Lug Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡱⠀⠀⠀⠀⢶⣖⣚⠡⠤⣄⠙⡄
⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⢠⠪⠈⠀⠀⠀⠀⠀⠀⢨⠃⠉⠉⠉⠷⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠳⢄⡀⢀⡤⢴⡑⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⠉⣙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠖⠚⣽⣴⡾⢟⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠶⡁⢗⠻⢦⠄⡀⠀⢀⠔⡝⡱⠑⢠⡾⠫⢌⣕⣣⡾⠛⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠊⠀⡇⠈⢆⠀⢑⡽⠖⠃⡼⠀⢧⣠⡟⡑⠣⢎⡾⠋⡔⠻⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣣⠃⠀⠀⢸⠀⠈⠞⠉⡆⠀⢀⡇⠀⠀⣿⣔⡶⠯⣿⢁⡞⠀⠀⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢧⠀⠀⠀⣸⡠⢼⡇⠀⢸⣿⣒⡀⠀⠘⡾⣄⣀⠤⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠀⠈⡇⠀⣿⢼⣓⢲⣄⣀⠉⠉⠛⠂⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢿⡒⠬⢼⠃⠘⠗⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠻⢦⣼⣆⣀⡴⠃⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⠀⠈⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢱⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠶⠄

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⠀⠀⠀⣠⣶⣿⣤⣤⣴⠟⢛⣴⣷⣶⣶⣶⣶⠖⢀⠔⢈⣤⣾⣷⣦⣀⠈⠛⢷⣌⢳⣄⡵⠶⠷⢤⡙⠿⡛⠿⢏⡑⢦⡄
⠀⠀⠀⠀⣿⣿⣿⣤⠞⠁⠀⠛⠋⠉⠉⠛⠛⠁⡴⠋⠔⣩⡤⠤⣬⣍⣙⡛⢲⣤⠙⢿⣧⡀⢶⣶⣤⣈⠂⠈⠢⡀⠹⣦⡀
⠀⠀⠀⠀⣿⣿⣿⠁⠴⠿⠿⠿⢿⣿⣿⠋⣠⠎⠀⠀⣩⣴⣶⣦⣀⠈⢹⡇⣾⣿⠀⠀⠹⣿⡶⣍⡻⢿⣷⣄⠀⢌⠳⣌⠁
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⠁⣰⠏⣴⣆⢾⡁⢾⣉⡷⠙⣷⡄⣇⢸⣿⡆⣦⣀⣘⣧⡈⠙⠲⠉⣿⣷⡌⢳⡜⠃
⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⡿⠃⣼⠃⣼⣿⣿⡆⠈⠳⣷⠶⢛⡿⠁⢸⡀⣿⣧⠛⡉⠉⢹⡟⠳⣄⠀⣿⣿⣿⡀⠟⠂
⠀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣤⣤⠀⣼⠃⣼⣿⣿⣟⢿⡄⠀⠀⢀⣿⠁⢀⡈⣇⠸⣿⢀⡽⣗⢾⣧⣄⡀⢸⡿⠿⣿⡇⢸⡄
⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⠇⢰⡏⢰⣿⣿⣷⠉⢉⣷⠀⠀⣼⣇⣻⣨⠣⠿⠀⣿⡌⣳⣯⣼⡧⢤⠄⣼⢡⠄⣿⡇⢠
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⠀⢹⡇⢻⣿⣿⣿⣧⠸⡜⣇⠀⣿⣿⡇⣿⢰⣶⣶⣿⣷⣿⣾⡏⢰⣶⡦⣽⡏⢠⣿⡇⢸⠇
⠀⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠃⠘⢷⣈⠻⢿⣿⣿⣷⣤⣿⡀⢿⣿⡇⣯⣬⣍⣭⣭⢹⣿⣿⠁⣿⣿⡇⣧⢀⣾⣿⠁⡿⠄
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣾⣿⡷⠄⠙⠷⣦⣉⠛⠿⣿⣿⣧⠸⣿⠇⠰⣞⠳⠶⠖⣾⣿⠃⣼⣿⣿⡇⣿⣿⡿⠃⣸⢧⡀
⠀⠀⠀⠀⣿⣿⣿⢋⣉⣉⣤⣤⣤⣤⣶⣦⣌⡉⠛⠶⣤⣄⣹⡆⠙⠛⡛⢮⡙⢶⣴⣿⠁⣾⣿⣿⣿⢧⡿⠛⢁⡜⠳⣄⠁
⠀⠀⠀⠀⣿⣿⣿⠘⠻⠿⠿⠿⠟⠛⠛⠋⢉⣀⠀⢰⣶⣄⡉⠙⠒⠤⠉⠼⠇⠀⣿⣿⣿⣶⣶⣿⠏⢠⡄⠺⣏⠹⣆⠙⠇
⠀⠀⠀⠀⣿⣿⣿⡀⢰⣦⣤⣴⣶⣶⣶⣿⣿⡿⠀⣿⣿⣿⣿⣿⣶⡶⢀⣤⠀⣾⠋⣉⡛⠛⢛⣡⡆⢸⡄⢢⡘⢧⡘⣧
⠀⠀⠀⠀⣿⣿⣿⣧⠸⣿⣿⡿⠿⠟⠛⠋⠁⡀⠀⣿⣿⣿⣿⣿⡿⢁⣾⣿⡆⢹⡆⢻⣿⣿⣿⣿⡇⢸⡙⢄⠹⡌⢳⣌⠃
⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⣀⣀⣠⣤⣴⣶⣿⡇⠀⣿⣿⣿⣿⡿⢁⣾⣿⣿⡇⠈⣧⠘⣿⣿⣿⣿⡇⢸⠷⡈⢢⠘⢆⢹⡇
⠀⠀⠀⢸⣿⣿⣿⠿⣇⢸⣿⣿⣿⣿⡿⠟⠋⢀⡄⠹⣿⣿⣿⠁⣾⣿⣿⣿⣷⠀⢹⣷⣄⣉⣉⣉⠀⢺⣇⠙⣄⠁⡄⢻⡇
⠀⠀⠀⠀⣿⣿⣿⠀⣿⡀⠛⠋⠉⣀⣠⣤⣶⡿⠉⠀⣀⣉⡛⢳⡜⢿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡄⢸⡍⢳⡈⢢⠈⢾⡇
⠀⠀⠀⢸⣿⣿⣿⠿⠛⡇⢰⣾⣿⣿⣿⡟⠉⠁⠀⣴⣿⣿⣿⣿⣿⣆⠻⠿⠟⣀⠾⢿⣿⣿⣿⣿⡇⢸⡟⢦⠹⣄⠱⣼⡇
⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⠀⣿⡿⠟⠁⣠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣴⣶⣚⣁⣀⣀⣀⣀⣀⣀⠁⢸⣷⡌⢣⡘⣿⡏⠁
⠀⠀⠀⢸⣿⣿⣿⣦⡼⠿⡇⢀⣠⣴⣾⠃⢀⡶⢤⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣹⠇⢰⣧⡙⣆⢻⡿⣧⡀
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣧⠈⣿⣿⠋⢀⣾⣿⣶⣬⣙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢸⣏⠳⣈⣿⢧⣀⡁
⠀⠀⠀⢸⣿⣿⣿⠷⠾⠟⠻⡆⢸⠇⢀⣾⣿⣿⣿⣿⣿⣷⣦⣍⡻⢶⣤⡀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢸⡟⣆⣻⣧⠀⠁
⠀⠀⠀⢸⣿⣿⣿⠀⠀⣠⠴⠇⢤⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⢡⡏⠈⠙⠳⢤⣴⠏⠀⠀⠀⠀⠀⢸⣿⣾⣧⣬⣍⣁⡀
⠀⠀⠀⢸⣿⣿⣿⢠⡞⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠉⠁
⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠿⠟⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠀⠐⠊⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⢿⣿⡎⢻⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⡿⠁⠀⠀⠀⠀⠀⡄⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣾⡾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣾⣿⠁⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⡀⠀⠀⢶⡦⠬⠉⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢻⢐⣋⠸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣼⡀⠀⠀⠀⠄⠀⡄⠀⠀⢀⣀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠘⠈⠠⢐⡏⢿⣟⣿⣾⢿⣿⣿⣿⡔⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⣠⠀⠀⠈⡀⠈⠀⢸⢰⠾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠀⠀⡀⠀⢀⠀⢟⠀⠀⢿⠃⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠇⠔⠘⡆⣿⡇⡴⣾⣁⢻⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⣰⠀⠀⠀⣦⠁⠀⣿⡁⠀⢻⡟⡾⣗⢁⠀⣄⠀⠀⠀⠀⠀⠀⠈⠀⢢⣤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡆⣄⢿⠆⣀⠀⣇⠆⢰⣿⠓⣶⣾⣇⡕⣺⡟⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠹⢾⡆⡁⡐⡯⠀⠈⢸⠆⠛⠙⠛⠠⣿⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⣆⣳⡰⠁⠳⠜⠃⠻⠇⠀⠀⠎⠄⠌⠁⡆⠀⣠⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣿⣟⣇⢀⣀⠀⣀⡄⢀⣀⠀⠀⢀⣠⣶⣺⣨⣯⣻⣳⡄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠐⣶⣶⡶⠒⢠⣎⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠹⣿⡟⡿⣝⣛⢛⣂⡀
⠀⠀⠀⠀⠀⠀⢄⡲⢦⣬⣂⡘⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠉⠀⠀⠲⠈⠀⢀⠤⠉⠜⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⢛⠉⣉⠈⠰⠢⠬⢕⣀⣈⣙⢯⠉⠉⠻⣿⣿⠟⡖⣥⣂⡀⠠⠰⠺⣿⣻⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠶⠞⠒⠾⠛⠛⠛⡛⠛⠿⠷⢦⣥⡴⣶⣿⣷⣦⣀⣤⣤⣴⣮⣽⣙⠒⠲⠶⠌⠧⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠛⡛⠻⠛⠛⠻⢿⣷⡶⣶⣶⣶⣶⣧⣄⢀⣈⠛⠛⠛⠛⠛⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠂⠉⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠿⠛⠛⠉⠉⣿⣿⣿⣎⡙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠿⠷⠶⠒⠒⠒⠒⠀⢀⣾⣟⠻⢍⠛⢿⣦⡈⠻⣿⣷⣄
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⡿⠶⠤⠤⠀⠀⠀⠀⣠⣾⣿⣿⣷⣍⠢⢄⡀⠈⠛⢦⣈⠻⣿⣷⡀
⠀⢠⣿⣿⡟⠉⠀⣽⣿⡟⠉⢈⣿⣿⡿⠛⠉⣿⣿⠟⢹⣿⡟⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣟⠿⣯⣙⠦⡙⠢⣄⠀⠉⠃⠈⠻⣿⣦
⠀⣾⣿⡏⠀⠀⣼⣿⡏⠀⠀⣼⣿⡿⠁⠀⣰⣿⠁⠀⣾⠀⣶⣾⣿⣿⡇⠀⣼⣿⣿⣿⢷⣦⡉⠳⣤⡙⠷⣄⠠⣌⠓⢄⡀⠀⠀⠈⣿
⢰⣿⣿⠀⠀⢠⣿⣿⠀⠀⣸⣿⣿⠃⠀⢸⣿⡏⠀⣸⣿⢠⣿⣿⣿⣿⡇⠀⢸⣿⣿⡟⢷⡉⠻⣦⣈⠛⢦⣌⠑⠄⠑⢦⡁⠀⠀⠀⣿
⣼⣿⣿⠀⠀⣾⣿⡿⠀⢠⣿⣿⣿⠀⢀⣿⣿⠃⢠⣿⣿⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⢿⣦⡈⠲⣌⠻⢷⣄⠉⠳⣄⠀⠀⠙⢦⡀⠀⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⣸⣿⣿⣿⠀⢸⣿⣿⠀⣸⣿⣿⣾⣿⣿⣿⣿⡇⠀⢸⣿⣿⡶⣌⠛⢦⡈⠳⣄⡙⠷⣤⡈⠳⣄⠀⠀⠙⢦⣿
⢹⣿⣿⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⠀⣾⣿⣿⢀⣿⣿⣿⣿⣅⣈⣁⣀⣀⣀⠀⠙⠻⢷⣌⠳⣄⠙⢦⡀⠙⢦⡈⠻⢦⡈⠳⣤⣶⣀⣿
⠀⠹⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠈⠛⢦⡙⠢⡄⠙⢷⣌⠻⣦⣼⣿⡿⠋
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠙⠂⠘⢦⡈⣿⣷⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣤⣄⣀⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣬⣷⣦⣼⣮⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠟⠛⠉⠉⠁⣸⣿⣿⣦⣉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⠛⠋⠉⠉⠉⠉⠁⣠⣾⣿⠭⡓⠉⠛⢷⣌⠻⣿⣧⡀
⠀⢠⣶⣶⡶⠒⢶⣶⣶⠞⢿⣿⣿⠿⢿⣿⣿⡿⣿⣿⠋⠉⠉⠀⠀⠀⣠⣾⣿⣿⣿⣷⢮⡑⢤⡀⠈⠳⠄⠛⣿⣦⡀
⠀⣾⣿⠏⠀⢠⣿⡟⠁⢀⣾⣿⠏⠀⢰⡿⠁⢰⠛⣣⣤⣴⣶⠀⢠⣾⣿⣷⣬⡙⢦⣙⠳⣌⠀⢈⠳⣄⠀⠀⠈⢻⣿
⢰⣿⡟⠀⠀⣿⣿⠀⠀⣼⣿⡏⠀⢰⣿⡇⢀⣿⠀⣿⣿⣿⣿⠀⢸⣿⣿⡷⣌⠻⣦⡈⠳⣬⡑⢦⡑⢬⡑⠀⠀⠀⣿
⣾⣿⡇⠀⢸⣿⣿⠀⢸⣿⣿⠃⠀⣿⣿⠀⣸⣿⠀⣿⣿⣿⣿⠀⠘⣿⣿⢿⣌⠓⢌⠻⢷⣌⠙⢦⡀⠀⠙⢦⡀⠀⣿
⣿⣿⡇⠀⢸⣿⣿⠀⣿⣿⣿⡄⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⡶⣌⠳⣄⠙⢦⡙⠳⣄⡙⢦⡀⠀⠙⢾⣿
⠹⣿⣇⠀⣼⣿⣿⣀⣿⣿⣿⡇⣸⣿⣇⣸⣿⣿⣿⣦⣤⣤⣤⣤⣄⠈⠙⠻⠌⠳⣌⠳⢦⡙⢶⣌⠻⣦⡙⢶⣷⣼⡿
⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⠑⢦⡙⢦⠙⢷⣌⣳⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠁⠈⣷⣼⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠾⠿⡾⠿⣿⡷⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⢉⡙⠋⢠⣶⣿⠷⣶⣶⣶⣤⡀⠀⣶⠙⠻⣿⣶⣤⣤⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⢸⣯⣤⣤⣤⣄⢰⣶⠄⣠⣤⣤⣤⣤⣤⡀⢴⣦⢀⣤⣤⣤⣤⣤⣄⢰⣶⠄⣀⣤⣤⣤⣽⡇
⠀⠀⠀⠀⠸⣿⠉⠙⣿⣶⣶⣶⡿⠉⠉⠉⠉⠻⣷⣶⣶⣾⠟⠉⠉⠉⠉⢿⣶⣶⣶⣿⠋⠉⣿⠏
⠀⠀⠀⠀⠀⢿⣇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣸⡿
⠀⠀⠀⠀⠀⠘⣿⡀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣠⣤⣄⣠⣤⣤⣤⣄⣤⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣧⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣉⠉⠓⠦⢬⣙⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⣀⠤⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠤⣈⠙⠲⣌⡛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠙⢦⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⠋⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⡐⠢⢄⠀⠀⠀⠀⠀⠀⠡⡀⢻⣷⡄
⠀⠀⠀⠀⢠⡿⠃⣠⠋⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠉⠉⠛⠛⠿⣿⣿⣷⣄⠑⢦⠀⠀⠀⠀⠀⠘⡄⢹⣿⡄
⠀⠀⠀⣰⡿⠁⢠⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠳⡄⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⣴⡟⠁⠀⡎⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢹⡄⠀⠀⠀⠀⠀⠈⣿⣿⣆
⢀⣼⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⢷⠀⠀⠀⠀⠀⠀⢹⡇⣿⣷
⣾⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⢿⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠈⢻⣧⡀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⡞⠀⠀⠀⠀⠀⠀⣿⣏⣿⡟
⠀⠀⢻⣷⡀⠀⣧⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣰⠁⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠹⣷⡄⠸⡆⠀⠀⠀⠀⠀⢆⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⢀⡴⠃⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠙⣿⡄⠙⣄⠀⠀⠀⠀⠀⠱⣄⠙⠿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣶⡿⠟⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠘⣿⣆⠈⢦⠀⠀⠀⠀⠀⠀⠙⠢⢄⡉⠙⠉⠉⠉⠉⠉⠉⠀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠢⣄⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣮⡙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢟⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣮⣝⡛⢶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⢿⣻⣽⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠙⠛⠿⢭⣍⡛⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠛⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡉⠛⠶⣌⡛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣈⠙⢶⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⠑⢦⡀⠀⠀⠀⠀⠀⠈⢢⠈⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⡿⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠀⠈⠉⠉⠛⠿⣿⣿⣷⣄⠙⢦⡀⠀⠀⠀⠀⠀⢱⠈⢿⣿⣄
⠀⠀⠀⠀⢀⣾⡟⠁⣰⠇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠱⡄⠀⠀⠀⠀⠀⠡⠘⣿⣿⣆
⠀⠀⠀⢠⣿⠟⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⢹⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⣠⣿⠏⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣣⠀⣧⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣧⡀
⠀⢰⣿⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣷
⠀⢸⣿⡄⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿
⠀⠀⠻⣿⡄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⡿⠃
⠀⠀⠀⠹⣿⣆⠀⠘⣇⠀⠀⠀⠀⠀⢀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣰⠇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠙⣿⣆⠀⢹⡀⠀⠀⠀⠀⠀⢣⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠘⢿⣧⠀⢳⡀⠀⠀⠀⠀⠀⠳⣄⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠳⡄⠀⠀⠀⠀⠀⠈⠳⣄⡉⠻⠿⣿⡿⢿⣿⣶⣶⡿⠿⠟⠛⠉⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣈⣁⠀⠀⠀⢀⣀⣀⣤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⠦⣄⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⡓⢦⣄⡈⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⣟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣬⣙⡻⢶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⡿⣟⣫⣵⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠀⣀⠄⠤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠒⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠊⠀⠀⠀⠈⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠬⠤⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠦⢄⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⣀⠀⠀⡀⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠊⠁⠀⠀⠀⠀⠉⠢⡀⠀⠀⡄
⠀⠀⢀⠂⣠⢀⠘⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣝⣁⡀⢀⠀⠀⠀⠀⣀⠀⠨⠢⣠
⠀⠀⠀⠀⠃⡟⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⡌⠉⠁⠈⠀⠀⠈⠁⠀⠙⠀⠀⠉
⠀⠀⠀⠀⠀⠱⠀⠀⠀⢱⠢⠤⠉⠂⠉⡤⢊⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⣆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢸⡀⢀⡀⠂⠀⠀⠁⡠⠂⠀⠀⡏⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠘⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⡆⠀⢣⠀⠑⠀⠀⠀⠀⠁⠀⡀⠀⠁⠀⢀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠸⢵⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠸⣇⠀⠀⠑⢕⠂⠀⢀⣀⣤⠊⠀⠀⢀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⣦⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⢄⡀⠀⠈⠉⡙⠉⢡⠃⠀⠀⠀⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠈⢒⡤⠊⠐⣠⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⢀⠠⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⡄⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠄⣴⡼
⠀⠀⠧⣀⢠⣀⢼⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠾⠛⠉
⠀⠀⠀⠀⠈⠉⠀⠉⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⡀⠔⠀⣀⠔⠉
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡲⠖⠉⢠⡔⠛⠁⠀⠀⠀⠀⠀⠀⢀⡐⢒
⠀⠀⠀⠀⠉⠙⠒⠀⠤⢀⠀⠀⠀⡀⠔⠀⠀⣠⠔⠁⠈⠑⠒⠢⠤⣀⡈⠀⠐⠒⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢈⠀⢀⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⡀⠝⠛⠒⠦⢤⣀⠈⢰⠀⠀⠈⠉⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⣞⡷⠊⠀⠀⠀⠀⣂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣄⡆⠀⠀⠀⢀⣻
⠀⠀⠀⠀⠀⢠⠶⢤⣆⠀⠀⢠⡽⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠒⠄⡤⠤⣀⠀⠀⠀⠀⠈⠓⠶⣦⣠⣸⠏
⠀⠀⠀⠀⠀⠰⡞⠉⣹⣆⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⢿⣿⣶⣦⣤⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠿⠿⠷⠶⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣰⣾⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠙⠛⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠿⠛⠁
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡀⠀⠀⠀⣠⣾⠟⠁
⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦⣠⣾⠟⠁
⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣧⡀
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠻⣿⣦⡀
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠙⢿⣦⡀
⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠛⠉⠉⠉⠀⠀⣄⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⣀⣤⠴⠶⠒⠀⠀⠀⠈⠳⡄⡄⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⠞⣉⡤⠖⣛⣻⣿⣂⠙⢦⠀⠙⣿⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡾⢁⡾⢉⡴⣫⣽⣿⣿⣧⠄⠈⣧⠀⠸⡿⡄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡾⢡⠏⡴⢋⡾⡽⣫⣿⠛⠁⠀⠀⢸⡄⠀⠇⠙⠆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⠁⠈⣼⢁⣞⡞⣽⡿⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠰⢤⣀⢣
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇⠈⣼⠹⣿⠁⠀⠀⠀⠀⠀⡿⠀⡴⢠⠆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⡼⠁⣼⣳⠃⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠞⢁⡼⣳⠏⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠙⢦⣄⣀⠀⢀⣀⡤⠞⠁⣠⠞⢡⡏⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢤⡀⠀⠈⠉⠉⠉⠁⢀⣠⠞⠋⢠⡟⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠉⢛⣒⣤⣤⠤⠄⠁⠀⠀⠠⠞⢁⣀⣀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡭⠭⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠉⢉⡉⢉⣉⣉⣉⣉⠉⣉⠉⣉⡉⣉⡉⢉⡉⢉⡉⢉⣉⠉⣉⠉⣉⣉⣉⣉⡉⢉⡉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣏⣉⣹⣿⠀⣿⠀⣿⡇⣸⡇⢸⡇⢸⡇⢸⣿⠀⣿⠀⣿⣏⣉⣹⡇⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣈⣁⣈⣉⣉⣉⣉⣀⣉⣀⣉⣁⣈⣁⣈⣁⣈⣁⣈⣉⣀⣉⣀⣉⣉⣉⣉⣁⣈⣁⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⡟⠛⣿⠛⠛⠛⠛⠛⢻⡟⠛⢻⣿⠛⠛⠛⠛⠛⠛⣿⡟⠛⢻⡟⠛⠛⠛⠛⠛⣿⠛⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣿⠀⠀⠀⠀⠀⠀⣿⣇⣀⣸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣟⣫⡭⠷⢖⣛⣛⣉⣉⣉⣉⣉⣙⣛⡒⠾⠽⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠾⣛⣭⠷⣾⣻⣭⠭⠶⠖⠒⠒⠒⠒⠲⠯⢭⣙⠓⠦⣌⠙⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣫⡶⣛⣭⠖⣛⣭⡴⠖⠚⣋⣉⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⡷⢛⡽⢿⡿⢟⣩⠶⣋⣭⠶⠒⣋⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣶⣿⣿⣋⣴⣋⣵⠞⣉⣤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⣦
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡟⣿⠉⠉⠉⠉⠉⠉⢹⠋⠉⢉⠉⢙⡉⢻⠋⡟⢻⢻⣿⣿⣿⣿⠛⠛⠛⠛⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠐⠀⠀⠀⠀⠀⢰⠄⠀⢠⠀⢈⡁⢸⠀⡇⢸⢸⣏⣧⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⢈⡀⠀⠀⠀⠀⢠⠄⠀⢀⠀⢈⡁⢸⠀⡇⢸⢸⠿⠇⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡏⢈⡁⠀⠀⠀⠀⢠⡀⠀⠈⠀⠸⠇⢸⠀⡇⢘⣴⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⡄⠀⠀⠀⠀⢈⠁⠀⠘⠀⠐⠂⢸⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⡄⠀⠀⠀⠀⢸⠁⠀⢸⠀⠰⠆⠈⣸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢸⡇⠀⠀⠀⠀⠘⠀⠀⢰⠀⢰⡆⢠⢻⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⡀⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢨⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢰⡇⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣧⣼⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⢰⡄⣸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠀⠀⠉⠙⢳⣦⣀⡀⠀⠀⢠⡀⢼⢸⡇⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣤⣤⣴⠞⠋⠀⠉⠙⠶⢦⣬⣄⣿⢸⣇⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠙⢿⣿⣿⣿⡿⠋⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠙⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠙⠿⠋⣠⣾⣿⠟⢁⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡈⠻⣿⣷⣄⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠆⢀⣀⣀
⠀⠀⠀⠀⣠⣶⣄⠙⢿⣿⣦⡈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢁⣴⣿⡿⠋⣠⣶⣄
⠀⠀⣠⣾⣿⣿⣿⣷⣄⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⣠⣾⣿⣿⣿⣷⣄
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢝⡦⠀⠀⠀⠀⠀⣶⣖⡂⠾⠦⠤⣄⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⠊⠀⠀⠀⠀⠀⠀⠀⠉⢻⠋⠒⡒⠒⠻⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡛⢢⡀⠀⠀⠀⠀⠀⠀⢠⢎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠤⠀⠁⠀⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠀⣠⠖⢰⡥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⠂⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡖⠭⣈⢍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⠖⠋⣯⣴⠾⡿⣟⡿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⠚⠱⠮⣢⣄⡀⠀⠀⠀⠀⢀⡤⢶⠞⡝⢧⣀⣠⣼⢟⡹⢛⡴⢋⣥⡿⠟⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡔⠉⢣⠀⢧⠀⠙⢏⣹⢦⠀⣠⠟⢠⠋⢰⠁⠀⣰⡿⣡⠊⡱⠛⣶⠟⠁⣠⣤⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠎⠀⠀⢸⠀⠈⢧⠀⢀⣹⠞⠋⠁⢀⡏⠀⠸⠦⣴⠟⣠⣁⡝⢃⣿⠋⢠⠞⠉⢿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢣⡇⠀⠀⠀⠘⡆⠀⠈⢿⠉⠹⡆⠀⠀⢸⠃⠀⠀⢰⣿⡒⠂⠾⢥⣿⠃⢠⠏⠀⠀⠈⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢻⡀⠀⠈⠀⠀⢹⠀⣀⣼⠀⠀⠀⣾⢷⣭⡭⠀⠀⠹⣦⣏⠀⠀⢀⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⠞⠋⠁⢻⠀⠀⢠⡿⢺⡒⣒⠀⠀⠀⠻⠽⣦⣾⡉⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡟⠚⢬⢭⣷⠶⢤⣄⠀⠀⠀⠈⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⣯⡓⢤⢠⣿⠀⠘⠟⠳⣦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢷⣬⣐⣿⠀⠀⣀⡼⠋⠙⠳⠶⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠉⠛⠿⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⣤



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠈⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⡬⠃⠀⠀⠀⠀⠀⠙⢿⣭⣍⣉⡙⠛⠲⢤⡀⠘⡆
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠭⠉⡉⠉⠈⢹⣦⡇
⢏⣨⠛⢆⡀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠆⠀⠀⠀⠀⠈⢉⠇
⠀⠀⠈⠉⠺⢕⡢⢀⢀⡴⠎⣺⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣒⠂⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠓⢮⣐⣋⠹⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⠞⣹⣃⣤⣶⣿⡿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣒⣊⡷⢮⣓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⡶⡋⠀⠀⣠⣿⠟⡻⢞⣡⠟⢓⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⢞⡜⠣⡄⠸⡌⠑⠮⡳⠦⢤⡀⠀⢀⣴⠟⣱⠃⣸⠀⠉⢛⣿⠟⢳⠞⣡⣞⣉⣼⠟⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣰⢫⠞⠀⠀⡇⠀⠹⡄⠀⠈⠲⣏⣽⡤⠾⠁⢰⠃⠀⣿⠀⢠⣾⠋⡰⠣⣴⢃⣴⡿⠁⢀⠴⣿⣦
⠀⠀⠀⠀⣠⠼⣱⠋⠀⠀⠀⢿⠀⠀⠹⣆⣠⣴⡞⠁⠀⠀⢀⡟⠀⠀⠉⢉⣿⠋⡺⠧⣼⣁⣼⡟⠀⣰⠋⠀⠈⣏⢠
⠀⠀⠀⠸⡡⣐⡇⠀⠀⠀⠀⢸⡆⠀⠀⢹⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⠀⣼⣿⣍⣛⠚⠳⢾⡟⠀⣼⠃⠀⠀⠀⢸⡞
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣧⠀⣀⣼⡁⠀⠀⢰⡿⣻⠤⠤⢌⠁⠀⢻⣴⣇⠀⠀⢀⣠⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣠⠴⠟⠋⠁⢸⡅⠀⠀⣸⡗⣿⠖⠒⠂⠀⠀⠀⢿⣹⣧⣾⡋⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⣇⠀⠀⣿⡗⠿⣭⡏⣳⣤⣤⣄⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢿⡗⣧⣸⠁⣿⡇⠀⢸⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣿⣄⢹⠉⣿⡇⠀⠀⠀⣨⠟⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⠙⠿⣦⣿⣇⣀⣤⠾⠋⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣄⠀⠀⠉⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢹⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣬⣧⣿⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣿⣿⣿⣿⣿⣿⣿⣶⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢤⣀⠀⠒⠶⢶⣶⣚⣿⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⠉⠐⢉⣤⡴⠶⠞⠉⠉⠁⠀⠀⠉⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠿⠋⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡞⠁⣠⡼⠛⠉⠉⠉⠉⠉⠙⠛⢦⣄⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠏⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢷⣄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣏⣴⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠃⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⣀⠀⠀⠀⠀⢘⣁⠀⠀⠤⠀⠀⠀⠠⠄⡶⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣧⠀⠀⠀⠈⢿⠛⠛⠒⠂⠠⢶⣶⡿⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣼⣧⡀⠀⠀⠀⠙⠶⠀⣀⠀⠒⠉⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣌⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠳⠟⡄⢻⡄⠀⠀⠀⠀⢴⠟⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠹⢿⡁⠀⠀⠀⠀⣰⣤⠀⠈⠑⢶⣄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⠀⠈⠙⣦⠀⠀⢠⡏⠈⢻⣆⡀⠀⢈⠿⠁⢸⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡇⠀⠀⠀⣤⣼⡇⠀⣼⠃⢀⠘⣿⡌⠳⣼⠃⠀⠀⢀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠉⠀⠀⠙⣦⡀⣰⠆⠉⠹⢾⠁⠰⠋⠹⠏⠙⣤⡟⠑⠦⣴⣎⢂⣼⠟⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣷⠀⠀⠀⢀⠀⢨⣧⡰⡦⠀⠙⠛⣄⡀⠀⠀⠛⠶⡶⠟⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣐⠂⠀⠘⠛⠀⠈⢹⣇⠀⠀⢸⣏⠙⢳⣤⡀⠀⠀⠀⠀⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⠖⠲⢆⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠈⣣⠀⠀⠀⣼⠉⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡷⠀⠀⠀⢸⡇⠀⠀⢻⣄⠀⠀⠙⠳⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠒⠒⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⠀⠧⠀⠀⠀⢸⡇⠀⠀⠀⠻⣄⡀⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣧⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠙⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⡆⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠇⠀⣿⠤⢤⠀⠤⠤⢤⡄⠀⠴⢄⠀⣀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡦⠀⠀⣿⠤⠀⠀⠐⠤⢤⡇⠀⠀⠨⠀⠆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃⠀⠀⠀⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢿⡽⠋⠀⠀⠀⠀⠀⠻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇
⠀⣶⣆⠐⠀⣿⠱⣦⠀⠀⠀⣠⠶⠶⣤⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇
⠀⡇⠈⠳⣄⠘⠂⠘⠳⣄⣀⣻⠀⠀⣽⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠲⡀⠀⠀⠉⠉⢛⠀⢀⠟⢷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿
⡘⢻⡛⠆⠀⠀⠈⠳⣀⠀⠀⠙⠉⠉⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿
⡇⠠⡿⠂⠀⣠⡄⠀⠈⠟⠀⠒⠐⠓⠒⠚⠛⠓⠚⢻⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠻⣄⠀⠀⠀⠉⣦⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠈⠳⣄⠀⠘⠏⠈⠿⠀⠢⠐⠀⠒⠀⠀⠀⠂⠰⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠉⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠳⠶⠦⠴⠶⠶⠶⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⠐⢂⣤⡼⠿⠞⣭⡔⠒⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⣀⣀⡐⠶⠿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⢶⣆⣉⣉⣉⣹⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠹⠉⠀⣄⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⣤⠴⠖⠯⢄⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡔⠋⠞⠀⠀⠀⢄⠀⠙⣄⡠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡙⠂⠀⠀⠀⠛⠂⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠀⠀⠀⠀⠀⠉⠒⣀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠒⢐⠒⢖⢇⠈⠣⣀⠀⠀⠀⢤⣀⠔⠁
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⢢⣀⣠⡑⢄⠈⠑⢄⠀⡄
⠀⠀⠀⠀⢠⠃⠈⢁⢂⡀⠀⠀⠀⠀⢃⠀⢱⠀⠀⣤⠋⢱⡎⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠆
⠀⠀⠀⠀⣨⠄⠀⠀⠀⠀⠐⠂⠀⠀⠈⡏⠉⠇⠀⠀⠀⢠⡽⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢁⡠⠖
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠀⠘⡀⠀⠀⠀⡇⠀⠇⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣠⣶⠁⣀⣄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠤⢤⡖⢮⠡⡀⠀⠀⠀⠀⣀⣤⡠⣤⡞⠋⢸⣷⣿⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⡇⠀⠈⠀⠉⠉⠐⠀⠅⢸⠀⢃⠀⠀⢀⡴⠭⡗⠊⠀⣧⡄⠸⠿⣻⠎
⠀⠀⠀⠀⠴⠥⠐⠛⠓⢎⠀⠀⣀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡀⢀⣸⢷⠀⠀⠀⠀⠰⣇⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡾⠋⢁⢼⠀⠀⢀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⡠⠖⢹⠀⠑⠂⡾
⠀⠀⠀⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠛⠒⠀⠄⣀⣀⣠⠞⠋⣀⣴⠋⠀⠀⠀⠳⠤⡄⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⣀⢀⣠⢀⣌⠀⠀⠀⠀⠀⠀⡙⣇⡤⠊⠁⡏⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⠀⠒⠒⠈⠓⢢⣄⠀⡀⡖⡇⣟⠒⠠⢤⡧⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢖⠟⢹⣠⡡⠗⠁⠀⠀⠉⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⣄⣐⠘⡄⠀⢀⣀⠀⠠⠄⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠏⠭⣵⣒⠛⠛⣂⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⡤⠔⠂⠹⠁⠀⠀⠈⣆⠌⠁⠀⠀⢱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡼⠟⠀⠀⠀⢁⠀⠠⡄⠀⣽⠐⠂⠀⣀⠎⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣉⣀⠀⠀⠀⠀⠈⠢⢄⡀⠔⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⠔⡆⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢺⠀⠑⠒⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⡆⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠁⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣷⣤⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡠⢦⡔⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢻⣿⣿⣿⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠰⣤⣝⠒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢟⡉⠛⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣽⡀⠀⠪⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⡤⠊⢀⠝⢛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠤⠤⠤⠤⠤⠄⢠⢬⡤⠒⣁⠤⠊⠀⣀⠜⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠨⠉⣁⠒⠋⠉⠁⠒⠒⠒⠀⠀⠤⠬⠛⠳⠤⠦⠴⠶⠤⠴⠃⠺⠓⠉⠀⠀⠀⠰⠁⡠⠰⢄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⣠⡃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠈⠢⠷⠀⠮⠚
⠀⠀⠀⠀⠀⠀⠤⠁⢅⣀⡸⢁⣱⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡜
⠀⠀⠀⢀⠴⠛⠚⠢⠤⠤⠤⠧⠸⠂⠤⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠜
⠀⠀⠰⠁⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠘⠆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠷⠰⣅⠧⠀⠀⠀⣌⣡⠄⣼⠀⠠⠀⠲⠀⠀⠈⠉⢳⠀⡖⠖⠲⠒⠂⠉⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠄⡁⠀⡆
⠀⠀⠓⠀⠀⠤⠤⠄⠀⠀⠀⣀⣘⠀⠤⠀⠚⠀⠀⠀⠀⡆⠀⢰⡇⠤⠅⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠴⠓⠉⠉⠉⠑⢦⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡠⠍⠘⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠒⢠⣦⠀⠀⣇⡀⠀⢠⠔⢀⣄⡀⢤⣤⡄⢀⠉⣙⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠱⣤⠀⠀⠄⠠⡆⠻⠒⠐⢾⣿⣿⠛⢲⠀⠄⠇⠀⠀⠀⠈⠾⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠀⠀⣤⡔⠀⢻⡇⡄⠀⠀⠀⠀⠀⢸⡀⠀⡀⡀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⣩⣭⣍⣷⠀⠘⣿⡇⠀⠀⠀⠈⠀⣸⡟⠏⠉⠉⠉⠉⡝⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⢹⣇⠀⢻⣿⡏⠋⠿⠿⠍⠹⡇⠠⠂⠀⠀⠚⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣤⣀⡠⡤⠮⣿⡖⢺⢺⡂⠀⠀⠀⠀⢸⣷⡁⢀⣀⣀⢀⣙⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆⢠⠀⢹⡧⠸⣿⠳⡙⠛⠓⢒⠾⣿⠀⠀⠀⠄⠀⠁⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢀⡄⠒⠀⠒⠒⠒⣒⡒⠒⠢⡤⠤⠴⢈⡆⠸⣷⠀⢻⠀⠈⣱⠊⠁⠀⣿⡀⢀⢋⡃⠀⡛⠁⣀⣐
⠀⠀⠀⠀⡔⠀⡇⠈⠀⠀⠀⠠⡉⣳⠀⠀⣥⠤⠴⡮⠿⠦⠟⠲⣾⠀⠀⣨⠀⠀⠀⣿⢀⣀⢀⣅⢰⡇⢠⠄⢨
⠀⠀⠀⢀⠁⠸⡅⢀⣀⠀⡀⢑⠬⣺⠀⢀⣗⠀⠀⣷⣒⣆⠀⡆⢹⡒⠚⠓⢺⠛⠒⣿⠛⢍⡉⠉⢛⡽⢛⠉⠉
⠀⠀⠀⡘⢀⢌⠆⣿⢸⠫⡢⡑⠥⡶⠀⢸⣿⠀⠈⠙⠉⠋⠀⠁⢸⠂⢀⣼⣾⣤⠀⣿⠀⠀⠈⢺⠁⠀⠸
⠀⠀⠀⠇⡭⡠⡍⣮⢸⢕⠌⠪⡑⡆⠀⢸⣿⣀⣄⣀⣤⣠⣄⣴⣾⣆⣀⣀⡀⣀⣀⣿⣤⢀⡀⣸⣀⣂⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡸⠓⠛⠋⠀⠉⠊⠛⠓⠛⣈⡁⠀⢸⣧⣨⣨⣬⠎⢏⢤⣮⣯⣭⣭⡩⢭⡽⠿⣽⣽⣽⣽⣽⣽⣽⣽
⠀⠸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡭⠭⠬⢭⠮⠮⠎⢬⢖⢒⣛⢛⠟⠑⣓⣱⢛⠳⡵⣜⣼⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⢘⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣬⣬⣬⣤⣵⣥⣿⣭⣭⣭⣭⣭⣬⡴⣥⣡⣎⣄⣕⣵⣡⡹⠀⠀⠀⢀⠔⠦⠠⢤⠖⡁⣤⠐⠇
⠀⠸⠀⠀⠀⢀⢾⡵⠒⠚⠺⢽⣄⠀⢠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡥⠛⠲⣿⡍⠉⠉⡟⣷⡄⠀⢀⠄⠉⠀⠀⠀⢀⠑⢄⠀⠀⠀⠀⡴⠲⠊⠒
⠀⠶⠂⠀⠀⣾⡞⢰⡟⠙⡷⠀⣿⣂⡀⢉⠁⢀⡀⣀⡀⣀⢀⣀⡀⡀⣀⣠⢿⢰⢟⡶⢸⢷⣠⠞⠀⠓⠃⠈⢠⠀⣀⠔⡐⠂⣈⡰⠊⠀⠀⠀⠀⠁⠂⠍⠓
⠀⠘⣛⡚⢛⡁⠘⣄⠙⠛⢁⡰⠡⠉⠉⠉⠉⠉⠉⠉⠁⠉⠀⠉⠁⠁⠁⠁⠘⢆⡉⡡⠖⠂⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⡈⠑⠠⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠾⢇⠸⠷⠾⠗⡸⠷⠀⠀⠀⠀⠀⠀⢠⠁⢠⣀⣀⣀⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⡆⠀⠀⠠⠀⢰⠉⠱⠀⠀⠀⠀⠀⢠⠀⠸⠟⠛⠛⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⣴⠆⠐⣦⡈⠒⠊⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠈⠛⠓⠔⠐⠢⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⡐⢛⣛⣛⡛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠎⠉⠉⠾⠅
⠀⠀⠀⠀⠀⠀⠓⠛⠃⠂⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⢠⢤⡖⠁⠒⠉⢲⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⠿⠧⢠⠭⠿⣷⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠚⠿⠛⠛⠛⠘⠿⠛⠂⠀⠀⠀⠀⠀⠀⠛⠺⣤⣤⣤⣤⠓⠛⠀⠀⠀⠀⠀⠀⢀⠈⠉⠙⠛⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠿⠇⠨⠇⠆⠿⠇⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣀⣑⣂⣒⣀⡿⠋⠁⠀⠀⠀⠀⠀⣠⣼⠟⠰⠆⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠙⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠤⣀⠤⠤⠤⠤⠤⠤⠤⢄⣠⠴⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⣰⡦⠄⠈⠀⠀⣈⣉⣉⡀⠀⠈⠀⠤⣶⡀⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⣴⠁⣤⣄⡀⠀⢰⡀⠉⠁⣠⠀⠀⣀⣤⡄⢹⣤⣈⣇
⠀⠀⠀⠀⠀⢠⡤⣾⣿⣷⡄⠀⠉⢀⣧⣄⠛⣯⣻⣷⡾⢗⣛⣓⠿⣶⣿⣫⡿⢁⣤⣇⠈⠁⠀⣴⣿⣿⡦⣤⡀
⠀⠀⠀⢠⣾⣿⡿⣿⣿⣿⣿⡄⢠⠏⣠⣶⣤⠀⠈⠭⠾⣿⣿⣿⡿⠮⠍⠀⢀⣴⣦⡈⢷⠀⣼⣿⣿⣿⡿⣿⣿⣦
⠀⠀⠀⡞⢈⠀⠁⢸⣿⣿⣿⣧⣸⠀⣾⠁⠙⠧⣠⡀⠀⠉⢻⠋⠁⠀⢠⣠⠞⠁⢹⣆⢸⣠⣿⣿⣿⣿⠀⡅⢸⠘⡇
⠀⠀⢰⡇⢸⠀⠁⢸⣿⣿⡯⣭⣭⣭⣭⣿⣶⣶⡇⢳⡶⠖⠛⠓⠶⣶⠇⢷⣶⣾⣿⣭⣭⣭⡭⣽⣿⣿⠀⡇⢸⠀⣳
⠀⠀⠘⣧⢸⠀⡇⠘⣿⣿⣿⣿⠙⣏⠹⣯⣭⣭⠴⠋⣠⠶⠟⠷⢦⡈⠳⢼⣼⣭⡿⢉⡟⢹⣿⣿⣿⡿⠀⡆⢘⢠⡟
⠀⠀⠀⠸⣤⣦⣶⣶⣿⣿⣿⣿⣼⣹⣦⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⣞⣻⣼⣿⣿⣿⣷⣶⣦⣦⡼
⠀⠀⠀⠀⠙⠿⢿⣿⣻⣿⡿⠃⠉⠉⠹⣄⡍⢻⣿⢶⣶⣤⣤⣤⣤⣶⠾⠿⣯⣅⡼⠋⠉⠁⠻⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⢸⣇⣾⣏⣀⠀⢐⢲⣶⠀⢀⣈⣦⣝⣿⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠉⠉⠛⣦⡟⠋⠉⠁⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄⠀⠐⠒⠂⠀⠠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠠
⠀⠀⠀⠀⠀⢀⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣀⡒⠀⠀⠀⠀⠀⠂⠀⠆⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢹⣿⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠛⢍⠉⠁⠉⠋⠉⠉⠙⠋⠉⠉⠉⠉⢉⠑⠒⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣶⣶⣶⣦⣤⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠈⠁⠒⢄⡀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡑⠄⣀⠀⠱⡀⣿⣿⣿⣿⣿⣿⣿⣏⠦⢄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣒⣲⣦⣾⣿⣛⣛⡻⣷⡘⠿⠿⠿⠇⢿⣿⣿⢿⣿⡿⡿⠟⠐⠿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢀⣀⠀⢀⣾⣿⣿⣿⡿⠛⠛⢷⡜⣷⣶⡶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁⢤
⠀⠀⠀⠀⠀⠀⠀⢻⣇⣸⣿⣿⣿⡿⠛⠛⠛⣿⣿⣿⣿⣿⡏⠀⠹⡿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠐⠈⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠟⠛⠃⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠠⠀⣿⢸⣿⣦⢤⣤⠤⡶⢶⣶⣶⣶⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣾⢣⣿⣿⠃⠀⠀⠀⠀⠀⠛⠿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠷⠛⠋⠁







⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠫⠁⢠⡇⢠⠄⠈⠟⡿⠛⠒⠀⠀⠀⠀⠀⠀⠀⢀⠴⠲⠶⢤⣄⣀
⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠀⣀⡄⢠⡀⠀⠀⠀⡀⠈⠓⣆⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢠⠏⠀⢀⣤⠴⢦⡤⠶⠞⠁⠀⠀⠀⠀⢃⡄⡀⠈⢑⣠⠄⠈⠃⠘⠁⠀⠀⠈⠁⢀⢠⡙⠀⠀⠀⠀⠈⠳⠶⢤⡴⠦⣤⡀⠀⠹⡄
⠀⠀⣠⠏⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡟⠁⠰⠟⠑⠆⢀⠆⢲⠀⠰⠂⢠⠄⠈⢯⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠹⣄
⠀⣰⠋⠀⠀⠀⡏⠀⠀⠀⠀⣀⣠⣄⠀⠀⡘⢁⠘⡀⠀⠀⠀⠀⠂⠈⡄⠰⠀⠀⠀⡀⠀⠀⢈⠁⡉⡂⠀⠀⣠⣄⣀⠀⠀⠀⠀⢹⠀⠀⠀⠙⣆
⣴⠃⠀⢀⠀⠀⢿⣠⡴⠖⠋⠉⠁⠈⠓⢦⠉⠇⠀⠙⠀⠀⢀⣬⠀⠀⠃⠀⠀⠠⠅⠀⠀⠐⠋⠀⡿⠙⡴⠚⠁⠈⠉⠙⠲⢦⣄⡾⠀⠀⡀⠀⠘
⣿⠀⢀⠌⠀⢀⠿⢅⡀⠀⠀⢀⣀⠠⠔⠘⣇⡆⠀⠀⠀⠀⠂⠀⠀⠞⠁⠘⠦⠀⠀⠐⠀⠀⠀⠀⢀⣹⡇⠢⠄⣀⡀⠀⠀⢀⡨⠿⡀⠀⠡⡀
⣧⢀⡞⠀⢀⠀⠀⠀⠈⠉⠛⠏⠁⠀⠀⣠⢇⠑⠀⠀⣀⠀⠂⠔⠀⠀⡆⢘⠀⠀⠆⠀⠀⣀⠀⠀⠀⡾⣅⠀⠀⠈⠹⠛⠉⠁⠀⠀⠀⡀⠀⢳⡀
⠙⠭⢤⣚⣁⣀⣀⣀⣀⣀⣀⣀⡤⠖⠋⠀⢸⠂⠐⠀⣈⡉⣉⠉⠃⠀⡿⣿⠀⠉⠉⡉⢉⡁⠀⠂⠐⠇⠈⠙⠲⢤⣄⣀⣀⣀⣀⣀⣀⣈⣓⡤⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢀⠀⢀⡀⠂⣀⢀⡆⢶⠀⣀⠐⢁⡈⢀⡀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⢂⣈⡤⠆⠈⠉⠃⠘⠈⠁⠰⢜⣁⡀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠤⠀⠐⠤⣤⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣤⣤⡄⣷⣶⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠛⠛⠉⠉⠁⠛⠛⠟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣛⣷⡄⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⣽⡿⠯⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⣀⠀⠠⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠒⠈⠁⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⡿⢻⡄⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠚⣿⡟⣻⠀⠀⣀⡄⣾⣿⣿⣿⣧⠀⠀⠀⠀⢰⣿⡿⢻⠉⡏⣻⢦⣀⠀⠀⡠⣴⣿⣿⣿⣦⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⣡⢼⢋⣿⣇⣹⠀⠈⠉⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠲⡶⠃⢸⠀⡇⣿⢸⠙⣦⣄⣧⢿⣿⣿⣿⣿⠎⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣡⠔⠋⠀⣜⣘⣛⣛⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠇⢸⠀⡇⠿⢸⠀⠇⢸⠈⣎⣻⣿⠟⡏⢩⠁⣿
⠀⠀⠀⠀⠀⠰⠊⣠⠔⠋⠀⠀⠀⠐⣏⣉⣉⣉⣹⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠹⣿⣷⣍⣸⣁⣅⣨⣀⣁⣏⣸⣉⣏⣿⣹⣁⣏⣷⡿
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣯⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠰⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⡗
⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠐⡏⠉⠉⠉⠹⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣀⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣻⢿⣿⣿⢵⣻⣿⣻⢽⣿⣟⣽⣿⣿⣿⣦⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢷⣻⣿⣹⡟⣿⣳⣿⣻⡿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣈⣽⣶⣷⣷⣾⣷⣿⣶⣿⣷⠷⡿⡶⣿⣿⣿⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡶⣲⡄⠁⣴⢶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣷⣗⢿⣿⣿⣿⣿⣯⣿⣿⣿⣷
⠀⠀⣴⣖⣦⣤⣟⣿⣿⡿⡙⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠘⢻⣿⣿⣷⣟⣿⣟⠟⠛⠿⠛⠛⠁
⠀⠀⢻⣿⣿⣟⣿⡏⣾⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠢⣺⣿⣿⣿⡏⠉⠁
⢸⣿⣏⣿⢾⣿⣿⣿⣿⣿⣝⣿⣾⣿⣿⣿⣿⣿⣿⣿⣀⣁⣀⣤⣾⣿⣿⣿⣿⣷⠠
⠀⡽⣿⣿⣹⣿⣿⡏⣿⣟⡿⡟⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀
⠀⠙⠻⠛⠋⠉⢿⣿⣿⣿⣿⢻⣟⣾⣿⣿⣿⣽⠟⠛⢻⣿⡿⠛⣿⣿⣿⣿⣿⠇⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠉⠉⠉⠉⠁⠀⠀⠈⠛⠁⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⣀⣠⠤⠒⡆⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡙⢇⠀⠁⠉⡠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠈⠢⠐⠉⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣀⡤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠛⢿⡇⠇⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡐⡍⢄⣀⠜⠀⠈⢄⠀⢀⢀⠀⠀⠀⠀⠙⢗⢆⠠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⣥⠀⠀⡎⢣⠀⠀⠢⠈⡻⠄⠀⠀⠀⠀⠈⢎⢇⠀⠀⠉⠉⠉⠛⠒⣦
⠀⠀⠀⠀⠀⠀⡰⠇⣰⠀⠀⡇⣠⠞⢇⠘⠦⠤⢒⣡⡂⢳⡀⠀⠀⠀⠀⠈⢎⢆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣰⠃⠀⠀⢧⠀⠹⡅⠀⠈⠒⠤⠒⡁⠀⠑⠘⡉⠀⠀⠀⠀⠀⠈⢎⢆⠀⠀⠀⣴⠃
⠀⠀⠀⠀⢠⡇⡤⠐⠃⠈⠢⣀⠈⠢⡐⠒⠉⢁⡤⠖⡚⠓⠲⠤⢌⠀⠂⠀⠠⢌⡊⠂⠀⠘⠁
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⡋⠉⠲⣌⣆⡀⠈⡇⠀⠉⠀⠤⢖⠂⠈⠙⠲⠀⢀⡿⢀⠔
⠀⠀⠀⠀⢠⡇⡀⢀⠄⡸⠂⠭⣑⡒⠯⢄⣀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⣠⠴⣋⠔⠁⢀⣀⣄⡀
⠀⠀⠀⠀⣼⠁⢀⡞⠀⢧⠀⠀⢰⠀⣉⡐⠢⠭⣑⡒⠤⠼⠧⠤⠴⢖⣋⠥⠊⠀⠀⢠⠋⢀⠀⠹⡄
⠀⠀⠀⢰⡇⠀⡾⣀⠈⠙⠲⢤⣾⡀⠒⠭⣝⡒⠦⣌⠉⠐⠒⣂⠉⠉⢀⡠⠐⠁⢀⡇⣰⠋⠑
⠀⠀⠀⢸⠁⣸⠁⣸⢳⠶⢤⣄⡀⠉⠛⠲⢤⣌⣹⠃⠀⠀⠀⠈⢧⠀⠀⠀⣠⠄⠘⠀⢿⡀⡀
⠀⠀⠀⡿⠀⣏⠀⠿⠬⠥⠀⠈⠉⠛⠲⠦⣤⣀⠉⠙⠲⠦⠤⠤⠾⠶⠒⢋⣡⡶⠃⢀⠈⠉⢁⡴
⠀⠀⠀⠛⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⠴⠶⠶⠛⠋⠁⠀⠙⠺⠷⠖⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⠶⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⣌⣴⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣴⣶⣶⣌⠆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣇⣀⣶⣼⣭⣏⣻⡦⣄⠀⠀⠀⠀⠀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣻⠀⠀⠀⠀⡨⠐⠒⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡴⠃⠀⠀⠀⠀⠖⠀⠀⠐⠊⠦
⠀⠀⢠⠒⠒⠤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⠖⠀⢲⡀⠀⠀⠁⠶⠀⠸⡠⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠢⠤⣄⣬⣝⣸⣿⣿⠿⣿⣿⠿⠿⠓⠚⢿⣿⣿⣿⣿⣿⣿⡿⢻⣤⣄⡠⠜⠀⠀⠀⣠⠞⢄⡸⣑⠦⢀⠀⠀⣰⠁⠾⡒
⠀⠀⠀⡰⢊⣤⣤⡀⠈⠉⠙⠦⣌⠙⠥⣀⣀⣀⣀⣐⣿⠿⣿⡿⠟⠉⢉⣀⡈⠑⡄⠀⢠⠊⠀⠈⠈⢈⠈⠀⠠⡟⠉⠑⠉⠉
⠀⠀⢠⠃⢸⡏⠉⠉⠀⠀⠀⠀⠀⠉⠓⠣⠤⠤⠯⠭⠓⠊⠁⠀⠀⠔⠉⠙⣿⡆⣹⠀⡑⢠⠀⠀⠀⠀⢠⡴⠢⡕⠄
⠀⠀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢿⡄⠀⣷⣶⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠟⠀⡇⣰⢀⠎⣃⠀⠀⠀⠈⡻⠤⡐⢄⠀⡄
⠀⠀⡠⠷⠤⠤⠤⠤⠄⠄⠀⠒⠆⠀⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠤⣤⣀⣰⠂⠃⠛⡎⠀⠈⠲⣰⠏⠀⠀⡁⠀⠉
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠉⢹⠒⠒⠒⠒⠒⠛⠛⡻⢹⠀⠀⠀⠀⠀⠀⢸⡦⠤⠤⠤⠤⠤⠤⢤⡖⠓⠒⠒⠛⣆⠀⠀⣶⠀⠀⢘⠒⠒⠒⣒⡂
⠀⠀⠀⠀⠘⡄⠀⠀⠠⠀⠀⢠⠇⠈⠉⠉⠉⠉⠉⠁⠉⡇⠀⠀⠀⠀⠀⠀⢸⠉⡿⣽⠄⢀⠙⠒⠁⠏⠀⠐⠏⠉⡿⠝⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠤⠊⠀⠉⠁⠀⠛⠒⠒⠚⠳⠶⠤⠟⠀⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠋⠁⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡋⠃⡀
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⢾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢂⢀⣀⣀⣀⡇⠀⡀⢀⢀⣀⡀⡀
⠀⠠⠀⠑⣀⠁⡀⠘⣮⠂⠈⢀⠌⠠⣅⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡡⣬⣴⣯⡥⠬⣷⡞⢿⣿⣿⣿⣿⣇
⠀⠀⢀⡀⠇⠀⠁⠂⣯⠀⢡⣿⡀⠒⣌⡴⠉⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠛⠒⢲⡲⠷⢶⡒⡶⠖⢆⣾⠿⡟⡟⠛
⠀⠀⠀⠀⠀⠐⠀⠀⠛⠒⠚⠒⠓⠂⠓⠛⠐⠓⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠚⠂⠘⠛⠛⠑⠀⠸⠴⠟⠓⠘⠀⠀⠂

⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢀⢀⡀⡀⠀⠀⠀⣠⣄⠀⠀⢠⡄⠀⢀⡄⠀⠀⣠⠦⡄
⠀⢰⣾⠿⣿⣧⠀⠀⠀⣠⣤⡀⢤⡄⠀⠀⠀⢠⣿⡿⢿⣶⠀⠞⠛⣿⣷⢀⠚⣸⠀⡇⣯⣷⠀⣼⣿⡟⠳⠄⠰⠇⠀⠘⠗⣀⢀⢼⡄⠁
⠀⠠⠀⠀⣿⣿⠚⠋⠉⠺⠿⠟⠟⠧⠈⠈⠉⢺⣿⡇⠀⠠⢀⠀⠀⢿⣷⡾⡀⠛⠻⠟⠛⣾⢲⣾⣿⠁⠀⡠⠀⠀⠀⣴⠿⢻⣾⠉⠓⢺⣇⡀
⠀⠀⠻⠷⡿⡿⠀⠶⣦⠀⠀⠀⠀⠀⣶⠦⠀⠸⡿⡷⡿⠃⠀⠠⣶⣾⣿⡇⠈⠲⠶⠶⢶⡏⠘⣿⣿⣶⡤⠀⠀⠀⡸⠁⠀⠀⣿⠉⠀⡊⠹⠄
⠀⠀⠀⣀⡀⠀⠀⣀⣿⡏⠉⠉⠉⢹⣿⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣸⡇⠀⢈⢈⣀⣀⠀⣀⣀⠃⠀⠀⢰⡿⣀⠈⠓⠚⠁
⠀⠰⠟⢻⣿⡀⡀⢈⣉⣀⣀⣀⣀⠈⠉⢁⣾⡿⠻⠇⠀⣠⣶⣠⣄⡄⠀⢠⡶⠀⣠⣶⣦⠀⠀⣼⣿⡿⡿⠘⡿⢿⣿⡅⠀⢸⣿⢘⣿⡀
⠀⠀⣤⣸⣿⡏⠉⠀⠀⢀⠀⠀⠀⠀⠈⠉⣿⣇⣤⡜⢀⡿⠟⣉⣉⡅⠘⢿⣿⣄⣼⣿⣿⠃⠀⡙⠟⠛⠦⣢⠞⠛⠟⡃⠀⠘⣿⡨⠿⣧⡀
⠀⠀⠉⠙⠋⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠿⢷⣄⠈⠠⠀⠀⠈⠛⣋⣀⣈⠁⠀⢸⣧⢕⢄⣊⠉⣃⢔⢨⣿⠀⠀⠈⠁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⢤⣄⣀⠀⠀⠀⢀⠀⠘⠓⢾⠝⠃⣸⢠⣤⣴⣿⣍⠙⢦⡀⠀⠸⣿⣷⣿⡿⠸⣿⣷⣿⣿⢠⣶⣶⣦⡀⢹
⠀⠀⠀⣠⡄⠠⣀⣀⡼⣿⡜⢀⣂⡤⠀⠀⠀⢘⣦⣀⣀⣼⣧⣄⣼⣿⣿⣿⡻⣿⣷⠀⢿⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⣿⣿⣿⠇⠿
⠀⠀⢸⣿⠁⠀⡏⠉⠁⠟⢻⣮⡽⣷⠉⠉⠂⠈⢿⠛⣟⠣⣄⣿⣿⣿⣿⣿⣿⣾⣿⣇⣾⠟⠀⠀⠀⠀⠀⣟⠻⣧⡀⠀⠀⠈⠉⠁
⠀⠀⢨⣿⠿⢿⣥⡀⢠⣺⣿⣿⣿⡝⣷⣤⣀⡄⠈⠳⡝⣇⣀⣉⣩⡼⣿⣿⣿⣿⡿⠋⣠⣴⣿⣧⡤⠆⠀⠮⠚⠷⢦⡴⠂⠶⢦⣿⣿⣷⣶⣶
⠀⠀⣆⡼⣧⣸⣿⡿⣿⣿⣯⣿⣿⡇⣿⣽⣿⣿⣷⡄⠘⢮⣿⠿⠿⠏⠁⠀⠉⠁⠀⠸⣿⣿⣿⣿⡇⠈⠲⢦⣶⣴⣾⠟⠳⠊⠸⣿⣿⣿⣿⡿
⠀⠀⢾⡁⠙⣛⣵⣾⣿⣿⣿⡾⠿⠚⢱⠹⣿⣿⣿⣷⠀⠀⢀⡀⢤⢠⣀⣀⣀⣠⢢⠀⣈⠛⠉⠀⠀⣤⣄⠸⣿⣿⣿⢠⣆⡄⠀⠀⠈⠙⠛⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⢻⣛⣻⠄⠈⠀⠘⢿⣿⡟⠁⠚⢹⣷⣌⣒⣈⠻⢋⣅⣺⣴⣿⠛⠂⠀⠀⠈⢬⡟⠙⠋⠙⠳⡌⠀⠀⠠⠄⠀⢄
⠀⠀⠀⠉⠁⠀⠘⢾⣿⡟⢻⠏⣹⡁⠘⠀⠀⠀⠀⠀⠀⢡⣼⣿⠽⣛⣈⡀⣀⣘⡫⢿⣿⣤⠄⠀⠀⢠⣿⠇⠀⠀⠀⠀⠛⠇⢀⠀⠸⠽⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠃⠩⠃⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠋

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⢶⣋⡉⠻⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢀⠀⠈⠙⢶⡴⣾⠷⠖⣠⡴⢖⣠⣦⠀⣀⣤⡄⢀⣤⣤⡔⢻⡟⠛⢛⣟⣀⣤⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠙⠷⠤⣤⠼⠃⠻⢤⠤⠟⠀⠘⠿⠧⠶⠏⠸⠧⠼⠳⢾⡇⠘⠷⠤⠜⠛⠁⠙⠶⠶⠄⠀⠀⠐⢯⣤⢿⣤⣾⠟⣧⣀⢾⣋⡿⢧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣹⠀⠀⡀⠀⠀⠀⠀⢠⡄⢠⡟⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⣾⠋⣰⣏⡿⠺⣯⡙⢢⡦⠀⡿⠀⣴⠀⣼⣿⠇⢀⣴⣶⠀⣾⠛⠃⣴⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠙⠷⠟⠷⠖⠻⠶⠛⠘⠃⠀⠛⠚⠛⠀⠙⠛⠚⠛⠁⠻⠆⠙⠓⠂⠙⠷⠦⣤⣤⡀


⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⢤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓



⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠀⣀⡀⠀⠠⠤⠌⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣴⠂⡌⢉⣁⣠⡶⠿⠛⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣴⠷⠋⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣠⠟⠁⣠⠶⠚⠉⠉⠉⠛⠷⢶⣄⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣼⠃⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣦⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢰⣁⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⠉⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢠⣤⠀⠀⠀⠸⣅⣀⠐⠀⠀⠊⠁⠒⢲⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠿⡄⠀⠀⠀⠻⡟⠛⠁⠀⠲⠶⣾⠟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣽⣦⣀⠀⠀⠈⠋⠰⠦⠀⠈⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⣌⣻⡿⢳⡀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⠉⢴⣯⣧⠀⠀⠀⠀⠚⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡌⠉⣿⠀⠀⠀⠀⡿⢷⣄⠀⠈⠑⢦⡴⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⣹⠆⠀⢠⡟⠀⢻⡷⢦⣀⡸⠁⢀⠀⡟⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠎⠙⠳⣄⠀⠀⡾⠛⢷⣴⠟⢀⡶⣤⡿⢀⡘⢇⣀⡄⢀⡘⢀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠳⡾⠀⠀⠀⠙⣆⠈⠀⠀⠀⠈⢿⡠⣄⠈⠻⢶⡟⠁⠀⠛⢶⣄⣠⠴⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⢳⡆⠈⠱⣿⠀⠀⠀⣿⠳⣤⣀⠀⠉⠉⠀⠀⡀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢤⠄⢤⠄⠀⠀⢙⠀⠀⠀⢻⡇⠈⠙⣷⠀⠀⢀⡰⠛⢦⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⢻⡀⠀⠈⠳⣦⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⢆⡀⠻⠀⠀⠀⠘⠃⠀⠀⠈⢳⣄⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠳⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠻⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠙⠳⡴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⢸⡷⠰⠆⠶⠂⠲⠦⠀⠾⠦⠄⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠃⠀⠘⠇⠀⠀⠀⠀⠰⠦⠄⠀⠠⠶⠀⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣼⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⡏⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣠⣤⡴⠛⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⢶⠆⠀⠀⠀⢸
⠀⢀⣄⣤⣄⣸⡧⠛⠀⠀⠀⠀⣀⣀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠘⠉⢦⡀⠈⡅⠻⣦⡀⠀⢟⠁⠙⢋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠙⢦⡀⠀⠈⠛⠶⠞⠀⢠⣷⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⣧⢴⡆⠀⠀⠙⠤⠀⠀⠀⠂⠀⠞⠀⠉⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡆
⠀⡇⢘⡧⠀⢀⡀⠀⢁⠖⠒⠒⠒⠐⠒⠐⠂⠈⠻⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇
⠘⢧⠈⠀⠀⠉⢹⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠳⣄⠀⠀⠞⠉⠿⠆⠰⠶⠶⠶⠶⠶⠶⠶⢾⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠳⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠳⣄⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⣤⣀⣀⣀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠉⠀⢻⠙⠛⠒⠖⠒⠛⠛⠁⠙⠛⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣿⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⢀⣀⣀⣰⣶⡤⠴⠛⠉⠘⠒⠭⣤⣖⣀⣀⡠⠤⠄⠠⠀⠤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠚⠛⠖⠒⠚⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⡛⢛⡛⠛⣛⣛⣛⣛⣄⣛⠛⣿⡛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣤⣭⣭⣭⣭⣿⣿⣿⣏⣉⣉⣿⣏⣉⣉⣿

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠉⢆⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⠊⠀⠀⡠⠓⠌⢆⠀⣠⠔⠁⠀⠉⠐⠒⠊⠉⠀⠀⡼
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⠜⠉⠂⣈⠜⠊⠀⠀⠀⠀⠀⣀⣠⠤⡶⢁⠞⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⠔⠁⠀⠀⠾⢥⣤⣤⠤⠐⠒⠊⢉⠇⢀⡜⡰⠋
⠀⠀⠀⢼⣁⣀⣀⠴⠚⠁⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⡸⢀⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢳⠀⣠⠠⢡⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⡤⠋⠁⡵⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢒⣲⠾⠤⠤⣗⣀⡴⠣⢥⡤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⣠⠔⠊⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠐⠢⣀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⡞⠁⡒⠉⠉⠉⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠙⠢⣄
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⢄⠈⠐⠀⠐⠒⠉⣀⣤⣔⢲⢲⠂⡆⡶⠀⠀⠀⣨⠆⢠⠐⠀⠀⠈⠳⣄
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢹⠒⠤⣤⢔⡉⠀⠀⠀⠉⠙⠚⠴⠁⣀⠤⠊⠁⠀⠲⠀⠸⠀⠀⢀⡈⠳⡄
⠀⣠⠁⠀⢀⠠⠄⠀⠒⠒⠂⠸⣇⠀⠀⠀⠈⠁⠒⠀⠀⠐⠒⠊⠉⠱⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠙⣆
⡌⠁⠀⢘⡳⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠘⣆
⠙⢆⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢫⠋⠀⠀⠀⠐⠢⢄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠢⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢤⣤⠄⠚⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠣⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠠⠓⠢⡈⠣⢄⠀⠀⠀⠀⠀⢀⡜⠈⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⢸⣅⢢⠀⠹⡀⡠⢿⠶⠦⠄⠐⢌⢣⡤⡾⠄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡁⠜⠤⡀⠏⡰⠃⠀⠀⠀⠀⠀⠂⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠃⠠⠃⠀⠀⠀⠠⠂⠈⠈⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⣀⣀⡀⠀⢀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄⡄⡄⠠⢀⣀⠀⠀⢀⣶⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠛⠃⠓⠃⠐⠈⠘⠋⠀⠘⢿⣿⣿⡇⢱⣴⡱⠂⠀⠀⠀⡤
⠀⠀⠀⠀⠀⡈⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⡄⠈⣛⢛⠒⠁⣠⠀⠀⠀⠀⠆
⠀⠀⢰⠀⠠⣤⡁⡋⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⡷⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠉⠀⠀⠄⠀⠦
⠀⠀⠀⠰⠿⠾⠏⠀⠀⣠⣤⣤⠀⠀⣾⣿⣿⣿⣿⣷⠤⣤⣶⠚⠂⠀⠀⣦⣤⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⣀⣟⠀⢼⣿⣿⡛⢿⣿⡧⠀⠠⡛⠂⠀⠠⠀⣿⣀⣈⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠈⠁⣠⣼⣿⣷⣶⣦⡀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⡴⠢⡤⠲⡄
⠀⠀⢀⣖⣠⣠⣄⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢇⠠⡧⢠⡇
⠀⠀⠀⡉⠹⠿⢋⠉⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠙⢦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠈⠉⠋⠉
⠀⠀⢸⣧⡌⡞⢸⢷⡿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣀⣱⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⢁⢁⡀⣠⣤
⣿⣶⣷⣶⣶⣶⣤⣴⣶⣶⣶⣤⣶⣶⣾⣿⣿⣶⣶⡶⠶⢼⣽⣿⣷⣶⣶⣶⣶⣶⡶⠐⢶⣶⠚⣶⣾⣾⣿⣿⣿
⠿⠿⠿⠿⠿⠿⢿⣿⠧⣿⡿⢿⠟⠿⠛⠛⠁⠛⠻⠃⠀⠈⠹⠟⠛⠿⠛⢿⠟⠙⠻⠶⠀⠙⠻⠷⠮⠹⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣾⣾⣷⣿⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⡟⠻⣧⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣽⡤⢤⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣶⡟⣿⢻⣷⡆⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠞⠀⠐⣶⣶⣾⣿⣿⣿⣿
⢿⡿⠿⣿⣿⡿⣿⢲⠿⠾⢿⣯⡖⣿⢿⡿⠿⠿⠿⣿⡿⠿⢿⠿⠿⢿⡿⠿⡿⣿⠀⠆⠰⢰⠿⣿⣿⠿⣿⣿⣿
⣸⣁⣀⣾⣼⣧⣽⣿⣁⣀⣸⣿⣥⣿⣤⣿⣀⣀⣀⣿⣇⣀⣘⣇⣀⣼⣁⣀⣿⣸⣏⣀⣈⣿⣥⣿⣿⣏⣉⣉⣿
⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢠⣀⢀⡄⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣛⣒⣂⡉⢹⣿⣽⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣵⣟⡿⣫⠍⠁⠀⠀⢼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣮⣵⣿⣿⡿⠋⠀⣠⣶⠈⣀⣾⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣀⡀⣠⡶⠟⣛⣯⣽⣿⠟⠀⢠⣾⣿⣿⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⣉⡵⠀⣯⣶⣜⡻⠽⢑⢿⡀⠀⣿⣿⣿⣿⣆⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⡟⠁⢀⣼⣿⠁⠀⠈⠻⣮⣝⡻⢾⠃⠀⠀⠘⣿⣿⣿⠏⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣸⣿⢟⣻⣿⣿⣿⣿⠁⠀⣾⣿⣿⠀⠀⠀⣘⢼⡻⣟⡳⢦⠀⠀⠀⠹⣿⡿⠀⠀⠀⠘⢛⣿⠁
⠀⠀⠀⠀⢀⣆⡿⣱⣿⠟⣿⣿⣿⣿⡇⢸⣿⣿⣿⡀⠀⠀⠘⢷⣝⢶⣍⠳⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⣠⢾⣿⣼⣿⢏⠀⢻⡇⣿⣿⡇⢈⣿⣿⣿⣧⠀⠀⠀⠈⢻⣷⣹⡇⠀⠀⢀⠀⣿⣿⣧
⠀⢀⣮⠵⣿⣿⣿⣏⣼⠀⠘⣷⠸⣿⣇⡾⠉⠻⣿⣿⣧⡀⠀⠀⢈⡟⢱⣷⣶⠾⠋⣸⣿⡿⠃
⠀⢀⡴⣞⣯⣿⢟⣽⣿⠀⠀⢻⠀⣿⡿⠁⠀⠀⢈⠛⠿⣿⣦⣀⠈⠀⣼⠋⠑⠶⣼⣿⣿⠁
⠀⣴⠿⣿⣛⣵⣿⣯⣝⠂⠀⢸⡆⠋⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠙⣿⡏
⠀⠙⠳⠾⠯⢭⡙⠻⢿⣷⡀⣸⣇⡆⠀⠀⣴⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣿⡇⠀⠘⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡝⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⠇⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣟⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁
⣿⡿⠿⡿⠻⢿⡿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡖⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⣴⣾⡛⠁⠀⠀⠀⠀⠀⠀⠀⣶⠗⠀⠀⠀⠀⢤⢰⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⣏⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣰⡆
⠀⠈⠳⣿⣷⠀⠀⠀⠀⣠⡴⣾⣽⣿⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⡌⠢⡄⠀⠀⠀⣸⣿⠻⢦⣠⠤
⠀⠀⠀⢸⣿⣇⠀⠀⠈⠁⠠⠇⠸⣿⣆⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⠀⠀⠀⠀⠀⠠⣇⠀⣾⡟⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⡏⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠋⠉⠳⣄⠀⠀⠀⠀⣸⣿⡿⠃⠀⠀⢀⣿⡏⠀⠀⠐⢢⡀
⠀⠀⠀⢠⡏⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⢠⡄⣢⠀⠀⠀⠘⣦⠀⠀⠀⠀⠈⠀⠀⢠⠞⢿⡟⠀⠀⠀⢠⡟⣻⡇⠀⠀⣸⣠⡇
⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢸⠁⠀⠀⠀⠈⣷⣿⡇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣶⠀⠀⠀⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣄⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⣿⠀⠀⠀⠸⣷⣄⣠⡼⠇⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡯⠉⠛⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⢠⡏⠀⠛⠀⠀⠀⢀⣾⣿⡏⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⣀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢈⣿⣿⠉⠁⠀⠀⠀⣀⣤⣹⣧⣤⠄⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡏⠈⢧⠀⠀⠀⠀⠀⠀⠈⠀⣿⣇⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⣾⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⢷⠀⠀⠀⠀⠀⠀⠀⡰⠋⢹⡇⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠇⠀⠀⠀⠀⣸⠉⢸⠇⠀⠀⣰⣟⣤⢓⡤⠀⠀⠀⡀⣀⡃
⠀⠀⠀⠀⣿⣴⣆⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⣿⣿⡿⠟⠁⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠈⢻⣿⣿⡆⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢠⡀⠀⠀⠀⠃⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⢀⡾⣿⢻⡄⠀⠀⠀⠀⠀⢀⣧⣤⡄⠀⠀⠀⠀⠀⢸⡀⣿⣀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⣿⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠸⢰⣿⡆⠇⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡇⠀⠀⠀⠀⢸⠉⡇
⠀⠀⠀⠀⠙⠃⠘⡿⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⢸⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⣸⠁⠀⠀⠀⠀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣇⠀⠀⠀⠀⠘⡇⠈⢻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡇⠀⠀⠀⠀⠀⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠀⠀⠀⠀⠿⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⡁⠀⠀⠀⠀⠀⠀⣠⣀⠐⡆
⠀⠀⠀⠈⡇⣤⡴⠗⠀⠀⠀⠀⠀⠀⠀⠀⣾⡬⠀⠀⠀⠀⠀⠐⣥⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣧⠖⠒⠆⠀⠀⠀⠰⣧⣼⡁⠀⠀⠀⠀⠀⠀⢳⣾⡶⠇
⠀⠀⠀⠀⢿⣿⣷⠾⠉⠓⠀⠀⠀⠀⠠⠶⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⡟⠛⠓⠈⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢰⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠋⢻⡄⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠆
⠀⠀⠀⠀⢸⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⣸⡇⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⢻⠉⣿
⠀⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⣿⠀⠘⠓⠀⠀⠀⠀⠀⠀⣿⠇⠀⠉⠁⠀⠀⠀⠀⠀⢸⠇⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉
⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠔⠉⠉⠀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣐⡀⠀⠀⠀⣰⠊⠁⠀⠀⠁⠀⠀⠉⢢⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠚⠀⡼⠋⠀⠀⠀⠀⠚⠄⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠻⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡈⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠦⠀⢀⡿⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⣄⣀⡴⢱⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⣸⣷⠶⣿⣥⣴⣄⣠⣤⣶⠃⠀⠀⠀⣿⡿⠆⠒⠀⢶⠟⠛⠉⠻⡁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣈⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⠀⠠⠒⠒⠚⠛⠛⠛⡛⢊⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⢉⡂⢀⣀⣁⣀⣀⣴⣤⣤⣤⣤⣀⣈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⢿⣿⣿⣿⣭⣭⣭⣥⣤⣤⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⠁⠙⠋⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢭⣀⡀⢀⢬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣾⣟⠀⠈⠁⠉⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣻⣧⡢⠀⠀⠀⠀⠀⠀⣿⣿⡇⠁⢠⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣟⡛⢿⡦⡀⢀⣀⠤⠊⢻⣿⣷⣤⠀⡾⢿⣦⣌⢀⢄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣩⡿⢷⣌⡛⢷⡼⠒⠛⠮⣻⣿⣿⡼⠁⢸⣿⣧⣄⣄⠞⠣⡘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣘⡤⢖⣉⣀⣤⣤⡴⠛⠁⠀⠀⠀⠀⠻⣿⢟⣵⣷⣾⣿⣿⣿⣿⣷⣄⠀⢐⣽⡙⠲⢤⣀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⢸⠀⣀⣀⣤⢶⣉⠉⠁⠛⣼⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣿⣿⣇⣀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣄⢚⣿⠏⠀⠀⠀⠹⡷⣄⣾⣿⡿⢿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡇
⠀⠀⠀⠀⠀⠈⠳⣤⣐⠒⢶⣿⡿⠿⣿⠀⠀⠀⠀⢰⠙⢮⡫⡉⠉⠢⣹⠿⢷⠲⢧⡀⠉⠉⠙⠛⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢸⡄⠀⠀⠂⠁⠀⢸⢿⣮⠢⠄⠀⠀⠀⢳⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⠀⠀⠀⠈⢈⠏⢷⠀⡄⡐⢲⣌⣇⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⢰⠇⣠⠀⠈⠀⣸⡇⡇⢣⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⣿⢰⡇⠀⢰⣆⣿⣷⡀⠀⡙⠛⢛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠻⠿⠴⠶⠿⠿⣿⣿⣿⣿⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣰⣶⣶⡆⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⠤⠀⠀⣴⣥⡀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠞⠁⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⢠⡄⠀⠀⡄⠀⠀⠀⠁⢶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠂
⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠰⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⢠⣴⠢⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠘⠠⣠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠄⡀⠀⠀⠀⡠⠢⠤⠁⠀⠀⠀⠒⠰⠀⠀⠀⠀⠄⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠊⠂⠀⡀⢈⠔⠀⠈⠢⡑⠢⠐⠍⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠑⠁⠁⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠏⡅⡊⠉
⠀⠀⠑⠂⠠⠤⠄⠀⠀⠀⢀⡀⠀⠀⠀⣀⣼⣠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠇⣇⡇⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠳⠿⠆⠆⠢⠄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡎⠀⠀⠀⠀⠀⢀⡄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⡶⣶⡶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠁⠀⡏⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢂⠘⡶⠉⠀⠀⠘⠃⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠑⣤⡟⠲⠀⠀⠀⠀⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠠⠤⠀⢤⣤⠁⠀⠀⠀⠀⡞⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠁⠀⠀⠀⠘⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠁⠒⠛⢛⠲⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⡄⠐⠒⡇⢀⡴⠃⠀⠀⠀⠣⡀⠀⡸⠀⠲⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠏⠉⠡⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢃⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠘⡄⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡇⠀⠀⠀⠀⠀⠀⡰⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢰⠀⠀⠀⢀⡰⠊⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡀⠀⠀⠀⡆⣠⠔⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢢⠀⡘⠑⠂⠤⠄⢀⣀⣀⠀⠀⣠⠆⠀⠀⠱⠀⠀⠀⠉⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⣄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣬⡷⢀⣀⣀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠎
⠀⠀⠀⠀⢠⠏⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⠶⠻⡆
⠀⠀⣠⠒⣇⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠈⢦
⢠⡖⠉⠈⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢨⠴⠂⣀⠀⠀⠑⢤⡀
⠣⣝⠦⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠵⠶⣤⣭⠆
⠀⠀⠑⢌⣦⡀⠈⢆
⠀⠀⠀⠀⠉⠻⢦⣌⡧
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣞⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⢀⣠⠎⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢠⣴⠟⠋⠉⠁⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡆⠀⠀⠀⣀⠀⠀⢻⣿⠀⣴⡟⠁⠀⠀⢤⣠⣴⣿⣦⣼⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⢿⣿⣿⣿⡶⠞⠋⠀⠀⠀⢸⣿⣿⣟⠀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣍⡉⠻⣿⣿⣿⣿⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣦⠈⣿⣿⣿⣿⢟⣥⣌⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠙⣿⠃⢠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⣿⣿⣿⡉⠻⣷⣄⡀⠀⠀⠀⢀⡀⠘⣷⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠘⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⣤⣤⣾⣷⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡝⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣿⡿⢫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣶⣶⣶⣶⡷⠂⠀⠀⢀⣿⠃⠀⣾⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢿⣿⣦⣤⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠉⠀⠀⠀⢀⣾⠃⠀⢰⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣴⣦⡀⠈⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠘⣿⣿⣿⣷⣝⠻⠿⣿⣿⣿⣿⣿⣅⣠⣤⣄⠀⠻⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠞⣁⠙⢿⣿⡿⠦⠴⣿⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠰⣶⣼⣿⣿⣿⣿⣿⣿⣿⠟⣵⣿⣿⣧⡀⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢸⣿⠘⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣿⣿⡆⠁⠀⠀⣿⣿⣿⣽⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⢹⣿⣿⣤⣿⢀⣿⣿⣿⣇⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢻⣿⠙⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠹⠄⠈⠻⣿⡟⠀⢸⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⢷⣌⠛⠀⢸⣿⣿⡿⠁⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣈⣿⣿⠁⠀⢛⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠇⠘⠰⠀⠂⠐⠀⠀⠐⠂⠀⠇⠀⠂⠀⠀⠂⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠃⠀⠂⠀⠈⠂⠁⠊⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠸⠟⠇⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⢦⡀⠀⢀⡴⠋⡁
⠀⠀⠀⣀⡤⠴⠶⣦⠤⣄⡀⠀⣠⠊⠾⣆⡀⠀⠀⠀⠀⠠⣼⣷⡀⢹⣿⡏⠀⣼⡧⠄⠀⠀⠀⠀⠀⣠⠴⠒⠲⡖⠲⠦⣄⡀
⢀⣴⣾⣿⣶⣦⣦⣿⣦⣦⣽⣾⣧⣴⣚⠛⢿⣆⠀⠀⠀⠀⠀⣈⡻⣯⡀⣹⢞⣁⠀⠀⠀⠀⢀⣤⣾⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷
⠸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣿⡇⡿⣿⠀⠀⠀⠀⠺⠟⣀⣬⠻⣦⡀⠻⠗⠀⠀⠀⠸⣿⣿⣿⡻⣿⡿⠿⠿⣿⣿⣿⠱⠋⠋
⠀⠀⠘⠿⠞⠁⠀⠀⠀⠀⠀⠘⠾⠞⠁⠐⠇⠹⠀⠀⠀⠀⠀⠀⠧⠏⠀⠈⠫⠀⠀⠀⠀⠀⠀⠈⠉⠳⠿⠃⠀⠀⠈⠿⠛⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣭⣭⣷⠾⡿⡿⢾⣭⣭⣝⡇⠀⠀⢰⣧⣰⠀⢠⣀⡄
⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡿⠆⠀⠀⠀⢷⢿⡏⣁⠛⠛⠛⢋⢩⡿⣼⠀⠀⠀⠀⢻⡞⠀⠈⣿⠁⠘⣶⠂⢠⣀⡀
⠀⠀⣾⠇⠀⣠⣴⣤⣤⣤⣤⣤⣤⣦⣶⣶⣶⠄⠀⠀⠀⠘⣏⣷⣉⣤⣤⣤⣉⣼⣳⠃⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠈⡏⠀⢦
⠀⣴⣿⠀⢰⡟⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⡶⠄⠀⠀⠀⠀⠘⢮⢿⣿⣿⣿⡿⣳⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⡇⠀⢸
⠀⠻⠿⠀⣿⡇⢀⣿⠀⣤⣤⣤⣤⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠈⠳⣝⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⣼⣇⠀⢠⢿⡄⢀⠿⡀⢀⢧⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠉⠀⠈⠋⠀⠙
⠀⠀⠀⠀⢀⣀⡀
⠀⢀⡴⠟⠋⠉⠙⠳⣦⡀⠀⠀⠀⢀⣸⣿⡇⠒⠒⢺⣿⣇⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⢀⣿⣁⣀⣤⣤⣤⣀⣸⣷⠀⠀⢰⣿⣶⠶⣶⣶⣶⣶⠶⣶⣿⡆⠀⠀⣼⣿⣿⣋⣿⣙⣿⣿⣇⠀⠀⠀⠀⢀⣤⣭⣿⣯⣤⡀
⠘⣿⠉⢻⣏⣋⣻⡋⢉⣿⠀⠀⢸⣿⠠⣷⠄⣿⣿⠠⠤⠄⣿⡇⠀⠀⣿⣿⣏⣹⠿⣏⣽⣿⡿⠀⠀⣧⣾⣿⣿⠏⠶⠹⣿⣿⣿⣆⣾
⠀⠙⣷⣿⠉⠉⢙⣿⡿⠃⠀⠀⢸⣿⣦⣭⣴⣿⣿⣦⣤⣴⣿⡇⠀⠀⠘⢿⣿⣿⣶⣯⣾⡿⠁⠀⠀⠏⠻⢿⣯⣤⣤⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃⠛
⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣼⣿⣭⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⡆⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⡀⠀⢠⣄⠀⠀⠀⠀⢠⡶⠶⠶⠶⢶⡄⣴⣶⣶
⠀⠀⠀⡘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⣷⠀⠀⠀⠀⠀⠀⢼⠁⢰⣆⠈⣧⠀⠈⣿⠀⠀⠀⠀⢸⡗⣶⠶⣶⢸⡇⣿⣿⣿
⠀⠀⠀⠱⡄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣷⡄⠀⠀⠀⠀⠀⠘⢧⣄⣠⡼⠃⠀⠀⢿⠀⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⢹
⠀⢀⣠⣴⣮⡛⣫⣶⣤⣀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠸⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠚⠛⠓⢾⡄⠀⠀⠀⠀⣿⣟⠿⣻⡏⠀⠀⢸⡇
⣼⣿⣿⣿⣿⡗⣿⠿⠿⣿⣿⡄⠀⠀⠀⢼⡟⠀⠀⠀⢻⡧⠀⠀⠀⠀⠀⠈⢹⡯⡅⠀⠀⠀⢸⠇⠀⠀⠀⠀⢿⣯⣾⣽⡇⠀⠀⠀⡧
⠿⠿⠿⠿⠿⠗⠿⠷⠶⠿⠿⠇⠀⠀⠀⠲⣤⡤⠶⠚⠛⠲⠦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⠶⠋⠀⠀⠀⠀⠀⠈⠉⢿⡉⠀⠀⠀⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠆⠀⣿⣿⣿⠀⠠⣿⣿⣿⠀⠰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⢘⣿⡏⠀⠀⢹⣿⡇⠀⠀⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⠛⠛⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⠛⠛⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠻⠿⠛⠀⠀⠀⠻⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢠⡞⢳⡄⡇⢸⢢⡞⣛⢰⣦⣼⢰⣦⣸⢸⣟⡃⢾⣓⠰⣟⡀⠀⡗⣴⠛⠃⡞⠛⣦⢺⣆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠋⠀⠙⠋⠀⠙⠚⠈⠃⠘⠘⠁⠙⠘⠓⠃⠙⠋⠘⠚⠁⠀⠃⠈⠛⠃⠙⠛⠁⠙⠈⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡲⣶⣄
⠀⠀⠀⠀⠀⡰⣽⣽⣿⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡾⣿⣽⡵⡀
⠀⠀⠀⣰⣿⡶⢿⠛⢻⣿⣿⣿⣞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣾⣿⣿⣿⠛⢻⠷⣾⣷
⠀⠀⠀⡇⢠⠁⢸⠀⢸⣿⣿⣿⣿⣽⣄⠤⠤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⢄⣼⣽⣿⣿⣿⣿⠀⢸⠀⢡⠀⡇
⠀⠀⠀⡇⠀⡆⠸⡀⠀⣿⣿⡟⣧⣤⣴⣶⣶⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶⣶⣶⣤⣤⡟⣿⣿⡇⠀⡘⠀⡆⠀⡇
⠀⠀⢸⣵⠀⣇⠀⡇⠀⢹⣿⣷⡻⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡿⣳⣿⣿⠁⢀⡇⠀⠇⢸⣽
⠀⠀⠀⠛⡆⠸⡀⠸⡀⠈⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⣿⣿⣿⡏⠀⡸⠀⡸⠀⡞⠃
⠀⠀⠀⠀⠘⣤⣷⣶⣷⣻⣞⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣾⣳⣷⣶⣧⡜
⠀⠀⠀⠀⠀⠘⠿⡿⣿⣷⣝⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⣵⣿⡿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⢺⠁⠀⠈⠉⠀⢐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⡶⠋⠀⣐⠀⢠⢔⣀⣀⣤⢼⡇⠾⠨⠀⡄⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⠁⢠⡟⠓⡿⠀⠀⠛⠀⠀⣿⣺⡇⣤⠤⢠⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⠐⠢⠌⠉⢘⠲⢤⣭⠐⠆⠀⠀⠀⣇⠀⠈⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠔⠠⠀⡠⠔⢺⠀⠀⠂⠀⠉⢽⡏⠁⠀⡶⠶⡴⡽⠄⣀⣴⣧⠄⠧⠅⠠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⡇⠀⠐⢰⠀⢀⡀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⢼⢯⢿⡆⠈⠁⢰⢒⠒
⠀⠀⠀⠀⠀⠀⡋⠘⠲⠚⠂⠂⠒⠚⠀⡋⣓⠐⠒⢶⣖⠒⠒⠲⡀⢤⠤⠀⠼⠗⠚⣇⣀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠓⡀⢤⢺⢠⣠⢸⠀⠀⠀⠀⡇⠀⠒⠀⠀⠀⡀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠏⠈⢀⠀⠀⠀⠀⠤⠀⢨⠤⠤⠤⠼⠦⠀⠠⠄⠧⠀⠀⠀⠐⠠⠁⠱⠦⠀⠸⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠟⣀⢊⣥⠦⠮⡾⡬⡤⢟⠛⠛⠛⡛⠛⠛⠛⢻⣷⠶⠿⡓⠔⣖⠶⠶⣶⡶⢒⣒⣒⢇
⠀⠀⠀⠀⠀⠀⠲⣋⠜⠁⠀⡀⢨⠯⠿⠿⠄⠀⠀⠀⠀⠀⠴⠿⠃⠄⡄⢨⠶⠁⢀⡄⢨⠿⠷⠷⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠠⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠐⠀⠂⠐⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣯⣶⠛⢿⣽⣿⡤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⣿⣿⣿⣿⣿⠛⢛⣛⠿⣿⣿⣿⣿⠈⠀⣋⡉⢸⡷⢺⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠟⢫⣿⣯⣭⣭⣴⣿⠿⣷⡄⠹⣿⣿⣆⠐⠉⠙⢺⠁⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣼⣿⣿⣿⣿⣿⣯⢻⣀⡤⠋⠀⣺⣿⣿⣿⣦⣤⡞⡆⢹⣿⡏⣻⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣮⣴⣾⣿⣿⣟⣛⣻⣿⣿⣆⢻⠀⠀⢉⣽⣿⣿⠿⡿⠟⠳⠟⣸⣿⡇⢾⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⢋⡭⠿⣿⢿⣿⠀⣦⣤⣤⣭⣛⣁⡸⠛⠫⠠⣶⡿⠝⣄⣸⣟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢠⣾⣟⠰⣾⣧⣿⣷⠘⣄⣀⣠⣾⣿⣷⣦⣶⣿⣟⢳⡄⢸⠇⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⠘⣦⣬⣦⣤⣭⣽⣿⣇⣸⣧⣿⣦⣤⣤⣤⣶⣿⣮⣧⢻⣼⠠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⢠⣝⢿⣿⣿⣿⣿⣛⣿⣿⣿⣿⡿⣹⠇⠦⢰⣠⡾⣿⣸⡟⢐⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⠄⢸⠙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣿⣛⠛⢣⣿⡇⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣿⠄⢸⣿⣉⣉⣉⣉⣿⣿⣍⣉⣉⣉⣉⣉⣉⣉⣉⣠⣴⣿⠿⡇⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡁⣸⣿⣿⣝⣋⡈⠁⣠⣽⠿⢿⣫⣿⣿⣯⣭⣩⣭⣾⣿⣴⡇⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣗⠂⢼⡇⣬⣭⣭⣭⣭⣭⣽⣭⣭⡭⠭⠭⠭⠵⢶⣆⢿⣿⣿⡇⠨⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠚⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣘⣿⣼⣿⢛⣇⠰⣾⣟⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣖⣓⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣑⡦⠴⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⢽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤
⣿⣯⣬⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠉⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⣀⣀⡀⠀⠲⡄⠀⢸⠀⠀⠀⠀⠀⠀⡠⠖⠚⠒⠢⠀⠀⠈⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⡟⠀⡼⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠘⠀⠀⡴⠃⠀⠰⠊⠀⢠⡇⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠳⡄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠁⠈⠦⠀⢀⡀⠀⠈⠓⠦⠄⠀⠀⠀⠁⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⣀⠀⠀⠀⠀⠉⢀⣀⡤⣴⠋⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⡸⠃⢀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣨⠛⠋⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠈⠉⠉⠀⢀⣜⣀⠀⠀⣠⠋⠀⠀⠀⢀⣀⣀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⣀⣀⠀⠀⠀⠀⠁⠀⢀⣠⣊⣀⣀⣀⣀⣀⣠⠤⠖⠋⠀⠀⠀⢠⡇⠀⠀⠀⢰⠋⠀⠈⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⢄⠀⠈⠉⠉⠙⡆⠀⣀⠴⠚⠉⠉⠉⢉⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠀⠀⠈⢦⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠦⣄⣠⡄⠀⢀⠞⠁⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡰⠀⠀⣀⡴⠋⢧⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠚⡇⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠶⠶⠶⠦⠤⠤⣄⣀⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⢤⣤⠤⠴⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠴⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⠀⠀⢹⣀⣀⡤⠤⠤⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠲⠦⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⠦⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⢿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⡿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠇⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠸⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡇⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⢠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣾⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣾⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣷⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠠⠤⣤⡤⢤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⡉⠠⠀⠀⢠⣿⠀⠘⣿⣧⡤⣾⡿⠛⣛⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠐⠈⠁⠀⠀⠒⠤⡐⠁⠀⣠⠟⠁⢀⠠⠋⠁⠀⠉⠻⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠈⠀⠀⢰⠀⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠀⠀⠐⠒⠁⠀⡄⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠁⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⣤⡄⠀⠀⡼⠀⠀⡄⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣄⠀⡀⢀⣤⠀⠀⠀⢤⣤⣤⣼⣦⡤⢤⣴⡾⠟⠊⠉⠁⠈⠙⠻⠄⠀⠀⠀⠠⡼⠃⠀⣹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣯⣤⣀⣀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⢠⣿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠶⠶⠶⠶⠾⠿⠛⠋⠉⢀⣀⣴⣿⣿⣤⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣦⡀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⠿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣤⣠⠙⠷⠶⠶⠒⠒⠚⠛⠛⠟⠛⠛⠿⢷⡶⢿⣿⣯⣟⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⠿⠿⣿⣾⣿⣷⣶⣶⠾⠿⠿⠿⢾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠀⠀⠀⠀⣀⣐⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣶⣤⣤⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣀⠤⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡽⠋⠁⡠⠞⡡⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⡴⠉⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⢱⠀⠀⠀⠀⡇⡀⠀⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣼⡀⠘⡀⠀⢀⠠⠇⠀⠴⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠺⠀⣀⡀⠶⠁⠴⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⠏⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠑⠲⢄⠀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠐⡆⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠐⠀⠠⠤⢅⣀⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠱⣆⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠱⣰⡻⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠘⠇⠀⠈⠀⠀⠀⢂⠉⠒⠤⢆⣀⣀⡤⠋⠀⠀⠀⠀⠀⠀⣏⢑⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⡂⠀⠈⡄⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠑⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠾⣹⣅⣲⠃⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⢨⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⠀⠌⠀⠨⣀⠈⠑⠒⠤⢤⣀⡀⠀⢹⣩⡦⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⡄⡀⡁⠀⠈⡌⢣⣀⡀
⠀⠀⠀⣀⠤⠐⠂⠉⡠⠄⠒⠋⠀⠀⠉⠀⠀⢀⢈⠓⠦⣀⠀⠀⢠⠋⠳⡼⠃⠈⠢⡀⠀⠀⠀⢀⡀⢠⠋⢸⣀⡇⣇⠀⠀⣟⢠⣅⣙
⠠⢖⢯⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡀⠀⠀⠉⠲⣧⠀⠀⠘⣢⠀⢣⡑⢤⡤⡞⢹⠟⡉⠤⣀⠀⢺⠁⢱⠴⠉⠛⢃⠀⠀⠐⠢⣀
⣧⡏⢇⢣⠊⠀⠀⠀⠀⢀⣀⣀⠀⠀⣰⣞⣁⡴⠊⢃⠀⠀⢀⠀⠴⣷⡄⠘⠻⠀⠀⢣⡈⡀⠧⠌⠸⠁⠐⠋⠑⢤⢥⡎⢱⠇⠀⠀⠀⠉⠁⠒⠀⠱⣄
⡇⠀⢸⣬⠀⠀⠀⠀⢰⢡⢦⠱⡀⠘⠯⢤⡭⠓⣦⠺⡄⠀⠘⡀⠀⠀⢸⠀⠀⠀⠀⠀⢫⡀⠀⠠⡖⠺⠇⢑⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢦
⢡⠀⢸⠹⢤⣀⣀⡀⣴⡇⠀⢡⢱⠀⠀⢘⡆⢘⣤⠙⢄⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠘⢠
⠀⣌⣫⠃⠀⠀⠀⠀⡟⡇⠀⠀⡆⠀⠀⠀⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⢛⡿⠿⠛⠻⠿⣿⣿⣷⡀⠀⠀⠀⠀⢸
⠀⡇⡘⢉⣷⠀⠀⠀⠰⢃⠀⢀⢇⠀⠀⠀⠀⠔⠛⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠰⡀⠀⠀⢰⠁⡰⠉⠀⢀⣀⣤⡀⠀⠹⣿⡇⠀⠀⠀⡸⠚
⠀⠐⠃⡆⠀⠑⢦⡴⢲⠩⣳⡪⠊⣠⠴⠶⡞⠢⣤⠅⠚⣆⠀⠀⠀⠀⠀⢠⣤⡦⠖⠻⠦⠄⠀⠀⢳⠀⠀⢸⢰⠁⠀⢠⢫⠉⠙⢿⡆⠀⢸⣇⠀⡖⠈⠉⢑
⠀⠀⠓⠛⠢⣀⢸⠃⠈⠓⠒⠒⢆⠻⠤⠴⠃⢀⣿⠀⠀⡏⠣⡀⠀⠀⠀⠀⢊⠹⡳⢄⠀⠀⠀⠀⢀⣇⣀⡸⢸⠀⠀⠀⡏⠀⢀⠀⡇⠀⢸⠎⠓⠓⠠⠤⠖
⠀⠀⠀⠀⠀⠀⠙⠧⠖⠀⠤⠤⠚⠒⠒⠒⠋⠁⢸⠀⠀⢻⣄⣸⢢⡀⠀⠀⠀⠣⡜⢦⡑⣦⣂⡜⠁⢩⡀⠀⠘⡀⠀⠘⣔⠀⣸⣰⠇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⢆⡀⠈⠙⠛⠒⠉⡦⣀⠀⠀⠘⡄⠙⢌⢿⠀⠀⠀⠱⣄⠀⠙⢄⠀⠀⠉⠉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠍⠒⠤⠤⠴⣾⡀⣀⣁⣀⣠⣼⣆⠀⢻⡦⠤⠤⡀⠀⠉⠓⠒⠓⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠓⠚⠓⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣽⣿⣿⣯⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⢿⣿⣿⠿⠟⠋⠙⠉⠙⣻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢟⡟⠙⠁⠀⢀⣠⣶⣤⣀⠄⠉⢹⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⣁⣠⣠⣴⣾⢏⣾⣿⣿⣆⠐⢁⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⡟⣼⣷⡟⣮⢿⣿⡅⢈⣻⣿⣞⣿⣿⣿⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣽⣿⣿⣿⣿⣿⡏⢁⣶⣾⣿⠇⣿⣍⢿⣿⣟⣿⣿⡤⠼⣿⣿⣿⣿⣯⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⡤⠾⠷⠒⠒⠒⠶⢶⣯⣭⣿⣿⣆⡀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢻⣿⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⣮⣶⣿⣿⣿⣿⣦⣤⣿⣿⣿⡟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⡟⠛⠛⠛⠋⠁⠀⠀⠀⠀⣸⣿⣿⣿⠢⣬⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⢷⣿⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⡿⠁⠀⠀⢀⣠⣴⣾⣿⣿⡻⠿⠿⠿⢿⣿⣿⣿⡏⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣚⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⣛⡧⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⡷⣷⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⣿⣻⣽⣿⣿⣿⠃⣰⣠⣤⡆⠙⣯⣿⣿⣿⣟⣿⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣎⣿⣿⣵⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠯⠛⠻⢿⡿⣧⣘⡻⢛⣣⣼⣿⣿⠟⠛⠿⣻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡿⢧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣀⠀⣀⣩⣽⣄⡀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣿⡇⠹⣷⠃⢸⣿⢏⡁⢠⣴⣿⣿⣍⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⢛⣭⣍⢩⣙⣿⣏⣺⣧⣼⣿⣄⣹⣿⡿⠓⣻⠿⠉⠋⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠈⣻⣵⣿⡏⢈⣻⣿⣿⣿⠟⣿⢛⣋⣽⣯⣤⣿⢯⣽⡿⠁⢀⠀⢠⡠⣞⣷⡀
⠀⠀⠀⠀⠰⡾⡿⠟⣿⣿⣿⠸⢿⣿⣿⠛⢻⠋⠉⢈⣁⠟⣉⠋⠈⠋⡀⠀⡀⢰⣧⣽⣌⡏⡇
⠀⠀⠀⠀⠀⠆⠓⢲⠏⣿⣿⣠⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡄⢄⢳⣼⢿⢿⣿⣇⡄
⠀⠀⠀⠀⠀⠄⠐⢺⡆⣿⡿⡿⠷⠢⠀⣿⣿⣿⣿⣟⣉⠙⣿⡏⡿⢿⣿⣮⡳⠬⡿⠿⣻⠏⡖
⠀⠀⠀⠀⠀⣀⠀⢸⣀⣿⣿⣿⣿⣿⣿⣟⣛⣿⢟⣛⡷⠲⣾⠿⠿⠌⠿⠿⠿⠿⠿⠿⠿⠯⠳⠿⠯
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⠟⠀⢂⡞⣿⡇⢀⣸⣿⣀⡄⠀⢸⣿⣿⣷⣶⠄⣰⣿⣂⡀
⠀⠀⠀⠀⠀⢈⣈⣉⣿⣿⣿⠿⣿⣿⣿⣿⣼⣶⣾⣿⣶⣿⣷⣿⣿⣿⠿⡟⠏⣶⣶⢰⢿⣿⣿⣿⣾⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠙⠛⠿⠿⠷⠽⣿⣿⣻⣿⣿⣿⣿⣿⣟⢤⢷⣿⠸⠿⡟⡟⢿⣿⣷⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠙⠿⠟⠻⡿⠿⢶⠾⡿⢶⣶⡾⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠐⢶⣿⣦⣀⠈⠻⣿⣷⡄⠀⠰⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠛⠿⣿⣷⣤⠀⠈⣻⣿⡷⠀⢈⣿⣿⡦⠀⠙⣿⣧⠀⠘⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⢿⣿⣶⠄⠀⣼⣿⠃⢀⣼⡿⠋⠀⣴⣿⠟⠉⢀⣤⣶⣿⠿⠃⠀⢹⣿⡆⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⠀⢀⣿⡟⠀⡸⠟⠳⠶⣾⣤⡈⠙⢿⣦⡀⠙⢿⣯⠉⢫⣤⡠⣴⣾⣿⠿⠟⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⣿⠀⢸⠟⠑⠛⠿⣷⣶⠀⢨⣿⠏⢀⣴⡿⠋⣀⣼⡿⠇⠀⢻⣷⠀⢱⣤⢠⣤⣴⣶⣿⣧⠀⢸⣦
⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⡆⠀⣿⠗⠾⠿⠿⣿⡆⢀⣿⣇⣠⡛⠓⢶⣮⡉⠲⣾⡙⢫⣤⠴⢾⠟⠛⠁⣈⣿⡆⠉⣍⠉⠉⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠸⣿⡇⣠⣷⣶⣶⡆⢰⣿⢥⣾⣥⣄⠉⣿⡗⢠⣿⠋⣠⡾⠟⣀⣽⡧⠈⣿⡞⢻⡛⠛⠁⠀⣿⡏⠻⡿⢿⣿⣿⠀⢠⣦
⠀⠀⠀⣤⣴⣶⣿⡆⠀⠿⢃⣀⣀⠀⣿⡇⣼⣷⣦⠄⣸⣏⣰⡛⠲⢯⡌⠻⣧⠘⣾⡉⢧⡴⠶⠿⠃⠈⣿⠘⠿⠿⠿⠇⢸⣷⣀⣀⡀⠀⣸⣿⠇
⠀⠀⠼⠛⠋⠉⢿⣿⡠⠿⠟⢻⣿⢀⣭⣤⣤⠀⡿⠠⠷⣦⠈⣿⣁⠞⢥⡚⢥⡾⢏⡡⡾⠗⢘⣗⢻⡛⠛⠀⣿⡶⣶⣦⣼⣿⠈⡻⠿⣿⣿⡟⠀⢠
⠀⠀⣀⣤⣶⡄⠈⢟⣠⣤⣦⠀⢿⣉⣁⠀⣿⠴⠷⣶⢀⣟⠺⣶⢉⣟⡰⢏⡼⣋⢼⢃⣹⡏⣏⢩⣼⡏⢹⠛⠻⠁⣼⣌⡉⠁⢠⣿⢄⣀⠈⠀⢠⣿⡆
⢀⣿⡿⠛⢿⣿⣄⠿⠛⠉⢿⡦⠟⠛⢻⡇⠶⠺⣇⣩⣭⠹⠇⢯⡙⢆⡿⣴⢣⢞⡼⢇⣷⢲⣛⣸⡦⣤⣼⠏⡷⢶⡿⢈⡿⠿⣿⡟⢈⡿⣿⣶⣿⡟⠁
⠈⠁⢀⣄⠀⠻⣇⣤⣶⣆⠈⣧⡶⣷⣈⡷⢾⣆⡷⢶⣸⣭⢹⡺⣍⡳⡗⣿⣮⣞⢽⣫⢖⣞⣩⢭⣴⡗⣤⣴⡗⣦⣤⣼⠗⣤⣀⢀⣼⣧⡀⠉⠛⠀⣠
⣠⣶⣿⣿⣧⡀⡿⠋⠉⢻⡷⢁⣤⡘⢧⣴⣆⡱⠶⡬⠶⢮⡳⡽⡷⣟⣾⣯⣿⣻⣽⣫⢏⠼⡡⠞⡦⣴⢣⣌⣰⢧⣉⠋⣠⡏⠻⣿⠟⠀⣿⣶⣄⣴⣿
⡿⠛⠁⠈⠻⣿⢀⣴⣷⣄⢹⠟⠙⢷⢉⣤⡙⡷⠷⣜⣙⡮⣝⣿⣽⣿⣿⣿⣿⣿⣿⢗⣽⣯⣽⡛⡱⢦⠞⣌⢉⣬⠛⢿⠟⢱⣦⡀⣠⣾⠀⠙⢿⡿⠋
⠁⢠⣾⣷⣄⠀⡿⠋⠈⠻⢇⣴⣷⣤⡛⣁⡙⣴⠳⢏⣬⣞⣻⣟⣷⣿⣿⣿⣿⣿⣿⣟⣿⣝⡺⣍⡝⢶⢾⡌⠛⣁⢷⣄⣴⡇⠙⢿⠟⠁⣿⣦⡀⢀⣤
⣴⣿⠟⠙⠿⣷⠁⣴⣷⣦⣸⠋⣠⣈⢳⠏⡙⢣⠟⠺⡴⢪⡞⣹⣻⣞⡯⣿⣿⣿⣟⢾⣞⢮⡹⢶⡚⠶⢎⠹⠞⢳⡌⠛⣁⢼⣧⣀⣠⣾⠈⢻⣿⣿⠿
⠸⠋⠀⣤⣀⠈⢹⡿⠁⠉⠛⢴⡟⠙⠻⢴⠟⠛⢼⢟⣓⣋⡭⠵⣪⢗⡯⡻⣿⢾⢯⣙⡬⣇⣛⡏⠷⢾⠹⡷⢾⡈⢿⠾⢻⡄⠹⠿⠛⢹⣦⠀⠙⠁⢀
⠀⢀⣾⣿⠿⣿⣾⡁⣼⣿⣶⣾⡁⣾⠷⢾⣠⡟⠛⠺⡏⣭⠧⢿⣱⡟⡵⢳⠝⣾⢹⣌⣳⢸⣇⣛⣛⢹⡦⠶⢺⣧⣠⣴⠺⣷⣀⣤⣶⠙⣿⣧⣤⣾⣿
⠀⢸⣿⠃⠀⡀⠉⠑⣿⠃⢀⣈⡙⡿⢀⣦⣤⣅⣸⡟⣋⣻⡘⣏⢡⡗⣩⡞⣱⠎⣽⡁⠿⡬⣽⠁⠿⢶⠆⣿⠀⢉⣉⣷⠀⠻⠛⠋⣵⡀⠘⠿⠛⠉
⠀⠀⠃⠀⣼⣿⣿⣶⣮⡀⣿⡟⠛⠿⠼⣿⠀⣤⣬⣧⢽⡅⢴⡾⢋⣱⡾⢓⡬⢓⡶⢉⣿⡀⠛⢶⠒⣾⠀⠛⠛⣛⠁⣿⣧⣴⣶⠊⣿⣷⣀⣠⣤⡖
⠀⠀⠀⢰⣿⡏⠀⠈⠉⠉⢿⡇⢰⣶⣶⣾⡄⣿⡀⢠⣶⠶⠞⢓⣈⡿⡄⢻⣦⣘⣳⠦⣭⠏⢹⡏⠐⠻⢿⡟⢹⣿⠀⠈⠉⢡⣷⠀⠸⣿⠿⠟⠛
⠀⠀⠀⠀⠻⠃⠀⣿⣿⣷⣾⣦⣸⣿⠀⢀⣤⣬⣧⡜⣷⡀⢺⣟⠉⣴⡾⠋⣠⣿⠃⢼⣿⣀⡙⢛⡿⢓⣿⠇⠸⠿⠿⢿⠋⢸⣿⡆⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣙⣀⠸⣿⡉⢀⣤⣴⡷⠖⠛⣢⣌⡿⠦⣈⣻⠷⢤⣬⠋⢹⣿⠁⠸⣿⣶⣶⡶⢴⣿⠀⠸⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⢻⣿⠿⠟⠛⠋⠛⢇⠀⢿⣧⠀⢰⣾⡟⠉⣠⣾⠟⠁⣰⣿⡃⠀⠿⢿⣶⣤⢄⣴⡇⠀⣿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⣴⣶⣿⡿⠟⠊⠛⣣⣀⣻⣷⣄⠈⠻⣷⣄⠈⠛⡿⠶⢦⣴⡎⠀⣼⣿⠁⠀⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢸⣿⣇⠀⢠⣶⣿⠿⠛⠁⣀⣴⣿⠟⠁⣠⣾⡿⠁⢠⣿⡟⠀⠐⠿⣿⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡄⠀⢻⣿⣄⠀⠺⣿⣿⡁⠀⢾⣿⣯⡀⠀⠛⢿⣿⣶⣤⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠦⠀⠘⢿⣿⣆⡀⠉⠻⢿⠷⠄⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠉⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠒⠦⣄⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢙⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡀⠀⠀⠀⠈⠙⢦⡀⠀⠈⣧⣤⣴⠶⠾⠛⣋⣍⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣶⣀⣀⣤⠤⠾⣓⣚⣭⣭⡄⡴⠺⡷⡐⠁⢠⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⠛⢋⣉⣉⣭⣭⣶⣶⣿⣿⣿⣿⣿⣿⡇⢇⣠⣧⠇⠒⠛⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⡤⢴⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣃⣤⣭⡷⠶⠛⣛⣫⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡳⢰⡃⠀⡷⡇⢸⣿⣿⠿⠿⣛⣛⣭⣽⠶⠞⣛⣋⠭⠥⠒⠚⠉⣉⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣷⠀⠳⠾⣗⣡⣬⡵⠶⣛⣛⣫⡭⢴⣶⣿⢻⣿⢀⣀⣤⣴⣶⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⠷⠶⢛⣋⡭⠵⠖⠒⠉⡏⡇⣴⣧⡾⠷⠞⣛⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⣀⣀⣤⡴⢖⣿⣿⣿⣔⣶⣶⣍⣙⠻⠿⠿⠛⠛⠁⠀⢀⣀⣤⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⠋⡟⠹⣷⣼⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⣦⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠉⠛⠛⠛⠛⣦⣄⢀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⣤⡦⠤⢀⣀⠀⠈⠻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠈⢻⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⢠⡴⢋⣽⣿⣷⣶⣾⣤⣄⡐⣄⣈⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠈⣿⣿⡧⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⣍⣙⠛⠻⠿⢿⣿⣧⢟⡷⠀⢀⡤⣀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣶⣼⣉⣷⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⣤⢍⣙⡒⠲⠼⡷⠋⣀⣴⠾⢃⣩⡶⠘⠛⠆⠀⠀⠀⠉⠙⠛⠿⠿⣿⡉⣏⣿⣷⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⠿⠀⠀⠘⠻⠽⠁⠒⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡛⢟⡛⠛⡟⣻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠤⢀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠔⢤⠊⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠃⠌⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⢒⠄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠁⠀⠀⠀⠀⠀⡆⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠐⠀⠀⠀⢰⡀⠀⠀⠀⠀⡠⠊⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠀⢀⠤⠊⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠓⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠢⡀⠀⠀⠀⠀⡐⠀⠈⠢⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⡄⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠁⠀⠤⢀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⢐⠂⠂⠠⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢆
⠀⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠑⠠⠀⠀⡔⠁⠀⠀⠀⠀⠈⠐⠂⠄⡀⢀⡌⠀⠀⠀⠘
⠀⠀⠌⠀⠀⣖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⢀⠀⠀⠀⠴⠁⠘⠂⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⡄⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠰
⠷⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠂⠀⠠⠛
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠛⠳⢦⡄
⠀⠀⠀⢀⣠⡶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⡆
⣼⠃⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠛⠛⠋⠉⠉⠀⠸⣇
⣿⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣤⠀⠀⠀⢸⡆⢸⡆
⢸⡇⠀⠀⠀⣀⣤⣤⠿⠀⠀⠀⠀⠹⠂⠀⠀⠈⠁⠘⣧
⠀⢷⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿⡀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠶⠶⣤⡀
⠀⠘⢷⡀⠈⠷⣤⣄⠀⠀⠀⠀⢀⣤⡴⠞⠛⢉⣿⠀⢸⡇⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠙⠢⣤⣤⣿⡀⠀⠀⠀⠀⠹⢦⣤⣤⠞⠃⠀⣿⠀⠀⣼⠃⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠈⢿⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡟⠀⠀⠀⣾⠁⠀⠀⠀⢻⡆⠀⠀⢸⡇
⠀⠀⣠⠀⠀⠀⠀⠀⣠⠿⠳⢦⣤⣤⣤⣤⡴⠾⢯⡀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⣀⡾⠁⠀⠀⢸⡇
⠀⣼⠋⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⣦⡀⠀⠹⣆⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣠⡟⠀⠀⠀⣸
⢰⡏⠀⠀⠀⢠⡟⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠷⣤⡀⠘⣧⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠉
⣼⠁⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡀⠀⠀⠀⠈⢹⡟⠛⠛⠛⢻⡍
⣿⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣨⣿⣄⣀⡀⠀⢸⡇⠀⠀⠀⠸⣇
⠉⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⡀⠀⠀⠀⢀⣴⠏⠀⠀⣴⠏⠁⠈⠉⠙⢷⣼⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⢦⣄⣠⡾⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣰⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠙⢷⣶⣤⣀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇⠈⠉⠛⠳⠶⣼⣧⣀⣤⠾⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠶⠶⠶⢾⡃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⢾⣋⣿⣍⣿⢶⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡆⠀⣼⣿⣿⣿⣷⠾⠿⢧⣼⡿⣶⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡄⢨⡷⠇⠀⠈⠯⠀⠀⠀⠀⡀⠀⢻⣿⣻⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠀⠀⠈⠀⠀⠀⠀⢶⣶⡄⠰⢛⢭⣀⠀⣿⡿⢿⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡻⠀⠁⠈⠁⠀⠀⠄⣠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠤⠆⠀⠀⠀⠀⢰⣏⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡛⠃⠀⠀⢀⣠⢾⣿⢹⣷⣦⡀
⠀⠀⠀⢠⠃⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣴⡶⠛⠁⠘⡇⣷⣼⢻⡇
⠀⠀⠀⣆⠀⠀⠀⡀⠧⠤⠤⠴⠖⠲⠶⠦⣤⡤⢤⡴⠒⠒⢦⣤⣠⣿⡉⠀⠀⠀⠀⠹⣷⣾⡛
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⢣⠀⠀⠸⣿⣿⠿⣷⡄⠀⠀⠀⠀⣸⠋⣣⡀
⠀⠀⠀⠛⠷⢾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣸⠇⠀⠀⠈⠉⢀⣤⡘⡆⣀⡤⠞⠁⣰⠃⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⣤⣤⣀⣀⣺⣿⡿⠁⠋⢰⡀⠀⠈⠛⠋⠉⣷⣿⡷⠒⢶⠀⡇⠀⢀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣟⣾⣣⡾⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠈⠳⢇⡴⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⡿⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠁⢻⣧⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠁⠀⢀⡟⠀⠀⠀⠀⠀⠤⠀⠀⣿⡆⠀⣀⣀⣀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠐⢤⠟⣷⠿⠽⠶⠮⢅⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⢆⢹⡆⠀⠀⠀⠀⠀⠀⣰⡏⣶⣤⣤⣾⣿⣭⣖⠤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⣄⠘⡀⡇⠀⠀⠀⠀⠈⠀⣿⠴⠛⠁⠀⠀⠀⠀⠙⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢀⡈⠣⠙⢣⠀⠀⠀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠉⢂⣀⣨⡤⠖⠊⠁⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠁⠀⠀⢀⡀⠠⠤⠤⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⠔⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠢⡑⠔⢄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⣉⠺⢑⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡵⢫⠄⠀⠀⢀
⡅⠀⢀⣀⠠⣀⠐⠪⢑⠲⢍⡒⠭⠲⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⠋⢈⠀⠀⠔⣠⠔⠁
⣇⠤⢄⣒⠮⣍⣒⣒⣲⣶⣭⣝⣓⣒⠦⢭⡠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠋⣠⠶⠛⠤⢂⠠⠔
⣿⡨⣥⣶⣶⣶⣮⣭⣭⣭⣭⣯⣭⣭⣿⣷⣶⣵⣤⣤⣄⣖⣶⣖⣴⢦⣮⡿⠋⠁⠀⠂⠩⣒⣲⣶⡖
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⣿⣿⠿⠟⠛⠛⠉⠉⠉⠻⣿⣿⣿⣷⣿⣧⣤⠐⠀⣂⣭⣭⠽⣷⡶⣶⣆⡀⠈⠀⠠⠤
⣿⣿⣿⣿⣿⣿⣟⣿⡿⠟⠋⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠙⢿⢿⢟⠿⣿⣿⣧⣾⣯⣭⣭⣟⡛⠻⢿⣶⡒⣀⢀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠎⡴⢁⢤⣼⢸⡤⣆⣦⣻⣿⣿⢿⣿⣯⣟⡿⣿⣿⣿⣋⣷⣯⣽⣻⣽⣔⡀
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠘⠘⢹⠘⠁⠻⡞⡟⡿⣿⢳⡌⠋⠙⠿⡾⣽⢿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣧⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠸⣷⡝⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⢠⠂⠀⠀⠀⠀⠀⣼⢟⣌⣽⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣧⣧⡄⡆⠀⡀⠀⠀⠀⠈⣷⣿⣿⣿⣿⠏
⣿⢓⣻⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣾⣿⣿⣿⣷⡁⡀⠀⢀⣼⣿⣿⣿⣿⠋
⣟⠯⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠙⠟⣟⣿⣿⣯⣇⣧⣶⣾⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠌⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠄⠀⠀⠠⠀⠀⠀⠄⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠈⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠐⠉⠃⢈⠘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡠⠊⠀⠀⠀⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠣⡀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠁⠀⠀⠀⡠⠈⠁⢀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠉⢦⡀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⠀⠀⠀⠐⠈⡀⠔⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡠⠁⠀⠀⠙⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⡀⠀⠀⢣⠀⠀⠀⠀⠈⠀⠀⣔⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⢃
⠀⠀⠀⠀⢠⠔⠒⠉⠀⠑⢀⡠⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡀
⠀⠀⠀⠀⡠⠏⠀⠀⠀⢢⡈⠢⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠞⠀⠉⠑⠔⠡⢀⡾⢧⡀⠀⣡⣀⡠⠤⡄⠀⠠⡤⢄⠀⠀⡀⠀⠀⠀⡀⠀⢠⠄⠠⠄⠀⠄⠠⡤⢤⡇
⠀⠀⡜⠀⠀⠀⠀⠀⠀⣸⠀⠀⢣⠀⠑⣤⠤⠤⠠⠶⠦⠰⠤⠀⠃⠄⠀⠀⠅⠤⠀⠴⠤⠠⠤⠠⠤⠠⠦⠞⠁
⠀⢠⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡄⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠁⠀⠒⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠠⠚⠁⠀⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⢸⡇⠀⠀⠀⠀⠀⠀⡌⠈⠁⠒⢲⠀⠀⠀⢐⠂⠂⠈⠁⢃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠁⠀⠀⠀⠀⠀⡰⠄⠀⠀⠠⢼⡄⠀⠀⢸⠤⠀⠀⠀⠼⡄
⠀⠀⠘⡄⠀⠀⠀⠀⠠⠸⣆⣠⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠉⠆⠨⠁⠀⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣿⣿⡇
⠀⠘⢿⣿⣿⡟⠻⣏⡉⠉⠉⣉⠉⠉⢉⣩⡟⠹⣿⣿⣿⠏⠁⠈⢻⣿⣿⣿⠙⣯⣉⠉⠉⣉⡉⠉⠉⣩⡿⠉⣿⣿⣿⡏⠁
⠀⠀⠈⠛⠛⠁⠀⠙⢿⣦⣸⣿⣆⣴⡿⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠈⢻⣷⣤⣿⣧⣤⣾⠋⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠃⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢤⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄
⠀⠀⠀⣴⣮⣿⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⣯⣶⡄
⠀⠀⢸⠉⡇⢸⠀⣿⣿⣿⣿⣷⡀⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠠⠤⠤⠄⣰⣿⣿⣿⣿⠇⢸⠀⡏⢹
⠀⠀⣸⠀⡇⢸⠀⢸⣿⡟⣧⣴⣶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠷⢶⣶⣤⡟⣿⣿⠀⢸⠀⡇⢸⡀
⠀⠀⢳⡆⢡⠀⡆⠈⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⡏⠀⡆⢠⠁⣾⠃
⠀⠀⠀⠱⡈⣆⣰⣠⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣰⣀⡎⡰⠁
⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿