢿⡀⠘⠀⣼⢸⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⢻⣶⣤⠤⠤⢄
⠀⠀⠐⢁⠤⠀⠤⣀⣤⣴⢿⣿⣤⡄⠀⠀⠙⣆⣼⠟⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣹⣿⣿⡮⠉⠉⠉
⠀⠀⠠⠡⢀⣨⣾⠋⠁⠀⠀⠒⠒⠲⠤⠀⠴⣻⡇⠘⠁⠈⢣⡀⠀⠀⠀⣠⣤⣤⣄⠀⢀⣼⡇⢸⡿⠋⠁⢀⣀⣠⠔
⠀⠀⢇⣶⠋⠹⠋⠀⠀⠀⠀⠀⣲⣶⣶⣾⣿⣿⡁⠰⣌⠀⠀⢻⡢⣀⠈⠛⠻⠟⠉⣠⣾⣯⣶⣿⠀⠰⣿⣉⡀⢀⡀
⠈⠒⣼⡇⠀⠀⠀⢠⠀⠀⢀⡞⡋⠈⠁⠀⠀⠹⢿⣷⣌⠙⢤⡀⠙⢌⠑⣦⣤⣤⣾⣿⡇⠘⠿⢿⣷⣤⣖⣁⣀⠲⡙
⠀⠀⡏⢿⣦⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠐⠒⠲⠖⠜⣿⣷⢠⢹⠀⠘⣇⠘⢿⣿⣿⣿⣿⣶⣷⣶⣔⠊⠉⠻⣿⣧⣱
⣿⢧⡥⠾⠋⣀⣀⠀⠀⢸⡇⠀⠂⢀⣀⣀⡀⠀⡀⠀⢹⣿⠿⠟⣠⣄⡿⠀⢸⢾⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣷⣿
⣁⣀⡙⠤⠾⣋⣼⣽⡓⣼⡇⢸⠠⣿⣿⣿⣿⣷⣄⠀⠀⠹⠒⠋⠀⠉⠁⣀⠘⣸⣿⣿⣿⣿⣿⣿⣿⡷⢻⢿⣿⣏⣁
⠉⠛⣷⣶⠞⠩⡟⠉⡛⠝⠀⠀⢧⠹⣿⣿⣿⡙⠿⣷⣌⡲⣤⣄⣈⠋⢉⡴⣳⣿⣿⣿⣿⣿⡿⣋⣼⠗⠈⡎⢟⣿⣿
⠀⠀⢨⡾⡇⠰⣀⡎⠉⠀⣀⣀⠌⢄⣿⣿⣿⣷⣾⠛⡿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⠟⣳⠏⠑⠀⠀⠀⠀⠙⢿
⣀⣹⣦⠤⠃⣰⡟⣠⣶⣿⣿⣶⣾⣿⣿⠏⠉⠁⠘⢯⣼⠷⣝⠻⢿⣿⣿⠿⢟⡻⠯⠓⠋⢸⢹⡀⠀⢰⣶⣶⣤⣤⣾
⡡⠚⢠⠦⢚⣭⣾⣿⠟⠛⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⣿⣠⡤⣵⣤⣈⣙⠓⠉⠀⢀⡠⠔⠁⣸⡇⠀⠈⢿⣿⣇⠈⠉
⠔⣁⣀⣤⣾⣿⣿⡟⠠⠀⢠⠕⠦⢀⠤⠤⠤⢤⡤⢚⡟⣿⠀⠀⣈⣙⠻⢶⣦⣴⣶⣶⣶⡾⠟⠁⠀⠀⠐⢻⣿⣷⡀
⣾⣿⣿⣿⣿⣿⠿⠁⠄⠀⠀⠀⠰⠁⠀⢰⠋⠀⠉⠉⢀⣿⠀⠘⡁⣀⣭⣓⠒⠉⠉⠉⠀⠒⠀⠀⢀⠠⣨⣿⣿⠟
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣷⣶⣶⣿⣶⣶⣶⣾⣿⣿⣶⣿⣶⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⣶⣤⣀⣀⣤⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠙⢿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣦⣭⣤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣈⡉⠉⠉⣛⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⣿⣿⣍⡀⠚⠻⠿⣿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢟⣠⣴⣾⣿⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣘⠿⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠛⠉⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⣠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣾⣿⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠹⣿⠃⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⣿⣿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣿⣦⣿⣿⣿⣿⣧⣿⣷⣶⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⣿⣿⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣿⣿⠁⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣿⣿⣿⣿⡆⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⣼⣿⣿⣿⣧⡼⠿
⠀⠀⠀⠀⠀⢰⣯⣄⣀⣰⣿⣿⣿⣧⢀⣠⣼⡆⠀⠀⣀⣤⠴⣿⣍⣀⢰⣿⣿⣿⣿⣇⣀⣩⣿⠦⣤⣄⠀⠀⠀⢠⣤⣄⡀⣾⣿⣿⣿⣇⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠈⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠁⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⡟⢿⣇
⠀⠀⠀⠀⠀⠀⢀⣰⠿⠀⣿⣿⣿⡁⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡹⣿⣿⣿⣿⢏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣥⣿⣿⣿⣿⣧⣼⡿⠄
⠀⠀⠀⠀⡴⢶⣯⣅⣀⣠⣿⣿⣿⣇⣀⣨⣽⣶⢦⠄⠀⠀⠀⣤⣤⣌⣷⣿⣿⣿⣿⣾⣡⣤⣤⠀⠀⠀⠀⠀⠀⣶⣶⣤⣀⣿⣿⣿⣿⣯⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⣶⠶⢶⠶⡶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣤⡄⡀⢠⡤⢄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠐⠀⢀⠀⠀⠀⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⠉⠀⠸⠀⠈⠉⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢢⣀⣀⣀⣤⠚⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠙⢦⠀⢠⠞⠁⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠲⢄⣳⣃⠴⠚⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⡇⠀⠀⡀⠀⠀⢸⠀⠀⠀⣀⠀⠀⣷⡀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣰⢿⢀⡤⠊⠁⠉⠉⠹⠉⠉⠁⠉⠢⣀⢸⢷⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢄⡀⠀⠀⠀⣰⣃⣾⣼⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣸⣇⣳⡀⠀⠀⠀⣀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⣀⠀⠙⣄⡠⠚⠉⠁⠀⠀⠈⠑⠒⠦⠤⠼⠤⠤⠖⠒⠉⠀⠀⠀⠉⠙⠦⣀⡜⠁⢀⡠⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠱⡔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡑⡴⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣜⢰⢁⡶⢠⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠢⣤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢧⠰⡄⢱⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠔⠂⠁⠲⢄⠀⠀⠀⠀⠀⠈⠒⢡⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡆⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠷⠤⠤⠴⠶⠶⠶⠦⠤⠴⠷⠶⠖⠒⠉



⣿⠿⠿⠿⡿⣟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣟⣻⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣄⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠉⠀⠀⣴⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⢌⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢪⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣆⢸⣿⡆⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿⣿⣿⣿⣾⠋⠀⠀⠀⠀⣾⣿⢀⣾⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣹⣿⣆⣿⣷⠀⠀⣀⡀⠀⠀⠹⢿⣿⣿⣿⣿⣿⡿⠋⠀⣀⡀⠀⢰⣿⣇⣾⣿⣱⣿⠇
⠀⠀⠀⠀⠀⠀⠰⣷⣮⣿⣿⣿⣿⣿⣿⣄⣼⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣿⣿⣄⣼⣿⣿⣿⣿⣿⣯⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣴⠀⠀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡀⠀⢸⢹⣈⡃⣀⣷⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠷⣟⡬⣃⣼⣿⣿⣿⣻⣽⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡯⣭⢞⢽⢳⣷⣿⣿⣿⣿⣿⡿⣿⣽⡀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢬⣾⣿⣿⠿⠿⠛⢿⣿⣿⣦⣿⣽⡄⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠻⣿⣿⣿⠃⢀⠀⠀⠀⢈⠛⠿⣿⣿⣿⣦⣇⡐⣀⢡⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢴⣂⡈⠁⠠⣿⣿⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⣿⣿⣿⣻⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠉⠉⠁⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢉⣹⠈⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡧⠂⠾⠿⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⣨⡀⠘⠇⠀⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠹⢟⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⣧⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⢹⡏⠩⠙⢳⡄⡨⢁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠈⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⢌⠸⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣆⠀⢿⢹⡇⠀⣠⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⡸⡄⢸⢸⡇⢰⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡙⢆⢳⣷⠼⠿⣧⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡉⠀⠄⡠⠀⠈⢧⣤⢤⡀⠀⠀⠀⠀⠤⣄⡀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣀⠀⣄⠀⣠⠞⠉⠒⠛⠠⠶⠶⠶⣦⣀⠙⢦⡀⠈⣷⡀⢸⡆⣷⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⣹⣽⡏⠁⠀⠀⢀⣰⣿⣿⣿⣿⣾⣿⣷⣆⢻⡄⢸⡇⢸⣇⡟⢠⡟⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⣴⣿⣿⣿⡟⠉⠈⠉⠁⠙⠻⣿⣿⢸⣷⣾⣿⢃⣾⢡⡿⢁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠋⢀⡀⡀⠀⠀⠀⠀⠈⠻⢸⣿⣿⢟⡼⣵⣿⡶⣋⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⡏⠀⠈⠋⠴⠚⢉⣙⠳⢦⠀⠀⠾⠋⣠⣾⡿⠿⠛⠉⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⢸⣿⣿⣿⠀⠀⠀⢠⠞⠉⠙⢻⣦⠀⠀⠀⠈⠉⠉⢹⡏⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⡼⢋⡛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠖⠒⢾⡇⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⣠⡄⠀⣠⣀⡀⠀⠀⣿⣿⣿⡇⠘⠷⡄⠹⡷⠀⠀⠀⢀⣠⣀⣀⠀⠛⠃⠰⣿⠃⠀⠀⢸⡇⠀⠀⠀⢠⣞⠁
⠀⠀⠀⢸⣿⣷⠀⢻⣿⣿⢀⣴⣿⣿⣿⡷⣄⠀⠀⠀⠀⠀⠀⢰⠊⠠⠌⢻⠄⠀⣀⣠⢀⠀⠀⢀⡿⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠸⣿⣿⣧⢸⣿⣿⣿⣿⣿⡿⢫⠞⠉⠀⠀⠒⠠⠄⠀⠸⢄⡛⣀⡼⠀⡾⠉⠤⢹⡄⣠⠞⠁⠀⠀⢰⠿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠋⢀⡟⠀⣠⣤⣭⣍⠉⠀⠒⠂⠀⢀⠀⠀⠀⠷⣌⡠⢾⡼⠋⠀⠀⣀⡔⠁⠀⠀⠀⠀⠀⣀⡤⣀⣤⣾⠠⠒
⠀⠀⠀⠀⠀⢀⣽⣿⣿⠿⠿⣠⡴⠯⠷⣄⠙⠿⠿⠿⠷⣄⠀⠀⠻⢎⣡⠇⠀⢀⣠⡞⠋⠀⠀⠀⢈⠟⠁⠀⣀⢴⡾⠔⠚⠛⠉⠁⠈
⠀⠀⠀⠀⡴⠋⢉⡠⠔⢀⣤⡉⠀⠀⠀⠈⠙⣒⣶⡶⠒⠋⠛⠲⠶⢶⣶⣶⣾⡿⢟⢻⡶⢤⣀⣀⠀⠀⠀⠈
⠀⠀⠀⢀⡷⠚⠁⣠⣶⣿⣿⣷⣤⣤⣴⣶⣿⣟⠉⠀⠀⠀⠀⣠⣤⣀⣙⣛⣛⣐⣁⣸⠃⣮⠟⡸⠓⡆
⠀⠀⠀⢸⣀⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⠀⠈⠀⣠⣄⠀⠀⢾⣏⠙⠺⣧⣔⣡⢞⡷⠀⠀⠴⢲⡿⠤⠶⡶⠧⠔⠒⠈
⠀⠀⠀⠘⠋⠀⢻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠏⠀⠀⢀⠀⠻⣭⡶⢧⣈⣒⠿⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢠⡧⠀⠀⠀⠀⡀⠀⠀⠀⠘⣆⠀⠀⠀⠉⠴⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉⠻⢿⣿⣿⣧⠀⠀⠀⢠⡇⢀⡤⠴⠢⡇⢤⣀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⡫⠁⡏⠀⠀⠀⢿⠀⠈⠙⢦⡾⠀⠀⠀⢀⣀⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢰⡟⠀⠙⠦⣄⣀⢀⣧⣀⣠⠞⠙⣦⣤⠚⠉⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠋⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⢛⣁⡤⠖⠋⠁⠈⠙⠲⠤⠤⠤⠤⠴⠋
⠀⠀⠀⠀⢀⣀⣀⡀⠈⠃⠀⠀⠀⠀⣀⡤⠖⠒⠚⠒⠒⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⣀⣤⠴⢺⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢁⠐⣿⣿⣿⣤⡄⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠑⡀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠐⡄⠀⠀⢻⣿⣿⣿⣿⠇⠘⠀⠠⠀⠠⠂⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠀⠀⠁⢀⠀⠢⠀⠈⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠠⠁⡐⢀⠔⠁⠀⠔⠁⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠈⠂⢀⠑⢄⠀⠄⠈⠂⠁⠀⠀⣸⣿⣿⣄⠀⠀⠡⠂⠀⢁⠀⠊⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⡑⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠊⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠩⠀⠀⠈⠀⠀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠁⠀⠀⠀⠤⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣐⠀⠈⠁⠄⠤⠄
⠀⠀⠀⠀⠠⠤⠄⠠⠤⠤⠤⠤⠀⠄⠠⠤⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠒⠂⢀⣼⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠁⢸⣿⣧⡀⠀⠂⠠⠀⡉⡁⠀⠀⠀⠀⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⢀⣴⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣯⠀⠈⢿⣿⣷⡄⠁⠂⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣤⣿⡿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣦⣤⣴⡀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⠀⠂⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⢼⡇⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣀⣀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⢸⣿⣿⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠀⣤⣤⠈⠙⢿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢀⣀⣠⣿⣿⣀⣀⡀⢻⣿⣿⣿⣿⠉⠈⠁⠉⢻⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡅⠸⠿⢿⣿⣿⡿⠿⠇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⣿⣿⠀⠀⢠⣾⣿⣿⣿⣿⣦⣤⣤⣤⣤⣼⣿⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⢁⣤⣤⣄⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣠⣤⣄⡈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡇⢰⣿⡟⠛⢻⣿⡀⢹⣿⣿⣿⣿⣿⣿⡏⢠⣿⡟⠛⢻⣿⡄⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠁⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⢀⠄⠂⠈⠁⠈⠉⠀⠢⢄⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢠⠀⣆⣔⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⢈⠆⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠄⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠊⠄⠀⠀⠀⠐⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠂⠀⢸⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣽⠟⢻⣏⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢁⢠⢴⠊⠐⢀⣨⠉⠕⡢⢦⢬⡉⠁⠀⢁⠅⢤⣀⠀⢂⢤⣄
⠀⠀⠀⠀⠀⢀⠌⢠⠔⠚⣉⡀⠀⢠⡖⠗⢒⣦⣐⣂⣁⢄⣩⡆⣤⢷⠖⠢⡆⣐⣀⢌⠑⢿⣆
⠀⠀⠀⠀⠀⡈⠀⣎⠀⠞⣁⡅⢀⠏⠘⢠⠋⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⡇⢡⠀⢈⠽⠂⢈⠈
⠀⠀⠀⠀⠀⠠⡀⠈⠓⠤⠤⣉⣀⢤⣀⡆⡄⠉⠙⠀⠀⠀⠐⢡⡴⠾⢼⡧⣤⣭⢥⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠀⠼⠯⣈⡻⠓⠲⣒⠲⠶⠴⠲⢾⠷⠶⠊⡀⠤⠄⠰⠼⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠉⠠⠾⠅⠠⠀⠀⠢⠸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄⠂⠐⢀⠀⢠⠀⠀⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠈⠀⠌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠀⢁⣀⡔⢐⠨⡀⠐⠒⠑⠀⢸⠀⣀⡄⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠉⠚⠛⠉⠄⠂⠔⠨⢘⠳⡦⠄⡠⠐⠜⠡⢐⠠⠀⠁⠈⠪⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡏⠩⠿⠿⠿⠿⠄⠀⠀⠀⠠⠶⠒⠒⠶⠄⢸⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠘⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢇⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠍⠍⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⠈⢉⠛⠛⠛⠷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⡌⠱⡏⢣⠋⢣⠀⠀⠀⠈⠙⠻⠶⠟⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠉⠓⢄⠀⢨⠀⠸⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢸⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢓⡂⢀⣿⣻⣦⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⢦⣄⡈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⣉⣉⣁⠟⠫⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⢸⣈⠔⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣾⡇⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠉⠁⠀⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠷⠦⠀⢀⣤⣶⣦⣄⠀⠀⠶⠟⠃⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⣿⠟⠉⠉⠀⣤⣭⣭⣭⣭⣭⣥⣄⠈⠉⠉⢿⡟⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⣛⠻⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠅⣿⣿⣿⣿⣿⣿⣿⡇⣿⣟⣻⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠾⣿⣿⣿⣿⢋⣭⣭⣥⣿⣿⣿⣧⣭⣭⣍⢻⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣠⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡇⣿⣯⣭⣭⣭⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠙⣿⣿⣿⣿⣶⣶⣶⡌⣿⣿⣿⠃⣶⣶⣶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⢃⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠛⠻⢿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠿⠿⢃⣾⣿⣿⣷⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠻⠿⢋⣾⣿⣿⣷⠹⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠶⠶⠶⠶⣶⣤⣤⣀⠀⣀⡀⠀⣠⣾⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡿⠛⠛⠛⠋⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠰⢟⣽⣿⣦⡀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡈⠁⠾⣻⣿⣟⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢠⡾⢿⣦⠀⠛⠹⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠠⣿⠁⠀⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠻⢷⡶⠟⠁⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣀⣀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠛⢷⣄⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣇⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠶⠾⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⣤⣤⡄⠀⣿⣿⠀⣤⣤⡄⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠟⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠾⠿⠟⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣂⠈⠲⣸⣀⣃⣀⣺⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣧⢫⡿⣽⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢉⡉⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣾⣶⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣤⠇⠸⣤⡀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⡇⠸⠤⡄⢠⠤⠇⢸⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣄⡀⠓⠚⢀⣠⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⣿⠀⣏⢙⠀⠀⡏⣹⠀⠐⡋⣹⠀⣿⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⢤⠀⠀⣿⠀⣠⢤⠀⠀⡤⣄⠀⢀⡤⣄⠀⣿⠀⠀⡤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠣⠼⠀⠀⣿⠀⠣⠴⠀⠀⠧⠼⠀⠐⠦⠜⠀⣿⠀⠀⠧⠜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡖⢲⠀⠀⣿⠀⡖⢐⠀⠀⡖⢲⠀⢀⡒⢲⠀⣿⠀⠀⡖⢲⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣉⣉⠀⠀⣿⠀⣉⣉⠀⠀⣉⣉⠀⠀⣉⣉⠀⣿⠀⠀⣉⣉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠧⠼⠀⠀⣿⠀⠧⠨⠀⠀⠧⠼⠀⠘⠥⠼⠀⣿⠀⠀⠧⠼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡖⠲⠀⠀⣿⠀⠀⠀⡴⠒⠲⠖⠒⢦⠀⠀⠀⣿⠀⠀⡖⢢⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠛⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣟⠙⠛⠋⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠠⡀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠆⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠐⠀⠂⠒⠂⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣤⣤⣠⣤⣄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣤⣤⡌⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢠⣀⣠⠀⡄⠀⣄⣀⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠢⠤⠴⠂⠀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣏⣀⣀⣀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠙⢦⣀⢀⣀⣀⣀⠀⢀⡼⠋⠀⠀⣠⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠙⢦⣀⠀⠀⢉⣙⠒⣄⠉⠛⠉⠀⠀⣀⡴⠃⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠓⠲⠦⣄⣈⠛⠒⠯⣌⣉⣁⣀⣠⡤⠶⢚⣁⣠⠤⠴⠖⠒⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠤⠤⠤⠶⠶⠖⠚⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠂⠀⠄⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠲⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠉⠉⠙⠒⠲⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠰⠀⠀⢀⡄⠀⠀⡖⠀⠠⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣄⣀⡈⠐⠂
⠀⠀⠀⠀⠀⠙⢆⠀⠗⠀⠀⠈⠂⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠒⠂⠀⣀⡀⠀⠐⠒⠈⠉⠉⠁⠀⠀⠉⠒⠦⣄⣀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢦⣄⣀
⠀⠀⠀⢀⠄⢶⡾⠩⢀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣴⣼⣅⡿⠁⠀⠀⠈⠒⠤⣀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡀⡸
⠠⠈⠋⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠁⠒⢤⡀⠈⠑⠤⣀⠀⠀⠀⡀⠠⠄⢐⡒⠄⠈⢹⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡇⣿
⠘⠂⠀⠐⠄⡀⠢⠀⠀⠀⠀⠀⠠⣄⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⢀⢠⠼⡖⠦⠩⣕⡚⣧⡀⠀⠀⠀⠈⠀⠀⠀⠀⡇⣿
⢳⡿⣷⡐⡀⠀⠑⠤⡈⠂⢄⠀⠀⠲⢍⡂⢀⣠⡀⠀⠀⠐⠀⣀⣠⣤⠦⠀⣞⣽⣿⠻⡘⣇⠈⠑⠲⢤⣀⠀⠀⠀⡇⣿
⠸⣿⣿⡷⣝⡦⣄⠀⠀⠁⠀⠈⠐⠄⠀⠀⠸⣿⣿⠂⠀⠀⢸⣳⣿⣯⡠⠀⣿⡿⠟⡁⡇⣿⠀⠀⠀⠀⠈⠙⠲⢤⣇⣻
⠀⠈⠉⠙⢶⣝⣪⠙⠢⣄⠀⠀⠀⠀⠢⡐⠀⠈⠉⠁⠀⠀⣾⡿⠻⣋⡀⡤⣛⡁⠁⠀⡎⢺⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠉⠻⣄⣸⣤⡁⠀⣀⡀⠀⠈⠈⠐⠀⣀⣀⣀⡏⠉⡉⡤⠀⠟⠉⠁⠀⠀⡇⣸⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⡀⠀⠈⠁⠛⠉⡇⢀⡀⡤⠄⠖⠚⡁⠤⠒⢉⣸⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣾⣿⣿⣧⠀⠀⠀⠀⠀⣿⠀⡀⠠⢔⡫⣥⣠⣴⣞⠁⣠⠟⠀⠀⠂⢄⠈⠓⢦⣀⠀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣷⣤⣀⠀⠀⠉⢁⣀⠤⣾⣿⠿⣛⡭⠖⠋⠁⣀⠀⠀⠀⠀⠉⠢⣄⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⡀⠀⠀⠀⠀⣀⡬⠖⠊⠉⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠑⢦⡀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠉⠁⠠⡀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠈⠳⣄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠈⣿
⠀⠀⢀⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⣶⣦⣄
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣬⣿⣷⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⠀⠀⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡟⠙⠛⠋⠀⠀⠀⠀⠈⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⠀⠀⠀⠀⢠⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⢸⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣠⣴⠿⠟⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠻⢷⣦⡀⠀⠀⠀⢸⣿
⣠⣶⣿⣶⣶⣶⡿⠁⣠⣶⠶⣶⣄⠘⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡟⠁⣠⣶⢶⣦⡀⠙⣿⣶⣶⣶⣾⣿⣦
⣿⡇⠀⠀⠀⣿⡇⠰⣿⠁⠀⢈⣿⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⣿⠀⠀⢸⣿⠀⣹⡇⠀⠀⠀⠀⣿
⠻⠷⠶⠶⠶⠿⣷⡀⠹⢷⣶⠿⠋⢀⣾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠾⢿⣧⠀⠻⢷⣶⠿⠃⢠⣿⠿⠶⠶⠶⠷⠿
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣶⠟⠁

⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⢸⠀⠀⠀⠀⣀⣄⠀⢸⣿⣿⣿⣿⣀⠀⠀⢸⡇⠀⠀⢀⣤⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣠⠶⢦⡀⢸⣿⣿
⢸⠀⠀⢠⣤⣽⣟⡁⢠⣿⣿⣿⣿⠷⠀⠀⢸⣇⣀⣠⣌⣿⣛⡀⠀⠀⣈⣿⣯⣄⠀⠀⠀⣠⣬⣦⣮⣴⠿⠉⠉
⢸⠀⠀⠈⢹⣿⠟⠃⢸⣿⣿⣿⣿⣀⠀⠀⢸⡇⠉⠉⣼⣿⡏⠙⠓⠺⠟⣿⣿⠙⠓⠀⣾⡟⣿⢸⣏⠁
⢸⠀⣤⡾⠟⢻⣆⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⡇⣀⣴⡿⢿⣦⠀⠀⠀⠘⣿⢿⡇⠀⠀⢸⣶⣿⣾⣧⡄
⢸⠀⠉⠁⠀⠀⠿⠇⠘⣿⣿⣿⣿⣿⠀⠀⢸⡗⠿⠃⠀⠈⢿⡦⠀⠀⢸⡿⠘⣿⠀⠀⣰⣿⡟⠉⣿⣿
⠀⠒⠒⠒⠒⠒⠒⠒⠒⠙⠛⠛⠛⠑⠒⠀⠀⠓⠒⠒⠒⠂⠒⠑⠒⠒⠚⠑⠒⠒⠂⠀⢻⡋⠁⠀⡟⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇⠀⠀⡗⠀⠀⠀⠀⣀⡀
⠀⢶⣿⣧⡀⠀⠀⠀⣀⡀⠀⣠⣤⡀⠀⣿⠀⡇⠈⠀⣀⣀⠀⠀⢠⣶⢄⠀⠀⢸⡇⠀⢸⡃⠀⠀⣟⢻⣦⣀⣐⣿⡿
⢀⣀⣙⢟⠁⠀⠀⠀⠉⠙⣻⣾⣟⠃⠀⣿⠀⡇⠀⠀⣿⣿⠀⠀⣨⡿⣧⣄⡀⢸⡇⠀⢸⠇⠀⠀⡯⠀⠉⢹⣿⣿⣦
⣿⡿⠟⠿⣿⡇⠀⣀⣤⣼⣿⠟⠙⢷⠄⣿⠀⡏⠀⠀⠀⠀⠀⠿⠏⣿⣿⡙⠋⢸⡇⠀⢸⡆⠀⠀⣏⠀⢀⣿⣿⣯⡿
⣿⡁⠀⠀⣸⡇⠻⠟⠉⠁⣿⠆⠀⠀⠀⣿⠀⣷⣶⣶⣶⣶⣦⠀⠐⣿⢿⣇⠀⢸⡇⠀⢸⠆⠀⠀⣏⢀⣾⢿⣿
⢿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠿⠇⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⠀⢸⡿⠘⢿⠇⢸⡇⠀⢸⡇⠀⠀⣇⣾⡏⠸⣿⡀

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢉⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠂⠀⠀⠘⠒⠀⠤⡀
⠀⠀⠀⢀⣠⡤⠟⠓⠛⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⢀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⡞⠀⠀⠀⣀⠤⠴⣶⣻⡯⠥⠼⠥⠀⠀⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⢧⠀⣀⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠟⠁⠀⠀⠀⢀⣀⣤⡤⠤⠐⠒⠐⣒⡛⠩⠤⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⢰⣼⡿⠷⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠂⠄⡄⠀⠀⠲⠀⠀⠻
⠘⡷⠖⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⢸
⣾⠁⠀⠀⣀⣠⠤⠶⣖⣛⣫⣭⠥⠦⠔⠂⠀⠈⠑⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⡈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢻⡀⠀⠊⢉⡴⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⢗⣽⠋⠀⠀⢀⣀⣤⠤⠶⠶⣲⠾⠿⠯⢥⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠃⢸⠇⠀⠀⢰⢟⠉⠈⠙⠻⣼⠃⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠈⠢⠤⠴⠋⢸⡀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣼⠄⠒⠚⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⢀⠇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸
⠀⠙⢆⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠂⠒⠀⠚⠀⠀⠀⠀⠤⠧⠠⠴⠒⡶⠁
⠀⠀⠀⢻⠛⠉⠉⠛⠉⠉⠙⠯⠉⠙⠯⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣓⠀⡇
⠀⠀⠀⢸⡆⠀⠀⠤⠤⠤⠤⠤⠤⠀⠄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠠⠤⠤⠤⠤⢤⣤⠶⡶⠶⠀⡇
⠀⠀⠀⠺⡉⠀⠰⠶⠶⠶⠒⣒⣒⣛⡫⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠀⠠⠤⠤⠴⣶⣾⣿⠀⡆
⠀⠀⠀⢺⡅⠐⠶⠶⠶⠚⠛⣛⣚⣓⡭⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⣀⣀⣀⣀⣠⣵⣶⣶⢶⠀⡇
⠀⠀⠀⢾⡅⠀⠶⠶⠶⠒⠚⢛⣒⣛⠻⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⣤⣤⡶⣶⡃⣠⠇
⠀⠀⠀⠈⠳⣄⡀⠒⠒⠒⢚⣓⣛⣾⣥⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠉⠭⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠉⡭⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣛⣛⣿⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠘⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠐⠇⠀⠀⠇⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣀⢰⡟⢻⡆⣀⡀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⣷⢰⡏⠉⠉⠙⢦⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣏⠉⠛⠃⠘⠛⠉⣳⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠙⣦⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡬⠟⠀⠀⠀⡴⠶⡤⠶⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢳⡴⢺⡆⠀⢷⡀⠀⣠⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠓⠚⠃⠙⠛⠁⠀⠀⠈⠳⠶⠶⠶⠶⠶⠶⠶⠶⠋⠈⠳⠶⠶⠶⠶⠶⠶⠶⠶⠞⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⠴⣶⡶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣶⡶⠦⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡴⣫⠞⠋⠉⠉⠙⠺⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠉⠉⠉⠙⠪⡹⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣼⢹⠃⢠⡞⠉⠙⢦⡀⢹⡼⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⢹⠃⢠⠞⠉⠙⢦⠀⢹⡼⣆⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠸⣆⠀⢀⣼⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠸⣄⠀⢀⣼⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠉⠉⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠉⠉⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠨⠠⢄⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠠⠀⠀⢸⠛⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠈⠁⠀⠀⡀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⢣⢀⠀⠀⣧⠆⠀⠀⢠⣯⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂
⠀⠀⠈⠿⡄⠐⠏⠀⠀⢠⠿⠃⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢃⠠⣲⠧⠒⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠴⠓⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢁⠀⠀⠀⠓⠈⠀⠀⠹⡀⢄⠀⠀⠃⠀⠁⠘⣦⠑
⠀⠀⠀⠀⠀⢀⣀⡀⠒⠆⠆⠀⠓⠀⠒⠀⣶⠠⢁⠰⡅⠀⠈⠀⠁⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⢈⣁⠐⠀⠀⠀⠀⠀⠀⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠈⣉⠉⠉⠙⠛⠧⠤⣄⣀⠀⠀⠀⠣⠀⠀⠀⠀⢀⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⠀⠀⠺⠿⣖⡀⠀⠀⠀⠀⠀⠰⠀⡀⣀⣤⡤⢶⣿⡿⠁
⠀⠀⠀⠀⠀⠄⠀⢁⠀⠻⢆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠃⠀⠘⣴⠀⠀⠂⠀⠠⠒⠒⠒⠂⠀⢀⣀⣾⠟⠀⠀⠀⣾⣿⠤⠐⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠃⠀⠀⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠉⢛⠀⠀⠀⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈

⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⣸⣯⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⣧⣀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠈⠉⠂
⠀⢸⣿⣿⣦⣄⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠘⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⣰⣿⣿⠸⣄⣀⡠
⠀⠀⢻⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⢀⣙⣿⣿⣿⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣏⣿⣿⡇⠀⠀⠀⢹⣿⣿⣿⣿⣷⣦⣀
⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠸⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⣀⣽⡄⠀⠀⠀⠀⠀⠸⣿⣿⣦⣙⣿⣿⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠪⡻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⣿⣿⡇⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣫⣵⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢿⣿⣿⣿⣿⣿
⢺⣿⣟⠛⠋⠀⠀⠀⠈⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠻⠿⠿⠛⣷⣦⠀⠀⠀⠀⢀⣀⠤⠤⣤⣤⣤⣶⣿⣿⣿⣿⣿
⠀⢙⣿⣷⡄⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠆⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⢀⣴⠃⠀⠀⢠⡖⠀⠀⠀⣴⠆⠀⠀⢀⡴⠁⠀⠀⣠⡎⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⣿⣿⣿⣿⣿⠿⠋⠡
⠀⢸⣿⣄⠀⠀⣿⣧⡀⠀⢸⣿⣄⠀⠀⣾⣧⡀⠀⢰⣿⣆⡀⠀⠀⢀⣤⣶⣿⣿⠋⠀⠀⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠸⣿⣿⣷⣦⣹⣿⣿⣷⣌⣿⣿⣿⣦⣸⣿⣿⣷⣤⣿⣿⣿⣶⣄⢾⣿⠿⢿⣿⣧⡀⠠⡈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠹⢿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠻⢷⣼⣿⣿⣷⠀⠑⢌⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣀⣬⣿⣿⣿⣧⣬⣿⣿⣿⣤⣽⣿⣿⣧⣬⣿⣿⣿⣤⣽⣿⣿⡇⠀⠀⠀⢹⣿⡟⠀⠀⠀⠑⣌⠻⣿⣿⣿⣿⡿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣼⣤⣽⣿⣿⣿⠃
⠀⣿⣿⣯⡀⢸⣿⣿⣅⡀⣿⣿⣿⣅⠸⣿⣿⣯⡀⢻⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠘⣿⣿⣿⡦⢻⣿⣿⣿⡜⢿⣿⣿⣷⠹⣿⣿⣿⡎⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄⡀
⢀⣀⣈⣹⣿⣏⣀⣈⣿⣿⣃⣈⣹⣿⣟⣀⣈⣿⣿⣇⣀⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣦⡀
⠸⣿⣿⡋⠉⢻⣿⣟⡋⠘⣿⣿⣛⠉⠻⣿⣟⡋⠙⢿⣿⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡿⠿⠃
⠀⢨⡽⠿⠇⠀⢬⠿⠿⠀⢠⣽⠿⠇⠀⢨⡿⠿⠆⠀⣽⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣾⠷⠶⠾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣶⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠶⠿⠿⠿⠶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣴⡾⠟⠉⢁⠤⢖⣒⣒⠒⠦⣄⣀⡀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠟⠛⠛⠻⣿⣾⣿⣅⣀⠀⡰⠁⠀⠀⢀⣽⠿⠿⠿⣿⣿⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣄⣀⢀⣼⠟⠁⠈⠉⠙⠽⡇⠀⢀⣴⡏⠀⠠⡶⢦⠺⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠈⢉⠛⢦⠀⠀⠀⠀⠀⢻⢇⡞⡻⠇⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⠷⣦⣄⠀⠀⠀⠺⣷⣤⣄⠀⠀⠀⠀⠉⢿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢱⠀⠈⠻⣏⡽⢷⣦⣼⣀⠈⠑⠶⣶⢶⠖⠋⠀⠈⠻⣕⢄⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡜⠀⠀⠀⣻⣿⠿⣿⣿⣿⣶⣮⣐⠾⣿⡗⠢⠄⣀⠻⠟⠒⠛⠐⠆⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠃⠀⢠⠎⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⣦⣝⡻⢶⣾⣷⣾⠀⠀⠀⠸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠠⠀⡞⠀⢻⣿⣦⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣶⣿⢻⣿⣯⣤⢄⠀⢣⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢿⣿⡿⣿⢿⣿⣷⣄⠀⢸⣿⣿⢿⣿⣿⣿⣿⠉⠻⢿⣿⣿⠘⣷⠈⣆⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡉⠻⣮⢣⢻⣿⣿⣦⣸⣿⣿⡀⠉⠻⣿⣿⣷⣄⠀⠉⠻⢧⣿⡀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠃⠀⢻⣇⣸⢏⣿⣿⣿⡉⠻⣿⣦⡀⠈⠙⢿⣿⣷⣦⡰⠎⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡀⠀⠀⠀⠀⠀⣸⣿⣿⣾⠟⣻⣿⣷⣄⣿⣧⣙⢦⡰⠆⢹⣿⣿⣿⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢇⠀⠀⠀⠀⣠⣿⣿⠿⡇⠀⠈⢽⣿⣿⣿⣿⣿⣷⣝⠶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠴⠚⠛⢷⣦⣄⣧⣀⡈⠒⠺⠿⣿⣿⣿⣿⣿⣷⣌⣀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣠⠀⠂⣈⢂⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠒⢀⠀⠒⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠠⠄⢀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡊⠀⠺⡀⢀⠗⠀⢑⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⢰⡦⠂⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡜⣏⠄⠂⠀⠘⠃⠀⠐⢄⢹⢧⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠢⡀⢀⠼⠔⠓⠢⠄⡀⠀⠀⢀⠠⠔⠚⠢⠧⡀⠀⠐⠘⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠑⠤⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⡇⠄⢆⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⡌⢁⢀⡀⡀⠀⠀⠀⠀⠐⣰⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠠⢀⠀⠀⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠒⠒⠚⠒⠚⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⣄⣤⡄⣠⠀⡄⣠⣀⠄⣤⠀⠀⠈⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⣠⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣡⣾⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⡤⠀⣠⡾⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣳⣞⣋⣐⣂⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⢀⣀⣀⣛⣛⣛⢛⡛⠛⢛⡛⢛⣛⣛⢛⣛⣛⣛⣛⡛⠛⡛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠋⠁⠉⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠭⠭⠭⠭⠭⠭⠭⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢺
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣻⣟⣿⣷⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿⣿⣷⣿⡇⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣽⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡠
⠀⠀⣀⣀⣀⣀⣀⣀⣤⣭⠤⠤⠤⠶⠖⠚⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠠⠄⠀⠀⠂
⠉⠉⠉⠉⠡⠒⠒⠒⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠲⠶⣦⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠂
⠀⠐⣿⣾⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡀
⠀⠀⢹⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢻⡇
⢰⣶⣶⣶⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣧⣤⣈⡁⠀⠐⠶⠶⣾⣧⣀⣀⣀⣤⣤⣤⣤⡶⡶⠶⢶⣶⢶⡶⣛⣿⣷⣦⣀
⠘⠉⠙⠛⠛⣛⣛⣛⡛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡈⠉⠙⠛⠿⠿⣿⠿⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠲⠶⠶⠆⠀⠀⠉⠛⠻⠿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠁⠀⠀⠀⠀⢈
⢠⣀⣀⣀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠘
⠀⠀⠈⠉⠁⠀⣿⠁⣐⡒⠲⠶⠤⠤⣤⣀⣀⣀⣀⣀⣀⡀⣄⡀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠓⠂⠀⠀⣿⡟⠉⠉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢡⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⢦⣤⣄⣀⡀⠀⠘⡇⠈⡏⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣀⠉⠙⠛⠲⠮⢽⣿⣛⣺⡷⠶⣧⣤⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡷⢲⣦⣤⣄⣀⡈⠩⢿⡟⠲⠶⢭⣭⣛⡓⠶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠿⠞⠃⣼⣿⣭⡽⢻⡶⢲⡦⣤⣄⣀⡈⠉⠙⠓⠲⢬⡟⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⢀⡾⠁⠈⠉⠀⢸⡇⢸⡇⣸⣿⣏⢉⣿⣷⡶⢶⣼⣷⠟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢸⡇⣨⡟⢹⡏⠛⠋⣠⠿⣿⣶⠟⠉⠓⠶⠦⢤⣤⣤⣤⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣁⣀⡀⠀⠀⠀⠀⠀⣸⡟⢹⡇⢸⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠉⠁⣿⠫⢭⣉⡉⠛⠲⣦⣴⣏⣹⡇⢸⡇⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣠⡽⠟⠿⢧⣤⣈⡉⠙⣻⡷⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠙⣷⣈⣽⠟⣷⣦⣤⣄⣉⣽⣟⠿⢶⣤⣌⣉⠛⠻⠶⣤⣄⣀
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⢷⣤⠿⠛⣷⡾⣤⣄⣀⠈⠉⠙⠓⠶⢤⣌⣉⡽⣷⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣿⠀⢸⡏⢹⣿⡶⢦⣤⣤⣀⡀⣹⣀⣿⠖⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⢾⡟⠛⠋⠻⣿⣏⣽⠿⢯⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢻⡇⢸⡇⠀⢀⡾⠁⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣿⠀⢸⡇⢸⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⣿⠀⣸⡧⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⣿⠋⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⠄⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⠶⠛

⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⣹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢀⡴⠋⢠⡾⠋⣵⡿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣤⠀⠖⠉⣠⠞⢋⣴⠿⣻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣆⠘⠁⣠⠞⢁⣴⠟⣡⣾⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠠⠋⢀⡴⠛⣡⡿⢛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠐⠋⢀⡼⠋⣠⡿⢋⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡉⢠⠞⠁⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠉⠛⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⠁⠰⠶⠶⣶⣶⣶⣶⣾⣷⣶⣶⡶⠶⠶⠄⠈⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡙⠓⠲⠶⠶⠶⠤⠤⠤⠤⠤⠴⠶⠶⠖⠚⢋⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠰⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⡀⠀⣀⠀⠀⠀⠀⣀⠀⢀⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢐⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢸⡇⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣠⣄⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⣸⣏⣹⣇⠀⠴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣭⣭⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣻⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢉⣉⣭⣉⡉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⣉⣉⣉⣉⣳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠟⠉⣤⠉⠻⣆⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⢹⡇⣿⠉⠉⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠸⠿⣿⠿⠇⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢷⣤⣉⣤⡾⠃⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣼⠇⣿⣤⣤⣤⣤⣤⣤⣤⣬⣿⣆
⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠿⢷⡄
⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣿⣶⣼⣷
⠀⠀⠀⠀⠀⠀⢸⣟⣻⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡿⢿⣷⣾⣿⣿⢿⣿⣿⣷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣾⣿⣿⢿⣿⣿⣷⣶⠶⢿⡄
⠀⠀⠀⠀⠀⠀⣿⣀⣴⣿⡿⠁⣠⣤⣄⠙⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⡟⢁⣤⣤⣄⠙⣿⣿⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠸⣿⣿⡿⠀⣿⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢹⡇⠸⣿⣿⡿⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣈⣉⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣶⣶⡶⠶⠶⠶⠷⠾⠷⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⢹⡇⠀⢸⡟⣿⡆⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⢸⡇⠀⢸⡇⣿⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣼⡇⠀⠸⠷⠿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⢻⡿⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⣠⡿⣱⡟⠁⢀⣼⢿⣶⣾⠛⣿⡄⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣯⣙⠻⣶⣟⣼⠟⠻⣶⡾⣣⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠉⠛⢿⣿⣿⠻⢶⣴⡟⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣿⣾⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣹⣿⣿⣿⡛⠛⣛⣛⡛⠛⠛⠛⣷⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⡇⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⡇⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠿⠷⡿⠃⠀⠛⠛⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣦⣀⠀⠀⢀⣽⠟⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠈⣻⣟⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣼⢿⡿⢶⡿⠃⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⢸⣏⣹⣧⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠶⠶⠶⠿⠿⠿⠿⠶⠶⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⣠⡾⠋⠀⠀⠘⢷⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⢀⣤⡾⠛⠁
⣠⣤⣤⣶⠶⠟⠛⠉⢀⣠⣴⠾⠛⠁
⠉⠁⣀⣀⣠⣤⡶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠄⠀⠀⣀⠀⣈⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠱⠁⠀⠀⠈⠀⠀⣿⣶⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠰⢄⣀⡀⠛⠀⠀⠐⣫⠁⠄⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠠⠒⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢩⡀⠀⠂
⠀⠀⠀⣇⠀⠀⠀⠀⢸⠁⡴⠀⠀⠀⠀⢀⡾⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠀⠀⠈⢢⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠧⠤⠀⡀
⠀⠀⠀⠸⡄⡆⠀⠀⢸⣼⡇⠀⠀⠀⢀⣾⣧⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇⢀⠎⠀⠀⠀⠈⡆⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠘⡡⠒⠀⠀⠀⠐⠒⠊
⠀⠀⠀⠀⠻⢿⡄⠀⠸⠃⡇⠀⠀⠀⡼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⢠⠀⠄⠰⠀⠀⠘⢆⢀⠀⠀⠀⢀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠁⠀⠀⡼⠁⢀⡠⡾⣃⡤⠖⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⣵⢪⠔⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠖⠁⡾⠛⠁⠀⢀⡀⣄⢀⣠⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣆⠀⠀⡠⠈⣇⢀⢴⢉
⠀⠀⠀⠀⠐⠂⠀⠠⠤⠀⠄⠰⠊⠀⣀⣀⡀⣀⠔⠊⠀⠖⠋⠉⣀⣀⢐⣍⠻⡆⢰⢦⡉⠀⡻⠃⢠⠊⢀⠸⠛⣧⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠤⡴⢠⡀⠀⠸⡆⠠⠤⠤⠄⢤⡇⢠⠹⠀⣣⣓⠈⡀⠉⠉⠡⢲⠀⠀⠈⠓⠚⠁⡱⡄
⠀⠀⠀⠀⠀⠀⣀⠀⣿⠗⠻⣁⠈⢣⢰⣄⣀⣠⠀⠄⠐⠒⠒⠚⠃⠘⢆⣚⡤⡼⡄⠁⡄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠇⠘⣄
⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠈⠄⠠⢨⣈⡀⠃⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡄
⠀⠀⠀⠀⠑⠒⠀⠀⠒⠒⠂⢄⠀⠀⢀⡀⠈⠀⠉⠉⠉⠹⠿⢿⣄⣀⣀⠀⠀⠀⢆⠀⠈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⡆⠀⠀⠈⠉⣒⣲⣤⣄⣀⠀⠀⠀⠈⠉⠉⠓⠂⠘⠄⠀⠀⠠⠀⠀⠀⠀⢀⣠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠹⡼⡆⠀⠀⠈⠉⠛⠷⢭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⡀⢤⣀⣴⣶⠿⠿⠛⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡄⠀⠹⡿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⣰⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⡇⣀⠀⠀⠘⡆⠀⠀⠀⣀⣠⣤⡠⢀⣀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⣠⣿⡿⠋⢀⣀⠤⠄⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⡿⡄⠀⠀⠈⠀⠀⠈⢉⠀⠀⠀⠀⠒⠭⢑⡲⢤⣾⣿⠃⠀⠀⠀⢀⣰⣿⡿⢖⡫⠥⠂⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⠹⡀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⢀⠙⠂⠨⣇⠀⠀⠀⠀⢘⢟⡥⠖⠉⠀⢀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠀⠐⠋⠀⠀⠀⠀⡿⠋⡀⠠⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⡏⠉⠉⢹⠀⠀⠀⠀⠀⠀⡟⠉⡉⢹⠄⠀⠀⠀⠀⣰⣶⣦⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⡠⠚⣦
⠀⠀⠀⠀⠀⣧⣤⣤⣼⠀⠀⠀⠀⠀⠀⡇⠻⠟⢸⡆⠀⠀⠀⠀⢿⣛⣈⢋⣛⣛⡀⠀⠀⠀⠀⢀⣾⡧⠞⠁
⠀⠀⠀⠀⠀⠧⠹⡏⠼⠀⠀⠀⠀⠀⠀⣷⣖⣒⣾⠁⠀⠀⠀⠀⠀⠙⠿⠾⠋⠀⠀⠀⠀⠀⠠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣈⣿⣯⣀⠀⠀⠀⠀⠀⣼⣿⣦⣴⠉⢱⠀⠀⠀⠀⠀⣴⡄⢰⣶⠀⠀⠀⠀⠀⢠⠾⠇⠀⠀⣿
⠀⠀⠀⠀⠀⣿⢋⣿⠉⠀⠀⠀⠀⠀⢈⠿⡁⢀⣵⡁⠀⠀⠀⠀⣤⣾⣶⢲⣷⣦⠀⠀⠀⠀⢸⠠⣟⢙⡆⣿
⠀⠀⠀⠀⠀⣿⣤⣯⣤⠀⠀⠀⠀⠀⢮⣀⠜⠻⣿⡿⠀⠀⠀⠀⠿⠿⠋⠈⠻⠿⠀⠀⠀⠀⠸⣄⣈⣉⣀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⣴⣦⣼⣧⣴⣦⡀⠀⠀⠀⢀⣴⣶⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣶⣷⡆⠀⠀⠀⠀⣰⣦⡿⢻⣤⣦⡀
⠀⠀⠀⠉⠈⢹⣙⣇⡯⠁⠉⠀⠀⠀⣴⣾⡸⣿⣀⣽⠇⠀⠀⠀⢠⠞⠉⢛⣿⠃⠀⠀⠀⠀⣹⣿⡷⢨⣿⣏⡁
⠀⠀⠀⠀⠀⠐⢦⡯⠃⠀⠀⠀⠀⠀⠹⠿⠟⠂⠉⠉⠀⠀⠀⠀⠘⠦⠔⠩⠶⠃⠀⠀⠀⠀⠹⠛⣷⣾⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⠀⠀⠀⠀⣶⡶⠲⠂⣠⣶⡄⠀⠀⠀⠀⠠⣞⠙⣷⠀⠀⠀⠀⠀⠠⡟⠁⠛⡆
⠀⠀⠀⠀⢀⡆⢠⡄⢰⡄⠀⠀⠀⠀⣿⣿⠙⢒⣿⣿⡇⠀⠀⠀⢀⡀⣙⣛⡋⣀⠀⠀⠀⠀⠀⠳⣄⡴⠣⡖⡆
⠀⠀⠀⠀⠘⢓⣤⠁⠸⠟⠀⠀⠀⠀⠛⠛⠀⠋⠛⠛⠁⠀⠀⠀⢸⡿⠿⠷⠞⠉⠀⠀⠀⠀⠀⠀⠙⢦⡤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠼⠧⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⠶⠶⠶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠤⠤⠤⠤⠤⣤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡞⢠⠇⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡼⢀⡟⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣀⣀⡼⠁⣾⣀⣀⣀⣀⣀⣀⣀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⢀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⣀⡤⣶⣒⣒⣶⢤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⡤⣶⣒⣒⣶⢤⣀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⢠⣞⡵⠋⠉⠀⠀⠉⠙⠺⣷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠕⠋⠉⠀⠀⠉⠙⠾⣷⣄⠀⠀⣿
⢸⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⣿
⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿
⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿
⠈⠛⠛⠛⠋⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠘⠲⣄⣀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣬⣤⡀⠀⢀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠐⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠉⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠀⠀⠙⠻⣦⠀⠀⢀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠙⣧⠀⠀⠛⠶⢶⡶⠶⠶⠶⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠻⢦⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠷⢦⣄⣀⣀⣀⣀⣀⣠⣄⣀⣀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠀⠀⠀⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⣿⣿⡷⣄⠀⠀⠀⠀⠀⠀⣀⡉⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⣀⠀⠀⠀⠀⠈⢷⠀⣹⣿⣷⠘⠳⠶⠶⠶⠞⠋⠏⣀⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⣠⣄⠀⠀⠀⠈⣰⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⣿⣿⠀⠀⢀⣠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡟⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣈⡻⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢈⣉⠁⡄⢈⣉⠉⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠃⠀⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠈⠉⢁⣠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣁⣀⣘⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢷⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠘⡀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣀⠇⠀⠀⠀⠀⠀⢸⡸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⡄⠀⠙⢤⣀⣀⣀⡀⣀⠜⠁⠀⣼⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⣀⣀⡈⠢⢄⣀⡬⠭⠆⠈⠀⣀⡤⠊⠀⣀⣉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠑⠲⠤⣉⡓⠒⣉⠭⠕⠒⠋⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠦⣄⡀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⣧
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠉⠉⠉⠉⠛⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠘⠇⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⡟⠁⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⣧⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠃⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣿⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⢄⣼⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣤⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠘⢷⣌⠳⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣀⣁⡀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣞⣉⣉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠛⠳⣤⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⢦⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡼⠋⠀⢰⠖⢲⡆⠈⠻⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃⣶⠶⠿⠀⠸⠷⢶⡆⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠿⠶⣶⠀⢠⡶⠾⠇⣸⠃⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⣄⠀⠸⢦⣼⠇⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢻⣄⠀⢠⡞⠉⣀⣀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢁⣀⠈⠻⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠿⠿⠀⣸⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠺⠽⠇⢀⡿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣀⣀⣀⣀⣀⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠒⠀⠀⠀⠐⠂⢨⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡦⠀⠉⠀⠀⠀⠈⠁⢠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣲⠤⣀⡠⢴⡋⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠀⠀⠈⠳⡀⡰⠋⠀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠉⠂⢵⠥⠊⠁⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⡰⢫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⡀⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⢠⡎⠉⠉⢢⠀⢀⠾⠓⠋⠉⠑⠂⠤⣀⣀⣀⡀⣀⡠⠄⠒⠉⠉⠓⠻⢆⠀⢠⠊⠉⠉⢦
⠀⠀⠀⠀⠀⡴⠋⠀⠙⢦⡰⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣠⠞⠉⠈⠳⡄
⠀⠀⠀⠀⢸⢱⢠⠂⡤⢸⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⣀⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⠠⡄⢢⢰⢹
⠀⠀⠀⠀⠘⢦⠈⢸⠱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⡤⠤⠒⠒⢄⡀⠀⠀⠀⠀⠈⠶⢹⠘⣠⠞
⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡇⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠷⠴⠒⠒⠉