Tolerance Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⠀⢀⠄⠰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡟⠀⠀⢠⡇⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡀⠅⠀⠀⡼⠀⢳⡜⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢿⡷⠀⠀⠘⠀⠡⠈⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢤⣅⣀⠀⢀⣆⠞
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢲⣳⣏⠣⡎⠁⠒⠃⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠐⠁⣀⠀⡜⠉⠀⢹⡓⡇⡀⣴⠀⠀⠀⡟⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⣈⣯⠂⡈⣃⡏⣇⢀⠀⡠⠞⠼⠒⢌⡓⠄
⠀⠀⠀⠀⠀⠈⡁⡀⣷⣿⠛⠃⠐⠀⠹⣄⠘⠆⡈⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⢀⠇⡸⠟⠁⠀⠀⠀⠀⠀⠘⠈⠀⡄⠀⠀⠀⠀⠡⠂
⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⢀⡴⢦⡀⠀⠀⣴⠲⠀⠸⣷⣿⡆⠀⠀⣿⣷⣄⠀⠀⢰⣿⣦⠀⠰⠀⣄⠀⠃⢀⣤⡖
⠘⠐⠀⠀⢠⠁⠀⠙⡠⢄⡧⠀⠀⣯⣴⠀⠀⣿⡛⠁⠀⠀⢹⢿⠇⠀⣦⢾⣿⠏⠀⠀⠀⣿⣷⡄⢈⣷⣿
⠀⠀⠀⠠⠀⠀⠀⠀⣠⢋⠀⠀⢀⡟⢣⠀⠀⠀⡀⠀⠀⠀⠛⠆⠀⠀⣧⢀⣽⣦⠀⠀⠀⢿⡿⠁⠀⠩⠶
⠀⠀⠄⠰⢀⠀⠀⠈⢹⠐⡄⠀⢸⡓⠈⠀⢀⢀⡀⠀⠀⠀⡆⢡⠀⠀⣿⠘⡇⢸⡆⠀⠀⡟⠙⡀⠀⢐
⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠘⡇⠀⠀⢰⡟⠁⠀⠀⠀⡇⠘⠄⠀⣿⠀⠃⠘⠇⠀⢰⡃⠀⠇⠀⡟
⠀⠀⢠⠀⢁⠀⠀⠈⠈⣦⣰⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⣥⠀⠀⠀⠘⢠⠀⠠⠀⠀⠀⠁⠀⠀⠠⠃
⠀⠀⡈⠀⠈⠀⠀⠀⠀⢹⠉⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠃⠀⢸

⠀⠀⠀⠀⠀⡄⠀⢀⠀⠠⡶⠦⣶⠀⠀⣶⣄⠀⠀⠀⢠⣶⡂⠀⢰⣶⣆⣀⠀⠀⠀⡀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠄⡀⠀⠀⣀⣰⠆⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠀⠀⠀⠉⢬⣥⠀⠀⣻⠿⠃⠀⠈⠉⠯⠿⠀⠀⠀⠂⠀⠀⠀⢿⡿⠂⠀⠀⠺⠟⠀⠀⠰⡝⠋⠄⠀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⠿⠿⢿⣿⡿⢿⣿⣿⠿⠿⢿⡿⠿⠿⣿⣿⣿⠿⣿⣿⡿⢿⣿⡿⢿⣿⠿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⢰⣾⠁⣴⣶⡄⢹⡆⢸⣿⣿⠀⠶⢾⡇⢰⠆⢈⣿⠃⣄⠸⣿⠀⣀⢹⡇⢸⠁⣴⣶⣶⡇⠰⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣄⠻⠟⠃⣸⡃⠘⠻⣿⠐⠗⢻⡇⢰⡄⠻⡏⢠⣥⡀⢻⠀⣿⣆⠁⢸⣄⠙⠿⢻⡇⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⣿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⠿⡿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⢠⣤⡄⠀⢀⣠⣀⣄⠀⠀⣄⠀⣄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⢠⣆⠀⠀⢀⣤⣀⣀⠀⠀⢠⡀⠀⠀⢠⣤⣠⡄⠀⠀⣶⣶⡀
⠀⠀⠀⠀⠀⣘⣿⣗⠀⠈⢉⣈⡉⠀⠀⠛⡀⠛⠀⠸⠷⡾⠺⠀⠘⢿⡭⠌⠀⠈⠻⡿⠇⠀⠈⠺⡿⠃⠀⢠⣼⣧⠇⠀⢀⠿⠿⠀⠀⠈⣷⣿⠇
⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⡙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⡿⢩⣤⣴⣶⡾⢛⡛⠿⢛⡛⠳⣦⣤⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⠐⣿⣿⣤⣿⣿⠆⣿⣿⣿
⠀⠀⢀⣾⣿⣿⣿⡿⢋⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣴⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣷⡘⢿⣿⡿⢃⣾⣿⡟⠁
⠀⢀⣾⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢸⣿⣿⣿⣌⠛⣡⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⠿⠟⠿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣦⣭⣭⣭⢁⣾⣿⣷⣌⠻⠏⣴⣿⣿⣦⡙⣿⣿⣿⣿⣿⣬⣉⣉⣥⣶⣷⣦⣍⣙⡉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣥⣤⠙⣿⣿⣿⣷⡘⢿⣿⣿⡟⣁⣭⣍⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢿⣿⣿⣷⡌⠻⣿⣿⣿⣆⠙⠋⣴⣿⣿⣿⡿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣉⣉⣉⣉⣉⣉⣉⢉⣴⣶⣤⠙⣿⣿⣿⣦⡈⢻⣿⣿⣷⠀⣿⣿⣿⠏⢠⣤⣤⡌⢿⣿⣿⣿⣿⠟⣋⣉⡉
⢻⣿⣿⣿⣿⠟⢛⡀⠻⣿⣿⣷⣌⠻⣿⣿⣷⣄⠙⠋⣉⣰⣿⠿⢁⣴⣿⣿⣿⠏⠸⠿⣿⣿⢁⣾⣿⣿⡇
⠘⣿⣿⣿⡇⢸⣿⣿⣧⡙⢿⣿⣿⣦⡘⢿⣿⡿⢁⣿⣿⣿⠋⣴⣿⣿⣿⡟⢋⣴⣶⣷⡆⢻⡈⢿⣿⣿⠁
⠀⠹⣿⣿⣿⣦⠹⣿⣿⣿⣆⠻⣿⣿⣿⠄⣴⣶⣿⣿⠟⣡⣾⣿⣿⣿⠟⣱⣿⣿⣿⡿⢃⣾⣿⣦⣍⠁
⠀⠀⠹⣿⣿⣿⡇⣈⢻⣿⣿⣷⡌⢉⣩⣼⣿⣿⣿⣿⣶⣿⣿⣿⡿⢃⣾⣿⣿⣿⠟⡁⢸⣿⣿⣿⠏
⠀⠀⠀⠙⣿⣿⡇⣿⣦⡙⠿⠟⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡿⢁⣾⡇⣸⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⠟⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⠟⣡⡘⠋
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠋⣥⣾⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠛⠛⠲⠶⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⠀⠀⢀⡾⠋⠁⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡾⠁⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠉⢀⣠⣀⡀⢀⣀⣤⡈⠛⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢰⡏⠀⠀⢻⡏⠀⠀⢻⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢀⡶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠈⠻⣄⠀⠀⣰⠟⠀⠀⢀⣾⠋
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣧⣼⠋⠀⠀⠀⠸⣇
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣤⣤⣤⣴⠛⠉⠉⠻⣦⡀⢀⣴⠟⠋⠙⠻⢦⡀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣤⡶⠟⠛⠻⢶⣤⣀⣀⣴⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⡀⠀⠀⠀⠈⢳⣾⠁⠀⠀⠀⠀⢈⣿⣀⣀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⣿
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠉⢷⣄⠀⠀⠀⠀⠙⢷⡀⠀⠀⣠⡞⠋⠉⠉⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠈⠻⣦⣴⠏⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡷⠶⠶⠶⠶⠶⠶⠶⠶⣶⡾⠛⠉⠛⢷⡀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢀⡾⠟⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⣠⣤⡴⡤⣾⠇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠙⢷⣄⠀⣀⣠⡟⠀⠀⠀⣴⠟⠁⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⣸
⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⡞⠋⠉⠙⢧⡀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⢻⡟⠉⠁⠀⠀⣠⡾⠃⠀⠀⠀⠀⢀⣾⠷⠶⠶⣤⠀⣾⡁⠀⠀⠀⢰⡟
⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠹⣦⣀⣀⣤⡾⠁⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢸⡇⠘⢧⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠘⢷⣄⠀⠀⠀⢘⡿⠉⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢀⡾⠁⠀⠀⠙⠻⣾⠇
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⡿⣄⠀⠀⠀⠀⠙⣶⣤⡴⠟⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⢀⣴⣿⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⡇⠘⢷⣄⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡞⠁⢸⡇⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠘⣧⠀⠀⠙⠛⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢀⡿⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⢀⣴⠟⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⠟⠻⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠈⠉⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⠶⠞⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠞⠋


⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⡷⠴⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⢾
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠀⡴⠐⢦⠀⡄⠰⡖⠀⠰⡄⠀⠴⠐⡄⡄⠀⠀⡖⠒⠀⠀⡔⠀⠀⠲⠆⢠⠐⢢⢀⠔⠐⢄⢠⠀⠄⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢗⠀⣠⠀⡁⠀⡇⢀⠧⠰⠄⠚⠘⡆⠃⠀⠀⡗⠀⠀⠀⠈⢲⠀⢨⠀⢸⠒⠇⠸⡀⠀⡸⢸⠛⣄⠀⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣿⣷⣶⣷⣶⣷⣾⣶⣶⣷⣿⣿⣶⣿⣿⣷⣿⣿⣶⣶⣿⣷⣶⣾⣶⣾⣶⣾⣶⣿⣿⣶⣾⣶⣾⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣀⣀⣀⡀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


⡿⠉⢹⡟⠿⠿⠿⠟⠿⡛⠿⠻⠟⠿⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⡿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⢿⣿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣉⣠⣿⠃⠀⠀⢻⣇⣈⣙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠙⠛⣿⡿⠿⠿⠿⠿⣿⡟⠛⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠻⠇⠀⠀⠀⠀⠿⠃⠀⠀⠐⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣤⣴⣦⣤⣀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠛⠛⠛⠛⠛⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠻⠿⣾⣿⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢠⣾⠟⠁⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⣇⣀⣴⡿⠋⠀⠈⠹⣷⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⢸⣿⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣷⣆⡀⣰⡶⠀⠀⠀⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠀⠘⣿⣄⠀⢠⣄⠀⢶⣆⠀⠹⣷⡀⣸⣿⠛⢻⣿⣤⣤⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⢿⣦⠈⢿⣦⣀⣿⡟⠛⠁⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣾⠟⠿⠟⠋⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⢿⣿⣿⡷⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⠠⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⠹⠟⢡⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣠⣤⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⣀⣤⣈⣶⣤⣿⣷⣿⣏⣹⣿⣥⣿⣿⣯⣤⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠄⠀⣀⣴⣬⣽⣿⣿⣿⣯⣹⣿⣿⣿⣟⣛⣿⡿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣾⣿⣿⢻⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠁
⠀⠀⢠⣤⣦⣄⡻⣯⣿⣶⡟⣿⣿⣷⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠘⠿⢿⣿⡿⠈⣿⣿⣿⣽⣿⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠐⠀⢶⣶⢢⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠛⠈⠉⣿⣿⣶⣿⣿⡟⢠⣾⣱⣿⣿⣿⡿⣿⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡿⣿⣽⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢨⢻⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣷⣾⣿⢏⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡇⢺⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⡟⠋⠁⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⢻⣿⣿⢯⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣾⣿⣿⣿⣿⣿⢯⡾⠃⣴⣾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄⣾⣿⣿⣿⣿⣿⣳⡟⠁⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠔⠀⠐⣄⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⡘⢂⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⡼⠀⠀⠀⠀⠀⠻⠀⠀⠀⢀⠈⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⢎⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣸⣿⠈⣇⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢢⠀⠀⢸⡏⠙⠁⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⣠⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢰⣿⡇⠀⠀⣸⠆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣄⣠⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⣿⡅⠀⠀⢻⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⣧⣌⣶⣦⡀⠀⠄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠺⣿⣝⢟⠛⢿⡆⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⢹⣿⡧⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠂⠀⠸⠋⠁⠀⢀⡠⠊
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣖⠂⠠⠄⠀⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠂⠐⠂⠠⠀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⣶⣦⣤⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠁⠀⠰⠈⠑⠒⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⣻⣿⣏⣿⣿⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠌⠀⢌⠀⠀⡀⠀⣀⠀⠀⠀⠈⠒⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣽⣾⡇
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡁⠣⡀⠪⣀⠕⢀⠈⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢷⣜⢿⣜⠿⣫⡿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠾⡉⠂⠈⢢⠘⡠⠊⢡⢄⡐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣱⣾⣻⣿⡟⣿⣟⣵⡞⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢂⠀⣦⠀⢄⠙⠁⠀⠀⠔⠁⡜⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣮⡻⣪⣾⣿⣿⣫⣿⢻⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠈⠁⠀⠀⠀⠀⡠⢠⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣾⣿⣿⣿⣿⣟⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⣀⠊⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠈⠻⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠤⡠⢀⠰⡩⠋⠿⡏⢹⠽⠿⡿⠿⠿⠟⠻⡏⣿⠿⠿⣿⠿⠿⣿⣿⠛⣉⣹⡏⣿⢿⢻⠉⡏⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠂⢴⠀⠁⠸⠀⡂⢰⠀⠁⠀⠰⠶⠀⠶⠄⡆⢰⠆⡁⠰⠆⡀⢰⡆⢸⣿⠷⢦⠙⡇⡁⢾⠸⠀⡇⡷⢌⠺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⢄⠒⠚⠵⣾⢦⢶⡾⣶⣶⣿⣶⣶⣿⣶⣷⣿⣶⣾⣷⣿⣷⣿⣿⣷⣶⣾⣷⣿⣾⣿⣿⣿⣿⣶⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⢄⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⡣⠀⢰⠊⡡⣤⣦⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⢔⠉⢉⡉⠉⣀⠀⠈⡇⠘⠿⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢟⣽⣶⣾⣿⣿⣿⣍⣛⣹⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠫⡈⠻⣉⠵⢤⠘⠛⠛⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⠿⣻⢿⡻⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⠖⢒⠶⠙⢄⠙⢤⠝⢀⠊⢱⣄⡔⢛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⢫⡿⣟⢯⡽⣪⢿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢣⠙⣄⠑⣄⠕⠋⠐⠁⡔⠁⣼⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡻⣝⣮⣿⣿⣷⢏⣾⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡧⠑⠃⠀⠀⠀⠀⣠⢪⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⡤⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⠜⠁⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣽⣿⣷⣟⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠜⠁⠙⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣚⣉⣉⣀⣀⡀⠀⠀⠈⠉⠛⠲⢤⡀⢠⠞⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⡤⠴⠞⢋⣀⣀⣀⡓⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠸⣅⠀⠀⣏⠀⠛⠀⣹⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠞⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣧⠀⠈⢧⣀⡼⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠸⡄⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⣏⠀⠀⢀⣉⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠓⣒⣾⠁⠀⠙⣤⠋⠀⠈⣳⣀⡀⠀⠈⠒⠚⠉⠈⠙⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣾⠁⠈⠳⣄⠀⠈⢳⣀⡴⠋⠈⠙⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⠶⠶⠶⢶⡞⠛⠳⡄⠀⠘⢦⡀⠀⢙⡆⠀⢀⡞⠛⠳⡄⠀⠀⢀⡴⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡴⠚⠳⡄⠀⠙⢦⠀⠀⢹⡖⠋⠀⡴⠋⠀⠀⣰⠷⠦⡄⣏⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠳⣄⠀⠙⢦⡀⠈⢳⠖⠋⠀⣠⠞⠀⠀⢠⠞⠁⠀⢠⠇⠙⠦⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣽⢦⡀⠀⢹⡖⠋⠀⠀⠀⠁⠀⢀⡴⠋⠀⠀⡴⢿⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢻⡀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠞⢁⡟⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡴⢯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠞⠁⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⢀⡴⠋⠀⠀⠈⢣⡀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠄⢠⠀⣀⠀⠀⣄⠀⠠⠀⠠⢠⡠⠀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠈⠀⠈



⠐⠐⠀⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠾⠿⠿⠿⠿⠿⠿⠷⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠿⣶⣴⡿⠁⠀⠀⠈⢿⣿⠋⠁⠀⠉⢻⣦⣶⠿⠻⠿⣶⡄⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣯⠀⠀⣀⡀⠰⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣷⣾⠟⠛⠛⢿⣿⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢸⣏⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠷⣶⡶⠟⠻⣷⣄⠀⠀⣠⣾⣿⣄⠀⠈⣙⣷⣦⡀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠉⠛⠛⠛⣉⣿⣿⣶⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣶⡾⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢿⣶⣶⣶⣶⣶⣶⣶⣾⡿⠛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠞⠛⠛⠲⠶⠶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⣁⣤⣴⣶⠶⢶⡶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢉⣴⠋⠁⠀⢸⡇⠀⠈⠻⣌⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠙⣧⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠓⠶⢦⣤⣤⣄⣀⣀⣸⣇⣀⣀⣤⣤⢹⡆⠲⠛⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⡆⠀⠀⠀⠈⠉⢹⡏⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⠳⣄⡀⢸⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢹⡄⢀⣴⠟⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠈⣿⣾⣧⠴⠶⠒⠲⠶⣤⣤⠶⠖⠒⠶⠶⠾⢷⣏⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⣴⡄⢀⡾⠋⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⢠⣦⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⣴⠏⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣴⠟⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⡴⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠐⢲⡄⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⡿⠋⠙⣧⣄⡀⠀⠀⠐⢦⡀⠀⠙⢷⣄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡏⠀⢀⡴⠋⠈⢹⣦⣤⡀⠀⠙⢦⣀⣠⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡏⠀⢀⣴⠟⠁⢈⣿⢦⣀⣸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠙⢦⡴⠟⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣥⣤⣥⣤⣤⣤⣬⣉⢻⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⢋⣤⣶⣶⡟⣋⣙⠛⣋⣙⢳⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿⣿⣿⡘⢿⣿⣿⡄⢿⣿⣴⣿⡿⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣆⠻⣿⠟⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣧⣉⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣌⠻⠿⠿⠿⠟⣩⣭⡙⢿⣿⢛⣉⣉⡛⢿⣿⣿⣿⣿⡘⢿⡿⠟⣛⣛⠛⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠆⠸⣿⣿⣿⣌⠡⣾⣿⣿⣿⠈⠿⠿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣷⡈⢿⣿⣿⣷⡘⢿⡿⢡⣾⣿⣷⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠟⣋⣀⡙⢿⣿⣿⣦⠹⣿⣿⣿⣦⠰⣿⣿⣿⠟⢁⣉⡛⢿⣿⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠘⢿⣿⣿⣌⠻⣿⣿⣷⣌⠻⠿⠟⣰⣿⡿⢋⣼⣿⣿⣿⢈⣿⣿⣿⠋⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢰⣾⣷⣌⠻⣿⣿⣧⡘⢿⣿⣿⠆⣶⣿⣿⠟⣡⣿⣿⣿⡿⢁⣤⣤⣌⢻⡸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣌⠻⣿⣿⣷⡘⢿⣿⣿⡆⢩⣤⣾⣿⡿⢃⣾⣿⣿⣿⢋⣴⣿⣿⣿⠏⣸⣷⣌⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⡘⢿⣿⣿⣦⠙⣋⣡⣾⣿⣿⣿⣶⣿⣿⣿⡟⣡⣾⣿⣿⡿⢃⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⣿⣦⡙⠿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⢋⣴⡿⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⠟⣡⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣼⡿⢡⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⠺⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⡿⠛⠁



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⠿⠿⡿⢿⡿⠿⠿⠿⠿⢿⡿⠿⢿⠿⢿⠿⢿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⠿⣿⣿⣿⣿⣿⣿⣶⠀⠐⠀⢶⠀⠀⠸⡇⠀⣀⠀⠀⢀⠃⠀⠘⠀⠀⠀⠀⠰⡶⠄⢀⣰⣿⣿⣿⣿⣿⡿⠟⠿
⣤⣼⣤⣽⣿⣿⣿⣿⣿⣿⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣶⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣧⣥⣤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠔⠢⢠⣔⠢⡀⠀⡠⠐⢠⡄⠀⠀⠀⠀⢀⡀⢄⠀⢀⠄⠀⠐⠒⡐⢂
⠀⠀⠀⠀⡐⡀⢸⠁⢈⠃⠀⢀⢀⣥⣾⣧⢘⡄⠀⣀⠀⡀⣐⣤⡀⠀⠀⣝⣛⡁⠀⠝⡄⠀⠀⠖⠀⢚⣒⠒⠄
⠀⠀⠀⠀⣴⡁⢸⡀⢀⢆⠀⢀⢭⢼⣟⣋⠋⠀⠀⣈⡐⡈⠵⡴⢀⠀⠀⠀⡡⠈⣆⣥⡁⠀⠐⠒⡃⠧⡦⠅
⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠐⠙⠀⠉⠀⠀⠀⠀⠉⠘⠒⠀⠀⠀⠀⠀⠊⠀⠐⠂⠁⠀⠀⠀⠀⠈⠀⠐⠊
⠀⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⢀⡔⠒⠂⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠤⠀⠈⢂⣇⠂⠀⠀⠀⠀⢈⣄
⠀⠀⠀⠀⣤⣿⣷⣿⣿⣤⠀⠀⠀⡒⢒⠐⠀⠀⠀⠅⠂⢙⡋⡐⠽⠀⠈⢰⡉⠉⢉⠆⠃⠀⢇⡸⢀⣠⣴⣆
⠀⠀⠀⠀⢠⡿⢿⡿⢿⠄⠀⠀⠀⠈⠐⢀⠄⠀⠀⠀⠠⡀⢀⠜⠀⠀⢈⠼⢍⠆⠊⣀⠂⠀⠀⢀⣰⣏⠈⣹
⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁
⠀⠀⠀⠀⢀⠊⢀⡀⠁⡀⠀⠐⠒⠒⣦⠊⠓⡄⠀⠔⠚⠄⢠⠒⠣⠀⠀⠰⠆⠪⠃⠖⠄⠀⠐⣈⠔⠊⠙⢦
⠀⠀⠀⠀⠈⣚⠁⠁⡁⡄⠀⠠⠄⡀⠛⠠⠤⠃⠀⣇⢀⡇⢸⡄⣸⠀⠀⢰⢂⡢⣄⠂⠀⠀⠃⡒⠂⠄⠊⠒⠂
⠀⠀⠀⢀⠔⠈⠳⠞⠑⠣⠀⠀⣴⣳⣟⢞⣷⡀⠀⢃⡲⠗⠺⢖⡈⠀⠀⠲⣶⣤⡼⠛⠀⠀⠀⢴⣀⠙⠠⠌
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⠀⠀⠈⢀⣀⡀⠀⠁⠀⠀⡀⠄⠀⠠⢀⠀⠀⠀⡀⢀⡀⢀
⠀⠀⠀⠀⣤⣬⠢⠔⣤⡄⠀⢼⢠⠔⠋⠸⣌⡃⠀⠀⡸⠛⡧⣠⣠⠀⢨⡀⠀⢒⡀⢠⡅⠀⢰⡠⢊⡑⢄⡡
⠀⠀⠀⠀⡳⡞⠀⠀⣞⢳⠀⠈⢿⠉⢠⢐⠔⠁⠀⠿⡿⡦⡼⢿⢿⠀⢀⢹⡈⠁⢉⡏⠀⠀⠿⣄⠋⠝⡠⢿⠁
⠀⠀⠀⠀⠈⠉⠇⠟⠉⠁⠀⠀⠀⠙⠡⠂⠀⠀⠀⠆⠰⠀⠰⠀⠆⠀⠈⠈⠁⠉⠹⠀⠀⠀⠐⠄⠱⠎⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡀⠀⠀⠀⠀⣾⠉⠙⠋⠉⣷⠀⠀⠀⠀⢀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣴⠟⢁⣿⣄⣀⠀⠀⠙⢷⣄⣠⡾⠋⠀⠀⣀⣠⣿⡈⠻⣦⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣴⠋⠀⢈⠉⢛⢶⡄⠀⠙⠋⠀⢠⡶⡛⠉⡁⠀⠙⣦⣬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠉⠛⣿⣹⠿⣦⠀⠀⣴⠿⣏⣿⠛⠉⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡶⠶⡈⣩⣼⠛⠀⠀⠛⣧⣍⢁⠶⢶⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢷⣄⠀⠀⠙⠾⠋⠉⢠⡶⠛⠛⢶⡄⠉⠙⠷⠋⠀⠀⣠⡾⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⣹⡷⠀⠀⠀⠀⠀⣿⠀⠰⠆⠀⣿⠀⠀⠀⠀⠀⢾⣏⠀⠀⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⠋⠀⠀⣠⢶⣄⣀⠘⠷⣤⣤⠾⠃⣀⣤⡶⣄⠀⠀⠙⢷⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠷⠶⢁⣙⢻⣤⠀⠀⣤⡟⣋⡈⠶⠾⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣀⣤⣿⣹⣶⠟⠀⠀⠻⣶⣏⣿⣤⣀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⠻⣄⠀⢈⣀⣬⠾⠃⠀⣠⣄⠀⠘⠷⣥⣀⡁⠀⣠⠟⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠻⣦⡈⣿⠋⠉⠀⠀⣠⡾⠋⠙⢷⡄⠀⠀⠉⠙⣿⢁⣰⠟⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⢿⣀⣠⣄⣀⡿⠀⠀⠀⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢳⡤⠤⠤⣤⣤⣤⣤⣤⡴⠋⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢠⠏⣠⠴⠚⠉⠀⣀⠀⠀⠙⢦⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠲⠀⣰⠋⠀⢧⣀⡤⠖⠋⠉⠙⠦⣄⠀⠙⣄⠐⠃⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣼⡁⡠⠤⡄⠀⠀⠀⠀⠀⠀⢀⠈⠙⠦⣌⡷⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣠⠛⢳⣤⣄⠀⠀⣀⠈⠙⢦⣤⠟⢠⡀⠈⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⣁⣠⠎⠀⡼⠲⣄⠈⠛⣦⠴⠃⠐⠒⢿⠒⠒⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⢾⣁⣼⣉⡷⠒⠃⡀⠀⠀⠀⢸⡆⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⣤⠭⠤⠤⠤⣧⠤⠤⠤⢤⡧⠤⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠓⢦⣠⠖⠛⠒⢦⠀⢻⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⢸⠇⢸⣄⣀⣀⣀⣇⣀⣀⣀⣸⣁⣀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⡉⠉⠉⡏⠉⠉⢩⠏⠉⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡴⠋⠀⠀⠙⠦⣄⡳⣄⠀⡇⢀⡴⢋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡢⠀⠀⠀⠀⠀⢀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠙⠶⡟⠠⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⡄
⠀⠀⠀⠀⠸⣷⡀⢀⢲⢴⡷⣧⣵⣄⣷⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡐⡆⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣰⣿⣷⠈⠉⢙⣟⠁⢤⣠⠶⠚⠉⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣙⡂⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣇⢷⢞⣩⠈⣻⡛⠒⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍
⠀⠀⠀⠀⣿⢿⢸⣷⢿⣽⣿⣄⠻⢷⡿⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠙⣧⠈⠛⠛⣻⣋⢻⢶⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣌⠁⠄⠀⠀⠀⠈⢰⠘⡦⣄⠀⣁⡭
⠀⠀⠀⠀⠀⣽⣧⣀⠀⠻⡙⠁⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣁⠀⠁⢀⠀⠀⣨⣴⠀⣻⡆⢀⠆⣼
⠀⠀⠀⣠⣾⣿⢾⣟⠳⡶⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂⢠⣤⣾⣿⣿⡗⠹⢁⣾⠗⢋⡠
⠀⠀⢸⣿⣿⣿⣆⠈⢹⣶⣿⡄⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⡀⢀⣀⡀⠀⠙⠿⠟⠛⠐⠏⡁⠈⠉
⠀⠀⢸⣿⣿⣿⣿⣷⡘⢻⣿⡀⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠋⠉⣀⣤⣄⣤⣠⣴⠶⢚⡀⠀⡐
⠀⠀⢸⣿⣿⣿⣿⣿⣷⣼⣿⣷⣼⣿⣿⣿⣿⣿⣿⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠙⢳⣿⣿⣦⣄⢉⠛⠿⠃⠒⢈⢀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠃⠀⠂⠙⠛⠿⠟⠛⠓⠒⠂⠐⠂⠀⠛
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢀⠀⡀⣀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠓⠛⠉⠉⠙⠃⠀⢘⣁⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠐⠀⠄⢰⡇⠀⠀⣀⡀⠀⠰⠿⠿⢾⣷⠟⠟⠃
⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠠⠀⠒⠾⠃⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠘⠿⠃⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⢠⢼⣿⢻⡇⠀⠀⠀⠀⢈⣀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠀⠀⠀⣷⢾⣿⣿⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠀⠀⠘⠋⠿⠛⠛⠚⠿⠾
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣷⣶⣦⡀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠈⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠎⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠛⠁⠀⠀⠀⠠⠄⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠒⠛⠚⠂⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⠑⠀⠀⠈⠉⠀⠈⠁
⠀⠀⠀⠀⣰⠻⡆⠠⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⣄⡀
⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⣿⣿⡟⠀⠀⠒⠌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⣠⠖⠂⠀⠀⠀⠀⠀⢀⡀⠀⠁⠈⠀⠀⠀⠀⠀⠲⢀
⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⢀⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡄⠀⠀⠀⣤⣴⡟⠻⣷⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⢀⢀⣃
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡏⠀⠀⠀⣿⠶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⢻⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⢷⡀⣿⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠲⡄⠀⠀⠀⠀⠀⢻⡆⠀⠀⢹⣿⠀⢠⡶⠛⠛⠛⢶⣴⠞⠛⠛⠳⣦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠳⣆⣀⡀
⠀⠀⠀⠀⣠⡴⣴⡇⠀⠀⣿⠛⢻⡆⠀⠀⠘⣷⠀⠀⠈⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⠀⠀⠀⠀⢠⡶⢶⣿⠀⠀⣿⠋⠙⣧
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⣇⣀⡀⠀⢻⡆⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⣀⣀
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡏⠉⢻⡆⠘⣷⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠉⢹
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠸
⢀⣤⡀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣼⠃⠀⢀⣤⣄⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿
⣿⠁⠹⣦⣿⠀⠈⠃⠀⠀⠁⠀⠘⠇⠀⠈⣧⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡅⠙⣧⣿⠀⠀⠉⠀⠀⠀⠀⠀⠛
⢸⡇⠀⢻⣿⠀⢀⣠⣤⣀⣀⣤⣤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⠸⣿⠀⢀⣠⣤⣄⣀⣤⣤⣄
⠈⣷⠀⠈⠿⢀⡿⠁⠀⠙⠋⠁⠈⢹⡆⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠻⠀⣿⠁⠀⠈⠋⠀⠀⢹⡇
⠀⢸⡄⠀⠀⠈⢻⣄⠀⠀⠀⠀⢀⡾⠃⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⡾⠃
⠀⠈⣷⡀⠀⠀⠀⠙⠷⣄⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣦⣠⡾⠋⠀⠀⢰
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⣌⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠈⠳⢄⠈⠓⠦⡀⠐⢤⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠈⢂⠀⠑⢦⡀⠈⢦⡀⠈⠲⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⣦⠀⠑⢄⠀⠳⣄⠀⠙⢄⠀⠙⢆⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣏⢳⡀⠈⠠⡀⠈⢲⡄⠀⠐⠀⠀⠁⠠⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢻⡄⠀⠑⠀⠀⠀⠀⠀⠀⢀⠀⢡⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢄⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⢳⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢾⣆⡘⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠈⠉⢿⣄⠀⠀⠀⠀⠀⠀⠀⠠⣍⡛⠿⠟⣡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠜⢿⡷⢄⣀⡀⠀⠀⠀⠀⢠⡝⢡⡞⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀⣈⠁⠈⢉⠁⢀⣈⢀⡰⢋⣴⠟⣽⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠫⣙⠳⣤⡐⣠⣬⡭⣁⣤⣾⠍⢀⡼⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣆⠀⠀⠀⠁⣿⢫⢾⣿⢃⣠⡴⠞⣱⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣌⠐⠀⠒⣺⣣⣏⣼⣿⣿⣯⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣿⣿⣿⠿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠉⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣙⣛⣉⣉⣩⣍⣤⣭⣍⣉⣉⣉⣭⣍⣙⠃⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢿⠁⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣊⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢃⠀⢀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠴⠖⠶⢦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠉⠉⠉⠳⣆
⠀⠀⠀⠀⢀⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣀⣀⡀
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢾⣁⣹⠷⣾⡷⠶⠶⠶⠶⠶⠶⢿⡷⠾⣇⣈⡷
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠉⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠉
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⢻⡄
⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣀⣠⠞⠙⣧⠀⠀⠀⠀⢻
⠀⠀⢹⡆⠀⠀⠀⠀⢀⣠⡤⠞⠃⠘⣧⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡇⠀⢀⡴⠞⠋⠉⠀⠀⠀⠈⢧⡀⠀⠀⢸⡇
⠀⠀⠀⢷⡀⠀⢠⠞⠉⣤⠀⠀⠀⢠⡌⢷⣄⠀⠀⣠⠏⠀⠀⠀⠀⠀⠈⣧⠀⣾⠀⢤⣀⠀⠀⠀⠀⣀⡤⠙⣦⠀⣼
⠀⠀⠀⠀⠙⣧⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⡄⠀⠈⠛⠀⠀⠛⠁⠀⢠⣇⡼⠃
⠀⠀⠀⠀⠀⠈⠙⣧⠀⣤⡾⠛⠛⢦⡄⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⠀⢠⣤⣤⡄⠀⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣝⠓⠲⠒⢛⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠛⠶⠞⠛⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⠀⠉⠉⠀⠙⠶⢤⣄⡀
⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠈⠛⢦⣄⠀⠀⠀⢀⣤⠀⠈⠉⠛⠶⣤⡀
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⣦
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⣿⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣾⣷⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣾
⣿⣿⣁⣍⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀
⠀⠀⠀⠀⠀⠙⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁⠀⠀⢀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⢀⠞⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⡸⠁⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢹⣦
⠀⠀⠀⡎⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⣰⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢿⡆
⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡷⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⡇⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣵⡄⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣷⠀⠀⠀⠀⢱⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠁⢷⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣆⠀⠀⠀⠨⣿⣶⠶⣶⠀⠀⠀⠀⠀⠀⣽⡃⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣀⣤⣄⣿⠇⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠙⢻⢦⡀⠀⠀⠿⠶⠶⠶⠶⠦⠄⠀⠀⠀⠀⣠⣾⠋⠹⣷⣄⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠑⠿⠷⣦⣄⣀⣀⠀⠀⠀⢀⣀⣠⣴⠾⠀⠀⠀⠀⠈⠪⣷⣦⣤⣀⡀⠀⠀⠀⠀⢀⣀⣴⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⣼⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⠀⢰⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣤⡤⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⣠⣿⡛⢴⠆⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⡀⣿⡟⢿⠀⠀⠀⣠⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⠘⡀⠐⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣹⠿⠛⣁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠄⢰⣶⣿⡨⠈⢀⢀⣘⣉⣘⣉⣩⣄⢁⣀⠤⢀⣀⣀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠉⠀⠀⠒⠛⠙⠙⠁⠀⠙⠛⠉⠉⠉⠉⠀⠈⠉⢉⠁⠀⠀⢀⡀⠤⠀⠂⡐⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⢠⡦⢶⡀⠠⢐⡃⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠓⣅⠐⢀⠀⠇⠂⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⢳⠤⡀⠙⠄⠘
⠀⠀⠀⠰⠀⠀⠀⠀⠈⠳⣶⡄⣂⡀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠈⠲⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⢷⠀⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠅⠊⣀⣤⣰⣤⣀⠀⠀⠀⠰⠀⠀⠀⠘⢢⣾⣿⣷
⠀⠀⠀⠀⠀⢀⣤⣴⡄⠊⢠⠀⠘⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣿⡸⠋⠀⢻⡷⠀⠀⡀⠀⠀⠀⡘⠻⠿⡟
⠀⠀⠀⠀⠀⠈⡛⠻⡇⢰⠅⢀⠄⢰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣿⠁⠀⠀⠈⠀⠀⣀⣁⡀⣀⣤⡤⠧⠤⠸⡆
⠀⠚⠛⠻⢶⣄⠀⠀⢇⠈⠀⠈⠄⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠐⢿⣿⣧⢋⠁⠄⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢉⠳⡀⠈⠀⡄⠀⡦⠐⠀⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠱⠀⡀⠀⠀⠀⠄⠁⠀⠀⠰⠁⠦⡀⡀⠀⢱⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢧⠀⠀⢀⢴⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⠀⡐⠀⠀⣤⣀⠐⠈⢩⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠓⠂⠁⠠⢮⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠤⠂⠀⠀⠠⠸⠛⡋⠐⠉⠀⠀⠸⠛⢂⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠒⠁⠀⠀⠚⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠀⠀⠀⠤⠈⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⣿⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣤⠺⣭⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠀⢀⡀⢀⠄⠠⠀⠀⠀⠈⢿⣾⢷⠀⠀⠀⢠⣴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣴⠀⠈⡷⣬⡅⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⠙⠒⠂⢀⣠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣦⣴⣦⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠐⠁⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠹⠆⠀⢸⣶⠎⠁⠙⣦⠀⡀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡤⠖⠋⠘⠀⠀⠙⠋⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠶⠈⠳⠿⠚⠋⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡤⠴⠖⠛⠁⢀⡤⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠀⢀⣀⣤⡴⠀⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⡀⠀⠀⠀⠀⠀⠠⠀⢀⢀⡀⠖⠲⠟⠗⠀⠀⠀⣀⠟⠁⢡⠀⠀⠀⠀⠀⢀⣿⣿⣷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢯⢹⡄⠳⣁⢠⣄⠰⠂⠁⠀⠣⠀⠀⠈⠒⢤⣀⠀⠀⠰⠛⠁⠀⠀⠈⣆⠀⠀⠀⠀⣼⣟⢉⣥⣀⡍⢿⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣎⢣⣙⢶⣛⠿⠃⠀⠀⠠⠄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠀⣀⣼⣿⣿⣿⣿⣿⣿⣦⢧⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠵⡘⢢⠘⠿⡹⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⡙⢣⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡾⢷⠀⠀⠉⠛⠿⢿⣿⣿⣿⢶⡝⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡆⢀⠀⠈⢷⡀⠀⠀⢀⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⡀⠀⠐⠀⠈⠛⠖⣠⣷⣿⣿⣿⣯⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠀⣠⡘⠋⠉⠉⠉⠻⣿⣧⣩⣉⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠐⠁⠀⠀⠀⠀⠀⠀⢸⡗⠀⠉⠃
⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⣷⣶⣿⠿⠄⠀⠀⠀⠀⠀⠀⠁⠲⠀⠀⠀⠀⠀⢈⡁⠄
⠀⠀⠀⠀⢰⣿⠟⠛⠁⠀⠀⠀⠜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⡞⠒⣦⣤⡌⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⢠⢞⣏⣀⠀⠀⢀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠿⣶⣤⣤⣤⣧⠀⠀⠀⠀⠀⢚⠿⡉⠙⠛⠉⠉⠻⣿⣷
⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⢒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠼⠀⠀⠀⠀⠀⠠⡘⠁⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠠⠔⠉⡀⠀⡋⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠘⢦⣤⠀⠀⠀⠀⠈⣏⢳⠑
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠰⢴⠒⠇⠄⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠓⠂⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠒⣈⠠⠁⠐⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢤⠔⠀⠀⠀⠀⡖⠛⠚⠛⠒⠒⠒⠒⠒⠒⠒⠀⠚⠒⡆⠀⠀⠀⠉⠠⠴⢈⣥⠩⠆
⠀⠀⠀⠀⠀⠈⢳⠊⠉⠀⠀⠀⢣⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠈⠉⠑⠒⢤⠗
⠀⠀⢰⠂⠙⠓⠚⠒⠒⠒⠒⠒⠒⠒⠛⠛⠚⠃⠀⠀⠀⠀⠐⠚⠋⠉⠀⠀⠀⠀⠀⠟⠀⠠⡌⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⠒⠒⠒⠒⠒⠒⠚⠀⠐⠂⢤
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⠀⢸
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠶⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣲⠀⠀⠀⠀⠀⢸
⠀⠀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⡉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢀⣴⡀⢢⣀⣀⡴⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠛⠒⠚⠋⠀⠻⢦⣭⡭⠴⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⣾⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣀⠤⠄⢀⡤⠀⢻⡸⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢃⡼⠃⠀⡴⠊⠁⠀⠀⢰⡏⠉⣿⠃⠀⠀⣰⠋⢀⡤⢦⣼⠀⢠⠀⠓⠦⣭⣭⡥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠖⠋⠀⠀⣼⠁⢠⡇⠀⠀⣾⠀⠀⡟⠀⠀⣰⠇⠀⡎⠀⠀⣿⠀⡘⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢠⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⡇⠈⢉⣿⢠⠃⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⡇⠀⢸⠇⠀⣿⠀⢰⠀⢠⡇⠀⣿⡀⠀⣿⠀⢸⠛⠚⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠾⢥⣀⠀⠈⣷⡀⠈⢀⣼⡇⠀⣾⡇⣼⡇⢀⡝⢷⣄⣀⣀⣸⣁⣠⠀⠀⣼⡯⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣬⢹⣆⠀⠈⠛⠛⠋⠘⠛⠛⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠁⠉⠀⠀⠀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢿⣿⣿⣷⡄⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣟⡟⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡛⣟⣷⣿⣿⣿⣿⣿⣷⠀⣀⣀⠀⠀⠀⢰⣿⣿⡟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠋⠃⠃⠘⢻⣿⣻⣿⣿⣿⣶⣄⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠘⠘⠚⢻⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣉⠉⠉⠉⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡈⠉⠉⠙⠒⠀⠀⠀⠀⠘⢧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡈⠉⠓⠲⠤⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣍⡉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⡟⢹⡷⠶⣄
⠀⠀⠀⠀⢦⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢀⣰⣟⠀⠀⣀⡠⠤⠀⠤⣤⡀⠀⠀⠀⣴⠋⢀⡞⠀⡼⠁⣸⠏⣷⡀
⠀⠀⠀⢠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠈⠉⠉⠀⠀⠀⠀⢠⡟⠛⢦⣀⡜⠁⠀⡜⠀⡸⠁⢀⠏⢀⡏⠙
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣸⠃
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠶⣴⠇⠀⡆⠀⣸⠁⠀⠀⠀⣠⠤⣄⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢸⠀⣸⠀⢠⠃⢀⠆⠀⡐⠁⠀⠈⠻⣧⣀⣤⡤⠶⠶⠒⢦⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠳
⠀⠘⢴⡇⢀⡎⢀⡾⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠙⠛⠷⠾⠷⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢀⡀⠀⠒⠦⣤⣍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡆⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣄⣀⠀⠈⠒⠒⠤⢤⣀⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡶⠦⠤⠤⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣠⢤⠤⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⣠⣦⠾⣠⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣯⣿⣗⠈⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢰⢸⣿⡅⢻⠀⢀⠀⠐⢫⣭⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠈⡁⢀⠀⠀⠀⣧⣿⢛⡉⣤⠇⡀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠈⠀⠀⠀⠀⣾⡿⣧⠠⠀⠀⡀⣀⣃⣉⣌⣁⣩⣴⢈⣀⡀⠄⣀⣀⣂⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠠⠈⠁⠀⠀⠒⠛⠙⠛⠋⠀⠀⠛⠛⠋⠛⠉⠉⡁⠀⠉⠉⠉⠉⠀⠀⠀⢀⠀⠠⠀⠐⢀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⢠⣌⣆⠠⠠⡀⡘⠀⠈⣬⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡐⠀⢄⠀⠨⣇⠄⠌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠉⠀⠈⢳⡀⠀⠙⢆⠀⠻⠀⡈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢦⣀⠉⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢄⡈
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⣹⠛⠃⠢⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡠⠫⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⢇⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⡈⠷⠀⢆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⣤⢰⣶⣷⡧⠄⣀⡀⠀⠠⠃⠀⠀⠀⠀⢱⣮⡻⣿⡧⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡊⠉⠤⠀⠈⠆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢳⢻⣠⠏⠀⠀⢽⠟⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠻⡁
⠀⠀⠀⠀⠀⠉⢈⡛⠛⣏⠀⢯⠀⢠⠀⢰⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⠀⠀⢀⣀⠀⠀⢀⣤⡤⠷⠄⠄⠳
⠀⠶⠚⠻⢷⣦⡀⠀⠀⠲⠒⠁⠀⠋⡂⠀⢱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢂⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⠋⠁⠄⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠁⠀⡀⢀⡄⠰⠀⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡄⠀⠀⠑⢠⠀⠘⠀⠀⠀⢀⢠⠃⠀⠐⠉⡝⢐⡈⠀⠀⡁⠒⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡑⢼⠀⢸⡀⠈⠃⠀⠀⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⠀⠁⢀⠌⠀⠁⠐⠉⢒⠀⠴⢇⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠳⠀⣀⡐⡼⠁⢀⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠶⠀⠃⠘⢷⣦⡁⢰⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⡀⠀⢀⡠⢊⠆⡜⠢⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠢⡀⠈⠁⠋⠁⠀⢀⡌⡡⠋⡁⠄⢀⠀⠀⠀⠀⢋⡀⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠐⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠉⠀⠀⠀⠀⠒⠀⠂

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

⠀⠀⠀⠀⢶⣶⡶⠀⠀⠀⣿⡇⠀⠀⠀⣶⣿⣷⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⣿⣿⡏⠀⠀⠀⢉⣿⣿⠅
⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢹⡁⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⢻⠏
⠀⠀⠀⠀⣠⣿⡄⠀⠀⢀⣾⣇⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢰⣿⣷

⠀⠀⠀⠀⢌⣷⣶⣇⠁⠀⠀⢤⣷⡤⠀⠀⠂⣿⣟⣿⡇⠂⠀⠘⢿⡟⠃⠀⠐⡀⠀⡴⠀⠢⣿⣿⣿⠔
⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠘⣟⠁⠀⠀⠀⠙⢿⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡍⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⣠⣿⣄⠀⠀⠀⢀⣼⣄⠀⠀⠀⠀⣼⣧⠀⠀⠀⣴⣷⡄⠀⠀⢠⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⣤⣤⣤⣴⡀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡐
⡀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠢⠀⣶⣫⠭⠾⡧⠀⠠⠋⠁⠒⠠⣀⠀⠔
⠀⠁⠒⡀⡀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠁⠻⠀⠀⠀⠐⡀⢈⡒⠒⠄⡀⠀⡋
⠀⠀⠀⠇⠀⢈⠐⡤⣀⠀⠀⠑⢄⠀⠀⠀⠐⣆⢀⣄⡀⣱⠋⠀⠀⢀⠞⠊
⠀⠀⢀⠂⠈⠀⠀⠟⠾⢉⠖⠤⠄⠈⠢⢀⣄⠭⢟⡾⠞⠉⠀⠀⢠⡃
⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣨⠡⡴⠋⠀⠀⠀⠀⢠⣗⣅
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡿⢋⣤⠂⢀⠀⠀⢀⢷⡥⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣶⡿⢃⣴⠏⣰⢃⠎⣜⢽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠱⢵⣯⣋⣼⣇⠎⡜⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠎⠀⠙⣆⠀⠘⡿⠯⣧⠉⠀⣠⣾⠟⠙⢆
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⢄⠀⠀⠸⠒⠈⠀⢠⣽⣶⢶⢿⣧⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⡀⠙⠉⠀⠀⠀⠈⠘⠉⠁⠸⠢⡳⣦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠁⠀⠀⠑⢍⠣⣠⡀
⠀⠀⠀⠀⢀⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡀⠀⠀⠀⠑⠈⠻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⣿⠿⠿⠿⠿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⣦⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣸⣿⣶⣤⣦⣤⣾⣿⠿⠀⠀⠲⢦⡀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠗⠀⠈⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠁⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⢉⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⡉⠛⠛⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠘⠛⠛⠛⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⡟⠛⠻⣧⣴⣶⣶⣶⣶⣦⣿⠉⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠉⢿⣆⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⣸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠒⠒⠒⠶⢾⡉⣿⣿⣟⡻⢿⣿⣿⣿⠟⣻⣿⣳⣿⣿⣿⠿⠦⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠤⠲⠤⠜⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣟⢛⣷⣿⣿⣿⠇⠉⠀⠀⠀⠀⠀⠉⠢⣴⡿⢿⣿⣦
⠀⠀⠀⠀⠀⡇⢠⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠉⠈⣽⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⣽⡿
⠀⠀⠀⠀⠀⠳⣄⣙⣋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢫⡻⢿⣿⣧⣴⣿⣿⠟⣹⡇⠀⠀⣼⣿⣷⠀⠀⠀⠘⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠻⠇⠀⠀⢀⣹⣷⣾⣭⣭⣭⣶⡿⠋⠀⠀⠀⢷⣶⡿⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣀⠀⠀⠀⠀⢀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⢀⡀⠀⠀⠀⠀⢀⣠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠈⠉⠓⠒⠶⣶⣿⣟⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡧⠀⠀⠀⠀⠀⠀⠉⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⣹⣿⣿⣿⣿⣿⣿⣿⣌⣵⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠛⢻⣿⣿⣿⣿⣿⣶⣶⣿⡍⠉⠁⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡇
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⢻⣿⣿⢱⣶⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣶⣾⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠁⠀⠀⢸⣿⣿⡇⠀⠀⢈⣹⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⣸⠁⠘⢦⡀⠀⠀⠀⢣⡀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠉⠀⢀⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠸⣄⠀⠀⠙⠲⠦⠤⠴⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠦⠼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⡀⡀

⠀⠀⢀⡤⠀⠀⠒⠀⠄⢠⠀⠀⠤⢠⠄⠀⠀⢄⠀⣠⠀⠀⠤⠀⢀⡔⠒⠒⠲⠀⠀⡤⠒⠒⡆⠀⠀⢀⠤⠒⠀⠀⠀⠂
⠀⠀⠅⡆⠀⢀⣀⣀⡼⢘⠀⠀⢸⠈⡀⠀⠀⠈⣞⢸⠀⠀⢸⢠⢔⡇⠀⡀⠀⡆⢰⢺⠀⠀⡇⠀⠔⡏⠀⣠⡖⡒⡲⠊
⠀⠀⡃⡇⠀⠸⠀⠁⢤⠱⠀⠀⢸⠀⡆⠀⠀⠀⢱⢨⠀⠀⢸⠸⢨⠀⠀⠇⠀⡇⢸⢺⠀⠀⠇⠀⠄⡇⠀⠘⠢⠠⠤⣀
⠀⠀⠄⡇⠀⢀⡄⣀⣼⠈⠀⠀⢸⠀⡇⠀⠀⠀⠀⣧⠀⠀⢸⡎⢸⠀⠀⠼⠀⢸⢸⢸⠀⠀⡀⠀⠣⢌⠢⠄⠤⣄⠀⠈⠆
⠀⠀⠂⡇⠀⢸⠉⠁⢸⢸⡄⠀⠼⠢⠧⠄⠸⡆⠀⠈⠀⠀⢸⡇⡏⠀⢠⠄⠀⠘⣴⢸⠀⠀⠧⠠⡄⣠⠈⡍⠱⠔⠀⠀⡄
⠀⠀⡁⢃⠀⢸⢀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣸⢇⣇⠀⣸⢸⡀⢀⣻⢸⣀⢀⣀⣀⣞⣅⡀⢀⢀⣀⣠⠞
⠀⠀⠶⠋⡬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠼⢷⠒⠁⠃⠈⠋⠀⠎⠀⠚⠙⠃⠊⠀⠐⠁⠘⠀⠁⠋⠃⠉
⠀⠀⢤⠀⠀⠀⣀⣀⠀⣀⠀⠀⡤⢠⣶⠟⠩⠅⠀⠀⢨⠁
⠀⠀⠁⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠌⠀⠉⠀⠠
⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⠀⠉⢉⡀⠀⠂⠙
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠶⠒⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢁⠀⠀⠀⠀⠤⠤⠄⠤⠒⠃⠒⠤⣤⡄⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢀⡤⠴⠒⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⡏⡄
⠀⠀⠀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣴⣀⠀⣈⡧⠀⠀⠠⠤⠬⠥⠶⠄⡔⣡⠵
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠁⠀⠀⠀⠈⠉
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠁⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠸⡟⠢⠀⠀⠀⠠⠀⡀⠀⡌⠡
⠀⠀⠀⠂⠀⡘⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⢈⠈⠀⠀⢨⠽⢇⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠬⡽⠶⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠒⠛⠀⠄⠄⣀⠀⠀⠁⠈⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⠕⣢⠄⡀⠀⠀⢸⣧⠤⠚⠁⠀⠀⠀⠐⠂⠀⢀⡉⠉⢀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣿⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⢿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡾⣶⡶⠶⠶⠦⠤⢰⡆
⠀⠀⠀⠀⠀⡸⠏⠉⠉⠁⠀⠀⢻⠿⠇⠀⠀⠈⠉⠉⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⠞⠉⠉⢻⠿⠗⠀⠀⠀⠀⢠⠂⠐⡀
⠀⠀⠀⠀⠰⠁⠈⠆⠀⠀⠀⠀⢸⣛⡇⠀⠀⠀⠀⠐⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡀⠀⠀⠀⠀⢸⣛⡇⠀⠀⠠
⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣭⠇⠀⠀⠂⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠸⣭⠇⠀⠀⠀⠘⠿⠿⠿⠿⠃
⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⢠⣿⡀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⣬⣭⣤⣤⣤⠁⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣭⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⡼⣭⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⡙⣞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⡉⣎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣭⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣭⣿⣦
⠀⠀⠀⠀⠀⠀⠔⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⠤⠴⠶⠶⠶⣶⣷⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠄⠀⠀⠀⠀⠸⠿⡟⠉⠉⠓⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣛⡇⠀⠀⠀⠀⢀⠜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠃⠀⠀⠀⠘⣭⡇⠀⠀⠀⠀⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠈⢤⣤⣬⣤⣤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⢷⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣳⢋⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣿⣭⣿⣧⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠿⠿⠿⢇⣀⣀⣀⡿⠿⠿⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⢊⣠⠄⠀⠀⠀⠀⣤⠀⢠⡖⠁
⣀⣀⣀⣠⠿⠿⠿⢿⣇⣀⣀⡼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠘⠊⠀⠠⠖⠁⠸⠀⠜
⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⡀⢰⣠⡄⠀⠀⠀⢀⡶⠀⠀⠀⢀
⠿⠿⠿⠿⣤⣤⣴⣼⠟⠻⡿⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠋⢀⡿⠋⠀⠀⠀⣠⡾⠓⠒⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣷⡆⠓⠉⢸⣾⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠃⣸⢁⣀⡄⠐⠾⣥⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⣀⣨⣍⠻⣧⡀
⣤⣤⣤⣤⠛⠛⠻⢯⣭⡀⢠⡞⠋⠀⠀⢠⠸⠦⠄⠀⠀⠀⠀⢰⠟⠛⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣼⣧⠾⠻⠛⢉⠛⢦⣿⡇
⣿⣿⣿⣿⠀⠀⢙⢺⡗⣶⣿⠃⠀⠀⢶⣘⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣀⣸⣿⣷⠘⠂⠀⠀⠋⣼⣿⣿
⠻⠛⢛⣻⣤⣤⣬⣟⡟⠻⢛⠀⢀⢳⢤⣉⠁⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠉⠙⠛⠋⠁⠀⠙⠀⠈⠉⠉⠁
⠀⠀⠀⢽⣿⣿⣿⣯⠄⣝⡡⢠⡙⠾⣖⠛⠆⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⠄⡀⠀⢠⣾⣷⡀⡠⠄⢀⣤⣦⣀⣀⣴⣦⣬⡀
⣤⣤⣮⣽⣿⡿⢻⢷⣶⣶⡴⣎⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠁⠨⠉⡉⠁⠀⢠⣿⣿⣿⣿⣿⡿⠛⢋⣁⡀
⣿⣿⣿⣿⠃⠒⣛⣾⣿⣿⠶⣏⡍⠁⠀⠀⡀⣀⣤⣶⣶⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠉⠉⢙⣿⣷⣾⣿⣿⣡⡽⢶⠫⢼⣿⣿⣿⣿⡿⣿⡿⢿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⣿⣿⣿⣶⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠛⠁⣿⣿⣿⣿⣡⡟⠤⠒⢀⣿⣿⣿⣿⠷⠋⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣦⡀⠀⢈⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⡀⡀⣀⠺⣿⡛⢒⢃⣾⠋⢹⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣾⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢨⣿⡿⢿⣿⠿⣿
⣿⣿⣟⣹⣇⡀⠀⢙⣯⣭⣉⠘⠃⠀⠀⢿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣦⡙⡋⠊⢣⠼⢾⠃⠈⠻⣿⣿⣿⣿⣿⣿⡼⢫⠄⡈⠛⠀⣬
⠉⠉⠉⢝⣿⣿⣶⡇⢬⡴⠀⠀⠶⠀⠀⠀⣇⠀⢺⣗⡔⠈⠃⣈⢄⠁⠀⣠⣶⡿⠏⣰⠰⡐⣽⣷⠆⠄⠰⠀⠕⠉⠛⠛⠟⠋⠁⠈⠛⢁⣤⣄
⠀⠀⠀⠈⢿⠿⠿⠂⢀⣀⠀⠠⡁⠄⠂⠀⡧⠀⢀⡀⠤⠤⠠⠤⠤⠣⠌⠛⠫⠐⠂⠓⢢⣿⡿⠛⠒⠒⠒⠒⠀⠀⠀⠤⠠⠀⠀⠀⠀⠘⠛⡩⠤
⣤⣤⣴⣶⣾⣶⣶⣤⣶⠛⠋⠀⢀⡀⠤⠂⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠛⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃⠂⠀⠀⠢⢤⠰⠦⠦⠴⠤⠄⠤⡤⡤⡴⡴⠀⠀⠀⠀⠀⠀⠈⠴⠤⠄⠀⠀⠀⠀⠈⣿⡟⠂⠋⡺⠓⢻⠛⡟
⠀⠈⢿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠠⠡⠀⠠⢀⣂⣮⡤⠧⠴⠠⠶⠯⠎⠁⠀⠀⠀⠀⠀⢀⡄⢠⠀⠀⠀⠀⠀⠀⠀⣷⣿⠁⠀⠀⠉⠑⠙⠈⠉
⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⡴⠋⠀⠀⢢⠀⠀⠀⠀⠀⠀⠸⠿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠋⠀⠀⠀⠐⠀⠁⠀⠀⠻⠿⠃⠈⠉⠀⡀⠀⣀⣀⣀
⠀⠀⠀⣠⣄⠐⢾⣿⣿⡽⢿⣿⣿⣿⣿⣧⠀⣀⢀⣀⣤⣷⡔⠲⣶⣿⣿⣿⠃⢠⣶⡄⢠⣤⣤⣤⣤⣤⣦⣶⣶⣶⣶⣶⣄⢸⣷⡀⢸⣿⣿⣷⣶
⠀⠀⠀⠙⢿⣣⣄⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣀⣼⣿⣿⡏⣰⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⡄⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢧⣡⠟⢍⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⣉⣉⣉⣉⡉⠛⠂⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣤⠶⣻⠿⠋⠉⡏⠙⠻⣿⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡼⠃⠀⠀⠀⡇⠀⠀⠈⢷⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣤⣄⣘⣁⠀⠀⠀⠀⡇⠀⠀⠀⡀⢷⠀⣤⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡉⠉⠉⠛⠛⠛⡟⠛⠛⠋⠉⠘⣇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢦⣀⠀⣸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⢀⣴⠞⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠙⣷⣿⣤⡤⠶⠶⠶⣤⣠⡴⠶⠶⠦⣤⣿⣶⡋⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⣴⡄⢀⡾⠋⠀⠀⠀⠀⢀⣴⠞⠁⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢠⣦⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣴⠋⠀⠀⠀⠀⠀⠀⢾⡁⢀⣴⠟⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣦⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠛⢶⣤⡤⣄⠀⠀⠀⠀⠀⡀⠀⠙⢦⡀⠈⠻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⡞⠁⠀⣸⠷⢶⡄⠀⠀⠙⢦⡀⠀⢹⣦⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⠞⠁⠀⣰⠟⠛⣦⡀⠀⢹⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠾⣇⣀⣴⠏⠙⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠖⠒⠚⠛⠋⠛⠛⠒⠒⠒⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠉⣀⡠⠤⠶⠖⠛⠛⠉⠉⠉⠉⠉⠉⠐⠒⠲⠤⢤⣈⡑⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⣀⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄
⠀⠀⠀⠀⠀⣴⠋⣠⠞⠁⠀⠀⠀⠀⢀⣀⡤⠤⠒⠖⠂⠒⠒⠒⠂⠢⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⡀
⠀⠀⠀⢀⡞⢁⡴⠃⠀⠀⢀⡴⠶⠛⣉⡤⠶⠒⠚⠛⠛⠙⠛⠛⠛⠓⠒⠢⢤⣀⡁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⡀
⠀⠀⢀⡞⢀⠞⠀⠀⣠⠞⢉⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘
⠀⠀⡜⢀⡞⠀⢠⡞⢁⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑
⠀⣼⠁⡞⠀⣰⠋⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣧⣀⢀
⢠⡇⢸⠃⢠⠇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠋⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠂⠄
⣸⢃⡟⠀⡎⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋
⣿⢸⡇⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⣿⢨⡃⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢿⢸⡇⠀⢿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⢸⡌⡇⠀⠈⢷⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠸⡇⣧⠀⠀⠈⢳⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢻⢹⡄⠀⠀⠀⠙⢦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠈⠆⢇⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠁
⠀⠀⠘⣮⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⣙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢄⡀⢫⠉⠀⠈⠙⠒⠦⡄⢀⣀⡀⠀⠀⢀⣀⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣧⠀⠀⠀⠀⠀⠀⠀⡹⢁⡴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣤⣄⣀⣀⣀⣀⣀⣀⠤⠤⠐⠒⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠢⣀⣰⣀⣀⠄⠀⢠⡆⠀⠀⠀⣀⣰⣀⡀⠀⠀⢀⠂⣀⠀⠀⢠⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆⠀⠀⠀⠀⣄⠀⠀⢀⣀⣠⢀⡀⠀⢸
⠀⠀⠈⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠈⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠉⣿⡏⠀⠀⢸
⠀⠀⠀⣿⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢸
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠈⠁⠀⠈⠀⡄⠀⠀⢀⣀⠐⠀⠂⠀⢀⡀
⠀⠀⠀⡄⡆⠀⡄⢢⠂⠀⠀⠀⠈⠤⡐⠐⡄⠆⡀⡀⢳⠆⠀⠀⣯⠀⠄⠄⡀⡁⢘⠄⠀⠈⠀⠀⠇⠀⠀⡀⢧⠄
⠀⠀⠀⢧⣧⡁⢈⣀⡹⠉⠀⠀⠈⠰⢓⣓⠈⡀⠉⡽⠞⠀⠀⠀⠘⢤⣄⣀⣈⣠⠎⠀⠀⠈⠻⣦⣭⣍⣡⠴⠟
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠋⠙⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘
⠀⠀⠠⢀⠁⠂⠀⠁⠠⠀⡆⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢀⡀⠰⠂⠂⠄⠑⢠⡆⢀⠀⢀⢤⠄⠅⡄⢠⢁
⠀⠀⠀⢸⡄⢡⡁⠌⠤⡴⠽⠀⠀⠈⠇⢸⠸⠀⣃⣈⣃⠀⠀⠀⢧⢁⣄⡤⡠⣀⣑⠠⠀⠀⢾⡴⠄⠄⣆⠸⠻⡐⠂
⠀⠀⠠⠌⢻⠴⢄⡴⠶⠃⠉⠀⠀⠈⠻⣎⣬⣉⣤⣸⠁⠂⠀⠀⠀⠋⠲⠰⠛⠈⠀⠂⠀⠀⠀⠈⠒⠴⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠸⢀⡃⠈⠀⠀⠀⡌⠀⠀⠀⢀⢀⠀⠀⡀⠀⠁⡀⠄⠀⡸⡁⡄⠀⠀⡀⠆⠤⠀⠀⠓⢂
⠀⠀⠀⢤⢰⠀⢀⡀⠑⢈⡖⠃⠀⠀⣼⠁⠔⠸⡀⡄⢸⡆⠀⠀⢄⡀⢀⠀⣀⠥⢹⠀⠠⠀⢣⡤⠄⠃⠐⠒⢸⠃
⠀⠀⢁⣠⣗⣩⣁⣠⠞⠋⠀⠀⠀⠀⠙⠮⣉⢉⣡⠤⠈⠀⠀⠀⠀⠹⣄⣤⡶⠶⠛⠁⠀⠀⠀⠙⠓⢚⠓⠛⠁
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⡀⠂⠀⠂⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠀⠀⠄⠂⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠂⠀⡀⡄⡆⠂⠀⡀⠀⠀⠀⠀⠀⠐⢀⠀⢂⠈⠒⠀⠀⡈⢄⠀⡀⠀⡔⠄⠀⠀⠀⠄⢰⢀⠂⠐⠀⢰
⠀⠀⠀⠠⡄⠁⠃⠆⠆⢰⠀⠀⠠⣄⠀⠤⠤⠜⠄⠌⢠⠃⠀⠈⣥⠨⠰⠡⠤⢤⡏⠀⠀⠢⣐⠀⠂⠂⠠⠐⣊⡜
⠀⠀⠀⠀⠘⠚⠳⠦⠞⠁⠀⠀⠀⠈⠙⠶⠷⣤⣤⠞⠉⠀⠀⠐⠊⠙⠒⣦⠔⠻⠀⠀⠀⠀⠈⠳⠤⠶⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⡄⠀⢀⣤⣤⣴⣖⣢⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢠⠆⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢗⣰⣾⣿⣿⣿⣿⡿⠛⢛⣻⣿⣿⣿⣿⣿⣿⣤⠉⠀⠒⠠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠋⣵⢿⣿⣿⣿⡿⠎⠙⠛⠛⠛⠛⠉⠀⠘⠉⠉⣀⣀⠀⠀⠀⠀⠐⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠀⠻⡟⠻⠿⣇⠀⢿⣿⣿⠿⠇⠀⠀⠐⠒⠂⠀⠶⠀⠀⠀⠀⣾⣿⣿⣷⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⣾⣛⣻⢿⣦⠻⣦⣄⠙⢃⣀⡀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢻⣿⣿⡿⠋⠀⣹⣿⠁⠠⣾⠟⣿⣿⣿⣿⣿⣹⣿⣿⣶⠀⠀⠀⡠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⣿⠁⠀⠀⠈⣉⣵⣾⠃⡖⠀⣿⣿⣿⣿⣿⢾⣿⣿⡇⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢸⡇⢰⣶⢠⣾⣿⣯⠋⠂⠇⢀⣿⣿⣿⣿⣟⢻⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡏⠈⢻⣿⣷⣶⡾⠿⠟⠻⠿⣶⠷⢶⣬⣽⣿⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠏⢀⠀⢳⣿⢋⠤⠒⠀⠈⠐⠈⠀⠀⢀⡙⠻⢿⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⣇⠀⣏⣠⡾⠋⢠⠀⠉⠛⠂⠂⠀⠀⠀⠈⠴⠂⣦⣌⡋⠁⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣿⢀⡾⠛⡁⠀⠌⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠉⣠⠃⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⣷⣶⠻⠃⡈⠀⠀⠀⢄⢀⠀⠀⠀⠀⣄⠀⠀⠀⠐⠶⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⠀⠀⣿⣶⣀⠀⠀⠈⠀⢤⣀⡀⠀⠀⠀⠁⠀⣀⡀⠀⠒⠘⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠰⣿⡆⠀⢿⣿⠋⠀⠄⠀⠈⠀⠙⠻⠶⢶⠶⠂⠻⣿⡟⣆⠀⠀⠈⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠆⠀⢸⣿⣇⢷⡀⣄⣐⣀⠀⠀⠀⠀⠂⢴⣶⣿⣧⠉⠂⢠⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣄⣠⣾⣿⣥⣤⣼⣿⣟⠉⠱⣦⣤⣠⣴⣾⡍⣹⣿⣤⠀⠁⣼⠸⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠟⣟⣟⢻⣿⣿⣿⣿⣽⣟⣃⣘⣋⠁⡼⠿⣿⣿⣿⡿⢻⠀⢂⡸⠶⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡝⠻⠿⣿⠶⠿⠛⢽⡟⢿⣿⡟⠋⠍⠉⣻⣿⢾⣿⣿⡿⣾⡄⢉⠒⢐⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣤⣤⣤⣤⠀⣸⠉⠨⣍⢉⠁⠀⠈⠛⠛⠚⠻⠿⠇⢻⡇⢠⡀⠀⣂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠸⠷⠞⠃⣰⠿⠦⠓⠚⠀⠀⠲⠶⢦⡀⠀⠀⠀⠀⠀⠱⣶⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣤⣤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣥⡀⢀⣀⣀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⣾⣿⣶⡤⠀⠀⠀⠠⢽⣦⣻⣿⢿⣿⣿⡛⠁⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⠀⠙⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡻⢀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣹⠀⢀⣀⣀⣀⣀⢀⢠⡄⠸⣥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⢫⠛⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⡄⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⠀⢸⣿⣿⣿⣧⠀⢀⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⠛⠀⠀⣿⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣯⣿⣵⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣛⠀⠀⠀⣿⣻⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠷⠀⠀⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣾⣂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢨⣝⣻⣧⣧⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⠃⠁⠀⠀⢇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠻⣿⡿⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣴⣶⢶⡄⠀⠒⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣉⠀⠀⠀⢈⣩⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⡀⠀⢹⣿⣿⠁⠀⣠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⡇⠀⠀⣿⣿⣯⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣭⣥⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣄⠀⢀⣿⣾⣷⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⡱⠀⠀⠀⠀⠀⢸⡉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡖⠒⠒⡆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⡤⠖⠒⠓⠒⢤⡀⣀⠴⢒⣛⠒⠲⢄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢠⠏⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⠀⠀⠙⢦⢳⡀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡇⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠐⠒⡶⠒⣶⡀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢰⠃⣸⠛⠛⠋⠉⠙⠓⠲⠌⢳⣄⣀⣀⣀⣀⣀⣀⡠⠚⠁⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡤⠴⠖⠋⠉⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡀⢀⡀⠄⠈⠁⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠸⡄⠹⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠰⠶⠿⠶⠿⠃⠈⠉⠙⠛⠒⠒⠒⠒⠶⠶⠒⠒⠒⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠨⠀⢸⠈⡆⠆⠙⢀⣚⡀⡧⡇⠄⠀⡇⠀⠀⢧⡀⠉⢰⢬⠀⠃⢑⢰⣄⢰⠅
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣷⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡇⠀⢸⡗⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⡇⠀⢸⡇⠀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⡇⠀⢸⡇⠀⣿⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⢻⠀⠀⡇⠀⠸⠇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠲⣄⣀⠀⠀⢻⠀⠀⣿⢀⡶⠋⠙⠲⠞⠋⠛⢦⠀⣿⠀⠀⠀⠀⡀⡶⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⡇⠀⡇⢸⣇⠀⠸⡇⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⣾⠉⡇⠀⣯⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⢸⡏⢹⡄⢷⠀⠀⠀⠙⢦⣀⠀⠀⣠⠞⠁⠀⡏⠀⠀⠀⣿⠀⡇⠀⣿⠀⡏⠙⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⡇⠀⡇⢸⡇⢸⡇⠈⢧⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⢰⠇⠀⣀⠀⣿⠀⡇⠀⣿⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢺⠉⢷⡇⠈⠁⠀⠁⠘⠃⠀⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡍⢳⣿⠀⠁⠀⠀⠀⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⣴⠚⠳⠖⠛⢦⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠿⢰⠚⠛⠞⠛⢳⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⣄⠀⠀⣠⠟⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠘⢦⡀⠀⣠⠞⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠞⠁⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢳⡀⠀⠀⠙⠚⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢰⠖⠒⠒⠒⠒⢲
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠲⢿⡉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⣧⠔⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⠶⠶⢦⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡖⠒⠛⢢⡟⠀⠈⠉⢉⡇⣀⠔⠋⠉⠉⠉⠉⠉⠀⠀⣠⢞⡋⠙⠛⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠐⠖⢰⢿⡆⠀⠀⠀⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⠷⢤⣀⣠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢠⣀⠚⢸⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⢷⠉⠁⡶⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢯⠷⣤⡾⡿⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠀⠀⠀⠰⠿⢞⣮⠁⣼⡙⠚⠛
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠁⠀⠈⠿⡧⣿⠃⠀⠈⠳⡄⠀⠀⣼⠉⠉⠉⠈⠉⡇⠀⠀⠀⡴⢿⣯⢀⣿⡽⢷⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣟⣿⠀⠀⠀⡀⠹⡄⠀⣇⣀⣀⣀⣀⣀⡇⠀⠀⣸⢁⠈⣿⠟⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡾⠀⠀⣶⠀⠀⠀⢀⡿⣺⡀⠀⢰⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣼⠁⠉⠀⠀⠀⣼⡆⢻
⠀⠀⠀⠀⢰⡇⠀⢰⢻⡆⠀⠀⢸⣏⣭⡇⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⣦⠘⠶⠶⢺⡇⡇⢸
⠀⠀⠀⠀⢸⡀⢤⡏⢸⡇⠀⠀⢸⡥⢶⡇⠀⠀⢿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⢸⠀⠀⠀⠸⣷⢃⡟
⠀⠠⠤⠤⠼⣧⠀⠹⣾⣧⡤⠤⠼⠿⠿⠷⠶⠶⣾⠆⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠜⣇⢱⡾⢤⡤⣤⡴⣣⠾⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠈⢷⣄⣸⣦⢶⣦⣀⠀⠀⣀⣀⠞⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⢷⣾⣍⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣴⡿⠿⠀⢸⣯⡙⣳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠀⣀⠀⠀⢡⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⠃⠀⠿⠂⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣁⣄⠀⠀⢀⡶⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢠⠖⠐⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢐⠚⢻⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣷⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⣻⣿⣿⣟⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⢀⣴⣴⣿⣿
⠀⠀⣾⠀⡀⢠⡀⠠⢹⡄⣠⡤⢽⣿⣿⣿⡿⣷⠆⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⣸⠋⠀⠀⠀⢸⣿⣿⣿⣿⠿
⠀⢀⣿⢀⣿⣶⣿⣶⣦⣻⣮⣻⣿⢫⡉⠙⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢛⡉⣙⠓⢂⡐⠀⢀⠀⠀⠀⠉⠉⠀⠁
⢠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣄⠐⢿⣿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠠⠄⠀⡰⣉⠎⢀⣾⡧⠀⠾
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⡶⠿⢶⣤⡴⡺⠛⢁⣶⣿⡇
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣡⣴⣿⢛⣿⠿⠛⠛⠋⠀⠀⠀⠈⢻⡏⣠⣶⣟⠵⠋
⡀⠀⠉⠉⠁⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣤⡄⠀⠀⠀⠀⠀⠀⠈⡿⠛⠋
⣿⣷⡀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣟⣿⣿⣷⣤⣄⣀⣀⠀⠀⢀⣠⠷⠚⠛
⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠀⠁⠀⠀⠀⠠⠀⠈⠙⠛⠉
⣿⣿⣿⣿⡀
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⢴⠻⠻⠃⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠀⠀⠀⠈⠻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⠀⠙⠆⠀⠐⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢳⡀⠀⠀⠀⣀⣤⡴⠆⠒⠒⠻⡆⠈⡆⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠔⠛⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠁⣠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠱⣄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠒⠂⠀⢀⡠⠄⠀⠉⠀⠀⠐⢮⡀⠀⠀⠀⠀⠀⠙⣤⡀⠀⠀⣀⠀⠀⠒⠁⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⡐⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠱⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣀⣤⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣆⠀⢷⣠⠤⠤⠤⠞⠛⠉⠀⠀⠀⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢦⠀⠀⢀⠀⠘⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣣⡀⠀⠂⠀⢈⡷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⢠⣠⠦⠗⠋⠁⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⠿⠿⠿⢿⡛⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣻⣛⣿⣟⣻⣛⣿⣛⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠰⠂⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣶⢰⠀⠀⠀⢰⠀⠀⢠⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⠐⣀⡸⠁⠀⠀⡘⠃⠀⠀⠀⣀⠀⠀⠈⠷⠀⠀⢠⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡉⠀⠆⠀⠀⢸⣀⢁⠁⠀⠀⠀⠁⠄⠀⠀⠀⡽⡇⠀⠀⢠⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⢆⡇⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢎⡵⠋⠀⠀⠀⠀⢀⠞⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠃⢈⠀⠀⠀⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⡸⠔⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠄⠀⠀⠀⢄⠻⠀⠀⠀⠀⠴⠀⠀⠀⢆⡀⠀⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⡇⠀⣷⡀
⠀⠀⠰⠘⡀⠀⠀⡸⠅⠘⠀⠀⠀⠈⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⣴⡧⠧⡖⠴⠆⠀⠀⢰⡇⠀⠀⠀⢤⠁⣸⣅⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠤⣀⡀⣠⠀⢀⡄⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡬⣿⠀⠀⠀⢸⡉⠋
⠀⠀⠀⠀⡱⢿⣿⠷⣴⠮⡇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡝⣿⡗⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⡉⠲⣈⣱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡇⠀⠀⠀⡇⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠁⠄⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠔⢩⠘⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢱⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢧⠀⠀⠀⡐⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣇⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⡁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⠌⠀⢤⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⣉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠰⠗⣼⣧⡀⠊⠀⢀⡒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⡋⠛⠃⠀⡤⠈⠁⡗⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠓⢀⠀⠀⠀⠷⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠤⠤⠈⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⡄⠠⢦⠀⠀⠈⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⣴⣄⢆⠀⠀⠀⠢⠀⠰⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢣⠐⡇⠀⠀⣀⡀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⢢⢰⠀⠖⠀⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⡨⠐⠂⠀⠀⠩⢣⣄⠈⣷⡘⢧⠀⠀⠀⠀⠀⠘⢧⡤⠌⠰⢷⣄⠠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡌⠀⠀⠀⠄⠀⠀⠈⠉⠷⣬⣷⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡂⢀⠄⠀⢿⣳⣴⠀⠀⠈⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠠⢊⠤⠀⠻⢿⠞⠄⠀⠀⠒⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⠀⣴⡿⢻⣧⠀⣰⡾⠻⣷⠀⢸⡿⠻⣶⡀⠀⠀⣿⡇⠀⠀⣿⡆⢸⣿⠀⢠⣾⠟⠃⢸⣿⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣤⡀⣿⡇⠀⣿⡇⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⢸⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⠀⣿⣇⢀⣿⠃⣿⣇⢀⣿⠇⢸⣿⢀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇⢸⣿⠀⢿⣇⠀⠀⢸⣿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠀⠈⠻⠟⠋⠀⠈⠛⠟⠋⠀⠘⠿⠛⠋⠀⠀⠀⠛⠛⠛⠃⠘⠛⠟⠋⠀⠈⠛⠻⠃⠘⠛⠀⠻⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⣶⡄⢰⣶⢠⣶⠀⣶⣶⣶⣦⢰⣶⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⡇⢸⡏⢸⣿⠀⠀⣿⡇⠀⢸⣿⣀⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠋⣿⣿⠇⢸⣿⠀⠀⣿⡇⠀⢸⣿⠛⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⢿⡿⠀⠸⣿⠀⠀⢿⡇⠀⢸⣿⠀⢼⡧


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⠀⢀⡀⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⢘⣿⠀⢸⣿⡆⢸⡇⠀⢸⣿⣿⠀⢸⣿⠀⠀⢰⣿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠦⠈⣿⠀⢸⡟⣿⣼⡇⠀⣿⡃⣿⡄⢸⣿⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⠀⢸⡇⠘⣿⡇⢰⣿⠛⢻⣇⢸⣿⣀⣀⢀⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠈⠁⠀⠉⠁⠈⠁⠀⠈⠉⠈⠉⠉⠉⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣯⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⢿⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠸⠃⣴⠋⢀⡠⠔⠒⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⡄
⠀⠀⠀⠀⠀⠁⢀⡞⠀⢀⣴⡿⠋⠙⠳⠀⢠⡶⠳⣦⠀⠀⣰⡾⠛⣷⡄⠀⠀⣶⠒⢦⣄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⣿⡇⠀⠀⠀⠀⣼⡇⠀⢻⡇⢠⣿⡇⠀⢸⣿⠀⢸⣿⡀⠀⢻⣧⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣿⡇⢀⣀⣀⠀⣿⡇⠀⢸⣿⢸⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⠈⣿⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠘⣿⣿⡇⠀⢸⣿⢠⣿⡇⠀⢸⣿⢸⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣸⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⣿⣧⢀⣼⡟⠀⣿⣧⢀⣾⡏⠈⢿⣷⣀⣿⡟⠀⠸⣿⣧⣾⠿⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠛⠿⠋⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣤⠀⠀⣤⠀⠀⣠⡴⠒⠂⠀⠀⣾⡄⠀⣠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠀⠀⣿⡄⢰⣿⡇⠀⠀⠀⢀⣿⣧⡾⠋⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢹⣿⠀⠀⠀⠸⣿⡇⠀⣿⠇⠸⣿⡇⠀⠀⠀⠘⣿⣿⠷⣦⡄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠛⠒⠂⠀⠀⠙⠿⠞⠋⠀⠀⠙⠿⠶⠖⠀⠀⠻⠟⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠳⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡄⠀⣴⠶⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⡿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢰⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠴⠒⢻⡟⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⣠⠇⢧⡀⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠏⢇⠀⢸⡇⠀⡼⠹⡀⠀⠀⠀⠀⠀⠀⣰⡱⠗⣡⢤⢤⡑⠷⣱⡀⠀⠀⠀⠀⠀⠀⣏⢐⠂⡉⠛⢉⠑⢏⡽
⠀⠀⠀⠀⠀⠈⣔⡎⠀⢸⡇⠀⢱⣦⠁⠀⠀⠀⠀⠀⠀⠹⡱⡦⡉⠘⠈⡡⡶⡹⠁⠀⠀⠀⠀⠀⠀⠀⠁⡈⠈⠉⠁⢱⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠈⣴⣯⣽⣦⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠳⡅⠈⡆⡎⠀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⢰⡋⢀⣠⠟⠃⢎⠥⠟⡆⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⡖⡖⠁⠛⢲⡀⠀⠀⠀⠀⠀⠀⡴⠛⠚⢢⠀⡰⠛⠚⢢
⠀⠀⠀⠀⠀⠸⣵⡿⠥⡄⠀⢐⠉⣱⠃⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣧⢦⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⣸⠀⣟⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠣⡀⢈⠆⢨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣰⣆⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⢯⣀⡼⠷⠶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⡼⢧⠿⢧⠿⢇⠀⠀⠀⠀⠀⠀⠀⢀⣬⣟⣓⣐⣚⣫⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣋⣁⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⡟⠉⢠⣶⣾⣿⣿⣿⣿⣷⣶⣄⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⠀⢸
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⢹

⠀⠀⠀⢀⡠⠠⠤⠄⠠⠤⣄⠀⣠⠀⠠⢤⠀⢀⡤⠤⠤⠤⡄⠀⢀⡄⠤⠤⢤⠀⠀⢀⡤⠶⠶⠶⢦⠀⠀⢀⡤⠠⠶⢤⠀⠀⠀⠀⣀⡤⠄⠀⠒⠀⠒⠂
⠀⠀⠰⠨⡅⠀⠀⠀⠀⠀⢸⡎⢼⠀⠀⠀⣔⠹⡀⠀⠀⠀⢸⣰⢙⡇⠀⠀⢸⠀⣰⢩⡇⠀⠀⠀⠈⡆⠀⡇⣇⠀⠀⢸⠀⠀⣠⢸⠃⠀⠀⣀⣀⣤⣤⡄
⠀⠀⢸⠀⡇⠀⠀⣎⣁⡵⠋⡇⢸⠀⠀⢰⣯⢸⡇⠀⠀⠀⠀⢿⢸⡇⠀⠀⢸⠀⡏⢸⠁⠀⢶⡀⠀⣇⠀⡇⡏⠀⠀⢸⠀⢰⠁⣞⠀⠀⢯⠒⠒⠒⠁
⠀⠀⢸⠀⡇⠀⠀⠁⠀⠀⢰⡇⢸⠀⠀⢸⣧⢸⡇⠀⠀⠀⠀⠸⣴⡇⠀⠀⢸⢠⡃⢾⠀⠀⡞⡇⠀⢻⠀⡇⡷⠀⠀⠘⡄⠸⡂⠱⡀⠀⠀⠉⠋⠐⠒⢤⡀
⠀⠀⢸⠀⡇⠀⠀⣠⠤⠤⣼⡇⢸⠀⠀⢸⡷⢸⡇⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⣼⠀⡟⠀⠀⠧⠇⠀⢸⡀⡇⣿⠀⠀⢨⠇⠀⠑⢄⡙⠢⠤⠤⢤⡀⠀⠀⢳
⠀⠀⢸⠀⡇⠀⠀⡟⠊⠈⢀⡇⣹⢀⠀⢸⠿⢸⡇⣀⣀⣺⡀⠀⠈⠁⠀⠀⢸⣿⢐⡇⠀⠀⣀⠀⠀⠀⣇⡇⣿⠀⠀⢸⣀⣀⡀⢀⣈⣉⣉⣲⡤⠜⠀⠀⣸
⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⡨⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠄⠀⠀⢸⣿⢸⠁⠀⢠⠇⡇⠀⠀⢿⡇⣿⠀⠀⠀⠀⠀⣧⢺⠀⠀⠀⠀⠀⠀⢀⣰⠃
⠀⠀⢸⢀⡷⠤⢤⡧⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⣤⠖⢖⡾⣯⠸⠤⠴⠞⡇⡻⠤⠤⡼⣣⠼⠤⠆⠶⢲⠾⢻⡸⠦⠄⠤⠴⣲⡶⠛⠁
⠀⠀⠈⠛⢒⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣥⡴⠚⠒⠊⣍⠉⠀⠈⠉⠉⠁⠀⠙⠈⠉⠉⠀⠈⠉⠁⠉⠈⠉⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⣔⠁⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣠⠂⣤⠾⠾⠉⠼⠀⠀⠀⠀⢴
⠀⠀⠚⠃⠀⠀⣀⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠀⠀⠤⠭⡶⢀⣨⠁⠐⠓⠀⠄⡀
⠀⠀⠆⡤⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⣀⣀⣀⠀⠐⠒⠒⠛⠈⠀⣀⠀⢤
⠀⠀⠚⣀⠂⠈⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⠀⠲⠂⠀⠒⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⠛⠛⠓⠂⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠤⠐⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⢤⣀⠒⠂⢠⠀⢀⣀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆
⠀⠀⠀⠀⠐⠀⠤⠀⠀⠉⠉⢛⡩⠡⠛⠉⠀⠀⠈⠙⠓⠾⢬⣀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⢀⠂⢀⣠⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⢀⢀⠀⠀⠀⠀⠉⠉⣩⢸⡀
⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⢤⣬⡇⣀⣀⣀⣨⡄⠄⠄⠄⡡⠠⣥⣍⣶⣦⠤⡄⣂⣠⣧
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠄⣀⠘⠂⠀⠒⠀⠀⠀⠓⠢⠄⡀
⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⠉⣲⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠋⠀⠀⠀⠈⢿⡉⠐⠏⠈⣐⠂⠐⠤⠄⣀⠀⡜⢀⠔
⠀⠀⠀⠐⠊⠀⢈⠓⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠠⠀⠁⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠈⠉⠒⠂⠀⣭⣹⣟⣲
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠓⣢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⣀⣴⢞⣁⡀⠀⠀⠀⠀⠀⢰⠉⠐⢄⡀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠔⠛⠋⠉⠋⠭⢉⣙⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⡧⡇⠀⠀⣠⠶⠛⠓⠠⢄⣀⠈⠉⠙⠒⠤⢤⣀⡠⠤⡤⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠩⠗⠂⠄⣀⠀⠀⢽⣧⠖⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢄⣀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣶⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣻⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣏⡴⠇⠀⠀⠀⠀⠻⢿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⡙⠉⠁⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣤⣤⣀⣀⠀⠙⠉⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠄⠀⠀⠀⠉⠀⠀⡀⠀⠀⡀⢀⠒⢤⡈⠓⢠⣶⣿⣿⡾⢓⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠴
⡗⢾⣿⣿⠷⣦⣤⣤⣤⣤⣤⣶⣦⣤⣴⣶⣟⡻⢤⠷⠄⣉⣹⡆⢛⣶⣿⡿⡶⢻⣿⣿⣷⣶⣿⣿⣿⣿⣧⡄⠀⠀⡰⣸⡆⣈⢄⠀⠀⢠⣤⡌
⠀⣤⣈⠙⣄⠈⠋⢁⣿⣧⣦⡀⠀⠻⢦⡀⠉⠀⠀⠀⠔⠉⠙⠋⢹⣧⠄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣤⡁⢧⢿⣾⡟⢻⣇⠀⠀⠀⠁
⠀⠈⠈⢳⠀⡀⠀⠀⠛⣿⡿⠁⠀⠀⠸⢷⡄⠀⠀⠀⠀⠀⠠⢴⠚⠁⠀⠀⢀⣀⣤⣤⡤⠶⡶⠶⠾⠟⠓⠻⠿⠿⠟⠺⠟⣧⡤⢛⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢀⢹⡦⠀⠀⠀⠀⢀⣼⠇⠀⠀⢤⣤⠀⠀⠀⠀⠀⠏⣸⣥⣶⡾⣛⠋⠙⢨⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠇⠀⠁⠐⠁⠈⢄⣀⠀⠰
⠀⠀⠀⠀⠀⠘⣷⢾⡄⠀⢀⣿⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠚⠲⣂⣴⣯⠾⠋⠀⠱⠈⠀⠀⢀⣤⣟⠛⠀⠀⠀⠀⣠⣫⢤⡖⠁⠀⠈⣏⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠰⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡋⠁⢰⡾⢻⡐⠂⣠⣔⣡⣿⣉⣠⡀⠀⠀⢼⣣⢶⣴⡗⢀⠀⠀⠏⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠰⠈⠋⠉⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣧⠘⢷⡶⣿⣾⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡈⢀⣾⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⡘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡙⠀⢄⠀⠈⠧⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⣿⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣦⠘⣷⣦⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⢿⣿⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣸⣿⡇⠀⠀⠀⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠈⠙⠿⣿⣦⣄⠀⠀⠀⠀⡰⠏⠁⠔⣿⡿⢃⣀⣠⣤⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠉⠉⠐⣶⣾⡦⠀⠀⠸⠋⠁⣞⣍⣹⣿⣧⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠈⠠⢠⣟⣿⡏⠹⣿⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⡏⠉⠁⡙⢦⡀⠀⠀⢿⠿⠿⢾⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢻⣿⡄⢀⡄⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⣢⣤⡴⣧⠈⠉⠛⠻⣿⡇⢏⣶⣶⣶⡿⣿⣆⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢸⣿⢿⣯⠎⢠⠤⠈⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠠⣤⡈⢃⣿⣿⣧⣽⠃⠘⢿⣷⠀⢀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡐⣷⣿⡀⠘⢷⣿⣿⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠉⠂⠈⠙⠻⠿⣷⡖⢸⣿⡇⠀⠀⢣⠀⣠⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠅⣺⣿⡿⣯⠀⠀⠀⠙⣏⣀⣀⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠒⠲⠶⣴⣿⣿⣿⣿⠇⠀⠀⢀⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⡇⠀⠀⠀⠀⢛⣙⣛⣿⣿⣿⣿⣿⣤⣭⠛⡟⢿⡿⣿⠛⠿⠛⢿⣿⣿⣿⣾⠷⣾⣿⣿⡟⢩⡿⠀⠀⠀⠘⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⡨⠇⠀⠀⣄⣛⣿⣿⣿⣿⠝⠿⠟⠛⠛⠛⠥⠍⡬⠙⡿⠒⣀⠀⣫⡵⠊⠀⠁⠀⢻⣽⣿⣠⠟⠀⢠⠏⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣅⡄⠀⢸⣿⠟⠛⠋⠉⢤⡄⠀⡔⠉⠠⠤⡶⠂⡛⢦⣴⣴⣿⣿⠃⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⣍⣱⠀⢸⠇⣀⣄⣘⠶⠞⠧⣀⣁⠀⠀⠀⠙⠀⠁⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣝⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⡀⠈⢰⣿⣿⣤⣀⢲⣿⣯⣿⣗⠀⠀⠀⡴⠀⠀⣿⡿⣻⣷⡄⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣯⠆⠀⣤⡸⠿⢿⣷⣲⣭⣿⣿⣿⠟⠷⠌⠉⠠⢀⠀⣵⣿⣿⣿⣁⣈⣾⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣿⡿⠂⣤⡙⢿⣛⣗⣹⣿⣷⠆⠠⠀⠠⢆⠱⡈⠛⠛⠿⠿⠿⠿⠟⠋⢩⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠐⠂⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢁⣾⣿⣟⣿⣿⣄⣸⣿⣾⣝⣿⣿⣿⣿⣞⠠⠖⠀⠈⠳⡌⠑⠒⠤⠤⠤⣤⣀⣠⠴⢢⠀⠻⡏⠁⠀⠁⢀⠀⠀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠹⠿⠿⠟⠛⠋⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠙⠁⠉⠀⠐⠒⠃⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣿⠿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣦⣤⣄⣀⡀⠀⠀⠀⣀⡀⢀⠀⠀⠀⣼⡟⠁⡸⠀⢸⠈⠹⠛⣼⣿⣿⣿⣿⣿⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⠌⠀⡤⠀⢨⣿⣿⣷⣤⣤⠈⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⣿⣿⣿⡄⢰⣆⠀⠀⠀⠃⢠⣾⣿⣿⣿⣿⣿⠄⠀⠙⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⣴⣇⡀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⡻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠰⠾⠛⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠋⠉⠀⠀⠁⠈⢻⣿⣿⣿⡿⠿⠶⠤⠴⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣄⣀⣤⠀⠀⠀⣀⡀⠀⠀⢸⣿⢹⡴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡇⠀⠀⠀⠀⠉⢻⣦⣼⣏⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣽⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣻⣆⠀⢀⠀⠠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣳⠀⠀⠀⠀⠀⣼⣿⡿⢸⣿⢹⣿⠀⣾⠂⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠳⣶⣶⣤⣾⣿⣿⣷⣾⡿⢸⣿⢰⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⣿⡇⠀⠀⠐⠿⠿⢿⡿⠿⠿⠿⠫⠛⠛⠧⠺⠷⠀⠉⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣾⣿⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⠀⠀⢠⡟⢡⡶⠀⠈⠛⠛⠛⠟⠋⠙⠁⠀⠀⣿⣿⡗⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⢾⣿⣷⠀⡿⢠⡟⠀⠀⠀⠀⠀⠀⠀⡦⢠⡆⠀⠀⠈⠁⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⣼⠃⣼⠃
⠀⠀⠀⠀⠀⣰⣾⣿⣷⣄⠀⠀⠀⢀⣴⣶⣶⣿⣿⣾⣿⣦⣄⠀⢁⡼⠃⠀⠀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⠛⠃⣿⣿⣿⣿⣿⣿⣿⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⣿⣿⣿⣿⣿⣿⣿⠘⠛⠃
⠀⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⢰⣿⡇
⠀⢠⣿⡇⢙⣛⣛⣛⣛⣛⣛⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣛⣛⣛⣛⣛⣛⣛⢸⣿⡇
⠀⠘⠿⠃⣿⣿⣿⣿⣿⣿⣿⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣿⣿⣿⣿⣿⣿⣿⠈⠿⠃
⠀⠀⠀⠀⣿⣿⣿⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⣉⣉⡛⠀⠟⣋⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⠛⠸⢟⣉⣉
⠀⠀⠀⠚⠛⠛⠛⠀⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠀⠛⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠊⠀⠁

⠃⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⢀⢀⠀⠠⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀
⣠⣤⣀⣀⣀⣰⣀⣤⣤⣀⣠⣤⣄⣄⣀⣠⣠⣤⣅⣀⣀⣠⣁⣈⣠⣀⣈⣄⣤⣤⣄⣀⣄⣄⣀⣢⣀⣤⣤
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠛⠿⠍⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠿⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣟⢰⣿⣿⣦⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡆⣻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣷⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⡄⠀⠀⢠⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠩⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣭⣭⣭⣭⣭⣭⡻⣿⣿⣿⣿⣿⣷⠦⠀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣸⣿⣿⣿⢣⣶⣾⡟⣭⡍⣩⣭⢻⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣛⣛⣛⣛⣛⣛⣛⣫⣵⣿⣿⣿⣿⣦⡙⣿⣷⡙⣿⣿⢋⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣌⣡⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣌⣛⠛⠿⣫⣭⣝⢿⡟⣩⣭⡙⢿⣿⣿⣧⡹⠿⣛⣭⣝⡻⠿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⣥⡙⢿⣿⣧⡘⣿⣿⢟⣨⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⣃⠹⣿⣿⣮⠻⣿⣿⣎⢡⣾⣿⣿⠃⡻⢿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⠶⠸⣿⣿⣌⢻⣿⣷⣜⠿⠟⣡⣿⠟⣱⣿⣿⡇⣿⣿⡿⣡⣶⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⣿⣿⣌⢻⣿⣷⡙⠿⠿⣰⣿⡿⣫⣾⣿⣿⢋⣴⣶⡎⣧⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡌⢻⣿⣷⡙⠿⠟⣰⣿⣿⣟⣴⣿⣿⠟⣵⣿⣿⡟⢡⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⣷⣝⠿⠟⣸⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⡿⢫⣾⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⡟⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠟⠉⠉⠉⠰⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣋⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠟⠛⢛⣛⣛⣁⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣭⣥⣤⣴⣶⣶⣶⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⡗⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⢸⣿⣿⣿⣿⣿⣿⡇⠘⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣷⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢈⣿⣿⣇⠀⠀⠀⠠⣿⣿⡧⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠻⠿⠃⠀⠀⠀⠀⠿⡿⠇⠀⠿⣿⠟




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠙⣆⠀⡐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠘⠛⠃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣤⣹⣦⣄⠀⠀⠀⠀⠀⠲⢯⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⡿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣫⣾⣿⡿⢢⣶⣤⡀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣷⡿⣾⢠⣿⣿⣿⡇⠀⠀⠀⡀⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣷⣤⣶⣄⣬⣪⣀⣤⣙⠿⣿⣿⣿⣿⡿⣻⣿⠉⢩⣿⣾⣿⡝
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠛⠛⠻⠛⠻⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⢷⣿⠃⣠⣾⣿⣿⣿⣏⡅
⠀⠀⠀⠀⠀⣠⡴⠿⠛⠉⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⣿⣿⣾⣷⣿⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠐⠻⠿⣿⣿⣶⣄⣀⣀⡀⠁⠀⢀⣀⣀⣀⣀⠀⠀⢀⣀⠀⢠⣟⣛⣛⠻⡆⠀⠐⣿⣿⣿⣿⣿⣿⠿⣾⣿⣿⣿⣿⣿⣿⢟⣻⣿⣿⡇
⠀⢰⣄⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣩⣿⣻⡇⢿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣟⡵⠋⠈⢉⣿⠇
⠀⠈⢊⡓⡆⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⡘⢬⣭⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⣾⠏
⠀⠀⢸⢸⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⠿⢿⣿⡿⠶⣤⡁⠀⠙⠿⡧⢀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⠦⠞⠛⠁
⠀⠀⠀⢟⣯⡣⢤⣀⠀⠀⢀⣠⢔⡋⢍⠉⠉⢁⠠⢶⣲⣭⣿⠁⠈⢿⠳⣤⡀⢠⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠿
⠀⠀⠀⠘⠮⡍⢲⣆⣭⣭⣽⣿⣿⣿⣶⣮⣭⣶⣿⣿⡿⡿⡌⠀⠀⢼⡇⢀⠋⠋⢿⣿⣿⡿⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣾⡒⡄
⠀⠀⠀⠀⠀⠈⢮⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡇⠃⠈⡄⠈⣷⠀⠀⠸⡿⢿⠟⠻⣾⣷⣿⣿⣿⣿⣿⣿⣿⠿⢟⠵⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢻⡈⠀⠈⠁⠈⣧⡀⢠⠀⡆⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⠂⠙⢧⣀⠴⠆⠀⠈⡷⠶⠶⣿⣿⣿⣿⣿⣿⣿⠟⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⣿⣿⣿⣿⣿⣿⣿⣿⡏⡆⣿⠇⠀⠓⢢⣄⡀⡼⣷⣶⣿⣿⣿⣿⣿⡿⢟⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⠿⣿⣿⣿⣿⣿⣿⣧⠃⢸⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠬⠍⢩⣙⡻⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⡿⣿⢟⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⢻⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠂⠶⠦⡀⠀⠀⠀⣤⠚⠛⠛⢦
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢸⠁⠁⠀⠰⠂⡇⠀⠀⣸⠁⠈⠀⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢈⣶⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡁⠈⠉⠉⠉⠉⢉
⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢲⡤⠒⠒⠒⠃⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠈⠲⡄⠀⠄⠀⠈
⠁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠤⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠹⠃⠀⠉⠠⠖⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠐⠚⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⣠⣤⣤⠶⠶⠚⠙
⠀⠀⠀⢀⡀⠤⠔⠊⠁⠢⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠉
⠀⠀⡄⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣀⣀⣀⡀⠀⠀⣀⠀⠈
⠀⠒⠐⠄⠀⠐⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⣠⡴⠃⠀⢀⢀⠐⠁⠀⠀⠀⠁⠁⠀⢦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠙⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⢀⢀⠆⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐⠂⠒⠢⠠⡀
⠀⠀⣠⠜⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠖⠃⠀⠘⠛⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠀⠀⢀⡔⠃
⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡰⠁
⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠊⠀⠀⠰⠦⠤⢀⡀⠤⠶⠚⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⣀⣀⠀⠠⠊
⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠂⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠐⠒⠀⠂⡜⠂⠀⠄⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⣤⠤⠄⠁⠀⠀⠀⠹⠉⣄⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒
⠀⠀⡸⠄⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠴⢆⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠐⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠄⠈⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠐
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡴⠚⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠒⠒⠀⠆⠠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⡀⠐⠙⠛⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⣶⡠⢄⡀⠀⠀⠀⠀⣴⠉⠉⠀⠈⠭⠌⠡⠄⠁
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠜⠠⠈⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⣠⠞⠂⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠐⠀⠀⠀⠀⠂⠀⠄⠀⡀⢀⠤⠞⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⡀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣶⣶⣶⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⣴⣶⣿⣶⣦⡀⣠⣶⣛⣷⡲⣄⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣿⢱⣶⣶⣶⣶⣶⣦⣄⣙⣛⣛⣛⣛⣛⡛⠁⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣠⣤⣶⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣇⣿⣿⣿⣿⣿⣿⣿⣟⣫⣭⣭⣭⣭⣭⣵⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠓⠁⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉