Duck man Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⢡⡮⢶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠂⠀⠀⠀⠲⢀⠁⠐⢀⡠⠞⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡆⠎⠐⠀⠀⠀⠀⠀⠀⢡⠞⠉⠀⢀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠃⠀⠀⠐⢡⠎⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠒⠀⠁⠙⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⡀⠀⠀⠀⠀⢐⣾⣯⣄⣀⡤⠔⠂⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠈⠁⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠂⣀⠀⢀⡼⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠀⠀⠀⠘⢊⣠⠋⠠⠤⠄⢠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠀⠀⠀⣀⠴⠂⣰⣎⣁⣶⣾⣶⠶⢿⣂⠉⢴⠆⠀⠀⡤⠊
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⠀⠤⠔⠊⢹⣄⠠⣏⢠⡟⠑⠦⢼⡤
⠀⠀⠀⠁⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠠⣏⠙⡗⠀⢀⡇⠀⠀⣀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠠⠿⠟⠀⠻⠿⠷⠴⠊⠳⠀⠈⠉

⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠛⠳⠯⣔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⣤⡀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡓⠂⣉⢲⡀⠀⠀⠀⠀⠀⠀⣠⣿⣥⡄⠄⣀⣀⣁⠐⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⣀⠉⠛⢶⣒⡐⢂⡈⠛⠛⠋⠀⠈⠉⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⠉⠿⠍⢶⣴⠈⠀⢀⣌⡀⠴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠈⠳⣤⡀⠀⠀⠀⠀⠀⡙⠒⠚⠛⠛⠉⠀⠄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⢹⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⣿⣿⣿⣶⣶⣆⢀⣀⣀⣀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢀⡀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⢠⡏⠀⠀⠀⠀⠀⠀⠉⣿⣿⠟⠁⡀⠀⠙⠀⠀⡾⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠀⢠⠀⢠⡄⠀⡀⣠⢄⡾⠀⣿⡟⠚⠇⠁⠀⠈⠉⠀⣷⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⠀⠀⠈⠀⣠⡄⠄⠀⠿⢃⠘⢃⣏⣼⣵⢂⣿⡇⢠⠀⠈⡆⢠⠀⠀⠀⠈⠋⠙
⢀⡀⠠⡄⠀⠀⠀⠦⠄⠀⠁⠙⠀⠀⠁⠀⠀⢠⡄⠀⠀⠀⠰⠿⢟⣸⣿⣇⣼⣀⣤⡰⢸⠀⠀⢸
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⠀⠀⠄⠤⠀⠀⠀⠀⠚⠓⠿⠿⣿⡿⠿⠟⡤⠤⠀⠰⠓⠀⣴⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠋⠉⠉⠙⠳⣄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⣀⣀⠀⠀⠀⢠⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⣀⣀⣀⣀⣘⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢻⣏⢳⡀⠀⣿⣿⣟⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠴⠶⡿⠯⠭⠥⠤⠤⢤⣬⣭⣳⣄⠔⢹⣿⣿⡿⠋⠀⠀⣿⠀⣧⠀⢻⣿⡉⢡⠃
⠀⠀⠀⠀⠀⠀⢸⠃⠀⣸⠃⠒⠀⠫⠀⡆⢃⡇⠀⠀⠧⠀⣤⣤⡟⠀⠀⠀⠀⣿⠁⣹⢀⣌⠉⢣⠘⣆
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⢠⣦⣤⠴⠢⠵⠄⡆⠀⠀⣠⣾⣿⣿⣧⠀⠀⢀⣀⢟⡀⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢷⣤⡼⢸⠈⢿⣷⣶⢏⡟⠁⢠⠞⠋⠉⠉⠉⣩⣷⠾⠟⠋⢊⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣠⣧⢀⣻⣦⣃⠈⠙⠋⣞⠀⠀⡇⠀⠀⠀⠄⡋⢉⢁⠠⣠⢖⣾⣛⣯⡉⠻⠿⠟⠟⠉⣡⠇
⠀⠀⠀⣠⠖⠋⠁⠈⠻⣿⡟⠈⠙⣻⠚⠉⠳⣄⢻⡲⠀⠀⠀⠀⠁⠀⠀⢀⣰⣋⣤⢝⣤⣤⣤⣤⡶⠞⠁
⠀⠀⡴⢯⣄⠀⠀⠀⠀⠈⠛⠓⢾⡁⠀⠀⢀⡤⣞⠉⠓⠦⠤⣤⣤⣶⣶⣦⣮⣥⣤⣦⣤⣤⣴⣿⣟⢦
⠀⢸⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⡟⠁⠘⡇⠀⠀⠀⡇⡂⠀⣩⠡⣬⠉⣤⠇⢉⣶⠉⢱⢸⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠸⡿⠣⠄⠠⡌⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠸⠠⢴⢤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⣰⡷⢤⣤⠞⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢷⣿⣿⠂⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠹⠏⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⢴⣄⣿⠿⠶⠆⠠⠶⠶⠶⢶⣶⣇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢰⡃⠀⠒⠀⣀⣸⣀⣀⣀⣀⢸⡋⠀⢠⠀⠀⠀⠀⠀⠷⠃⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣸⡇⢠⡄⡛⠿⢻⠛⠛⠛⠛⢻⣇⠀⠀⢱⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣥⠞⠉⢁⣿⡇⣿⣷⣾⣶⣤⣤⣴⣼⢯⡀⠀⢀⣿⣶⣷⡄⠀⠀⠀⠀⢸⣿⡄⠀⠀⢀⣠⣴⣶⡶
⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⣠⣾⣭⡧⣍⣩⣭⢈⣉⣀⣉⣹⠀⢧⣀⣾⣿⣿⢿⡗⠀⠀⠀⠀⠘⣿⣿⡄⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣆⣤⣾⠾⡿⢿⡗⠿⠿⠿⡿⠿⠟⣿⣿⣤⣤⣽⣿⣿⣙⠟⣇⢁⠀⠀⠀⢀⣾⣻⣿⡟⢯⣿⠏
⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⣿⣿⢵⣶⣾⣷⢿⣶⠾⠿⣿⣿⣿⣿⣿⣿⠉⢀⠋⢉⡀⠀⠀⠘⢇⢹⣿⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⣿⡯⣯⣽⠂⢀⣀⡏⠙⠻⢷⡄⠀⡼⠀⠈⣧⡄⠈⡇⠀⢠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠌⣛⣛⠿⠛⠋⠤⠞⢛⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⣿⡇⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⠚⠋⠉⢶⣦⣤⡴⠿⠏⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢯⢾⣶⠶⠦⢂⠤⠶⣦⠤⠤⢾⡀⠀⠀⠀⠀⠀⠀⠀⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⢶⡴⠖⣺⡏⢲⣶⠶⢒⣒⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠲⡶⠞⢻⠀⠸⡟⠿⠦⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠛⢒⣊⡉⡏⠀⠀⡟⠛⢖⣠⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇⠀⠐⡇⣭⣵⣦⡆⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠶⠀⠇⠀⠀⢀⠆⠀⠀⠀⠢⡰⠦⠠⠻⡁⠀⠀⠀⠀⢀⣅⠤⠍⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢆⡀⢀⡀⠀⠀⠀⠀⢆⡀⠄⠘⠒⠛⠒⠀⠉⠀⠀⠀⠞⠁⠋⠉⠗⠊⠲⠄⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⠀⠀⢀⠄
⠀⡠⡄⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⣸⠷⠄⢱⣀⠀⠁
⣠⡱⣕⢼⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠈⠁⠠⠍⠉⠁⠒⡄
⠑⠬⠼⠧⢈⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣄⠀⡳⣠⠔⢚⢦⠀⢀⡜⠁
⠀⠀⠀⢰⢸⠀⠀⠙⢕⢴⡇⢀⣰⠳⡀⠀⠀⠀⠀⠀⠀⣸⡈⠧⠞⠤⡞⠁⠀⠀⠘⣧⠎⠀⠀⠄
⠀⠀⠀⠈⠉⠀⠀⢠⡾⠋⠰⡜⢀⣁⣱⠤⠔⠒⠂⠉⠉⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⣇⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡱⠀⠀⡇⡼⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣧⣀⣠⢴⡏⢀⡟
⠀⠀⠀⢀⣀⡴⠔⠉⠀⠀⢰⠀⢷⠸⠴⠼⠭⠗⣦⣄⠀⠀⢀⡀⠤⠐⡲⢀⠃⠀⠉⠉
⠀⠰⣏⡁⠀⠀⠀⠀⠀⠀⣆⢀⣘⣆⣀⡤⠔⠒⠛⡯⠙⡉⠘⣱⢊⢷⠃⢸⠇
⠀⠀⢇⠉⠒⠤⣀⠀⠀⠀⢄⣠⢮⠘⢷⠋⠓⢤⡜⠁⠑⡿⠐⣗⢡⠇⠀⠀⡆
⠀⠀⠀⠑⢄⠀⠀⠙⠲⢴⣏⠁⠘⢆⣬⡆⠀⠀⠀⠀⢀⢴⠊⠉⣻⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⢢⠴⣇⠀⢙⠆⠀⠀⠀⠺⠞⠀⠀⠸⣄⠀⠀⣷
⠀⠀⠀⠀⠀⢴⣉⡀⠀⠀⠀⢘⣶⣟⣀⣮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠉⠵⠍⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠒⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⡤⠶⠀⠼⠀⠠⠄⠠⠄⠀⠀⢀⠲⡰⡆⠤⠽⠏⠀⠀⠀⠀⠀⣀⣏⡀⠜⡈⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠸⡖⢂⡀⠜⠢⠟⠶⢖⡚⠂⡀⠉⠀⡖⠁⢚⠛⠧⠖⠲⢌⠙⠄⠈⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢴⣋⠙⡆⠀⠀⠞⠁
⠀⢮⢳⡐⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡰⠯⠍⠁⠾⠶⠤⣀⡀
⣞⠢⠑⣕⠃⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⠖⠀⢀⣜⠁⠀⠀⢀⠇⠀⠠⠤⠄
⠈⠉⠉⠉⡂⡖⠛⢷⡦⡀⢀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⡍⢳⡀⠔⡡⠊⠉⠎⢳⠀⣠⠏
⠀⠀⠀⠰⣇⡇⠀⠀⠙⣎⡿⠇⢠⠼⣉⢇⠀⠀⠀⠀⢀⣀⣠⠼⠄⠛⠛⠈⡻⠁⠀⠀⠀⢸⡖⠁⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠸⡞⢠⣇⠼⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠤⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠀⢀⠇⢸⠀⡤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢤⣤⡤⢾⣃⣀⠏
⠀⠀⠀⠀⣀⣠⠶⠊⠁⠀⠀⢸⠀⢸⠐⠷⠧⠧⠽⢳⣦⣀⠀⠀⣀⣀⠤⠔⢲⠆⢸⠀⠀⠈⠁
⠀⠰⣎⡉⠁⠀⠀⠀⠀⠀⠀⣌⣀⣈⣇⢀⣀⠤⠖⠚⢻⠯⢗⠉⡘⣹⢎⣷⡎⠀⢸
⠀⠀⢧⠉⠓⠤⣀⠀⠐⠀⠀⣀⢀⣬⡼⢯⡞⠓⠦⣀⡞⠉⠱⣷⠰⡧⣊⡜⠀⠀⢸
⠀⠀⠀⠱⣄⠀⠀⠙⠢⢄⣠⡟⠉⠸⣇⣀⣇⠀⠀⠉⠀⠀⠀⣥⠖⠛⢺⠑⠀⠀⢸
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⠹⣄⣠⡞⠉⠑⣄⠀⠀⠀⠀⣞⡸⠀⠀⠸⡌⠆⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡸⠧⠀⠀⠀⠀⢨⣁⣿⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠼⠃
⠀⠀⠀⠀⠀⠈⠓⠒⠲⠶⢴⣲⣶⡶⠿⡲⠾⠿⢒⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⠖⣷⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⠀⠀⠈⠙⣙⡗⠲⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣄⠈⠁⠀⠀⢨⡏⠁⠀⠀⠀⠈⠉⠒⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠢⣀⠈⠘⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⢸⢃⠀⠀⠑⠀⠀⠠⠴⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠘⠀⠀⠀⠀⠀⢐⡄⠃⠹⠄⠠⠒⠒⠂⠀⠅⠀⠀⠃⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢢⡀⠀⠀⢠⣻⡛⣦⣀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣔⣠⡴⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠹⢦⣄⣿⣬⠼⠟⠉⠈⠀⠀⠀⠸⠟⠀⠀⢰⠦⠶⠖⠛⢛⠊⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⢀⣠⡆⢠⠁
⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⢀⠤⠴⠿⠻⠃⠌⠀⠀⠐⠒⠒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⡝⠢⣐⡀⣀⣀⣀⣀⣇⣀⣀⣀⣋⡟⠀⠀⠀⠀⠀⠀⡠⣶⡄⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠙⠍⠀⠀⠀⠀⠙⠻⣿⠙⣛⣿⣿⣩⣟⡙⠈⣀⣀⠀⠀⠈⠂⠀⠀⠈⠁⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡠⠰⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧⣤⡾⣿⡿⢿⣶⠀⠀⠀⠀⠀⠀⢶⣆⢀⣀⠀⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡞⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅⠈⠵⠼⣻⠀⠀⠀⠀⠀⢠⢞⡭⠚⣭⠛⢷⡶⢊⠀⠁⠀⠀⠀⠀⠀⠀⠀⡸⠟⡓⢄⡀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠒⠀⠀⠁⠉⢉⠀⠀⠀⠀⠀⠣⣿⠃⠀⠉⠀⠾⣻⡯⠼⠃⠀⠀⠀⣠⢤⣴⡶⠀⣠⢤⡀⠿⠓⣶⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠿⣿⣷⣤⣀⣀⠠⠀⠀⠀⢄⠀⠀⠀⠸⡚⠀⢠⡴⠛⠁⠀⠀⠀⠀⢀⡌⣰⣾⣿⣇⠀⠀⠀⠰⠀⣸⢇
⠀⠀⠀⠀⠀⢰⠋⠀⣀⡀⠀⠉⠉⡉⠀⠀⠈⢧⠈⠀⠀⠀⠀⢳⠀⠈⡇⠀⠀⠀⠀⠀⢀⠎⣸⣿⠋⠙⢿⣦⣀⠀⠠⠴⠋⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⡀⠀⠀⠐⢳⠀⠀⢸⡆⣧⠀⠀⠀⢸⡃⠀⢼⡄⠀⠀⠀⣰⠏⢠⣿⡟⠀⣀⢛⠎⠁⠀⠀⡰⠂⠀⠈⡀
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢿⣁⠀⠀⠐⣫⠆⠦⣼⣿⣿⠀⠀⢀⡾⠉⠓⡼⢧⠀⠀⡼⠇⡴⠿⢿⡇⢠⡠⠊⠀⠀⣠⠎⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⣧⠀⠀⡴⠋⠀⠀⠈⠙⠿⣀⣤⠾⠀⠀⠀⠀⠈⠳⣼⡅⣾⠁⠀⠀⢳⣀⠇⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠿⣿⣿⠁⢀⣼⣿⡷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠋⠀⠀⠀⠈⣿⣤⣶⠟⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠈⢳⡀⠚⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣃⡀⢀⣽⣿⠁⠀⠀⢀⡴⠳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠹⣏⠈⠁⣷⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡁⠈⡏⠈⢿⣄⣤⠞⠉⢣⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣄⠀⣨⢶⣶⣿⡅⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢤⣼⠶⣯⣅⠞⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣼⡟⠘⣿⣿⠈⠻⣟⢁⠞⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡀⠀⠀⣿⡿⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⣿⣷⡀⢿⣿⠀⣀⣥⠆⠀⠀⠀⢸⣿⡿⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⣧⣿⣿⡇⠀⢰⣿⣇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢸⣿⣿⠋⣾⣿⣠⣋⣟⠀⠀⠀⠀⣸⣿⡇⠀⢠⠄⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣷⣆⣸⣿⡇⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢸⣿⠟⠸⠛⢿⣿⣵⣿⠆⠀⠀⢀⡿⣿⠇⠀⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣷⣶⣦⣤⢿⣼⣯⣀⠀⠀⢼⠃⣿⠀⡄⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⠏⠉⠻⢯⡿⣏⡈⠀⠀⠀⢀⣰⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⢻⠀⠀⠸⡁⠀⠀⢠⡆⡿⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⢠⠇⠀⠀⠀⠀⠀⠀⣽⡿⠟⠛⢙⠇
⠀⠀⠀⠀⠀⠀⠸⣟⢛⣧⡼⠀⠀⠀⡄⠀⠀⠈⡇⣿⠀⠀⣤⢠⣶⣶⣿⡁⠋⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⡄⠀⣀⣧⠀⠀⢀⣿⣭⣶⣿⣿⣼⡏⣠⣿⣧⠰⡴⠃⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢧
⠀⠀⠀⠰⣶⣴⣶⣼⣽⣿⣿⡇⠐⢿⣏⠄⠀⠸⡿⢿⣿⣷⣾⣿⣿⣿⣿⡿⠳⣷⡀⠀⣀⣀⡀⠀⠀⢸⡇⠀⠀⢀⠆
⠀⠀⠀⠀⠈⠙⢿⣿⣿⡿⣿⣦⣤⣀⢹⠀⠀⠀⡇⠀⠀⠈⠉⢹⡽⣌⠁⣀⣀⣹⢟⠢⣄⡉⠛⠿⢶⣾⣿⣷⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣷⣶⣾⣧⠒⠚⠛⣿⣛⡙⠄⠛⣀⣬⡍⠒⠤⡤⠀⠹⠆⠀⠀⠀⠈⠙⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠉⠙⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⡏⢨⡆
⠀⠀⠀⡠⠒⠉⠁⠀⠉⠉⠒⢄
⠀⠀⣰⠁⠀⢀⡴⠒⠲⢤⡀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢿⠀⠀⠸⣇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⠛⠻⣷⠆⠄
⠀⠀⠈⢣⡀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⣤⠀⣏⣀⢨⡀
⠀⠀⠀⠀⠈⠓⠦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣾⣿⣶⣾⠃
⠀⠀⢀⣄⡀⠀⢀⣸⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠋⠁⠀⠀⠀⠆
⠀⠀⢸⡇⠉⠉⠉⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢰⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠘⠧⠤⣀⣀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⢠⣧⢸⠀⠀⠀⠀⠀⠀⠀⠳⣦⣤⣀⡀
⢠⣾⣿⣻⣿⣦⣼⠀⠀⣸⣀⣴⣾⣿⡄⠀⠀⣠⣾⣿⣿⣿⡟⡀⢸⠻⣌⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠿⣿⣷⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⣠⣼⣿⣿⣿⣿⡿⠀⠳⣀⠓⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⣿⣿⡄
⠙⣛⣫⣭⣽⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣧⣤⣤⣬⣷⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿
⠸⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⠿⢿⠿⠿⠛⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡁
⠀⠀⠀⣾⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣦⡀⠀⠀⠀⠀⡠⠒⠀⣰⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣶⣤⣀⣜⣀⣤⣾⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡼⠁⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠀⠀⠘⠃⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢈⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠒⠚⢭⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡠⢤⡀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⢀⡀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⢸⡁⠈⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶
⣿⣿⣤⣶⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣶⠊⣽⣆⠀⠻⣳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠬⠤⠵⣄⣘⣿⠶⠄⠀⠑⠫⣓⠤⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠉⠛⠛⠀⠀⠀⠆⠀⠑⠈⠑⢤⣀⠀⠀⠈⠁⠀⠋⠉⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠥⠈⠉⠀⣨⢀⠀⠀⠀⡀⠀⠀⠀⠑
⠀⣶⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⠤⣄⠀⠀⠀⠸⠁⠀⠸⠀⣠⠁⣠⣾⡶⠀⣷⣦⣐⠀⠀⠀⣀⡠⠀⠀⣀⡤
⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠐⣛⣍⣉⣩⣹⣶⠀⢇⠀⠀⠀⠐⣠⣾⡟⣻⠿⠀⠀⣼⣷⣾⡷⢤⣶⣄⠀⢶⠊⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠈⡛⣿⣿⡗⠀⠌⠀⠰⠷⠀⣹⣿⢻⢱⡟⠀⠠⣟⣿⣿⣿⣦⢙⣿⣦⠈⢂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠛⣄⠄⢉⠛⠃⠀⠀⢀⣙⣓⣬⣿⣿⣾⣿⡇⡠⠴⠿⠿⢿⣿⣿⣿⣦⣽⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠤⠤⠤⠀⠤⠀⠀⠀⣂⣦⠔⢻⣶⣬⣀⠉⠙⡻⣿⣿⣿⣿⠿⣧⡠⣀⣶⣄⠤⡴⠒⠖⢨⡁⠀⠄⠨⠁
⠀⠀⠤⡄⠈⠁⠀⠀⢶⣦⠤⠱⠀⢰⣻⡆⢘⢿⣿⣾⣭⣕⡲⣿⡻⠟⠿⣔⣿⢈⣠⣞⣿⣷⡙⠀⠀⠈⠁
⠀⠀⢀⠃⡜⢦⠀⣐⣙⠁⠀⠀⠴⠋⣻⡇⠈⢠⢿⡿⢺⣿⣿⣿⣧⣄⢢⡌⠹⢯⣿⣿⣽⣿⣿⡆⠀⢀⠌
⠀⠈⠿⣄⠈⢚⣆⡠⣿⠀⠀⠀⠄⠰⠟⠇⠔⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠉⠛⣿⣿⡿⣷⣁⡀⢤⡀⠀⡀
⠀⠀⠀⠈⣙⣻⣭⣶⣿⠏⠀⢀⡔⠀⡘⣀⡀⠰⠀⠀⠄⠙⡩⠊⠉⠀⠘⡆⠠⡄⠒⢈⣻⣿⣯⣿⣿⣿⣦⡲⠀⢱⡇
⠀⠀⠀⠉⢤⣟⣿⣿⡟⡄⠀⠸⠀⠸⣿⢟⡷⣦⣤⣀⣠⠊⠀⠀⠀⠀⢀⠤⠀⠀⠁⠖⣹⡯⢙⣿⣿⣿⣿⠏⠀⢊⡄
⠀⠀⠀⢀⠒⢶⣶⣼⢡⠈⠀⠀⣠⠘⠁⠋⠞⢟⡿⠃⠽⢟⣷⣿⣢⠆⠀⠀⠀⠂⠀⠀⢁⣤⣾⣿⣿⡿⠋⠀⠰⡋
⠀⠀⠀⢘⣿⣿⣴⣷⠈⠀⠀⠀⠑⠇⢀⡤⠖⠉⠴⠀⠰⣋⠘⣿⢿⣴⡄⢀⠀⠀⠀⠠⢟⣻⣿⣿⣟⣀⣀⠀⣠⣥⡄
⠀⠀⠀⠀⠉⠫⠍⠀⠀⢀⠀⠀⠰⡃⠀⢀⠰⢀⠄⢀⣾⡯⠸⣳⢟⢏⣴⡞⢀⣀⣴⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⠉⠑⢿⣇⠀⠈⠀⠀⠃⠇⠀⠹⠀⠀⠀⡉⠋⠀⠰⣫⣧⡞⠘⣏⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠁⠀⢀⠃⠉⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⡸⠁⠀⠀⣴⣿⣿⢣⣿⣛⣽⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠁⣠⡇⠀⢰⡄⠀⠀⠀⠀⠀⢰⣷⣿⢁⣴⣧⠐⢛⣿⢇⡟⣡⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠁⠈⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⣛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠈⠳⡄⠀⠀⠀⣰⣾⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⣀⣠⠤⠤⠤⠽⠤⣀⠈⠙⠛⠋⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣊⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⢿⣿⣶⠶⠶⣶⡶⣶⠾⠃⠀⠀⠀⠀⠀⣰⣿⣿⣄⡀⣀⠀⠀⠀⠀⢀⣤⣚⣵⠞⠋
⠀⣠⠴⢦⡀⠀⣀⣀⣿⣿⠈⣿⣿⣶⣴⠟⣧⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⣤⣾⠟⢿⡏
⣿⣧⢀⢠⡱⣎⡁⣙⢿⣿⣴⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣻⡵⣧⡾⠁
⣿⣿⣿⣿⣿⣿⣿⣾⡿⢿⠙⣿⣬⣙⣛⣿⣿⠃⣴⣆⠀⣾⡇⠀⣴⡀⣰⣧⣶⡿⢟⣥⡟⣷⢀⡟⠀⣴⡄⢀⣾⡄⢰⣷⢀⣿⣆⣰⣆⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣏⠀⠈⠻⣿⣇⠀⢸⣿⡟⢺⣿⣿⣾⣿⣿⣾⣿⣿⣽⠟⠉⠀⢸⢘⡇⠘⢰⣷⣰⣿⣷⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣿
⣿⣿⣿⣿⣿⣟⠀⠙⠳⢤⣀⡈⠛⠷⣼⣿⣥⠀⣹⠿⣿⣿⣿⣿⠾⠋⠀⢀⣠⡾⠋⠈⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠉⠛⠲⠦⠤⣬⣭⣭⡭⠿⠿⠿⠛⠁⠀⣻⠟⠀⠁⠀⣀⡴⠿⢿⠃⠀⣴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣠⣾⣏⠀⠀⣞⡀⣸⡇⠀⠀⠀⢀⣀⣐⣶⣦⡤⠄
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣀⣀⣠⡿⣿⣿⣆⢰⣿⣿⠻⠿⡦⠀⠀⠀⢈⣉⣹⣿⣶⠄
⣿⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⣿⣿⣿⠀⢠⠃⠀⠀⠈⠙⣛⣿⣿⣤⡀
⣿⡏⠀⠀⠀⠀⠀⠀⣷⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⡟⢠⡏⠀⠀⠀⠈⠻⢿⣿⣯⣅⣀
⣿⡇⠀⠀⠀⠀⢀⣼⡿⠞⠋⠉⠈⣷⠀⠀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⠀⠀⠀⠠⠶⢶⣿⣿⣿⣛⣉⠁
⣿⡇⠀⠀⡴⠚⠉⠁⠀⠀⠀⠀⠀⢸⡶⠛⢩⣭⣹⣿⣿⣿⣿⡏⠻⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢤⣶⣾⣿⣛⣉⠉
⣿⣇⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣟⣋⠉⠁
⣿⣿⡀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣍⣉⡁
⣿⣿⣿⣦⣤⣤⣶⣿⣿⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⣯⣉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⡟⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢬⠋⢠⠒⠒⠖⠹⡞⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⣭⡠⠔⠂⠴⠷⢄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢸⡉⠓⠒⠲⢺⡟⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠞⠀⠳⢄⣀⣠⢞⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠋⠉⠀⠈⠙⠢⢌⣑⠀⠤⠤⠈⠉⠉⡀⠉⠑⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠥⢤⣀⣀⣀⠀⠀⠈⠑⠒⠒⢐⣀⣀⣐⣚⣭⠥⢶⡖⠛⠲⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⣍⢉⡏⠑⠒⠂⠉⠉⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠶⢶⣀⣴⠾⠻⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠳⣤⣤⣄⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⢻⡁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡾⠁⠈⠙⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⡀⠀⠀⠀⠙⢆⣠⠾⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⣷⠶⢤⣤⣈⠁⠀⡀⢬⣧⣤⣤⣀⣀⣀⣀⣠⠤⠖⢻⠀⣠⠖⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡏⠀⢀⡿⠈⠛⢲⣽⣿⣉⣀⣀⣀⣀⣀⣀⣀⣠⣤⢾⣆⠻⣶⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⠇⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠘⣟⠛⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠁⠀⣿⠀⠀⢸⡟⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠀⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⠀⢠⡇⠀⠀⢸⠁⠀⠀⠈⠙⢦⣄⣤⣀⣴⠞⠋⠁⠀⢿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡇⠀⠘⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⡟⠀⠹⣧⠀⠀⠀⠀⢸⠀⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢠⡿⠀⠀⠀⠹⡆⠀⠀⠀⢸⠀⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⡾⠀⠀⠀⠀⠀⡼⢡⠆⠀⠀⢠⣿⡇⠀⠀⠀⠀⠹⣄⠀⠀⢸⡄⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣰⠏⢸⡇⠀⠀⠀⠀⠀⠹⣆⠀⠏⢳⣼⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣼⠁⠀⠀⠀⠀⣿⣄⣀⢀⡼⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢈⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⡟⠀⠀⠀⠀⠀⡿⠉⢻⡟⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⣿⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢸⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⢷⣄⡀⣀⣴⡿⠀⡾⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠾⣧⢸⡇⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡟⠀⠈⠙⠉⣼⠃⢸⡇⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⢹⠀⢿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠾⠧⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣤⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⣤⣽⣧⣤⡀⣀⢀⡀⣄⠀⣤⣤

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⡄⠀⠀⣠⣤⡀
⠀⠀⠀⠀⡄⠀⠀⡴⢓⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣁⣄⣸⠤⠀⠻⠖⠉⠘
⠀⠀⠀⠀⠘⠋⣠⠂⠒⠐⡃⢀⡀⡀⢀⣠⣄⣰⣠⠄⠀⠀⠀⠉⠀⠀⠠⡂⠶⡾⡋⢒⠀⠀⠲
⠀⠀⠀⠀⠀⠀⣡⣀⣀⣀⡏⠀⣠⠄⠊⣹⠩⠔⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣌⡉⠁⠈⠁⢦⣤⣆
⠀⠀⠀⠀⠀⠀⠻⠿⠛⠉⠎⠉⠠⡤⠎⢀⠜⠁⠀⠀⠀⠀⠀⠀⡤⡤⠶⠊⠉⣗⡐⠾⠁⠆⠈⠛
⠀⠀⠀⢰⠐⠄⠀⡡⠄⣤⠘⡛⠒⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⢺⣏
⠀⠀⢠⡧⠀⠠⠀⡀⠀⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢰⠀⠊
⠀⠀⣾⣀⣄⡠⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢉⡜⠃⠀⢀⠴⢶⣶⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡄⠀⢀⡀⡀⣀⠀⠀⢀
⠀⠀⢸⡷⠄⠘⠁⠀⠀⠈⠳⡤⣚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠋⠞⠛⠑⠃⠴⢣⡾⠂
⢀⣠⣿⡿⠁⣀⠀⠀⠀⠀⠀⠹⣿⣧⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢂⠀⠄⠀⠀⣠
⠈⠉⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡠⠐⠟⠠⣀⠀⠀⠀⠀⠀⢠⣶⠒⢲⣶⠒⢲⣆⠀⠀⠀⠀⠀⠰⠀⣀⡀⠰⠀⠀⠀⠀⠀⢀⡤⠖⠚⠒⢦⡀
⠀⣴⣾⣶⣶⣶⣶⣾⣶⡄⠀⠀⠀⢿⣏⣀⣸⣿⣀⣈⡿⠇⠀⠀⠀⠀⣶⢲⣾⣷⣶⡆⠀⠀⠀⠀⡞⢠⢠⣄⡄⣤⢹
⠀⣿⡏⠍⢉⡉⠍⡙⣿⡇⠀⠀⠀⢸⡏⢹⡏⠉⣿⠉⣿⠀⠀⠀⠀⣠⣿⣴⣿⣿⣿⣧⡀⠀⠀⠀⢷⡈⠉⠉⠉⠉⡸
⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⣿⣿⠇⠀⢿⣿⣿⡄⠀⠀⠀⢧⣶⣶⣶⣶⣶⣦⠇⠀⠀⠀⠀⠙⠶⠤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠛⠻⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢦⡀⠀⠀⠀⠀⠀⠀⢀⣎⡏⣻⣳⠀⠀⠀⠀⠀⢠⣸⣭⠆⣯⢿⣠
⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢤⣼⣥⣤⣤⣤⣝⣤⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠶⠶⠆⣶⠰⠶⠶
⠀⠀⠘⢛⡛⠛⣛⠋⠀⠀⠀⠀⠀⠈⣷⣺⣒⣗⣺⣲⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⢸⣿⡇⣿⢰⣿⣷
⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠸⠾⠧⠧⠿⠞⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠼⠀⠀⠀⠀⠘⠿⠇⠿⠘⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣉⠀⠁⠈⠀⠉⠈⡁⠀⠀⠀⠀⢸⠀⣿⢸⢸⡆⣿⢸⠀⠀⠀⠀⠀⡿⣶⣶⣤⣭⣭⡷⠀⠀⠀⠀⠀⡠⢪⢭⡙⣦
⠀⠤⠰⣒⠄⢀⣠⠄⠄⠀⠀⠀⠀⢸⠀⣿⢸⢸⡇⣿⢸⠀⠀⠀⢀⣶⣶⣶⣶⣶⢾⡍⡇⠀⠀⠀⣿⣾⣶⣾⣶⣷⣾
⠀⠒⠀⣊⠖⣯⡒⠐⠂⠀⠀⠀⠀⢸⠀⣿⢸⢸⡇⣿⢸⠀⠀⠀⢠⣿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠽⠀⠀⠉⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠿⠒⠒⠲⠞⠛⠃⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡀⢀⡀⢀⣤⣤
⠀⣴⡶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠘⡇⣤⣤⣤⠀⠀⠀⠀⢠⣮⣤⣤⣿⣧⣤⣬⣧⠀⠀⠀⢹⠁⢹⡏⠈⠻⠟
⠀⢸⣧⣇⣆⣇⣸⣿⣶⣐⣆⠀⠀⠀⠀⠀⠷⢿⣿⣿⡇⠀⠀⠀⢸⡇⡀⠀⡉⢁⠀⠀⢸⠀⠀⠀⠈⠀⢈⡁⠀⣈⣁
⠀⠿⣿⣿⠿⠿⠿⠿⠿⣿⣿⠧⠀⠀⠀⠀⣞⣻⠜⠒⠀⠀⠀⠀⢸⡃⠏⠘⠞⠚⠃⠃⢸⠀⠀⠀⠰⣤⡞⠁⠈⠙⠋
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⢛⣉⡩⠤⠤⠤⠤⠤⠤⢍⣉⡛⠻⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⣡⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣌⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢁⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⡈⠻⣿⣿⣦
⠀⠀⠀⠀⢀⡾⣿⡿⠁⡰⠛⡽⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣸⡟⠀⠈⢿⠻⢦⠈⢿⣟⢷⡀
⠀⠀⠀⠀⡼⠱⢹⣀⠜⠀⠀⠀⠀⠀⢻⣿⠛⠛⣿⠛⠛⣿⡟⠛⢻⡟⠛⢻⣿⠃⠀⠀⠀⠀⠀⠱⣀⡏⠂⠁
⠀⠀⠀⠈⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⡶⣶⣿⣷⡶⣿⣷⣶⣿⣷⡶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠁
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣽⣇⣀⣿⣇⣀⣿⣁⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣟⣉⣿⣏⣹⣿⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⣇⣠⣴⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠉⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣦⣄⣸⡇
⠀⠀⠀⢀⣤⣾⢿⡿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⣷⣾⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡟⣶⣤⡀
⠀⠀⠀⢿⣿⣿⡎⢧⠀⠃⢉⣊⠣⠉⢈⣁⢃⡘⠐⣈⠙⣉⢊⠁⣂⢂⣘⢓⣀⠋⢉⡋⣉⢡⠀⠀⡾⢱⣿⣿⡿
⠀⠀⠀⠈⢿⣿⣿⡌⢣⠀⣿⡿⣥⣼⡟⢉⣟⢹⣿⣿⣾⣷⣿⣿⣿⣿⡯⢿⣽⣿⡟⠉⣿⠿⠀⡼⢡⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣄⠡⡙⠛⠀⠀⠙⠋⠙⠋⠉⠉⠁⠉⠋⠉⠙⠙⠛⠉⠈⠁⠙⠋⠙⢛⠊⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠼⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⢿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⡿⠿⠛⠛⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⢸⠉⠩⡍⢉⢉⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⣣⡇⣠⠼⠃⠀⠀⠀⠲⠲⡄⠀⠀⠀⠾⠗⠀⠀⠀⠀⠀⠀⠀⢠⡼⣛⣭⣷⣮⣍⡻⣤
⠀⠀⢸⠀⠀⠹⡿⡿⠀⠀⢼⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⣶⡶⠒⠒⢲⠀⠀⠀⠀⠀⢻⣶⢶⡶⢶⡶⣶⡶⣶⡆⠀⠀⠀⣾⠁⣿⠛⡟⡛⢻⡟⢸⡆
⠀⠀⠸⣤⣤⣤⣤⣤⣤⣤⡾⠀⠀⠀⠀⠀⠀⠘⡿⠿⠿⣿⡿⠻⠿⡟⠀⠀⠀⠀⠀⠀⢿⡾⢳⡞⢳⡞⢶⡿⠀⠀⠀⠈⣿⠀⣿⣚⣛⣷⣸⣇⢸⡏
⠀⢀⣾⣿⣿⣿⠛⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠻⡛⠿⠿⢻⠟⠁⠀⠀⠀⠀⠘⢧⣝⠿⢿⠿⢟⣩⠿
⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠳⠶⠶⠿⠷⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠋⠉

⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤
⠀⠀⡏⢰⣶⣶⡆⡶⢦⡖⠘⡇⠀⠀⠀⠀⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠶⠶⠤⣤⣤⣤⣤⠀⠀⠀⠀⣾⣿⣿⣿⣿⣋⢀⣈⣿⡆
⠀⠀⢧⡘⠷⠛⠳⠃⠘⢀⣰⠇⠀⠀⣠⠤⣤⠤⣿⣶⠶⠦⠤⠄⠀⠀⠀⠀⠀⣿⠁⣴⡶⠶⠶⢤⣤⣝⣛⣀⣤⠀⠀⢸⣿⢫⠶⠶⠶⠶⠶⠶⠶⠶⡄
⠀⠀⠀⠉⠉⠁⢻⣶⣭⡍⠁⠀⠀⠀⠓⠶⠋⠐⠊⠻⣆⠒⠀⠒⠐⠂⠀⠀⠀⣿⠇⡟⠁⠀⣴⡋⢿⡄⠈⠻⣿⠀⠀⠈⣿⢸⣷⣶⣶⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠈⠿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠓⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⣧⣄⣘⣯⡞⠁⢀⣴⣿⠀⠀⠀⠀⢸⠀⠶⠶⠶⠆⠠⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠉⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣿⠀⠀⠉⢓⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠈⠳⢤⣘⣉⣉⣭⢹⡇⠀⠀⠀⠀⠀⠸⢋⣴⠏⠛⢿⡿⠋⢸⡇⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⢀⡟
⠀⠀⠀⣠⠞⣡⠀⠉⠉⠉⢸⡇⠀⠀⠀⠀⢀⡴⡏⠚⢷⡾⠋⠀⠀⢸⡇⠀⠀⣾⢣⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣤⣤⢀⡾⠋⠀⠸⣧⠤⢤⡀
⠀⠀⢾⡁⠞⢿⣷⣤⢀⡴⠋⠀⠀⠀⠀⠀⢸⣷⣖⡾⠋⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⣻⣿⣿⣿⣟⡽⢒⠶⣄⠀⠀⠀⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⢦⡀⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣿⣿⣿⢸⡑⢾⠗⣸⠀⠀⠀⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠙⠋⠀⠙⠒⠚⠁⠀⠀⠀⠘⠛⠛⠘⠓⠒⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⡞⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⢀⠴⠛⠛
⠀⠀⠀⡾⢻⡟⠿⠟⣿⢻⡆⠀⠀⠀⠀⠲⢠⡶⠶⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⠀⠀⠘⣇⠀⣀⣠⣴⣦⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⡇⠈⢁⠀⠀⠉⢸⡇⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⠀⣷⡤⠤⡄⠀⠀⠀⢠⡿⡜⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡟⣛⣛⡛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⡇⠐⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⣿⣶⣶⣿⠀⠀⠀⠈⢧⢛⣽⣛⠿⢟⣻⡄⠀⠀⠀⠀⡇⠉⠉⠁⢠⣦⠀⣶⠀⡇
⠀⠀⠀⡇⠀⠈⠛⠋⠀⢸⡇⠀⠀⠀⠀⠀⠸⢿⠫⢻⠶⠶⠶⠶⢿⡟⢝⡿⠇⠀⠀⠀⠀⣏⠶⣸⡇⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠇⠀⠿⠀⡇
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⢒⡒⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠎⠀⡀⠉⠁⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠃⣎⢨⠆⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⡀⠀⢎⡁⠀⠀⠀⠀⣶⣾⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣷⣶⠊⠉⠁⠀⠀⠀⠈⢻⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⠐⡤⠅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢄⣊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡞⢀⣀⠠⠤⠠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣉⣙⡽⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡴⡶⠉⢀⣠⣤⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⢺⣸⠀⠀⠀⠀⠀⢀⡴⣿⡿⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠰⣰⡹⠂⠀⠀⢀⣴⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣹⢗⡆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠲⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣾⣇⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢲⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢲⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠐⠎⡞⠁
⠀⠀⡠⢒⠲⢈⠉⠉⠙⠒⠒⠦⡤⢄⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡔⡸⠁
⠀⠪⡞⠕⠠⣀⣦⢄⣀⣀⣀⣰⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⡠⢱⣧⣤⣤⣄
⠀⠀⠉⠘⠊⠀⠀⠀⠈⠁⠀⠳⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠋⠀⠀⠀⠃⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⢉⣉⡉⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣰⢠⡆⣰⠀⣆⢰⡄⣆⠀⠀⠀⠀⠀⢰⣦⣤⣤⣶⣴⠂
⠀⠀⠀⠀⠀⠸⠻⠀⣾⣷⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠃⡘⠁⢹⣄⠛⠈⢣⠘⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠔⠂⠈⠁⠀⠀⠀⠀⠀⣯⣭⣽⣯⣭⣭⣿⣿⡇⠀⠀⠀⠀⠀⣇⣀⣀⣁⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠭⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿


⠀⠀⠀⠐⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⣶⣖⣶⣶⣶⢰⠋⠉
⠀⠀⠀⠀⠀⠻⠟⠀⣴⢲⣦⠀⠀⠀⠀⣭⣤⣼⡟⢻⣷⣤⣬⡅⠀⠀⠀⠀⠀⠹⠿⠿⠿⠇⡟⠀⠀⠀⠀⠀⠀⠀⠘⠙⠈⠉⠸⣶
⠀⠀⠀⠀⠀⣤⢠⠀⢿⣠⡿⠁⠀⠀⠀⣟⠛⠻⢯⡽⠟⠛⢻⡁⠀⠀⠀⠀⠀⠀⣛⡛⢛⣏⠁⠀⠀⠀⠀⠀⠀⢠⣠⣀⣀⣀⣀⣀⣠
⠀⠀⠀⢠⣤⣤⢠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠉⠙⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⠿⠏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣃⣈⣇⣀⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡾⣵⡋⣝⣷⣹⡀⠀⠀⠀⠀⣀⡔⣀⣾⠉⣩⠉⠉⢹⡆⠀⠀⠀⠀⢩⢯⣯⣭⣿⣽⣽⡍
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠁⠀⠀⠀⠀⠀⢷⡻⣝⣢⡿⣸⠁⠀⠀⠀⠀⢿⣿⡿⠿⠷⠷⢶⣥⡾⠀⠀⠀⠀⠀⠘⡟⡅⣧⡟⣓⣹⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⡠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣄
⠀⠀⠀⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⢻⡆⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⡟⠛⢛⠛⡟⠛⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⠀⡀
⠀⠀⠀⡇⠤⠨⠠⡇⢽⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⢶⣟⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣛⡃⣸⣷⣿⡄
⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠿⠟⠿⠛⠿⠿⣿⠆


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢖⣋⠭⠭⢗⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠔⠋⠤⠤⢀⠀⠈⠪⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡴⠃⠀⠀⠀⠀⠀⢡⠀⠀⢡⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⠞⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢧⣀⣀⣴⣿⣥⠀⠀⣀⡀⠀⠀⠉⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣈⡥⠼⠛⢋⣉⣁⣀⣶⣉⣑⣦⡀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⢙⠏⠛⣉⣱⢀⠄⢉⠙⣿⣙⠻⣷⣌⠓⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡻⠘⣤⠿⠟⠺⣐⣿⣷⠏⠉⡲⠀⡹⢧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⡷⠚⠁⠀⠀⢀⣉⠉⠙⠀⡴⡁⢀⣛⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣀⣀⣀⠀⢉⡷⠀⠈⠛⢱⣯⣭⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠙⠓⠛⠉⠉⠉⠉⠛⠻⣥⣒⣒⠒⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢶⠠⠤⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣂⠀⠀⠈⠙⠂⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⡄⣀⣀⡀⠠⠄⠀⠀⠀⣀⡞⢙⠂⢤⡥⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢒⡦⡤⠶⠦⣬⠰⠚⠁⡙⠻⢦⣤⣧⡴⠞⢢⡀⢄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉⠜⠁⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣄⠀⠀⠀⠙⠺⢦⡀⢆⠀⡄⠀⢻⠳⣦⡀
⠀⠀⠀⣀⠴⠚⢋⣁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡹⣄⠘⢧⡈⠙⢦⣄
⢀⡴⠋⠁⢠⡴⠋⠀⠀⠀⢰⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢳⡦⠿⠃⠀⠙⢷⣄
⣿⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠙⣧
⠹⣦⢠⡿⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣼⠃⠀⠀⠈⢷⡀⠀⠸
⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠁⠀⠀⠀⠀⢸⣧⠀⢠
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣤⡾
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢟⢦⣷⡀⣠⠀⠀⠀⠀⡠⠄⠀⣀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢆⣻⠀⠉⢷⡘⠀⠀⣰⡟⡗⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⡰⠊⡿⠀⠀⠈⢿⣧⡾⢟⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⡇⠧⣇⠀⠀⠀⣼⠷⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣦⣤⣀⡀⠀⠘⡶⢷⡀⢰⡀⢹⣆⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠢⢍⡛⠻⠾⣶⣮⣷⣌⣷⣈⢻⣅⣉⠲⡀⠀⠀⠀⠀⣀⣀⣀⣤⣴⡶⠞⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠐⠂⠤⢤⣉⣉⢉⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣉⣉⣡⣤⡴⠾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣤⣄⣀⠀⡔⠁⣾⠉⣿⠉⡟⡏⢙⡟⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠛⠻⠿⠿⣶⣿⣤⣿⣥⣾⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣤⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⢀⣾⠃⠀⠀⠀⠉⡏⠉⠉⠉⠋⣻⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠉⠁⠀⠀⠀⠀⠀⢻⠦⣤⣦⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣛⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣁⣩⠀⠀⠀⠀⠀⢰⠋⣀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⣷⠀⠀⠀⠀⠀⡆⡀⢳⡄⠉⠶⠤⠄⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠄⠀⠀⠀⠉⠡⡷⠽⠀⡌⠤⣄⢤⣤⣤⢤⢤⣼⣼⢢⡀⠀⠁⢶⣶⣶⣶⣶⣶⡆⢀⣚⡠⡄⣌⡂
⠀⠀⠀⠀⣀⠀⠀⠀⠤⠘⠀⡄⠾⠟⠳⠆⠀⠀⠀⡏⢁⠢⠁⠀⠀⠈⠿⠛⠛⠛⠛⠀⠀⡜⠁⠽⠏⡀
⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⡇⠘⡃⠛⠘⠛⠃⠻⠈⢹⠀⠀⠀⠀⠀⢸⡿⠉⢽⡇⠀⠀⢢⣄⡀⣴⠃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣸⡟⠛⠛⠒⠚⠀⡄⠤⠀⢀⠀⠀⠀⠀⠀⠀⣀⡈⢀⡀⠈
⠀⠀⣡⠤⠤⠤⢬⡀⠀⡀⠀⠀⠀⠀⠦⠥⠈⠄⠀⠀⠀⠀⠑⠚⠘⠐⠲⠃⠀⠀⢀⠀⠫⠠⡀⠀⢃⣨⠇
⠀⠀⡇⣀⣝⣉⠘⠐⢣⡁⠀⢀⣥⢧⠀⡄⠀⣠⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠐⠉⢀⣄⠑⡮⢷⠻
⠀⠀⡇⣻⣏⣿⢰⠀⠀⠀⠈⡂⠉⠈⠀⠴⠾⠓⣷⠤⠀⠀⠀⢀⣰⡈⠁⣦⣀⠀⠐⠛⠛⠀⠇⠟⠁
⠀⠀⠃⠛⠛⠛⠐⠀⠀⠀⠀⠳⠄⠐⠈⠃⠀⠀⠈⠀⡆⠀⠀⠀⣂⣒⣴⣀⠃⠀⠀⠀⢀⡀⠀⠠⣄
⠀⠀⠃⠀⠐⠁⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⢐⠢⠀⠀⡀⠒⠒⠂⠒⡒⣒⠒⠀⠐⠂⢀⡈⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠤⠀⢈⡔⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡏⠃⠀⣀⠀⠀⠀⠳
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣰⣀⣦⠀⡤⠀⠈⠈⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⡤⡀⡄⠐⠆
⠀⠀⠀⠀⣀⢀⠀⠀⢸⡁⠀⠀⠀⣀⠟⣠⠚⠆⠀⠀⠀⠀⠈⠀⠀⠀⢤⣄⠲⡢⢀⣰⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⢀⣀⣉⣉⡝⠀⠃⠀⠀⠀⠃⠐⠀⢰⠀⠆⠍⡍⠲⡇⣟⠋
⠀⠀⣀⣀⠀⠀⢀⣀⣀⡄⠀⠀⠀⠀⠈⠑⠒⠃⢀⡀⠀⢀⡀⠀⠀⠤⠤⣄⣣⠠⠛⢀⣀⣄⣀
⠀⠀⠀⠀⢠⡆⢠⠀⢸⠖⠶⠦⡀⠀⠀⣠⡔⣪⠥⣦⢤⡐⢈⢢⠀⠀⠀⠀⠀⠦⠖⠇⡁⡀⠄⠘⠦⠤⠂
⠀⠀⢀⣀⣘⠀⠙⠀⢸⡇⣀⢀⠙⡄⠈⠻⢽⡃⠈⠙⠀⢹⠉⡢⠄⡀⡂⠀⠉⡖⠂⠀⠴⢻⠃⠀⠀⠐
⠀⠀⠹⣅⡽⠁⠀⠀⠈⠉⣏⢹⠟⠁⠀⠀⠈⠓⠤⣀⣀⢈⡠⠁⠈⠀⠀⠀⠀⠁⠀⠉⠐⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠁⠀⠀⠰⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⢤⣀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠘⢫
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣀⠀⠀⠉
⠀⠀⠀⠐⠢⠤⡤⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡛⠦⡔⣀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣷⣦⣭⣡⢀⣼⣿⣀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠀⠀⢀⣤⣾⣿⣏⠻⣿⣷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡡⠀⠀⠀⠀⠀⠀⠙⢻⣯⣤⣤⣶⡿⠛⠛⠛⢿⣿⣿⣿⣿⡀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠤⢶⣶⡆⣤⣄⠀⣠⣴⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣭⣽⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⣿⡀⠀⠀⣀⣠⣤⣸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⣠⣴⠎⠀⣼⣿⣿⣿⣿⣆
⠀⢫⡗⣋⣩⡛⣿⣯⡶⠶⡶⣠⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⣠⣴⣿⠛⠅⢀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠁⠊⠉⠀⠀⠀⣾⣾⣿⣿⣿⡟⠁⠀⠀⠀⢀⣰⣶⣿⣿⣿⠗⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢧
⠀⠀⠀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣀⣠⣴⣶⣿⣿⣿⣿⣷⣶⡤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠺⣶⠆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠇⠀⣸⣿⣿⡟⠋⠉⠀⠉⠉⠛⠛⠛⠛⠓⠛⠛⣿⣿⣿⣿⣿⣿⣭⣿⣿⣧⣽
⠀⣠⣤⣄⣠⣴⣿⣿⣷⣬⣣⣶⣿⡏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡶⠁
⠀⢶⣬⣙⠛⠋⠀⢈⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⣿⢡⣦⣿⣿⣿⣿⣿⡿
⠀⢼⣿⣿⣿⣄⡴⣻⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣍
⠀⠘⢿⣿⣿⣿⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠨⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⠤⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠓⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠿⠿⠿⠿⠿⠿⠿⠖⠾⠶⠿⠿⠿⠿⠿⠿⠗⠒⠒⠒⠲⠷⠿⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⡾⠛⠻⣦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣷⡶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⣀⣠⡿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⣵⠟⠁⠀⣿⠀⠈⠻⣮⡉⠛⠷⣤⡀⠀⠀⠀⠻⣦⣈⣉⣉⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣼⠏⠀⠀⠀⣿⠀⠀⠀⠘⣷⡀⠀⠈⠻⣦⡀⠀⠀⠀⠉⠉⠉⠉⢹⡏⠉⣿⠉⠁
⠀⠀⠀⠀⠀⠀⣠⡿⠻⠶⣤⣄⣼⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⣧⣠⣤⡶⠞⠻⣦⣤⣤⣤⣤⣤⣤⣼⠇⠀⣿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢩⡿⠙⠛⠛⠛⠛⣿⠛⠛⠛⠛⠋⢻⡏⠀⠀⠀⠀⠹⣧⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢻⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⢾⡷⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢈⣷⣀⣤⣤⣤⣤⣿⣤⣤⣤⣤⣄⣼⡇⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠙⣧⣴⠶⠛⠋⢻⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢩⡟⠙⠛⠷⢦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢻⣄⠀⠀⠀⣿⠀⠀⠀⢠⡿⠁⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡻⣦⡀⠀⣿⠀⢀⣴⠟⣁⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠿⠿⠿⠛⠛⠉⠁

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

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣌⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠟⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⡁⡀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁⢀⣀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠁⠀⢠⣿⣿⣿⣇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠁⠈⠙⢿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠉⢀⣠⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠈⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⢿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⢀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠙⠛⠛⠛⠛⠛⠉⠁⢀⣿⣥⣤⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⢻⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣉⣉⣹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⡀⠀⣀⣀⡀⠀⣀⣀⣀⠀⣀⣀⣀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠿⠿⠇⠀⠿⠿⠇⠀⠿⠿⠟⠀⠿⠿⠿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣤⠀⣤⣤⡄⠀⣤⣤⡄⢠⣤⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⠿⠀⠸⠿⠇⠀⠿⠿⠃⠸⠿⠿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢰⣶⡄⢠⣶⡆⠀⣴⣶⠀⣰⣶⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠉⠁⠀⠉⠁⠀⠉⠉⠀⠉⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣈⠙⣿⣿⣿⣿⡟⢉⣀⡈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠇⢸⣿⣿⣿⠀⢾⣿⣿⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣁⣤⣿⣿⣿⣿⣷⣄⣉⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠤⢾⣿⣿⢩⣤⣾⢡⣦⡙⣿⠀⢿⠁⢹⠇⠸⡏⠈⡏⠠⢼⡇⠴⠌⡟⢡⣦⡿⠠⠴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⣛⣻⣶⣷⣌⣛⣻⣌⣛⣡⣇⣰⣀⣇⣸⣀⣇⣰⣄⣇⣘⣛⣇⣰⣄⣧⣈⣛⣧⣘⣛⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉⠉


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⠛⠋⠉⠉⠉⠉⠹⠿⠿⢿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢰⠟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣿⣠⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣼⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣠⣤⣶⡿⠛⠁⢴⡆⠙⢿⣦⣄⠀⠀⠀⠀⠘⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⣿⡟⠛⠛⠛⠻⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⢿⣿⡿⢿⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⢋⣡⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⣶⣶⣶⣶⣤⣴⣶⣶⡾⠿⠟⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠒⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠒⠀⣀⠑⢦⡀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠓⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠉⠀⠡⠂⠀⠀⠈⠁⠺⡄⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢆⡀⠁⠀⠀⠀⡇⢀⠔⠉⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⣈⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢟⡆⡁⠠⠊⠀⠀⠀⠀⠀⢀⡤⠠⢀⠀⠀⠀⢱⠀⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠎⠀⡠⠔⢢⠀⠀⠀⢀⡎⠀⣸⣿⡇⠀⠀⠔⠀⠀⠘⣄⢀⡰⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠜⠀⢸⠀⣼⣿⣇⠀⠀⠘⠤⠤⠛⠋⡁⠀⢎⠀⠀⢀⠀⡇⣌⠁⠠⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠚⠈⠐⠚⡫⠷⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁⡇⠀⠈⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠠⡲⠄⠀⠀⠘⣇⣀⡰⠖⠺⢔⠢⠄⣀⠀⠀⠀⢠⠒⠒⠦⣴⠚⠑⠲⡌⠲⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡀⠐⠹⣄⣠⠔⠊⠘⠀⣧⠀⠀⠀⠁⠀⠀⠀⠀⠐⠃⠀⢄⡀⠀⠱⠀⠀⠁⠀⢻⠳⣄⣠⠘⠛⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡽⢄⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⢄⡀⠈⠃⠀⠀⢀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠛⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠃⠀⠀⠀⠂⠐⠁⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠡⡀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢆⠀⣇⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉
⠀⠀⠀⢀⣀⣀⡠⣜⡄⠀⠀⢀⣼⡀⠀⣦⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⢠⡟⠳⢤⣀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⣌⠉⢰⠋⢀⠴⠚⠉⠀⢳⡀⠌⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡸⢿⣄⠀⠀⠀⠁⠀⠙⡄⠆⠀⠀⠀⠀⠀⠠⠤⠃
⠀⠀⠀⠀⢠⠏⢻⠀⠀⠀⠀⢀⠀⠉⢻⠇⠀⠀⠀⠀⠀⠀⢳⣄⠀⠸⠀⠀⠀⠀⠀⢠⠞⡅⠀⠙⠳⢤⡀⠀⠀⠀⠘⡀⡄⡸⠀⠀⠁⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠃⠄⢤⠀⠘⢄⠀⠀⠀⠀⠀⠀⠈⠓⢆⠀⠀⠀⠀⣴⠋⢰⡤⢄⠀⣠⠀⢘⡄⠀⠀⠀⡷⢡⠃
⠀⠀⠀⠀⠈⢆⣀⠀⠀⠀⠘⠶⠊⢸⠇⠀⠈⠑⠤⣀⣀⠀⠀⠀⣄⠈⡀⠀⠀⠀⠿⣄⠘⠞⠀⠇⢉⠀⠊⠀⠀⢀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⢈⡅⠒⠤⠤⠤⠖⠋⡰⠀⠀⠀⠀⠀⡏⠑⢤⠀⠈⡆⠈⢄⡠⢄⡀⠀⠀⠀⣠⠇⠈⠳⠤⢤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⢄⣠⡤⠞⠁⠀⠀⠀⢀⠼⢃⡤⠾⠒⠊⠁⡼⠀⠉⠐⠛⠛⠋⠉⠓⠀⠀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⣁⣀⣀⣀⣀⣠⢴⡏⣠⠀⠀⢀⡴⠖⠾⣥⣀⠀⠑⣤⣀⠀⢀⣀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠘⣧⠘⢆⡀⢟⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠂⠂⠀⠀⠂⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠋⠈⠀⠑⠘⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⡀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠙⢦⡀⠀⠈⣯⠻⣆⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠻⣆⠀⢻⡀⢹⡆⠀⠀⠀⠀⣠⡟⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠳⠾⠁⠀⠛⠛⠓⠶⠶⠋⠀⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⢹⣷⠀⠀⠀⠀⣾⡏⣹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣹⣶⣿⣿⣷⣫⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣷⣿⣿⣿⣿⣿⣿⣿⣷⣌⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⣞⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢸⡏⣵⣾⠿⢿⣷⣾⡿⢿⣿⢿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣮⢹⡇⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⠳⠶⠶⣾⡿⢻⣟⣛⣻⣿⣿⣿⣿⣿⣟⣛⣛⣛⡿⢿⣷⠶⠶⠞⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠮⠭⣭⣭⣭⣿⣽⣯⠭⠿⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣤⡤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠐⣿⡍⠉⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣤⢤⣴⢶⡤⣤⡄⠀⠀⠀⠀⢀⣤⡞⣗⣦⣀
⠀⢿⡇⠀⣼⡇⠀⠀⠀⠀⠀⢴⣷⣶⣲⣶⣻⠿⢿⠆⠀⠀⠀⡟⡟⣿⢻⢻⡿⠀⠀⠀⠀⠀⢹⣹⡏⢈⡏⣿⠁
⡄⠾⣷⣒⣿⠇⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢧⡇⣿⢸⢸⣧⠃⠀⠀⠀⠀⣸⣹⣇⣈⣇⣿
⠈⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠘⠛⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠋⠁⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠂


⠀⢀⡰⠆⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣾⡿⠴⣶⡎⠁⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⣦⠀⠀⠀⠀⢀⡀⠀⢶⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣅
⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⡎⠑⠀⠀⢸⡜⠀⠀⠀⠀⣴⡧⣴⡟⣷⡠⣷⡄⠀⠀⠀⠀⠀⠀⢈⠛⢁⣠⠤
⢀⣿⣇⣀⣿⡇⠀⠀⠀⠀⠀⠀⠧⣥⣬⣥⣞⠇⠀⠀⠀⠸⣿⣄⣿⣧⣿⢧⣼⡿⠀⠀⠀⠀⡀⠀⠐⠒⠚
⠈⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⡀⠀⠀⠀⠀⠀⣼⠟⠁⢶⠼⠇
⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢀⡴⣾⣵⣯⡻⣇⡀⠀⠀⠀⠰⣤⣶⣦⣶⣤⡴⠀⠀⠀⠀⠀⠀⢰⡄⠀⣦
⣶⣶⣾⣿⣧⣶⣶⠂⠀⠀⠀⠋⣾⣿⠁⢼⣿⣿⠙⠀⠀⠀⠀⢸⡇⣿⣇⣿⡄⠀⠀⠀⠀⢀⣠⢿⡿⣾⣿⣇⣀
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣍⣴⣿⣿⠀⠀⠀⠀⠀⢸⡇⣿⡇⣿⠁⠀⠀⠀⠀⣿⠅⢸⡇⠈⣿⠀⣿
⠈⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠈⠀⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⣾⣷⠀⣠⡄⠀⠀⣠⣤⠀⠻⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠈⠀⠀⠈⠀⠀⣰⡟⢸⠁⠀⣴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣿⠀⠀⣠⣤⠀⠀⢀⠀⠀⠀⢀⣿⣡⠏⠀⣼⡇⣰⠃⠀⠀⢀⡀⠶
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣸⡏⠀⠀⣿⡏⠀⠀⣯⠙⢦⢀⣸⣿⠁⠀⢠⣿⠖⠁⢸⣿⠀⣸⡇⠀⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⣿⡇⠀⢸⣿⠇⠀⢀⣤⡍⠹⡏⣿⡇⠀⠀⢸⣿⠀⠀⣸⡿⠀⣿⡇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⣿⣧⣤⢼⣿⠀⠀⣾⡿⠀⢀⡇⢻⣿⠀⠀⣼⣿⡀⢀⣿⣇⡼⢹⣷⠾⠛⠛⠛⠃
⠀⠀⠀⠀⠀⢠⣄⠀⣀⣤⢾⣿⡇⠀⢸⣿⠀⠀⣿⣷⠀⣸⠃⠘⢿⣦⣼⠃⠻⠿⠟⠘⠛⣱⢻⣿⠀⠀⠀⣠⣤
⠀⠀⠀⢠⡄⠘⠿⠟⠋⠁⢸⣿⣇⠀⠈⣿⣧⣼⠋⠛⠟⠃⠀⢀⣀⠉⠁⠀⠀⠀⠀⠀⣸⠁⣼⣿⠀⠀⠀⠉⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠈⠉⠁⠀⠀⠀⠀⣴⡏⢸⠀⣴⠚⡆⠀⠀⠀⢻⣴⣿⠏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⢰⣿⢀⡞⢸⡏⢀⠇⢀⠀⠀⣀⠉⠁⠀⠀⠀⣶
⠀⠀⠀⠀⢐⣾⣿⠁⠘⠃⠀⠀⣀⡴⢾⠃⠠⡖⠦⡄⠀⣸⣿⠊⠀⣿⡷⠋⠀⣿⡇⢨⣿⠀⠀⠀⠀⣹⣿⣯⡀
⠀⠀⠀⠀⠈⠈⠃⠀⠀⢠⣴⠟⠋⠀⣼⡄⠀⣹⡒⠺⡖⣿⡇⠀⠀⣿⡇⠀⢸⣿⠁⣼⣿⠀⠀⠀⠈⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⡇⣼⣿⠁⠀⡇⢸⣿⠀⢰⢻⣧⣀⣾⣿⡼⢉⣿⡶⠶⢶⡄
⠀⠀⠀⠀⠀⠰⣿⡇⠀⠈⠉⠀⠀⠀⣿⡇⣿⣿⠀⢀⡇⠈⠿⣶⠏⠀⠙⠋⠀⠀⡴⠋⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠀⢰⡿⠛⠃⠀⣿⡇⠘⢿⣦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⣿⡇⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢻⣤⣾⠟⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠴⠟⠁⢀⡀⠀⣶⣦⠀⠀⠀⢀⣠⡆⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡉⠁⠀⠀⠀⡨⢿⡿⠄⠀⠀⠠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡉⠁⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⠶⠿⠷⢿⣄⠀⠀⠀⠀⠀⠀⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⣷⣤⣤⣀⡀⠀⠀⠙⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⢿⡟⠀⠀⠀⢀⣀⡀⠀⠈⢷⡀⠉⠙⠳⠶⠚⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⡯⠁⠀⢸⠁⠀⢀⡾⠉⠀⠉⠳⣄⠈⢷⡀⠀⠀⠀⠀⠀⠿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⡇⠀⣠⡾⠿⢿⣿⡷⢦⡄⠀⠀⠀⢸⠀⠀⡾⠁⠀⠀⠀⠀⠘⢦⠈⢧⠀⠀⠀⠀⠀⢤⡤⢤⣌⢷
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⡟⠀⢠⡀⠹⣧⢎⡴⠂⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠀⠀⠀⠀⠙⣦⠈⠻
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⠀⢀⡼⡇⠀⣿⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⢀⡿⠀⢿⡀⠁⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢿⢸⡇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⡞⢻⣿⣆⢻⡀⠀⠀⠀⠈⠓⢦⡀⣷
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧⣼⣿⣿⡌⡇⠀⠀⠀⢀⣀⡀⢻⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢹⠀⣠⡾⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⣇⢀⣀⡀⠀⠀⠀⠀⡴⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇⠸⣾⠏⠀⢀⣤⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⢿⡏⠉⠙⢦⣄⠀⠸⠃⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⡟⠀⣰⠛⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡐⠦⣀⡀⠀⠀⠀⠰⣄⡀⣀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠁⢠⡇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⣄⠀⢀⣤⠴⠖⠒⠛⠻⠿⢥⣌⣣⡀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⡾⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠀⠒⠚⠁⠀⠀⠀⢀⡼⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠁⠀⠄⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣴⠏
⠀⣠⡴⠶⠖⠛⠛⠓⠲⠶⠦⢤⣴⡚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⡾⠃
⣾⠁⠀⠀⢀⣀⣠⣀⣀⣀⠀⠀⠀⠉⠙⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠋
⠻⣤⣀⣐⣋⡡⠄⢀⣩⡽⠿⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⣴⠟⠁
⠀⠀⠈⢩⣏⣀⣀⣀⣀⣀⣀⣀⣀⡀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢋⣠⡶⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠙⠛⠶⢦⣄⣉⠙⠛⠛⠲⠶⠖⠚⠛⠋⣁⣤⠶⠛⠁

⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⡠⠂⢦⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⡀⠀⠀⠀⠀⢀⡴⠲⣄
⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣻⡟⡟⣿⠀⠀⠀⠀⠀⠛⣿⣿⡿⠃⠀⠀⠀⠀⠘⢇⠙⠟⠛⠁⠀⠀⠀⠀⠈⠧⣤⣞
⠀⠰⢿⣿⠗⠀⠀⠀⠀⠀⠘⠷⠧⠧⠏⠀⠀⠀⠀⠀⠀⠛⠉⠓⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⢀⣰⣶⣶⣶⠖⠀⠀⠀⠀⠀⠰⠩⠡⠂⠀⠀⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣤⡤⢤⣤⡀⠀⠀⠀⠀⠀⠁⠀⠈
⢸⣿⣿⣿⣿⠿⠀⠀⠀⠀⠃⢃⢘⣘⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠿⣿⣌⣸⣿⠟⠀⠀⠀⠀⠀⠀⠛⠁
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⢀⣄⣤⣤⡄⠀⠀⠀⠀⠀⢰⠛⠛⢿⡆⠀⠀⠀⠀⠀⢳⡰⣶⣿⡀⠀⠀⠀⠀⢰⣯⣴⣿
⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣯⣭⣯⡇⠀⠀⠀⠀⠀⠘⢿⣶⡾⠃⠀⠀⠀⠀⠀⠀⣷⡭⠒⠃⠀⠀⠀⠀⠸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡀⣠⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⣀⣤⣀
⠀⣤⣤⣸⣿⡇⠀⠀⠀⠀⢸⣿⡄⣿⡏⠀⠀⠀⠀⢿⣿⣿⣿⣇⣈⡆⠀⠀⠀⠠⠀⠀⠀⠸⠃⠀⠀⠀⠀⢸⣻⡛⣟
⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠨⣿⠇⣿⡇⠀⠀⠀⠀⠙⠞⠛⠛⠛⠟⠃⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟


⠠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡤⠤⠀⠤⣄⠀⠀⠀⠀⠀⣠⠤⢤⣄⠀⠀⠀⠀⠀⢰⠒⠒⠐
⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣶⣴⣾⡏⠀⠀⠀⠀⠀⠿⣧⣼⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣀⣀⣈⣁⣉⣀⣀⣀⣀⣀⣸⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣙⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⣰⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣠⡿⠉⢿⡄⣴⡟⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠋⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⢸⡿⠋⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣴⣶⣤⡀⠀⠀⠹⣧
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⠿⠛⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠈⢻⣦⠀⠀⢻⡇
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠃⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣀⣤⣄⡀⢻⣇⠀⠘⣿
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀⣴⣿⣿⣶⣽⣷⡄⠀⠀⠀⠀⣿⡇⠀⣾⣿⣿⣿⣿⣮⣿⠀⠀⣿⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⠁
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣷⠈⢿⣿⣿⣿⣿⣿⡟⠀⢠⣿
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣧⡈⠛⠿⢟⣿⡿⠀⣀⣼⣧⣤⣤⣤⣄⣀⡀
⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠈⠛⠿⠿⢛⣿⠇⢀⣠⣴⡾⠟⠛⠛⠻⣷⣿⣿⡾⠟⠛⠋⠉⠉⠉⠉⠉⢙⠻⣷⣄
⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠉⠻⣷⣤⣤⣴⣾⣿⡾⠟⠩⢀⠀⢁⣠⣴⡾⠟⠋⡀⣀⠉⠂⠀⠀⠀⠀⠀⠀⢈⡆⠈⣿⠄
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠻⣶⣤⣤⣤⣴⡾⠿⠛⡋⠁⠀⣀⣈⣤⣶⡿⠛⠉⠀⠀⠉⠉⠀⠀⠀⠀⠐⠀⠒⠀⠒⠂⢁⣴⠟
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣿⡇⠀⠀⢀⠀⠁⠀⠀⠀⠀⠐⠂⠀⠐⠻⠿⠿⠛⠛⠉⠀⠀⠀⠤⢠⡄⢀⣀⣤⣶⠶⠿⠟⠛⠛⠛⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢹⣧⠀⢈⠊⢁⣵⣿⡐⠠⠠⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠼⠶⠤⣂⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠻⣦⡀⠘⠛⠋⡙⢻⣷⣶⣶⣤⣤⣤⣄⣀⣀⣉⣥⣤⣴⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡈⠛⠷⢶⣶⡁⢂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣼⣿⣦⣀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠎⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠩⠉⠉⡉⠉⠉⠠⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣶⣶⡶⠿⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⢻⣉⡛⣻⣟⣹⡿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⠋⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⢸⡇⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡟⠹⣆⠀⠀⠀⠀⠀⢀⣴⣋⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⣦⣀⣠⣤⣤⠶⠶⢶⣶⠀⠀⠀⠀⠀⠀⢠⡾⢁⡞⠀⠀⢹⡄⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⣿⠀⢠⣶⡶⠶⠶⢯⢤⠜⠀⠀⠀⠀⢻⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢻⢠⡟⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⣾⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⢸⣿⠁⠀⠀⠀⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠉⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣸⡟⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡄⠀⠀⢠⠟⠀⣀⡤⢖⠟⠳⣄⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢸⠀⠀⠀⠀⢠⠞⢹⠀⠀⠀⠀⠀⠉⠉⠉⣋⠀⢀⡾⠖⠋⠁⢀⠎⠀⠀⠀⠙⠦⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣸⠀⠀⠀⣴⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡸⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡞⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠘⣇⠀⠀⢀⠎⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⠀⠀⠀⠀⠀⢹⡀⠀⢀⣀⣀⣠⠏⠀⠀⠘⢤⣤⣮⣤⣤⣤⠀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⡟⢯⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢾⣏⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⡿⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢇⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⡀⠀⣰⠃⠀⠀⣀⡀⠀⠀⠈⢧⡼⠃⠀⠀⠸⣟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⡇⣰⢃⣠⠔⠋⢹⠉⠓⠦⣄⡈⢷⡀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣧⢟⠁⠀⠀⠀⡏⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢸⣼⠀⠀⠀⢸⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⣿⠀⠀⠀⡟⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠀⠀⢰⠃⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢹⠀⠀⡏⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⠀⣸⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⢀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⡢⠀⠀⠘⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠐⣵⡏⢀⣇⠂⠀⠀⠀⠀⣸⡟⠀⠀⣶⠆⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⢸⡿⠀⡜⠁⠀⡠⣐⣄⠀⢻⣇⠀⠀⣿⠀⠀⣴⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣼⡧⢪⠒⢻⡆⣼⢏⠤⢅⢸⡇⠀⠐⣿⠀⢸⣿⠀⢰⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢸⣧⠃⢀⡿⠁⣿⢸⣄⢸⢸⡇⠀⠀⣿⠀⠸⡿⠀⢸⡇⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣸⠁⠀⢿⣄⣠⡇⢸⣇⣀⠀⣿⠀⠀⠻⠞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣧⣀⡾⠀⠉⠉⠀⠀⠁⠀⠀⠻⠶⣖⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠈⠋⠀⠀⠀⠀⡀⣿⠃⠀⠀⠀⡞⢤⡹⢀⣿⠁⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⣄⣠⠄⢀⢀⠀⠀⢱⣿⠀⠀⢀⡀⠳⠄⣠⠶⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠄⠉⣿⠀⣾⠋⠹⠀⢹⣿⠀⠀⢸⡇⠀⠀⢰⡗⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⢸⣿⢸⡇⢸⠛⡆⢸⣿⡇⠀⢸⡇⠀⠀⣻⣷⠀⣿⡀⠀⠀⢾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢼⣟⠘⣧⠀⢀⣇⢸⡟⠀⠀⢸⣇⠀⠀⠙⠟⠊⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣸⡏⠢⡈⠙⠛⠁⠈⠛⣒⠚⢸⡧⠀⠀⠀⠀⠀⣾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣦⠴⡻⠁⠀⠙⠲⣦⣀⣀⣀⡟⠀⠈⠳⠖⢁⡤⢤⢰⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣦⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠁⠛⢈⣇⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⣒⣴⣔⡠⠀⠀⠀⠀⠈⠓⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣛⡣⡀⠀⠀⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣏⢶⠝⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠥⡾⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠫⠍⠓⠒⠒⠉⠽⣒⢤⡀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠠⠈⠑⢆
⠀⠀⠀⠀⠀⠀⣠⢫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢦⠀⠀⠀⠀⢐⠀⠀⠀⠂⠆⠐⠀⠀⢃
⠀⠀⠀⠀⠀⣸⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⢳⠀⠀⠀⢸⠀⠀⠀⠀⣷⠀⠀⠀⡘
⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⣄⢀⡾⢦⣀⡴⠛⠀⠀⠀⠎⡇⠀⠀⠈⠢⡀⠀⠠⠤⠀⢀⡰⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡌⢉⢰⡆⣭⢰⡇⠀⠀⠀⡀⡇⠀⠀⠀⠀⠈⠐⠒⠒⠂⠁
⠀⠀⠀⠀⠀⢇⢇⠀⠀⠀⢸⡇⣿⢸⡇⣿⢸⡇⠀⠀⢠⢱⠃⠀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠘⢮⢢⠀⠠⠬⠥⠍⠌⠥⠍⠌⠁⠤⡠⣣⠏⠀⠀⠀⠀⠀⢀⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠓⢝⡢⢄⡀⠀⠀⠀⢀⡀⢤⣪⠞⠁⠀⠀⠀⣴⣄⣠⣿⠟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠯⠭⠭⠖⠚⠉⠀⠀⠀⠀⣠⣾⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡫⠊⠹⠇⠀⣼⠟⠛⠛⠛⠛⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⢀⣴⣿⠏⠀⠀⢸⡏⡭⠭⠭⠭⠭⠭⡟⢻⠭⠭⠭⠭⠭⢭⢹⡇
⠀⠀⠀⠀⠀⠀⢀⡔⠉⢀⠀⠒⠦⡄⠈⠳⡄⠀⠘⢻⠁⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡀⠀⠀⣄⣀⣠⣤⣼⣧⣧⡤⢤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⢌⠁⠁⠈⠀⠀⢘⠇⠀⠀⠈⠳⡤⠴⠦⠻⣿⣿⡟⠉⠉⠉⣀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⡜⠂⠒⡄⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⠤⠿⢿⣷⣶⠚⠉⠉⠁⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣁⣃⣀⠴⠋⠀⠀⠀⠀⠘⠻⡀⡀⠀⣀⣀⣼⣿⠿⠶⠒⢢⣀⣀⣀⣀⣀⣀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⣿⡿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⣀⡄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡄
⢻⣿⢽⢽⢽⣿⠃⠀⠀⠀⠀⠀⢴⣙⡿⢷⣷⣿⠀⠀⠀⠀⠀⠀⣿⡆⣿⢸⡇⠀⠀⠀⠀⠀⣸⡟⠟⠿⡟⣿⡀⠀⠀⠀⠀⣀⣰⣶⣶⣶⣶
⠈⢭⣭⡭⣭⠟⠀⠀⠀⠀⠀⠀⢠⣟⠋⠿⠶⠾⠃⠀⠀⠀⠀⠀⠋⠁⠉⠈⠃⠀⠀⠀⠀⠀⠘⣷⣾⣶⣷⡟⠀⠀⠀⠀⣾⣤⣿⣿⣿⣿⣿⠨⠄
⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠛⠉


⠀⢴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⠀⡔⣫⠕⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⣀
⠀⡶⠳⠶⣶⣼⠃⠀⠀⠀⠀⠀⣿⣿⡏⢦⣹⣿⣧⠀⠀⠀⠈⢦⡘⣃⢜⠞⣷⣆⠀⠀⠀⢠⣶⣾⣷⣶⣮⣙⡳⠀⠀⠀⠀⠶⠶⠶⠶⢶⡾⠇
⣰⣷⣶⣶⣿⣿⣦⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⠛⠓⠀⠀⠀⠀⠀⠈⠀⢿⣧⣾⠟⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠠⠭⠥⠘⠋⣠


⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡜⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠢⠀⠀⠀⠀⠀⠀⢠⣞⣽⡿⢙⣶⠀⠀⠀⠀⠀⠀⢤⣥⣧⣥⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄
⢿⣻⣟⣿⣻⣟⡿⠀⠀⠀⠀⠀⣾⡛⠛⠛⣛⠻⡇⠀⠀⠀⠀⢸⣿⣷⣴⡿⣿⠃⠀⠀⠀⠀⢸⣿⡷⠰⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⢳⠄
⠸⣽⣧⣿⣼⣯⠇⠀⠀⠀⠀⠀⠻⣴⣾⣧⣽⣶⠇⠀⠀⠀⠀⠀⠙⠷⠷⠟⠁⠀⠀⠀⠀⠀⠸⣿⡧⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠫⠔⠋


⠀⠀⡴⠑⠵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃
⠀⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⡆⠀⠀⠀⠠⣶⣂⣸⣿⣏⣹⡷⠀⠀⠀⠠⣶⣂⣙⣟⣉⣹⠷⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⡆
⠀⣿⡐⢒⣽⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠶⠿⠟⠋⣈⣿⡇
⠀⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠿⠿⠃⠀⠀⠀⢹⣿⡟⠁⠀⠀⢸⣿⠉⠉⣠⠞⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣾⣷⠀⠀⠀⢰⣿⣠⠞⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠟⠿⠆⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠿⠿⠿⠿⠿⠇


⠀⠀⠀⠀⠀⢀⣻⣿⣿⡀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⢀⣿⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢠⣤⣤⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣠⣬⣿⣥⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠼⠿⠿⠟⠿⠿⠄⠀⠈⣿⠁⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣶⣄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⣤⣾⠤⠄⢼⣦⡄⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠸⠿⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⢀⣶⡀⠀⠀⠀⣴⣶⣤⣶⣦⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠐⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠈⣿⣿⡟⠁⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠔⢿⣿⡟⠆⠀⠀⢠⣿⣼⣷⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⣶⡆⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠛⠉⠛⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⢀⠎⠏⣠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡱⡜⠼⠊⠄⠈
⡄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠠⠀⢈⠓⢤⠠⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⢢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠘⠚⢸⠃⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢀⠈⠀⢀⣀
⡇⠀⠀⠓⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⣡⠔⠋⠀⠀⠀⢀⣄⢄⣤⠆⠀⠀⡠⢐⡄⠠⠤⠐⢂⡡⠔
⡇⠀⠀⠀⠀⠀⢀⢹⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⡈⢀⢀⣰⣴⣶⠏⠐⠫⠇⠆⠀⢠⠳⠎⠔⠀⠀⢀⠦⡑⠄
⡇⠐⠁⠢⠔⡔⠆⠈⡠⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⢀⣸⣿⡿⠛⠋⠂⠀⠀⠐⢐⠄⠀⠑⣦⣀⣠⠔⠋⠀⠈⠚
⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡃⠀⢊⠟⠲⢄⠀⠀⠒⠀⠀⣩⣇⠀⠀⠈⠢⣌⠑⢄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢢⡼⢂⢄⠀⠀⠀⠀⠱⡌⠒⠆⡇⠀⠀⣀⠨⠷⢂⣙⡦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⡆⠀⠈⠀⠜⠆⠀⠀⠀⠀⠀⠀⠀⠠⣴⠅⠒⡡⠔⠚⠉⠁
⡇⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠂⠀⠒⡞⢳⠀⠀⢀⠿⠀⠀⡇⠀⠀⠀⢀⣠⣤⣸⠀⢠⠤⡄⠀⠀⢱⠀⠘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⢰⠁⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⠈⠙⢄⢠⣤⣬⡆⣧⣸⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠆⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⠀⠈⢣⠙⠛⣡⣧⠉⡀⠀⠀⠡
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⢄⢶⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠓⢲⢊⠩⠀⢀⠀⠀⠀⢂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠧⢠⠄⠃⠀⠐⠂⠀⠀⠓⠦⠖⠃⡆⢸⠀⡇⠀⠨⠀⠀⠀⠀⡀
⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢁⠀⠀⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠀⠄⠀⡇⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠄⠀⠣⣤⠆⠀⠠⠤⠠⠀⡅⠀⣇⠤⠼⠀⢻⠒⠒⠠⡤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⠉⠱⣄⠘⠠⠴⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠀⠀⠀⠀⠀⢀⠌⣰⠉⠉⠉⠉⠁⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⡁⠄⢠⠃⡔⠁⠀⠀⠀⠀⢀⠎⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⢣⠈⢦⠀⠀⠀⣰⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡐⢣⠀⢠⣉⠉⠉⠉⠉⠑⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⢀⣴⣶⣆⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠻⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢻⣇⠀⠀⠀⣿⣯⠀⠀⠀⣹⡇⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣽⣿⡁⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⢾⣿⣶⣶⣾⠋⠹⣷⣶⣾⣿⡇⠀⠀⢀⣼⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⣶⣿⠏⠹⣿⣶⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣦⣀⠀⠙⠻⣿⣿⣶⣾⣿⡿⠟⠋⠀⣀⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣶⣤⣛⠻⠿⠿⠟⣃⣤⣶⣿⡿⠛⠋⠉⠁
⠀⠀⣰⣶⣤⣤⣾⣿⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠉⠻⣿⣷⣦⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣦⣴⣿⡿⠟⠋⠁
⠀⠀⢻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠿⣿⣶⣤⡀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣾⡿⠛⠉⠀⠀⠀⠀⠉⠛⢿⣷⣦⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠉⠛⢿⡿⠛⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⢠⣤⣤⠀⠀⠀⠀⢠⣿⣦⣀⣤⡟⢳⣤⣀⣴⣿⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⣾⣿⣿⣿⣧⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣼⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣧⡀⠀⠀⠀⠹⣿⢿⣿⣤⣴⣿⡿⣿⠋⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠙⠿⠿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠿⠿⠟
⠀⠀⠻⠿⢿⣿⣿⣦⣄⠀⠀⢿⣾⣿⡿⣿⣿⣷⡏⠀⢀⣠⣴⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡟⠉⠀⠉⢹⣿⠉⠉⠉⢹⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣬⡻⣿⣿⣿⡿⢟⣥⣶⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣴⡟⠹⣦⣤⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣯⣽⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣿⣦⣦⣿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⣷⣿⠿⣿⣿⡿⣿⣾⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⠿⠋⠁⠀⠀⠈⠙⢿⣷⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⡿⠛⠁⠀⠻⢿⣿⣿⣿⣿⣿⠿⠀⠈⠙⢿⣿⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠒⠛⠛⠛⠓⠒⢲⠖⠉⠉⠓⣄⡠⠴⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⣀⣀⡤⠤⠴⠖⠒⠒⢾⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⢀⡠⠴⠚⠉⠁⠀⠀⢀⣀⡤⠤⠤⠼⣆⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⢰⠉⠢⡀⢀⠔⠉⠀⠀⠀⢀⠴⠚⠁⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⢀⣀⡠⠬⠦⡀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠙⢇⠀⠀⢀⣠⣾⠥⣄⠀⢀⡠⠖⠉⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠈⠓⠚⠉
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠠⠐⠉
⠀⢀⣀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢀⡠⠈
⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⡰⠋
⠈⠉⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⠞
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠱⡃
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢆⣠⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠛⢫⡿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠁⠀⠘⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⣤⠶⠒⠛⠛⠳⢦⣀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠀⢀⣤⠀⠀⠀⣤⡀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠰⣏⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⣨⡷⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠉⠙⠛⠿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠿⠋⠉⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⣠⡴⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣠⡶⣶⡄⠀⠀⣴⡟⠁⠀⣿⡇⠀⠀⠀⣀⣤⡀⠀⠀⣰⡿⠃⠀⠀⢸⣿⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣠⡾⠋⠀⢸⣧⢀⣾⠏⠀⠀⠀⣿⣇⣤⣤⣼⠟⠙⢿⣦⣾⠟⠁⠀⠀⠀⢸⣿⣾⡿⣷⣄⠀⣀⣀⣤⡿⠋⠀⠀⠀⠀⢸⣿
⠠⣦⣠⡾⠋⠀⠀⠀⠀⠿⠟⠁⠀⠀⠀⠀⣿⣿⠏⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠈⠻⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⢒⣿⣶⡖⠒⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⢤⣀⣚⣉⣉⣁⣀⣀⣀⡤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⡀⠀⠉⠉⠉⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⣍⣛⠛⠟⠛⠛⠛⠛⠛⣉⣽⣆⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⢶⠆⡣⣈⠉⠛⠛⠛⠛⠛⠛⠛⠋⣽⢡⡇⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡀⠀⠴⠛⠸⡗⠮⢭⣝⣓⣒⣒⣶⡾⠿⠋⣿⠈⣇⠤⢤⠤⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡳⢬⣉⣉⠉⠉⠀⣵⠢⢤⣤⣀⣀⣀⣠⣤⣤⣶⢿⡄⠘⣆⠀⠀⠾⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢛⡉⠙⠛⠛⠛⢻⠻⡿⣤⣀⣀⣀⠈⠉⠉⣉⣀⣠⣤⣷⡄⠘⢦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠚⠂⢃⣀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠃⠙⢦⡀⠙⢂⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠚⠛⠋⠁⠀⠀⠀⠙⠉⠉

⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⢸⣏⣉⣩⠩⠃⠀⠀⢸⣿⣿⣿⣿⣆⣠⡀⠀⠀⠀⠀⠀⡈⣄⣀⡄⠀⠀⠀⢠⣶⣿⣾⣶⣆⠀⠀⠀⢰⡭⣌⠱⣘⡆
⠀⠀⠀⠀⠀⠈⢻⡗⠓⣶⠂⠀⠀⠘⠻⠛⠛⠛⠛⠻⠋⠀⠀⠀⠀⠀⠁⠋⠉⠃⠀⠀⠀⠘⡿⣿⣾⣿⠗⠀⠀⠀⠈⢷⣤⡢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠚⠓⠂

⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠐⠑⡀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣴⣿⣿⣶⣷⡄⠀⠀⠀⡢⠕⢚⡉⠀⢠⠀⠀⠀⠸⡈⠇⡨⠰⣦⡀⠀⠀⠠⢖⣋⣵⣾⡆⠀⠀⠀⠰⣶⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠧⠀⠸⠇⠀⠰⠀⠀⠀⠀⠈⠉⢾⣧⣽⠇⠀⠀⠀⣿⣿⣿⡧⣾⠀⠀⠀⠀⢸⡏⠓⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠐⠶⣶⠞⠀⠀⠀⠀⠀⠀⠀⡀⠚⠢⡀⠀⠀⠀⠀⢀⣀⣛⣢⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢛⣿⣌⣹⣿⡃⠀⠀⠀⠛⠚⣟⠚⠣⠀⠀⠀⠀⠀⣿⢿⢿⣿⢿⡇⠀⠀⠀⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⣼⢋⣠⡝⣇
⠀⠀⠀⠀⠀⠸⠿⠿⠼⠼⠁⠀⠀⠀⠀⠀⠧⠀⠐⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠙⢤⣁⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠠⣿⣧⠄⠀⠀⠀⠀⠀⠀⢺⣿⠓⠀⠀⠀⠀⠀⠀⢤⣿⡧⠀⠀⠀⠀⠀⠀⠐⢺⣿⠂
⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⢹⠀⠀⠀⢰⣶⣦⣴⣶⡆⠀⠀⠀⠀⣶⣶⣭⣶⣶⠀⠀⠀⠀⣶⣶⣤⣶⣶⡆⠀⠀⠀⣠⣶⣬⣵⣶⣄
⠀⠀⠀⠀⠀⣿⣿⣶⣤⣴⣿⠀⠀⠀⠸⠿⣿⣿⣿⠇⠀⠀⠀⠀⠿⣿⣿⣿⡿⠀⠀⠀⠀⠿⢿⣿⣿⠿⠁⠀⠀⠀⠸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣀⠀⠀⠀⢰⠛⠿⠿⠛⠆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⢠⣤⡾⠃
⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠸⠄⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡿⡟⡏⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢛⡛⢇⠂⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⡇⡇⡇⠇⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣑⣊⡌⣶⡎⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⣿⢿⣿⣶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⣶⣾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣾⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟


⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣭⣥⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠇⠀⠀⢯⡃⠀⠀⠻⠿⠇
⠀⠀⠀⠀⢰⣷⢸⣿⣿⡷⢶⣦⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣦⠀⠀⠳⠃⠀⢀⣾⣿⡆
⠀⠀⠀⠀⢸⣿⢸⣿⣿⣬⡟⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠀⠘⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠘⠿⡾⠿⢿⣿⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣀⢧⡀⢸⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠉⠈⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠻⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠉⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢴⠶⠆⠀⠀⠄⠁⠴⠾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣶⣶⡀⠀⣀⠀⡾⣷⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⡁⠈⣻⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠑⠛⡠⠊⠀⠈⣢⡏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⣛⣿⣿⣿⡷⠟⣉⣠⣤⣄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠂⠒⠈⠀⠀⠔⠉⢀⡀⢀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣽⣷⣬⣤⣶⣿⣿⣷⣶⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠐⠤⢕⠠⡄⠤⠐⠈⣱⠀⢠⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⣀⣪⡠⣀⣀⠟⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢛⣛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠚⣎⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡗⢤⣀⠀⠀⣀⣴⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⢄⡀⠀⢀⣀⠎⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠈⣉⡭⠿⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠤⠇⢀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣠⡞⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣠⠊⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣇⡀⠀⢀⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡮⠊⢰⠀⠀⠀⢀⠬⠀⢀⠀⡀⣠
⠀⠀⠀⢠⣄⠒⠒⠚⠙⢿⣿⣿⢷⡟⠋⠁⠉⠉⣉⣽⡏⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⡤⠼⠖⠂⠁⠀⠉⠀⣉⠔
⠀⠀⠀⠀⠈⠉⠁⠶⣀⣈⠉⠙⠛⠿⠿⠿⠛⠛⠿⠒⠀⠄⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠂⠐⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠄⠤⠀⠠⠀⠐⠒⠀⠁

⠀⠐⢠⣀⠀⠀⠀⠀⠀⡎⠀⠐⠒⢀⠐⠀⠀⠑⠤⠀⠁⠠⠘⠁⠀⠀⠠⠀⠒⠩⠀⠀⠀⡀
⠠⠀⠸⠍⢴⠀⠀⠀⠀⡇⢀⠤⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠠⠄⠒⠁⠀⠀⠀⢤⠀⣀⠀⠀⡀⠠
⠂⠊⣈⠓⠚⠀⠀⠀⠀⡇⠈⠈⠀⠁⠀⠀⠀⠀⠐⠀⠀⡠⣠⠜⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠊⠀⠀⠐⠒⢛⢌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⣀⠀⠀⠀⠀⠫⠋⠛⠛⠛⠛⠛⠛⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⢤⣤⡤
⠀⡤⠀⠀⠁⠀⠁⠀⠀⠈⠳⠤⠤⠤⠴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢻⢛⠛⡟⠀⠀⠀⠀⠀⠀⢦⠠⢠
⡸⠁⠀⠂⠰⢰⠂⠀⠦⢰⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢰⣹⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢉⣈⣀
⠆⠀⢰⠀⡀⣿⠀⢸⠀⡀⠁⠀⠀⠀⠀⠠⡤⠂⠀⠀⠀⢀⣤⠀⠀⠀⠀⠹⠀⠀⠀⠴⠀⠀⠐⠀⠀⠀⠀⠀⠲⠂
⠀⢀⠀⠁⠁⠍⠉⣉⢀⢡⠉⡠⠀⠀⠀⠀⠀⠀⢳⠀⣠⡾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢀⠄⠒⠂⠀⢄
⠀⠈⠀⠒⠀⢀⣀⢀⠘⡇⢰⠀⠀⠀⠀⡄⠀⢀⣬⣾⠏⠀⠀⠀⠀⠀⠈⣁⠀⢲⠀⠀⠀⠬⠀⠀⣰⠃⠀⠀⠠
⠀⠀⠚⠀⠀⠊⠃⠘⠀⠓⠘⠐⠂⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⡀⠀⠻⠟⠉⠓⢤⡀⠀⠀⠀⢻⠀⠀⠐⠦⠄
⠀⠀⠀⠀⠀⣠⠤⢄⣀⠀⠀⠀⠀⠀⠀⣸⡲⠀⡀⠀⡤⠔⠂⢀⡠⠦⠤⣠⣤⠆⡀⠀⠈⠂⠀⠀⠈⠷⣄⣈⣉⣠⠔
⠤⠀⠤⣤⠚⠁⡠⠀⠈⠃⠀⠀⠀⠠⡄⠈⠁⠀⠀⡸⠁⠀⠀⠀⠙⠒⠐⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠉⠋⠉⠉⢧⠀⣀⣀⢔⠄⠀⢠⣄⡀⠀⠀⠀⠀⣰⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠉⠑⠢⢥⢫⢳⡄⠀⠀⠀⢀⠈⠀⠀⠁⠂⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀
⢀⡰⠤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠀⠀⠉⠉⠀⠀⠀⢴⡏⠀⠤⠄⠀⢨⣦⡀⢢⠀⠀⠀⠀⠐⠒⠰
⡁⠀⠀⠀⣉⠲⢀⠀⠀⠀⠈⠐⠐⠒⠒⠐⠄⠀⠀⠀⠀⠈⠀⠐⢠⠄⡄⣄⠀⠀⠅⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠰
⠀⢙⡖⠉⠀⠀⠂⠀⢂⡀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠰⠤⠤⠬⠤⠤⠤⠤⠤⠀⠁
⠈⠁⡇⠀⢀⠠⠀⠀⠀⠀⠈⢃⠄⠒⠐⢠⡎⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡄⠀⡀
⠀⠁⠂⠘⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠤⠀⠄⡄⣀⡄⣀⠄⠠⠀⠄⡄⠀⠀⠀⠀⠀⠛⠁⠀⣹⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡑⢋⠑⢫⠂⠰⠂⠀⡁⠀⠀⠀⢀⣤⣶⡤⠴⣧⡀⠀⠀⣤⡀⠀⠀⠈⠑⠢⡄⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⡈⠀⢰⡆⠀⢸⡇⠀⠂⠀⠈⠐⢤⠔⠊⠁⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⢀⡶⢰⡆⢲⡄⠀⢀⣰⣁⣑⣄⠀⠀⠐⡦⣤⣤⣤⠀⠀⢠⣮⣤⡷⠀⠀⢀⣚⣊⣑⣓⡂⠀⠀⣤⠄⢠⡆
⠀⠀⣟⡟⢻⣛⡇⠀⠘⣿⣿⣽⣾⠁⠀⠀⠘⡟⢛⠛⠀⠀⢸⣿⣿⣿⠀⠀⠀⣾⣾⣷⣿⡇⠀⠀⠁⠠⡄
⠀⠀⠉⠁⠈⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡄⠀⠀⠀⢠⢠⣄⡄⣤⡄⠀⠀⠾⠥⣤⣤⠄⠀⠀⠀⠴⡗⣄⠀⠰⠖⢀⣄⠒⠆⠀⠀⠀⣠⡄
⠀⠀⣸⣹⣿⣿⡀⠀⢸⢸⣿⡇⣿⡇⠀⣾⣿⣿⣿⡆⠀⠀⣿⣿⣿⡿⣇⠀⠰⠆⠈⠊⣄⡀⠀⢠⣾⣷⣿⣧⡀
⠀⠘⠝⠿⠛⠟⠇⠀⠘⠈⠀⠁⠉⠃⠀⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠘⠇⠛⠁⠀⠈⠉⠉⠋⠙⠁
⠀⢀⣤⠤⠤⠤⡄⠀⠀⣠⠀⢀⢀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⣤⢤⣤⠀⠀⠀⠀⠀⡀
⠀⢸⠀⠻⠗⠀⡇⠀⠸⣯⠁⠚⡞⠀⠀⠀⢧⢀⣽⠀⠀⠀⢸⠢⣤⢼⠀⠀⢰⡿⣿⢿⢿⡇⠀⠀⢀⡄⢀⡀
⠀⠈⠉⣹⣿⡉⠁⠀⠀⠙⢷⣶⡄⠀⠀⢰⣶⣝⣴⣦⠀⠀⢸⣦⣠⣼⠀⠀⠈⠻⠟⠛⠛⠁⠀⠀⠠⡜⠭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⢚⡟⢢⠀⠀⢰⣾⣶⣾⣷⡄⠀⠀⢀⣷⣆⠀⠀⠀⣀⢠⣾⣁⡀⠀⠀⣶⣶⣶⢰⡆⠀⠀⡟⢻⣿⣿
⠀⠀⣷⣀⣃⢴⡇⠀⠈⢿⣿⣿⠟⠀⠀⠀⣾⠿⣿⠄⠀⠀⣿⢸⣿⣿⡇⠀⠀⠻⢿⡿⠘⠃⠀⠀⣯⣭⣭⣿
⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⠤⣤⠤⢀⠀⠀⡤⢠⠄⣶⡆⠀⠀⣤⣤⠆⠤⡀⠀⠀⣤⣄⣀
⠀⢸⣯⣻⣿⣆⣄⠀⠀⠀⢿⣿⡇⠀⠀⠏⠉⠛⠁⢹⠀⠀⠃⢘⠀⣘⡀⠀⠀⣿⣟⠂⢁⠃⠀⠀⡌⣄⡄⡄
⠀⠈⠛⠉⠉⠑⠉⠀⠀⠀⠿⠏⠁⠀⠀⠣⠁⠩⠩⠽⠀⠀⠱⠋⠀⠉⠁⠀⠀⠈⠻⠄⠃⠀⠀⠀⠂⠐⠒
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠊⣁⡀⡙⡄⠀⢠⢋⣀⢈⠑⠀⠀⢀⣰⢿⣻⣷⠀⠀⢀⣴⡿⣹⡦⠀⢀⣠⣈⣁⣂⡀⠀⢀⣔⣈⣁⣀
⠀⠀⠦⣀⣀⡠⠃⠀⠘⢄⣀⣀⠴⠀⠀⣾⣯⣾⠟⠁⠀⠀⣿⣯⡾⠋⠀⠀⢸⣿⣭⣭⣽⡇⠀⢸⣿⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠙⠒⠦⣀
⠀⠀⠀⡴⠋⢀⠴⠚⠉⢉⣩⠙⠢⡄⠈⢣
⠀⠀⡼⠀⣰⠃⠀⠀⢸⡉⠉⠂⠀⠘⡆⠀⣇
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣚⣋⣉⣉⣉⣛⡒⠲⢤⡀
⠀⠀⢣⠀⠹⣄⠀⢳⠖⠚⠁⠀⣠⠎⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⡞⠭⣴⣀⣠⠖⠛⢳⠾⢙⡆⠘
⠀⠀⠈⢷⣄⡈⠙⠒⠲⠖⠒⠋⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣈⡙⠒⠒⠒⠒⠒⠒⠚⣉⣤⣾
⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⠤⠤⠴⠶⠶⠤⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⠔⣒⡏⠉⣩⠭⠿⣗⣲⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡈⠺⠭⢿⣟⣁⣀⣤⠥⠴⠚⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣀⣀⠉⠙⠿⣿⣿⣿
⠀⢀⣠⠤⠖⠒⣒⣒⡒⠒⠦⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡴⠋⠁⠀⠀⠉⠲⣄⠈⢿⣿
⡔⠁⡔⣊⣩⠉⢉⡽⠛⢯⡭⠖⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⣠⠄⠀⢀⣠⣀⠈⢆⠈⢿
⣷⣄⡙⠒⠮⠭⠭⠤⠤⠤⠖⠚⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠈⠹⣄⣠⠎⠀⣸⠶⢸⠀⢸
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠣⡀⠀⠀⠀⠀⠁⢀⡞⠀⡸
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠉⠛⠋⠉⠉⠉⠀⠘⢿⣷⣄⠉⠒⠤⠤⠤⠔⠋⠀⡴⠃
⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠉⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠿⠿⠿⠿⢿⣿⣶⣤⣤⣄⣸⡇⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⠇⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠉⠀⣀⣴⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠟⠋⠁⠹⠃
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⡀⠀⢀⣤⣾⡿⠟⠋⠁⠀⠻⣿⣿⣦⣠⣴⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣴⣿⣿⠟⠁⠐⠉⠉⠻⢿⣿⣿⣷⣦⣀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣶⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠻⠟⠓
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠦⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠠⠤⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⣴⣄⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠉⠛⠂⠀⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠈⠉⠙⠛⠛⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠁⠁

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


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

⠀⢠⡀⠀⠀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣐⡉⠄⡿⢃⣀⡄⠀⠀⢀⠠⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡗⢦⢀⣧⣀⡀⠃⠀⡄⡀
⠀⠀⣉⠱⠰⠞⢫⠁⠀⠳⢮⡛⠾⠲⠁⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⢇⣴⠈⠜⢾⠉⠵⣮⡁⣀⠙⠀⠴
⠀⠀⠁⠀⠈⠶⣘⣄⡶⢈⡄⠱⢦⡁⣢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢓⠚⢻⡴⢠⣥⣄⢠⠂⣩⣿⡶⠛⠁
⠀⠀⠀⠀⠶⠃⠉⡤⠀⣿⡃⣀⠃⠝⠉⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠐⢀⣦⠃⡊⠁⠠⡦⣀⠫⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣷⠁⠀⠁⠀⠀⠀⠉

⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡦⡀
⠀⠀⠈⠅⡆⠀⣠⠞⠁⠀⡀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡾⠥⢀⢀
⠀⠀⠀⢘⣧⢀⡡⡨⣩⡌⣣⡾⣃⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠄⡸⡅⠒⠀⣼⡱⢋⠀⣀⣈⠻⠃
⠀⣀⠀⢸⡿⡯⢻⠀⡉⣞⠷⢻⠇⡐⠳⣤⠀⠀⠀⠀⠀⠀⠙⠋⠁⢎⠊⡟⠑⠷⣍⢨⠙⢢⢢⡜⠋
⠀⠀⠀⡮⠳⠵⠛⠴⠇⣈⠚⣵⡔⡒⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⣤⠄⣈⠁⠸⣾⠆⣯⡏⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁⠀⠶⠘⢱⡖⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⣠⡀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄
⠀⠉⠁⢁⢟⣣⡄⠠⣝⠄⠀⠀⣤⠤⠞⠁⡀⣴⠀⠀⠀⠀⠀⠀⠀⢠⠤⣴⣒⣦⣤⠌⠀⠀⢀⠈⠑
⠀⠘⠃⠀⢞⡯⡀⠄⠘⠀⠐⠙⠶⣤⡁⡜⠟⠀⠀⠀⠀⠀⠀⠀⠤⢀⠟⠭⠅⠀⠀⢦⡘⢯⣄⣁
⠀⠀⢶⠄⠁⠏⣛⠰⣞⠋⣠⣠⣊⢨⠟⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠖⣸⡐⣿⡄⠻⢃⠙⠰⠺⠯⠤
⠀⠀⠀⠀⠀⠀⣈⠁⠂⠹⠉⠡⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠀⠯⠵⠊⢀⠸⢟⡀⠰⠆⠤⣤⡀⡃
⠀⠀⠀⠀⠀⠀⠉⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠾⠟⠛⠛⠛⠛⠻⠷⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠋⠛⠛⠛⠉⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠧⠤⣀⣠⣶⣶⣄⣀⡘⠇⢹⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠶⠛⠻⣿⣿⣿⣿⡿⡟⢿⠀⣹⠘⡏⠉⠙⠓⠶⠶⣤⡀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠐⣉⣉⣿⠿⣮⣹⢏⣶⠈⣠⡏⠀⡇⠀⠀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⣷
⢸⡟⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⢀⣴⣶⣿⢄⠛⠓⠚⠋⠀⢹⡁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢻
⣾⠇⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⢀⣠⣿⣿⣿⣿⡆⠉⠀⠀⠀⠀⠘⣇⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⣠⣾⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣷⢤⣤⣀⡀⠀⢀⣼⠛⠳⢶⣿⣄⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⢠⣾⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⣷⣿⠟⠀⠀⠀⠈⠹⣷⢦⡀⠀⠀⠀⠀⢻
⢿⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢸⢹⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠹⡆⠉⠂⠀⠀⠀⢸⠀⠀⠀⢰
⢸⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼
⠀⢿⡜⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢹⣿⣿⣿⣿⣿⠠⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⡿
⠀⠈⣷⡘⠿⠿⠿⠿⠛⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⠀⠀⣠⢠⡏⠀⢠⣿⠁
⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣵⣿⠁⣠⡿⠁
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⢹⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠾⢷⣾⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠭⠪⠗⠻⠪⠕⠷⠏⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⢒⠒⡒⢒⠒⣒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⣦⣿⢸⡆⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⢸⠿⠘⠁⠏⠘⠁⠐⣸⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠸⠥⠥⢭⡽⣥⠭⠬⠽⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⣭⠓⢀⡀⠈⠉⢩⢔⠦⡀⣀⣰⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠣⣾⠵⠃⠻⡀⠀⠸⢬⠽⠣⠥⡧⢽⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢯⡽⣼⠼⡽⠀⢙⠆⠀⠒⢢⠐⠒⡗⢲⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⠤⡀⠰⠊⠦⣄⠨⠝⢨⠇⣉⣏⣽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠐⠚⠉⠀⠈⠉

⠀⢀⡀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠠⠠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠈⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⡀⠁⠀⢀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⠀⠢⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⠤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄
⠀⣀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⡀⠐⢀⡀⠉⠈⠀⠀⠠⠀⠐⢂⡀
⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠔⠠⠀⠀⠀⡆⠂⠒⠀⠒
⠰⠀⠀⠀⠀⠱⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠠⠀⠃⠀⠀⠁⢤⠀⠀⣄⠀⠀⠀⠮⠔⠀⠀⢹⠨⠅⠏⠭⢨⠁⠀⢁⠀⠀⠉⠀⠀⠁⠀⠀⠇⠀⠰⠀⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠂
⣀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⡀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⢀⠀⠀⠀⠀⡀
⠁⠘⠁⠐⠀⠀⠀⠀⠀⣀⡀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⢡⡀⠀⠀⠀⠀⠀⠀⢄⠐⠄⠀⠅⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠒⢾⠀⠁⢸⠁⠀⠀⠀⠀⠀⠀⠊⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡒
⠀⠀⠀⠂⠀⠀⠀⠀⠈⠏⠉⠉⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣀⡀⠀⠀⢀⢀⣤⠀⡀⠀⠀⠠⣄⠀⠀⠀⢀
⡆⠀⠐⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠠⠀⠀⠀⠋⠉⠉⠉⠉⠉⠃⠀⠐⠁⠉⠐⠉⡈⠀⠀⠀⠀⢄⠀⠠⠂
⠓⠀⠚⠀⠀⢀⡀⠀⠀⠀⠈⡶⠒⢐⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠖⠶⠀⠀⢀⠈⠁⢙⠀⠃⠃⠀⠀⠀⠀⢰
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠁⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠰⠆⠀⠘⠀⠀⠈⠉⠭⠉⠍⠈⠀⠀⠀⠑⠛⠾⠛⠂⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠅⠀⠀⠀⠀⠈⠀⠀⢡⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⢁⡀⠀⠀⢀⡀⠀⠠⠠⠤
⠀⠲⠐⠀⠀⠄⠀⠀⠀⠀⠀⢸⡤⢤⡀⠀⠀⠐⠐⠂⠒⠐⠀⠀⠀⠀⠠⠤⠀⠸⡂⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢠⠗⠰⠋⠀⠀⠀⠀⡇
⠀⠀⠤⠀⠀⠀⠀⠀⢘⠐⣶⠈⡃⠀⠁⠀⠀⠀⠠⠓⠁⠠⠀⠀⠀⠄⠀⠀⠀⠄⠰⠀⠀⠢⠄⠠⠰⠤⠅⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠠
⣀⣀⣀⣀⡀⠀⠀⠀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡿⠀⠀⠀⠷⢠⠄⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢲⢲⠀⠀⠀⠀⠈⠉⡄⢀⠀⠀⠀⠀⠀⠀⠨⠄⡀
⠀⡆⡆⣰⠸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡁⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠠⠀⠀⢰⠀⡄⠀⡇⠀⠀⠀⠀⠀⠢⡀⠀⠂
⢄⠃⠃⠘⠈⠈⠀⠀⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡂⠰⠆⠐⠀⠀⠀⠀⠘⠀⠃⡀⠃⠐⠀⠀⠈⠀⠀⠈⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠉⠸⣷⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⣤⣤⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⣩⡟⠀⠀⠀⠀⠀⣉⣿⢶⣄⣠⣤⣴⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣀⣾⠋⢠⡶⠶⣦⣄⣾⠋⣠⡾⢋⣡⣤⠶⠿⠀⠈⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⡀⠉⠁⣴⠟⠀⠀⠀⠉⠁⣼⠏⠙⠟⠁⠀⢰⡆⠀⠀⠻⣦⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⠀⠈⠻⢦⣤⡶⠶⣦⣴⣯⣤⡀⠀⠀⢸⡟⠛⠃⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣦⡀⣰⠶⠶⠞⠛⠶⣼⣇⢸⡇⠀⠀⠈⠛⢻⡇⠀⠀⠘⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣾⠃⠻⣦⠀⠀⠀⠀⠈⢻⡄⠻⣆⠀⠈⠹⡟⠁⠀⣠⣄⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣘⡷⠀⠀⠻⣤⣄⠀⠀⠀⠀⠻⣶⣟⠀⠀⣤⡶⠶⠟⣉⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡄⠀⠀⠀⢹⣿⠶⠀⢀⣠⡾⠛⠛⠶⠿⠶⠶⠛⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠷⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢀⠸⠈⠀⠁⠈⡀⠀⠅⠈⠈⠀⠠⠁



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

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⣄⢠⡀
⠀⠈⠻⢬⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢆⣀⠀⠀⠀⠀⣀⠀⢀⡀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⢀⠄⠀⠀⣏⣩⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠢⣄⡄⢐⠀⠀⠀⠀⠀⠉⠀⠀⠀⡤⣰⡃⡿⠁⢠⣤⠆⠀⠀⠀⣠⣼⣲⣆⡀⠀⠁⠀⠈
⠀⠀⡠⠒⢰⠄⢨⣤⡾⠛⠃⠞⠃⢄⠀⠀⠀⠀⢀⠠⣀⠀⢽⠟⠁⠀⠀⠀⠀⡀⠔⠚⠉⠈⠉⠻⣷⢀⣦⠀⢰⡐⡄
⠀⢠⣠⠏⠈⠀⠘⠔⠀⠀⠀⢤⣄⡀⠈⠲⡄⠀⠀⠓⠂⣴⠿⢀⠀⠀⠀⡔⠀⢀⣠⣤⣀⠀⠀⠀⠀⠻⢦⠃⠈⠛⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢠⢾⣿⠛⠀⠀⠈⠀⡀⠀⠈⠁⣨⠟⠁⠀⢈⣀⣾⣽⣋⣉⣁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠻⢿⣿⠶⡀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⢻⣸⡓⠓⣚⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⣷⢿⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⣇⠀⠀⠛⠭⠭⠼⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠀⠀⠀⠠⠃⢀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠈⠛⠲⠇⠀⠀⠀⠙⠢⠄⣀⣀⠤⠄⠀⠀⠤⠐⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠉⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛
⠀⠀⠀⣞⠱⠆⠀⠀⠻⣥⠀⠀⠀⡠⠀⠀⡞⠀⠘⠀⠀⠀⠀⠀⡁⠀⠀⢀⡀⠀⠀⢢⡄⠀⠀⣄⢄⣀⣀⡀
⠀⠀⠀⠙⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⢿⣆⠌⠿⠂⠋
⠀⠀⠀⠀⠀⠉⢹⠆⠀⠀⠀⠀⠀⡴⠋⠀⡀⠀⠀⠀⣠⡤⣰⡄⡇⠀⠀⠀⡇⠀⠀⠀⠘⠷⣄⣀⠘⠓⠂⠀⢰⣦
⠀⠀⠀⢸⣡⠄⠃⠀⢀⡤⠀⣠⠞⠀⠆⠀⠿⠀⡇⠀⠷⠖⠉⠀⡇⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠉⠓⢦⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⣤⠀⠀⠀⡍⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠐⠀⠀⠉⠳⠶⡦
⠀⠀⠀⠀⠀⣤⠞⠁⠐⡆⠀⠀⣿⠀⠀⢠⡁⠀⡇⠀⠻⣙⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡆⠀⠂⠀⠀⢳⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⡇⠀⠆⢀⡆⠀⡄⢰⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠱
⠀⠀⠀⣉⠠⠤⠧⠤⠶⠧⠴⠷⠛⠒⠖⠾⠷⠖⠇⠀⠀⠀⠀⠀⠛⠛⠉⠋⠙⠛⠛⠛⠛⠒⠛⠒⠓⠂⠀⠄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⣤⠀⠀⠀⠀⠀⠚⠶⢤⡀⠀⠀⣀⠠⡄⠀⠘⣃⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢸⣷⠀⢸⣤⣴⣶⡆⠠⣤⢐⣏⠁⠀⠀⠋⣰⠃⠀⡰⠣⠁⠀⠀⠤⠤⠚⠃
⠀⠀⠀⠀⠀⣄⠳⠊⢁⡐⣛⠒⢕⣫⣾⢿⡏⠀⣼⠏⠉⠿⠀⠀⠀⠙⠉⠀⠀⠈⠉⠀⢀⡜⠁⠀⢀⠀⢦⠀⠀⠀⢀
⠀⢉⣈⣐⣒⡏⠀⠐⣟⣛⡂⠈⣿⠅⠀⠜⢉⣷⠶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⡄⠰⡏⠁⣸⠂⠀⠀⢠⡀
⠀⠘⢂⣀⠀⣿⣀⣀⡿⠖⢀⠀⠹⡄⠋⣀⠈⠛⠐⠉⠁⠀⣀⣈⠀⠀⠀⡀⠀⠀⡉⠁⠀⠀⠈⠂⠤⠴⠋⠀⣠⡎⠈⠁
⠀⠐⠊⠁⠀⡴⢿⣧⢿⡶⠀⠀⠑⢠⡎⠀⣀⠀⠐⢀⣠⡞⢹⢿⣐⢰⡻⠏⡙⠿⣆⠀⠀⠀⢌⠀⠀⢠⠴⢚⢧⠠⣄⠈
⠀⠀⠀⠲⡄⣀⡢⠥⠀⠀⢀⣚⠫⠓⠛⠉⠀⠀⠠⠄⠈⠫⠛⢲⣬⣿⣏⣈⣧⠴⠀⠀⠝⠓⠀⠀⠀⢐⠒⠀⠾⠀⠉
⠀⠀⠀⠀⠈⠉⠛⠒⠒⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⠶⢧⣿⠖⠖⡂⠢⠀⠀⠀⠀⠀⠀⠈
⣠⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣛⣿⢉⡛⣛⡛⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣠⠊⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⠤⢀⡀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣠⠖⢋⠡⣐⠒⠀⠈⠉⠉⠀⠀⠀⠁⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢀⡴⢋⣄⣈⠔⠋⠀⠈⠉⠉⠉⠉⠉⠉⠐⠒⠢⢬⡀⠑
⠀⠀⠀⠀⠀⢀⠄⠀⡞⢠⠞⡠⠁⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡈⠳⣼⡀
⠀⠀⠀⠀⢀⠀⠤⠄⢧⢉⠜⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠘⡗⢒⠢⢄
⠀⠀⠀⢠⠊⠀⠀⠀⠘⡏⠀⠀⠀⠀⣴⣿⠷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠦⠀⠁⠀⢹⠀⠆⠀⠱
⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣤⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡇⠀⠀⢸⠀⢸⠀⠀⡀
⠀⠀⠀⢣⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⢿⣿⣿⠇⠀⠀⠠⠀⠠⠀⠀⠀⠻⣿⠿⠁⠀⠀⣸⠁⣌⠀⡰
⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⣄⣈⠁⣉⣠⡆⠀⠀⠀⠀⠀⠀⢠⢇⣀⡠⠜⠁
⠀⠀⠀⠀⠀⠈⠙⠶⠦⠾⣄⢀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⡠⠋⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠣⠤⠜⠁⠀⠀⠀⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⢃⣈⣙⣒⠦⠤⣤⣄⣀⣀⣤⡤⢤⡖⠋⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⣡⠀⠀⢆⠀⠢⠀⠔⠀⡄⠀⢫⡖⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠂⠤⠀⠊⡀⠀⠀⠣⣀⣀⠔⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠡⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠠⢄⣀⠀⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣓⠄⠰⡱⠙⠦⠠⢀⣩⠤⢀⠝⡦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠁⠀⠀⠉⠛⠒⠀⠒⠛⠋⠉⠁
⠀⠀⠀⣀⠤⣆⡒⠺⡺⢻⡅⡀⠲⣦⠴⠀⠀⠁⢐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⢤⣀⡀⠀⢀
⠀⠠⠎⠘⠒⠛⠒⠬⠠⣈⠽⠛⠚⠁⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⠀⠀⢀⠖⠀⠀⠠⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⠔⢠⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠿⣥⠭⡅⠂⠀⢀⢄⠞⠁⠀⠀⢀⠀⡂⠤⣀⠠⡀⢀⣼⠇⠒⠒⠀⣲⣤⣤⢀⣀⣠⡷⢶⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠲⡀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠐⠃⢈⡠⠤⢍⡿⢀⠾⢇⡀⠀⠀⠈⢚⣈⣡⠁⡠⢄⡀⠹⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⣩⠔⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠢⠀⠀⠀⠉⠀⠀⡀⠞⠩⠝⡟⡳⠁⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢆⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠸⠀⠀⠀⠀⠀⠊⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠢⢧⣀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠥⠠⠄⠀⠀⠀⠐⠒⠦⠒⠂⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠕⣒⠀⠀⢰⠽⠒⠰⡦⠤⣔⢃⢷⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠒⠒⠒⢀⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⠚⠉⠁⠊⠀⠑⠫⣢⡰⠂
⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⣀⡠⣄⠀⠺⠄⢲⣆⣀⠨⣟⠐⠯⠕⠶⠤⠤⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢥⣀⠐⣀⠀⠀⠈⠑
⠀⠀⠀⠠⡊⠁⠀⠏⠃⠁⢠⣄⡜⢢⠀⠈⠀⠀⠀⠈⠁⠛⠬⠤⠤⣥⡤⠤⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠉⠀⠉⠐
⠀⠀⠀⢠⡁⠂⢅⠱⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡖⠚⠒⠲⠄⡀⠄
⠀⠀⠀⠈⢏⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠯⠷⡷⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠉⠊⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆⢁⣠⠬⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠄⠠