Beds Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⡉⢉⣉⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠠⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⠶⠶⠶⠖⠲⣩⢕⡬⠵⠤⠤⠤⠤⠴⠶⢆⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢲⠇⠠⠤⡄⠀⠀⠙⡇⠀⠀⠀⣀⠄⠂⠁⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠸⢄⣑⡒⣃⣀⡤⠚⢀⡠⠒⠉⠀⢀⡠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠞⠁⣠⠇
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣥⠤⢤⡔⠊⠁⠀⢀⡤⠒⠉⠸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡡⠊⠀⢀⡔⠁
⠀⠀⠀⠀⠀⢻⠛⣋⣉⣉⣉⣉⠭⠤⠤⠤⠤⠔⠒⠒⠲⡀⡇⣀⠴⠊⣁⣄⣀⣀⣀⣀⣉⣙⣒⣒⣒⣚⣛⣒⣒⣒⣒⠛⠒⠒⠒⠲⠴⢒⡿⠋⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠘⡇⡇⠀⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⢀⡠⠗⠋⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠉⠑⢒⡴⠋⠀⠀⢀⠞⠧⠇
⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⠛⠤⠴⠃⣀⠴⢚⣩⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⢻⠸⠤⠤⠤⠔⢒⣶⣖⡯⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡵⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⠜⡡⠊⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⣸⠷⠊⡝⠡⠒⠒⠒⠠⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⣛⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣱⠴⠋⢰⠁⠀⣠⠋
⠀⠀⢀⣠⠔⠋⠁⠀⠘⢇⡀⠀⠀⠄⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡀⠀⠁⢀⡇⢀⠞⠁
⢠⠶⠯⠤⠤⠶⠶⠒⠒⠒⠛⠛⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡍⠢⠤⠊⡴⠋
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠎
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡔⠁
⠀⣧⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⢲⡖⠛⢲⠛⠛⠋
⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⢷⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠛⠲⠦⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠦⠠⠤⢄⣀⡈⠉⠙⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣀⠦⡠⠦⠀⠀⡈⠉⠒⠢⠄⣀⠈⠉⢳⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠥⢀⣀⠆⠎⠀⠀⠀⠀⠀⠑⠦⠀⠀⠀⠉⠒⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠑⠂⠤⢀⠀⠀⠈⠐⠤⢄⡀⠀⠀⠠⠂⠉⠈⠁⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠑⠒⠶⠞⢓⠈⠀⠀⠁⠓⡀⣿
⠀⠀⢀⣠⣤⠶⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠉⢢⠀⠂⠀⡇⣿
⠀⠀⢾⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⡆⠀⠀⠀⠀⠀⠇⣿
⠀⠀⢸⠫⣉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⣀⢴⠀⣿
⠀⠀⢸⠀⠦⡉⠐⠠⠈⠑⠂⠄⣀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⠜⢀⡇
⠀⠀⢸⡄⡓⠤⣀⠀⠀⠈⠐⠠⢄⡉⠒⠎⢱⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠏⠻⣤⡼⠃
⠀⠀⠘⣇⢸⡷⢤⣉⠒⠤⣀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠉⠋⠁⠀⠉⠛⠶⣤⣉⠒⠤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⡇⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣆⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⠿⠛⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⠀⠀⣠⣴⣦⣀⠀⠀⠀⢸⡇
⢀⣤⣤⣤⣤⣤⣤⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⣼⡏⠀⠀⢻⡆⠀⠀⢸⡇
⣾⣿⣿⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣼⠇⠀⠀⢸⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣋⣀⣀⠀⢸⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⠉⠙⠻⢾⣧⣀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢶⣤⣰⣾⠋⠁⠀⠀⠀⠒⠂⠀⠀⣀⣨⣿⡗
⣿⣿⣿⠀⠀⣀⣠⣴⠾⠛⠋⠁⠀⠀⠈⠉⠛⠿⣶⣤⣄⣠⣤⣶⠾⠛⠋⠉⠉⠛⠳⢶⣤⡀
⣿⣿⣿⢰⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠋⠉⠉⠁⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠛⠳⢶⣤⣄⡀
⣿⣿⣿⣾⡿⠿⠷⣦⣄⣀⣤⣴⠶⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣄⡀
⣿⣿⣿⣁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣄⣀
⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⣀⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠋⢻
⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⢸
⠀⢸⣿⣿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠸⠿⠿⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣴⡶⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠋⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣧⣀⣤⣤⣴⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣈⣉⣉⣉⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠒⠤⢤⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠋⠉⠉⠉⠙⢶⡨⠒⠚⠒⠒⠛⢋⠙⠛⣦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢩⡇⠐⡒⡂⠀⠀⣨⠇⠀⢀⡠⠔⠈⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠓⠦⠭⠥⠔⠚⣁⠤⠚⠁⠀⣀⠤⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡠⠊⠁⣠⠎
⠀⠀⠀⠀⠀⣤⣶⣩⠤⠤⠤⠤⠤⠤⠖⠒⠒⢒⣒⣒⠓⡏⠀⠀⣠⠔⠊⠁⠀⠙⠢⢤⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣤⡷⠊⠀⢀⡔⠁
⠀⠀⠀⠀⠀⢻⢠⡖⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠈⣇⡧⠖⢉⡤⠖⠒⠒⠒⠐⠲⠶⠶⠀⠠⠶⠶⠒⠶⠶⠦⠤⠤⠤⢄⣉⢉⡵⠋⠀⠀⣠⠋⡇
⠀⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠰⡏⠉⡳⠀⠀⣀⠴⠚⣁⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⠞⠉⠉
⠀⠀⠀⠀⠀⠀⡇⢇⠀⢀⣀⣀⣈⣉⡤⣖⡭⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠟⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⣷⠈⠉⠉⣁⣤⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡴⠃⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢸⣠⠔⡾⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠢⣀⡴⢁⠎⠀⢀⡴⠁
⠀⠀⠀⢀⡠⠖⠋⠀⢸⠀⠀⠀⢀⡀⠠⠤⠤⠤⠤⠄⠤⠤⠤⠤⠖⠒⠂⠉⠉⠁⠀⠀⠀⠒⠒⣒⠒⠒⠚⠃⠀⡸⠀⣠⠋
⢀⣤⣞⣉⣀⣀⣀⣠⡤⠵⠶⠤⠤⠤⠤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⡟⢦⣀⡴⢁⠜⠁
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡴⠋
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠞
⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⣶⠶⠶⡶⠒⠚⠁
⠀⠈⠉⠙⡇⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡇
⠀⠂⠀⠀⢿⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠛⠿⣿⣷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠉⠓⢮⣝⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⢉⣽⠷⡾⠶⣦⣄⡀⠀⠀⠀⠈⠙⠦⣝⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠻⢯⣀⠀⠀⢰⣿⡁⠁⠀⠀⠀⠈⠙⠳⣦⣄⠀⠀⠀⠈⠳⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠈⠓⢤⡈⠙⠦⣄⠙⠻⣦⣄⡀⠀⠠⣲⠽⢚⣿⣇⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠙⠲⣄⡉⠛⠾⣾⣷⠾⠋⠁⠈⠙⠦⣄⡀⡇⣿
⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⣀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⣈⡽⣧⣿⡀
⠀⠀⢠⣿⣅⠀⠀⠀⢠⣴⠛⠁⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠙⠲⣄⡀⢀⣠⠴⠋⠁⠀⣿⢻⡇
⠀⠀⢸⡇⠈⠓⢦⡀⠘⣿⡷⣦⡁⠢⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣠⢿⡏⠁⠀⣠⡴⠚⠁⢸⡇
⠀⠀⠘⣷⡀⠀⠀⠉⠳⢤⡙⠾⣾⣷⣌⢐⣤⣤⣶⣿⠟⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⡇⢸⣥⠶⠋⠁⠀⣀⣴⠾⠃
⠀⠀⠀⣿⠙⠲⣄⡀⠀⠀⠙⠲⣌⡛⢿⣷⡿⠟⠋⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⢀⣤⠟⠋⠀⠀⣠⣴⠟⠋
⠀⠀⠀⢿⣄⠀⠈⠙⠲⣄⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⣀⡴⠚⠉⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠙⠻⣦⣀⠀⠈⠙⢦⣀⠀⠀⠈⠳⣄⣠⠖⠋⠁⢀⣠⠶⠋⠁⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠈⠓⢦⡀⠀⢸⠁⢀⣠⠖⠋⠀⠀⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠉⠳⢾⡶⠋⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⢸⠀⣀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⣶⠚⠛⠻⣶⡀
⠀⠀⣀⣤⣶⡾⠟⡟⠉⢹⡇⠀⠀⣿⠀⠀⠀⢹⡇
⢠⣾⡿⠋⣹⠀⠀⡇⠀⢸⡇⠀⠀⠋⠀⠀⠀⠉⠃
⣿⣿⠀⠀⣿⠀⠀⠗
⣿⣿⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⣿⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⠿⠿⢻⡏⠛⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⡏⠁⠀⢸⠀⠀⢸⡇⠀⣿
⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⣿⠀⠀⢠⡇⠀⠀⢸⠀⠀⢸⡇⠀⣿
⣿⣿⠛⠛⠿⢿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⠀⠀⢸⡇⠀⠀⠸⠀⠀⢸⡇⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⢀⣸⣧⣶⡿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⣇⣀⣤⣾⠿⠟⠋⠁⢸⡇
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣷⣶⣦⣤⣀⣀⠀⠀⠀⢸⡇⠀⠀⢀⣧⣤⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣷⣶⣾⣧⣴⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⡅⠈⢓⣒⣒⣒⣒⢒⡀⠉⢉⠉⠉⡀⠀⠀⡀⡀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠒⠒⠒⣒⣲⠋⢀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠸⠀⠀⠀⠈⡀⠀⠀⠈⠀⠃⠀⠀⠀⡅⠀⡅⠀⠀⠘⢁⠉⠀⠀⢀⠁⠀⠀⠀⡈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠸⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⡇⠀⠀⡄⠀⠀⠀⠀⠸⠀⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣠⡄⠼⠀⠀⠀⠤⠸⠄⠠⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠥⠤⠄⠒⠒⠒⠂⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠐⠒⠒⠒⠂⠤⠤⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠰⣁⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⢄⡱
⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠸⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣰
⠀⠀⠀⠀⠀⠈⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⣀⣨⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⠴⠤⢦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⠤⠤⠤⢤⣀⠀⠀⣀⣤⠤⠤⠤⢤⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡇⠀⠀⠀⠀⠀⢸⣀⣀⡇⠀⠀⠀⠀⠀⢸⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠓⠲⠶⠶⠶⠖⠚⠁⠀⠓⠲⠶⠶⠶⠖⠚⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⡀⢄⠠⠶⠚⠛⠛⠒⠒⠂⠤⢀⡀⡠⠤
⠀⠀⠀⢄⡠⢐⠅⠊⠀⡠⠀⠀⠀⠁⡀⡀⣀⣀⡤⠤⠭⢿⡀⡀
⠀⠀⠀⢀⡟⡕⢀⡠⠆⠒⠂⠂⠉⠉⠙⢧⣥⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠐⢸⡷⠅⠩⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣓⢀⠀⡀⠠⠤⠄⠀⠗⠀⠀⠤⣀
⠀⠀⠀⠈⠁⠀⢀⠒⢤⣀⡀⠄⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀
⠀⠀⢞⠔⣄⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢄⡀
⠀⠀⠀⢰⣄⡙⠲⣄⡄⢀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠠⠅⠂⢄
⠀⠀⠀⠘⠛⠟⠦⣄⠉⠪⠔⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡈⠑⠃⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠄⡠⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⠤⠤⢖⣒⣂⡺⠤⠴⢶⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠐⠫⢒⠤⠀⠥⠤⠆⢆⣒⣚⠍⠬⡤⠒⡒⡊⣈⡉⣩⡮⠴⠴⠞⠛⠿⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠉⡖⠟⠁⣡⣫⣠⣬⡴⠴⠶⠾⠟⠉⠉⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣁⣼⠯⠭⠭⠅⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⠒⠒⠦⢄
⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⣁⣠⠤⠴⡶⡖⠒⠒⢢⠈⡇
⠀⠀⠀⠀⢀⠔⠉⣠⠔⡞⡏⠁⠀⠀⠀⡇⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⡞⠀⡞⠁⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⣀⣸⠧⣇
⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⡇⢀⣀⣤⡴⠷⠛⠋⠉⠉⠓⢌⠓⢤⡀
⠀⠀⠀⠀⡇⢸⣇⡤⠴⢷⣟⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠈⣓⣤⡀
⠀⠀⠀⠀⡇⢺⠧⡀⠀⠈⠫⡻⣕⣄⠀⠀⢀⣀⠤⢖⣪⠭⠖⠚⠉⠁⠀⠉⠳⢤⡀
⠀⠀⠀⠀⡇⢸⠀⠈⠲⣄⠀⠈⠙⠛⠒⣋⡥⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⡇⢘⡦⣀⠀⠀⠙⠢⣄⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⣀⣠⣤⣄
⠀⠀⠀⠀⡇⢸⠃⠈⠓⢤⡀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⢉⣁⣀⣀⠀⢳
⠀⠀⠀⠀⠙⠚⠀⠀⠀⠀⠉⠲⣄⠀⠀⠈⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⣁⡤⠴⣒⣾⠹⡿⠟⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠈⠙⠶⣄⡀⠀⠀⢰⠋⢀⠴⣺⡏⣷⠶⠛⠋⣹⠀⡧⠒⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠈⠙⠶⣤⣿⠀⣼⠋⢉⡇⡷⠖⠚⠉⢹⠠⡇⣀⡼⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠈⡟⠀⡟⠉⠉⡇⣷⣀⡤⠴⢚⣛⡭⠴⢲⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⡇⠀⣧⠤⢒⣓⡯⠴⠒⠋⠉⠀⠀⠀⠘⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠤⣉⡑⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠓⠒⠲⠤⢌⡉⠒⠢⢬⣉⡒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢴⡯⣄⡀⠀⠀⠀⠀⢀⣨⡷⠤⠤⠤⣉⡉⠒⠤⢍⣙⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠈⠙⠺⠷⣦⣤⣠⣿⠋⠀⠀⠀⠀⠀⠈⢓⣤⣀⠀⠙⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠭⣖⣤⣄⣀⣴⣊⠥⢖⣫⠽⠲⡇⢸
⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣥⡔⠊⠁⠀⢀⣠⠇⢸
⠀⠀⠀⢰⠉⠓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣇⡠⠔⠚⠉⠀⠀⢸
⠀⠀⠀⠈⡦⢄⡀⠀⠉⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡅⠀⠈⠑⠢⢄⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠈⠉⠒⠶⠚⠉⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⣀⣸⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⡀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠈⠉⠉⠁⠀⠀⠈⠉⠉⠉⠃⠐⠂⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⢸⠀⠐⢐⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣶⣶⡷⡶⠶⣿⣷⡶⠒⠞⠛⠓⠶⠶⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠙⠐⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⢀⠀⠂⠀⢛⡆⠀⠀⠂⠀⠀⢸⣿⡆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣿⠁⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⠐⠠⢴⡁⠀⠀⠀⠀⠀⢸⣿⣷⣀⢐⠀⢀
⠀⠀⠀⠀⠀⠃⣿⠀⠁⣀⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣣⣄⡈⠀⠀⠐⡄⠀⠈⠁⠀⠀⠀⠰⠂⣸⣿⣷⣶⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⢰⣋⡄⢐⡃⣤⠄⠀⠀⠀⡀⣿⠇⠹⣿⡿⠿⠟⠛⠿⠿⡤⣄⠀⠀⠀⠀⢰⣦⡀⠈⠀⠙⢏⡷⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⢿⡇⠀⡐⢚⢿⣏⠲⣶⡿⢷⡀⠀⠀⠀⠀⠀⠈⠓⠈⠁⠀⠈⢻⣿⣾⡆⠀⢚⡟⠀⠄⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡟⣻⡧⣪⡟⠳⡦⠵⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠃⠈⢡⡀⠇⠀⠰⠴⣦⢄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠎⣇⣿⣼⡇⠀⠁⣀⠁⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡄⠀⠀⠙⢀⠀⢁⠄⠀⣯⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⢚⢦⡐⣽⢯⣧⣀⠀⠀⠀⣀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠸⠃⠀⣀⡴⢿⣿⠏⠀⠤⠤⠄⡄
⠀⢀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠩⣹⣵⣄⠤⡼⠂⠀⠀⠧⠀⡜⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠛⠀⠀⡀⣤⠐⠉⢹⣿⡉⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⢃⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⢿⣣⠀⢹⡄⠀⠀⢿⣓⠞⢁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠲⠄⢀⠜⠛⠿⢦⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⠀⠐⠄⠹⣷⣿⣾⡿⠀⠀⠀⠀⠀⠀⠠⠈⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⣐⡀⢼
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠏⠀⠀⠒⠶⡆⣀⠀⠂⠀⠀⠀⠀⠀⢀⠰⣿⣿⣿⡇⣤⣀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠿⠉⠹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⡠⠎⠀⠀⢻⣿⣧⣸⢽⣿⠀⠀⠀⠂⠀⠑⢄⠙⠒⠯⣤⣼⣿⣄⠀⠀⠀⠈⠷⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠘⡁⠂⢀⣶⣲⣦⡀⠈⣿⣿⣿⣿⣼⣝⣢⣤⡀⠂⠤⠦⣹⣞⣰⣶⠿⣿⣿⡀⠀⠀⢀⠖⠚⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠶⠿⠶⢷⣶⡿⡿⠿⠮⢇⠤⢈⣉⣹⣿⠿⢿⠿⠿⠿⡿⢿⣒⠒⠻⠟⠛⠛⠛⠉⠀⠁⠀⠸⠰⠷⠂⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⣀⣀⣀⠴⠊⠀⠈⠑⢦⣀⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉⠉⠑⠆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⠤⠤⠤⠤⣴⣾⠃⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⣤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⣻⠋⠛⠛⠻⠷⠶⣤⣤⣬⡇⠈⠙⢦⡴⠋⠛⠛⠿⠷⠶⠦⣤⡇⠀⠙⢶⣶⠶⠶⠶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⢛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡖⠶⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣮
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢸⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡽
⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠉⠉⠁⠀⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⡤⣤⠼

⠀⠈⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇
⠐⠚⠀⠀⣀⠤⠒⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣤⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⡇
⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠃
⠀⢸⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⢸
⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣅⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⣤⠓⠀⠀⠂⡀⠂⠀⠀⠐⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠋⠀⠀⠠⣠⠆⠀⡀⣿⠀⡂⠁⠀⠀⠀⢠⡦⠀⠀⠀⠀⣀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠐⠁⢂⣀⠠⠀⡀⠠⡤⡍⠀⠀⠀⠀⠤⠀⢀⡄⠀⣥⣒⣬⡭⣭⠴⣀⢀⠲⡀
⠀⠀⠀⠀⢀⠔⠀⡛⠋⠁⡴⠁⠐⠉⠉⠁⠀⠀⠈⠁⠰⠃⠈⠉⠉⠉⠁⠈⠈⠙⢄⠱⠼⢇⠉⠛⠘⣤⠈⠱⡌
⠀⠀⠀⡠⠃⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠃⠀⠀⠈⠇⠀⠘⢦
⠀⠀⡰⢁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠰
⠀⠀⠀⠆⡐⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠰⠀⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⡀
⠀⠀⡀⢸⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢲⠇
⠀⡄⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠧⠤⠼⠤⠆⠀⠩⣤⣦⣤⣤⣤⣤⣴⣴⡦⠀⠀⠀⠀⠐⠂⠐⢒⡀⠀⠀⠂⣀⣄⣲⣚⣛⡛⠀⣒⣲⣂⢀⠄⡄⠀⠍⠃
⠀⠘⡀⡀⠐⠀⠀⠺⢿⣿⣯⡥⠭⠽⣾⣿⣿⣗⣲⡀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉⠁⠁⡌⠍⠀⠀⡉⡁⠤⠈⡀⠀⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠨⠉⠀⠀⢀⡊⢬⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢸⢀⠌⠍⠘⠥⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⣀⠈⠤⡅⡸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⢋⠰⠧⡅⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢿⡘⠈⠰⡎⠉⢛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠪⢤⣜⡂⠀⠀⠀⢄
⠀⠀⠀⠀⢤⡀⠀⠀⠠⡘⢀⠙⠶⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠠⣄⣤⣤⣠⣀⣒⣀⣲⣾⣿⣼⡟⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠶⠖⠒⠲⠶⠦⠩⠁⠂⠈⠁⠚⠛⠛⠻⢐⠛⠛⠛⠛⠻⠛⢙⣡⠟⠁⠀⠉⢨⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢤⣀⠀⠀⠀⢀⡠⠀⠂⢀⠢⠤⠘⠢⠄⠀⠀⣂⡾⠛⠁⠀⠀⠐⠾⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢆⠀⠀⠀⠀⠀⠀⠙⠒⠰⠄⣌⡀⢠⣄⣠⡄⠀⡐⠚⠉⠛⠊⠁⠀⠀⠀⠀⠀⠀⠛⠛⠓⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠂
⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⣴⠛⠛⣦⠀⠀⢸⡇
⣾⣿⡿⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣽⠀⠀⢸⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣃⣤⡀⢸⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⣁⠈⠙⠻⢷⣄
⣿⣿⡇⠀⠀⠀⣀⣤⡴⠞⠛⠛⠾⣿⣤⣀⠀⠀⢉⣀⣤⡴⠶⠿⣥⣀
⣿⣿⡇⣴⠞⠋⠉⠀⠀⠀⠀⠀⢀⣀⣭⡿⠛⠛⠋⠉⠀⠀⢀⣀⣤⠽⠿⠶⣤⣄⡀
⣿⣿⣷⡿⠿⢶⣤⣀⣠⡤⠶⠛⠋⠉⠀⠀⠀⣀⣠⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⣄⡀
⣿⣿⣧⣀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⡀
⣿⣿⣿⣿⣿⣷⣦⣤⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠁
⠀⣼⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠿⠿⠃⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⡇⠀⠀⣀⣀⣤⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⢟⣛⣛⣛⢿⡿⣛⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⣶⣿⣿⣿⣿⢰⡆⣿⣿⣿⣿⣶⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⣿⣾⣿⣿⣷⣾⣷⣾⣿⡿⠷⠿⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢫⣭⣭⣽⣿⣶⣶⡾⣟⣻⣽⣶⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣷⣝⠿⢟⣻⣽⣶⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⢁⣀⣀⡀⠤⢤⣄⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⢀⣠⡤⠖⠚⠋⠉⣽⢸⠀⠀⠀⠈⡇⢸
⠀⠀⠀⠀⠀⡴⠁⢀⡴⠊⢹⢰⡀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢠⡇⠀⡞⠀⠀⢸⠈⡇⠀⠀⠀⠀⢹⢸⠀⣀⣠⣤⡷⢾⡀
⠀⠀⠀⠀⢸⠁⢠⡇⠀⠀⢸⠀⡇⠀⣀⣠⡤⠾⠚⠛⠉⠉⠀⠙⠲⡌⠳⢄⡀
⠀⠀⠀⠀⢸⠀⢸⠇⣀⣠⠼⣾⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠉⠳⣄⡀
⠀⠀⠀⠀⢸⠀⣼⣏⠁⠀⠀⠻⣻⢝⠦⡀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣋⡥⠴⠒⠛⠉⠙⠷⣄⡀
⠀⠀⠀⠀⢸⠀⣿⠈⠑⢄⡀⠀⠈⠲⢽⣿⣄⡤⠴⢒⣫⠥⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀
⠀⠀⠀⠀⢸⠀⢿⡀⠀⠀⠙⠢⣄⡀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⢸⠀⢸⡟⠦⡀⠀⠀⠈⠙⢦⣼⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⣤⡤⠴⠖⠲⢦⡀
⠀⠀⠀⠀⢸⠀⢸⠁⠀⠈⠳⢄⡀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⣀⣠⣤⣤⡀⠀⢷
⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠋⠉⢀⣠⠤⠖⣚⣿⢹⡷⠟⠋⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⢀⡞⠁⢀⡤⢲⡏⣯⣴⠶⠟⠋⢻⢸⣧⠤⠒⣯⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠉⠻⢦⣄⡀⣸⠀⢀⡿⠾⢻⡇⣿⣀⡤⠴⠒⢻⢸⡇⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠉⠻⣿⠀⢸⡧⠤⢺⡇⣿⠀⠀⠀⢀⣸⣼⠷⠒⣋⣩⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⣼⠀⢸⠀⠀⢸⡇⣿⠴⠒⣋⣩⠤⠖⠚⠉⠁⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡿⠀⢸⠖⣋⣩⠭⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣤⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢻⣿⣿⣿⣟⣿⣿⣿⣟⣻⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣷⣿⣿⣷⣿⣇
⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣭⣿⣿⣟⡸⣿⣶⣿⣿⣹⣿⣿⣿⣭⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣿⣏⣿⣿⣏⢿⣶⣿⣏⡀
⣸⣿⣿⠛⣟⣙⣿⣿⡟⠻⣿⣿⣿⣴⣶⣼⣿⣿⣿⣿⡿⢟⣹⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢯⣿⣿⡛⣿⣿⣷⣾⣶⣿⠿⣿⡇
⣿⣿⡿⣿⣆⣽⣿⣿⣷⣾⣿⣿⡟⣿⣿⣿⣿⣅⣸⣿⣷⣾⣦⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⣿⣿⣿⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠉⣹⣿
⣟⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣝⣻⣿⣷⠏⠀⢠⣿⡿⠻⠳⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡏
⣿⡿⣿⣷⣿⣿⣽⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⣠⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣶⣿⣛⣻⣍⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⠿⣿⣿⣷⣄⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣾⣷⣤⣽⣿⣿⣿⣿⣿⣿⡇
⣿⠟⣻⣭⢻⣿⣿⢋⡽⣟⠻⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣧⣶⣿⡿⣿⠧⣿⣿⣟⣳⣶⣾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣏⣡⠘⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣽⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⣯⣄⡤
⡄⠀⠀⢈⡇⠀⠀⠘⣿⠃⢀⡄⡠⠲⠄⠀⠁⠠⡀
⠇⠀⠀⠈⠃⠀⠀⠀⣿⠀⣿⣿⣷⣄⡀⢠⠀⠀⠀⠀⠒⠤⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⣤⣤
⢦⣠⢴⣬⣁⣀⣀⣀⢹⡇⢸⣿⣿⡻⣇⣘⡆⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠀⠈⠀⣀⣴⣶⣖⣶⣆⠀⠻⣹
⣿⢿⣿⣟⠿⠤⠤⡧⢺⣷⢸⣷⡟⣿⠏⠈⣷⡄⠀⠀⢸⠎⣧⡀⡀⠀⠀⠀⠀⠘⣿⠯⢍⣛⡟⢛⣷⠀⢸
⣿⡿⣿⣟⢻⣶⣾⣿⢹⣿⡘⣿⣷⣿⣶⣾⣏⠙⠂⠀⢠⣶⡎⠋⠺⢦⣄⡀⠀⢠⣿⣀⣸⣿⣄⠀⣿⣤⠻
⣿⣧⣿⠇⠐⣿⣿⢿⣿⣏⠁⠙⠿⣿⣿⣿⣿⣭⣁⣬⣍⣩⣉⣁⡐⠶⠶⠦⠀⣿⡤⢤⣾⣷⠀⢶⣿⣿⡃
⣿⣟⣿⣷⣾⣿⡇⢸⢻⡇⠉⠙⠒⣮⣭⡟⠛⢛⡙⠋⠉⠛⠛⠛⠒⠀⠀⣀⡀⣸⠁⠰⣾⡟⠧⣘⢿⡟⠧
⠻⢿⣿⣿⣿⣿⣷⡞⣸⡇⠀⠀⠺⣿⣿⣷⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⣾⠿⠟⠃⢰⣤⣾⣷⣴⣾⢓⠿⣖
⠀⠛⠳⠿⣯⣭⣿⣿⣿⣿⣟⣷⣄⣼⣧⣾⣼⣿⣿⣷⣦⣤⣤⣶⣶⣾⣿⣷⣶⣶⣾⣿⣻⣝⣻⣷⣧⠀⠉
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⡀⠘⠟⠈⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣤⠀⠀⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⣇⣠⠀⠀⠀⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⡄⡀⠀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⠙⠃⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⡄⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⡀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⡀⠶⣾⠀⠀⠀⢀⠤⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠏⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠊⠁⠀⡇⡨⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣃⣰⣖⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⣠⣚⡂⠠⠀⠄⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣶⣶⣦⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⢸⣿⣿⡏⢭⣭⠉⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠱⠶⠎⡁⠀⠀⠀⠀⠀⠐⠒⢲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣒⢒⡚⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣛⣛⠃⠀⠀⠀⠀⠀⠀⡐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡶⢴⣀⣀⣶⣦⣶⣶⣶⢀⣴⣶⣾⣿⣿⣿⣿⣿⣃⠀⠀⠀⢀⣀⣀⣴⡿⣿⡿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠁⠑⢀⣽⡿⢿⣿⣿⣿⣿⣿⣿
⢨⢤⣔⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⡉⠄⠀⠒⣚⠿⠿⠿⠃⠤⠤⠀⣀⠐⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⣴⣮⣷⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡶⠾⣿⣶⡶⣶⣶⡆⠀⠀⣠⣌⡙⠶⣌⠋⠐⠀⡀⠀⠀⠀⠀⠀⠸⣿⡟⣯⡽⢿⡿⢿⡿
⣿⣽⣗⣻⣧⣾⠟⠉⠉⣹⣿⣿⣿⣿⣿⣿⠟⠩⢀⣤⣤⣤⣤⣤⡀⠀⢙⠀⠀⠂⡠⢀⠀⠂⠀⠤⠀⠀⡑⠀⠀⠀⠀⠀⠀⢿⣿⠋⠁⢸⠃⢀⣀
⣿⠿⠟⠋⠉⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠿⣿⣿⣿⡿⠟⠣⢤⣀⣠⠶⠆⠀⢀⡔⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠈⢃⣀⣠⣴⣿⣥⡄⠀⣀⡍⠀⠀⠀⡆⠘⠆⢰⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⣿⡿⠿⡿⠿⣿⣿⠏⠀⠀⠀⠈⠀⢀⡆⠐⢁⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠤⠤⠤⢤⣤⠴⢂⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡀⠀⠀⠀⠀⠛⠀⠈⠀⠲⠃⠁⠀⠀⣧⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠐⣀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛
⠀⠀⠀⠀⠙⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠸⠃⢠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠙⢿⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠚⣵⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⢈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣷⣶⣷⣶⣶⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⡦⠀⠀⠀⢀⠜
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣴⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣶⣶⣤⣄⡐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⢴⣖⣒⣆⣈⠓⣒⣶⣺⣿⡛⠋⠁⠀⠀⠀⠀⠒⠠⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢖⣽⡿⠻⢩⡽⠿⣿⣷⠂⣀⠈⠀⠀⠠⠀⠀⠰⠾⠆⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣿⠛⡿⢻⡷⣀⣸⣷⡃⢙⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣤⣤⣿⣿⣇⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣾⣿⣿⢿⣿⣠⣿⡛⣿⣯⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡉⢉⠉⠉⣈⠻⠿⣿⣿⡾⡿⠁⢀⣘⣯⣙⣛⣷⡀
⠀⠀⠀⢀⢿⣿⡟⣫⣾⣿⣽⣭⣭⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣧⣤⣠⣤⣤⣤⣶⡛⠻⠷⢿⡤⢍⠁⣯⡿⢿⣿⡇
⠀⠀⠀⢸⣿⣿⡷⡗⣿⣿⣿⠺⡥⣿⣇⣤⣤⣤⡄⠤⣤⢤⣠⣤⣬⡤⢤⠄⠔⣤⠀⠿⣿⡇⠖⣾⢾⣿⠂⠀⠻⠤⣞⡁⡇
⠀⠀⠀⢸⣿⡿⠿⠁⣿⣿⣿⣧⣤⣿⠿⢿⡒⠒⠒⠔⠂⠒⠢⡶⠶⠶⠶⠶⠚⠞⠻⠛⢿⢳⣶⣾⣿⣿⣦⠀⠰⣠⢋⣠⡇
⠀⠀⠀⢸⣏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣮⠙⠛⢛⡦⢖⣤⣶⣿⣿⣎⣥⡀⠀⢀⠄⢠⡾⣽⣿⣿⣿⣿⡃⡆⠀⠾⠻⣿⣧
⠀⠀⠀⠸⠇⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⣷⠶⣴⠿⠏⢉⢿⣿⠻⣿⣿⠙⠀⠠⠞⣻⠌⣿⣿⢸⣿⣿⣧⡅⠀⠀⠀⡿⢻
⠀⠀⠀⢸⣵⠀⠀⠀⢸⣿⢿⡟⣻⡇⠈⠻⣿⣶⡏⠰⠞⠁⣼⣿⡄⠀⠉⠁⠂⠀⠈⠀⠀⢸⡿⣿⣿⣿⢻⡁⠀⠀⠀⠻⣦
⠀⠀⠀⢰⢻⠄⠀⠀⠠⢹⢸⡇⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢻⣿⣿⢾⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠘⢸⡀⠀⠀⢸⢸⣽⡇⣾⡇⠀⠀⠀⠀⠀⢀⣀⢰⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣺⢻⡟⠈⠓⢐⣐⣴⣆⣉
⠀⠀⠀⠀⠈⢇⠀⠀⣼⣾⣿⡇⣻⣇⣀⣠⣖⠀⠁⠀⠘⠣⣿⣿⠀⠀⠟⠑⠂⠀⠀⠀⠀⢸⠃⣇⠠⠉⠂⠀⣿⣿⣿⣿⣻
⠀⠀⠀⢸⡄⢘⡄⣀⣿⣿⣿⠀⣿⢿⡿⣷⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⣿⡀⠁⠀⢀⣀⠉⠉⠛⣿⡅
⠀⠀⠀⣞⣃⣸⣿⣿⣿⣿⣿⣀⣿⣻⣷⣿⠀⠀⣀⣀⣀⣀⣉⣁⡀⣀⠀⠀⠀⠀⠀⠈⠀⢸⣇⣿⣥⣶⣾⣻⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠋⠫⠿⣿⣿⣿⣿⢾⡉⡇⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣷⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡳⢾⣿⣿⣿⡿⢿⡿⣧⣿⣿⣉⠄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⠅⣿⣿⣿⣿⠗⠊⠁⢹⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⡑⢿⣿⣿⠁⠀⠀⠀⠈⠿⣿⠣⡀⠀⠀⠀⠀⠘⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠋⠉⠉⠉⠉⢿⣿⠋⠁
⠀⠀⠀⠀⣈⢼⣿⣿⣦⣤⢠⣇⣉⣀⣻⣿⣿⣀⠀⠀⠀⣠⣿⠖⠀⢄⣠⡤⠒⠒⢛⣹⠟⠛⢻⠁⠀⠀⠀⢀⡀⣻⡏
⠀⠀⠀⠀⢿⡘⢿⠿⣿⡇⢸⣯⣥⣼⡿⠛⠻⠛⠿⡖⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⣾⠀⠀⠀⠀⠀⠐⣿⣿⡄
⠀⠀⠀⠀⢼⡷⢦⣠⣤⣽⣿⡛⠛⠉⠉⢈⠛⠛⠛⡁⠒⠘⠓⠘⠛⠐⠚⢙⡻⠇⠉⠒⠐⠊⣿⠀⣤⡤⠴⢲⣿⣿⢿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣉⡟⢚⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣯⣭⣵⡾⢿⣭⣟⣋⡿⠇
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠹⢧⠈⠈⠟⠋⠋⠻⢛⡛⠉⠙⠻⠛⠿⠻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢂⣸⡄⠀⠄⠀⠀⠒⢂⡄⠀⡀⢠⣴⣶⡖⣢⣀⣈⣭⣿⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⡗⡇⢸⠻
⠀⠀⠀⠀⠀⠀⣂⣤⣾⠿⠉⠁⠐⠒⠈⠉⠀⠀⠈⠛⠋⠉⠛⣟⠛⣟⠋⠉⠉⠙⠛⠛⠟⠛⠹⣿⣿⣿⣿⣁⣿⣇⣇⣨⡼
⠀⠀⠀⠀⠰⠛⠉⡀⠠⠂⠠⠂⠀⣠⣀⣠⣀⣀⣀⣤⡀⣴⣶⡀⠀⣸⣆⣤⣀⣀⣀⣄⢀⡀⠉⠉⠙⠻⡟⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⠠⠰⠐⠀⠁⢀⠀⠀⠈⠉⠀⠒⠰⢭
⠀⠀⠈⠱⢅⠁⢀⣐⠥⠤⠤⠤⠲⣖⡒⠋⠉⠉⠉⠁⠐⢄
⠀⠀⠀⣿⣜⠐⠉⡀⠀⠀⠀⠀⠀⡈⠙⠦⠀⠀⠀⢀⡀⠀⠦⠤⠀⡀
⠀⠀⠀⠋⠀⠀⠂⢀⡀⠀⠀⠒⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠀⡀
⠀⠀⡂⠦⣀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀
⠀⠀⠀⣦⣌⠑⠦⡄⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐
⠀⠀⠀⠉⠉⠛⠢⣄⠑⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠈⠈⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠀⣘⣁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠁⠈⠴⡄⣀⠀⠀⣀⣀⡀⠤⠤⢀⣐⡂⠦⠍⠯⠖⣲⣚⣪⡭⢭⣟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⠈⠐⣤⠖⠐⡢⠛⣙⣃⣥⣥⠴⠶⠚⠚⠈⠛⠚⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣃⣰⣾⣫⠍⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢛⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡉⠉⠉⠓⠒⠦⠤⣄⡀⠀⠈⠉⠛⠷⢦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠧⣄⠗⡢⠒⠒⠒⠠⢀⠀⠈⠉⠒⠢⢄⡀⠀⠈⢹⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⠛⠛⠥⢀⡀⣰⠀⠊⠀⠀⠀⠀⠀⠀⠉⠲⠦⠀⠀⠀⠈⠉⠒⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠣⠤⢄⡀⠀⠀⠀⠈⠑⠼⣀⡀⠀⠀⠀⡔⢁⠄⠂⠈⠙⠂⢀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠀⡀⠀⠀⠉⠑⠒⠦⠧⣨⣤⣤⣤⠤⠤⠈⠉⠒⠤⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⢄⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⣠⣶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣀⠀⠀⠀⠑⠆⠈⠀⠀⡁⢸⠃
⠀⠀⠐⡏⣉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢹⡄⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣿⢉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣠⠁⣸
⠀⠀⠀⣿⠈⡍⠓⠢⢄⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠊⠁⣸⠀⣿
⠀⠀⠀⣿⠀⣍⠐⠀⠀⠈⠐⠠⢄⡈⠉⠒⠤⣄⣀⡤⣒⠀⢀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣷⢶⣟⠁⢀⣿
⠀⠀⠀⣿⠐⢬⡙⠲⢄⡀⠑⠢⢄⠈⠑⠢⢄⡀⠙⠀⢸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡴⠏⠀⠙⠿⠟⠃
⠀⠀⠀⢸⣇⢰⡟⠷⣦⣈⠑⠢⢄⡀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠛⠶⣤⣈⠙⠲⠤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⢸⠀⠀⠀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⣷⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠈⠁⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡄⠀⠀⠀⣠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠋⢁⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣦⣄⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠢⣀
⠀⠀⠀⠀⠀⠀⣀⠸⣿⣦⡀⠀⢀⡀⢴⠦⣰⣄⠀⣠⠀⡁⢙⣷⣤⢀⣀⠀⠀⠀⢠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⢻⣾⠀⣼⣿⣿⡗⣿⣿⣿⣿⠀⠘⣿⠂⠀⠀⠀⢸⣿⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠟⣿⣆⣄⣤⣤⡘⣧⢰⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⣷⣡⠄⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠉⣩⡼⢼⣿⣷⢃⡿⣾⣯⣿⣿⣿⠻⣿⠛⡟⣿⣿⣭⠆⠀⢀⡀⢒
⠀⠀⠀⠀⠀⠀⠘⠻⡆⠀⠃⠨⣿⣿⠞⠱⡏⣼⡇⡿⢻⠀⣾⡆⢿⡘⠿⠡⠞⡀⠛⢳⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣯⣤⡴⠟⣷⣼⣏⣉⣻⣀⣿⡶⠾⢿⣷⢤⠀⣃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⢰⣷⡄⠀⢦⡙⣌⡛⠇⠙⢋⣸⣿⣿⣿⡛⠿⣦⠞⣱⡦⠀⣿⣦⠠⣫⡀
⠀⠀⠀⠀⠀⠀⢨⢠⣿⣧⠀⠇⠉⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣶⣿⣿⣿⣿⠀⣿⣏⢸⣭⡁
⠀⠀⠀⠀⠀⠀⡸⢾⣻⣿⣠⣏⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⣿⣿⣿⣇⣿⡅⢨⣿⢇
⠀⠀⠀⠀⠀⢠⣇⣻⡏⢿⣮⣿⣿⣿⣿⣯⣿⡿⠋⣉⠈⢻⣿⣿⣆⣿⣿⣿⣿⡿⠀⣼⣿⢸⡇
⠀⠀⠀⠀⠀⢻⢻⣿⣿⣾⣿⣿⣿⣿⡿⢿⠟⠁⠖⡄⠈⠀⠈⢿⠿⢿⣿⣿⣿⡇⠈⢿⣿⡇⠽
⠀⠀⠀⠀⠀⠸⡿⢿⠿⣿⣿⣿⣿⣿⡇⣿⡆⠀⢰⣿⠆⠀⠀⢻⡖⢺⣿⣛⣿⡇⠀⣼⣿⣷⣇
⠀⠀⠀⠀⠀⢶⣷⢸⣷⣾⢏⣘⣛⣡⣤⡟⣀⣀⣤⣁⣀⣠⣤⡼⣧⣤⣭⢿⣀⡹⣰⡾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢘⣿⢿⣿⣿⠶⠶⠶⠲⠿⡇⠲⠿⠇⢴⠇⠀⢿⡿⠿⠿⠟⠢⠴⠿⠏⣿⢻⣿⡿
⠀⠀⠀⠀⠀⠘⢻⠘⠈⣿⠀⠀⠰⡄⢸⣿⠟⠀⣷⡏⠀⠀⢸⣿⠿⠃⠀⠠⣄⠀⠀⢿⣿⡏⠗
⠀⠀⠀⠀⠀⣰⡈⠃⠀⣿⠀⢨⠀⡇⢸⡟⠀⠀⣸⡅⠀⠀⢸⣿⠀⠀⢀⡁⠻⠀⠀⢸⣿⡗
⠀⠀⠀⠀⢠⡟⢻⠇⠀⣿⠀⢻⠀⡇⢸⣧⠀⠀⢸⡀⠀⠀⢸⣿⠀⠀⠀⠇⣶⠀⠀⠈⠙⠁
⠀⠀⠀⠀⣸⡿⣿⠃⠀⣿⠀⣸⠷⠇⢸⣿⠃⠀⠈⡁⠀⠀⢸⣿⣃⠀⢀⡖⢿⠀⠀⣷⣿⠂⠘⡀
⠀⠀⠀⠀⣿⣿⣧⣄⢠⣿⠀⠿⠿⠇⣼⣿⣁⢀⣞⣓⠀⣀⣸⣿⡧⠆⠸⠿⠿⠀⢸⣿⣷⣧⢾⣧⢠⣆
⠀⠀⠀⢰⣏⡟⠉⠁⢀⡀⠀⠄⠀⢰⣿⣀⣀⢤⡽⡏⠉⢩⣽⣇⢀⣠⠀⠠⠤⢴⣿⣿⠉⠁⠀⣹⡄⢻
⠀⠀⠀⣦⡟⡛⠉⠉⠁⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠨⡏⠀⠀⠀⠀⠀⢠⣿⣟⠉⠁⠁⢹⡷⠈
⠀⠀⢨⡉⣻⡷⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢤⡄⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠛⣧
⠀⠀⠈⠛⠻⠀⠀⠀⠀⠐⠛⠛⠛⠛⠇⠀⠀⠲⠾⠅⠀⠐⠾⠀⠀⠀⠚⠛⠛⠛⠛⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡉⢠⣶⡆⢀⢀⠀⣴⣶⠀⡄⡄⠰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⣿⣿⠀⡇⡇⢰⣿⣷
⠀⠀⠀⡆⠀⣠⣴⣤⣤⣄⠀⡆⠀⠀⢰⠀⣰⣤⣤⣄⣆⡆⠀⠀⠀⠀⢀⢰⣾⣿⣿⡜⠄⣴⣿⣿⣧⠀⢹⣼⣿⣿
⠀⠀⠀⣿⡾⠿⠛⡿⠿⠛⠟⠃⠠⠤⢶⡿⢿⡯⠙⠛⣹⠃⠀⠀⠀⠀⠠⢾⡛⢙⡟⠁⣱⠚⣛⡟⠁⠀⡶⠽⡟⢻
⠀⠀⣐⠁⠀⠀⢰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠐⠛⠻
⠀⢸⣿⣷⣄⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣧⣤⣤⣿⣀⣀⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⡄
⠀⠘⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⣿⣟⣻⠿⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⣇⢀⠈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡉⠁⢀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢹⠀⠀⠰⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠀⡄⠀⠀⠸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠐⡆⢸⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⡆⢠⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠃⡆⠀⠀⠀⠀⡇⢠⠀⠀⠀⡆⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⣇⠀⢸⠀⠀⢀⡀⣇⠀⠀⠀⢀⡇⢘⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠛⠛⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣉⣈⣉⣈⣉⣉⣉⡉⠉⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⠁⣀⡠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠂⠠⠤⠤⢄⡀⠱⡀
⠀⠀⠀⠀⠀⠀⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⢀⣀⢀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⡈⠓⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠠⠆⠀⠀⠀⠀⠠⠄⠉⠉⠑⠲⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠁⠀⠀⠀⠀⠐⠃⠀⠀⠀⡀⠀⠘⠀⠀⠀⣀⠀⠒⠀⠀⠀⢈⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠏⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠉⠀⠀⢀⣠⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣴⣾⣉⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⡀⠀⠈⠀⠀⢠⣴⣶⢿⣟⣽⣿⠟⠁
⠀⠀⠀⠀⠀⢀⠉⠙⠻⢽⣳⢶⣁⠀⠀⠀⠀⡀⠘⠋⠀⢀⣀⣀⠀⠀⠈⠀⠀⢠⣴⣾⢿⢻⢛⣼⠿⠋
⠀⠀⠀⠀⠀⠸⣴⣄⡀⡇⠈⠹⠹⢽⣗⡶⣏⠀⠀⠀⠀⠀⠙⠁⠀⢀⣀⣤⣾⡽⣏⣿⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢣⣇⡄⠀⠈⠈⢹⠳⢯⣖⡶⣄⡀⠀⢀⣤⣾⡽⡞⠃⢹⣸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢗⡄⡀⠀⠘⠸⢹⠳⠯⣟⣿⢾⣇⣧⣷⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠣⢆⡄⣀⠐⢰⠀⣿⣸⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠟⣼⣴⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠈⠁⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠒⠀⠐⠒⠀⣾
⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣙⣿⣁⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣤⣼⣶⣒⣂⠀⣀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣴⣶⣿⡟⡄
⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣾⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣾⣟⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⡉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠉⠙⠶⣭⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⢘⣇⣠⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⠑⠒⠦⠈⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡤⠖⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⡟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠤⠤⠤⣄⣀⣀
⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⣠⡠⠶⠒⠉⢉⣁⣀⡀⠴⠶⠶⠶⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠐⠒⠶⠶⠶⠦⣀⣀⣉⠉⠑⠲⠶⣤⣀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⢸⡏⠀⠈⡷⠀⣠⠔⠋⣁⣤⠔⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⢤⣤⡈⠛⠦⣄⠀⣿⠁⠀⢹⡆
⠀⢸⡇⠀⠀⣿⠎⣡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢦⡈⢳⣾⠀⠀⢸⡇
⠀⢸⡇⠀⠀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠀⠀⢸⠃
⠀⢸⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢸
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠁⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢨⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⢸⣧⣀⣠⣿⣀⣀⣀⣀⣀⣠⣿⠇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣼⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⢹⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⣀⣠⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⣀⣀⣹⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠐⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠴⠶⠶⠂⠛⠛⠒⠒⠶⠶⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣿⠀⢸⡇
⠀⢸⣧⣄⠰⠶⠒⠶⠶⠶⠶⠶⠖⠒⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠛⠶⣾⡇
⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠖⠒⠈⠉⢉⣉⣁⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⣉⡉⠉⠁⠒⠲⠶⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⣁⣀⣤⡤⠄⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠠⢤⣤⣀⣉⠛⠲⠤⠀⠀⠀⠀⠀⠀⠀⢸
⡁⠀⠀⠀⠀⢀⣴⠞⣉⡴⠶⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⣈⠐⣄⡀⠀⠀⠀⢘
⡀⠀⠀⠀⢀⣿⣥⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⣿⠀⠀⠀⢸
⣤⡄⣀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⢀⡀⢼
⣿⣀⣀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣀⣀
⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⣿⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣿⠀⠀⠀⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡆
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⠀⠀⢀⡀⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⡀⢀⣤⣤⣤⣤⣤⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⡖⢢
⠀⠀⠀⡇⢸⠀⠀⣀⡤⢤⡀
⠀⠀⠀⡇⢸⠀⣀⢇⡰⠊⢉⡱⠾⢦⣀⠀⢀⠤⢤⠖⠚⡏⠉⠉⠉⠉⠑⠒⢤⡀
⠀⠀⠀⡇⢸⠉⠈⠙⡇⠐⢯⠀⠀⠀⠈⣷⣿⢧⠀⢳⡀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⢄⣀⣀⢠⢲
⠀⠀⠀⡇⢸⡀⠀⠀⠳⠤⠤⠧⢤⣤⣚⣙⣯⣀⣳⠖⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣸
⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⢲⡖⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠸⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⡏⢸⠉⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⡇⢸⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⡏⡏⢸
⠀⠀⠀⡇⢸⠀⠧⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠃⡇⢸
⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠭⠤⠤⠤⢤⠌⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠉⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠴⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⠄⢃⠀⠀⠀⣠⡄
⠀⠀⢐⠒⠀⠀⡴⠶⠖⠀⢄⠀⠀⠀⠀⠀⠣⣀⡼⠄⢐⡀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠊⠀⡉⠉⢱⠊⠀⠀⠀⠔⢀⡈⠨⠁⠀⠀⠀⠀⢱⠀⠧⠳⠇
⠀⡆⢷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄
⠀⠃⢀⡀⠘⡀⠀⠸⡁⠄⡀⠀⣀⠠⠃⠀⠀⠀⡠⠀⠀⠀⡀⢠
⠠⠀⢸⡇⠀⠈⠢⠤⠈⠈⠅⡀⠈⢂⣀⠐⠀⠀⠀⠈⠀⠀⠀⠂
⠀⡆⠈⢡⣄⠀⠀⠀⠀⠀⠀⠈⠲⢀⡊⠉⠉⠁⠒⠒⠒⠀⡰⠠⠤⠄⠀⢠⠤⢠⣖⠂⠀⠀⠐⠂⠀⣤⢀⣀⣀⣀⣀
⠀⠘⢆⠀⠙⠷⠄⣤⡀⠀⠀⠀⠀⠀⠉⢢⠞⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠨⡑⢦⣀⡤⠒⣂⠥⠀⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢱⡶⠄⠤⢬⡙⢠⣄⠀⠀⠀⣠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡡⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢆⠂
⠀⠀⡔⠉⠀⠀⠀⠀⠈⠢⡉⡠⠒⠉⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⡞
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⢀⡇
⠀⡠⢁⣀⣀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⡇⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠃⠘⠇
⠀⠉⠈⠈⠀⠉⠢⠤⠤⠒⠉⠀⠀⠙⠒⡇⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠉⠀⠀⠀⠀⠀⠈⠉⠒⠂⠠⠄⠀⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠒⠢⠤⠤⠒⠒⠚⠒⠤⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣇⡀⢀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⢳⣼⠋⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⠃⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠆⠀⠒⠒⠒⠒⠒⠚⠛⠚⠓⠒⠒⠒⠒⠒⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣠⠴⠒⠛⠛⠋⠉⠛⠛⠲⣄⠀⠀⠀⠀⣠⠶⠚⠛⠉⠉⠛⠛⠓⠶⣤⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠳⢤⣤⣤⣤⣤⣤⣤⣤⣤⠞⠉⠉⠉⠉⠹⢦⣤⣤⣤⡤⣤⣤⣤⣤⠼⠋⠉⣩⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⣠⠞⠋⢿
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⢀⣠⡴⠋⠁⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⣠⠶⠋⠁⠀⢀⣤⠞⠋⠀⣧
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠀⢀⣠⡴⠞⠉⠀⠀⢀⡤⠞⠋⠀⠀⣠⡴⢻⡄
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉⠁⠀⣀⣤⠶⠚⠉⠁⠀⢀⣠⠴⠛⠁⠀⢀⣠⠴⠛⠁⠀⣨⣇
⠀⠀⠀⠀⠀⠀⣟⢀⣀⣀⣀⣤⣤⡴⠶⠒⠛⠋⠉⠀⢀⣀⣠⣤⠶⠚⠋⠁⠀⢀⣀⣤⠶⠛⠉⠀⠀⣀⣴⠞⠋⠁⢀⣠⡶⠛⠁⣻
⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠌⠀⠀⠉⠉⠙⠛⠳⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡏⠉⠒⠒⠦⠤⢄⣀⠀⠀⠉⠙⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣇⣀⠦⣒⠶⠒⠠⠀⡀⠈⠉⠑⠲⠤⣀⡀⠀⠈⠙⡟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠻⠿⣉⡀⠀⣠⠁⠞⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠉⠑⠢⢤⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠿⢅⣀⠀⠀⠀⠀⠈⠙⠣⣼⠀⠀⠀⠀⠀⠠⠂⣀⠄⠒⠒⠦⠀⠀⠀⠀⠰⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠑⠒⠄⣀⠀⠀⠀⠉⠓⠢⠤⣄⣇⢰⠁⣀⣀⣀⣀⠀⠓⠤⣀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⢁⡀⠀⠀⠀⠀⠉⢣⢸⣿
⠀⠀⠀⢀⣠⣤⣴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠀⠀⠀⠈⠓⡄⠀⠊⠀⢸⢸⡇
⠀⠀⠀⣿⠛⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠶⡄⠈⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⣿⠞⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⡎⢸⡇
⠀⠀⠀⣿⠘⡶⢄⣈⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡤⠔⠚⡇⢸⡇
⠀⠀⠀⣿⠀⠳⠄⡀⠉⠂⠄⡀⠉⠒⠢⢄⣀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠜⠁⢸⡇
⠀⠀⠀⣿⠀⡛⠦⣀⠀⠂⠄⡀⠉⠒⠤⣀⠈⠙⠲⡞⠙⣆⠠⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠏⢻⣦⣤⣿⠁
⠀⠀⠀⢹⡆⠙⣦⣄⡉⠲⠤⣀⠉⠂⠀⠈⠉⠒⢤⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠟⠛⠁⠀⠀⠈⠉
⠀⠀⠀⠘⢷⣴⠟⠉⠛⠷⣤⣈⠙⠲⢤⣀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣈⠉⠓⠦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣀⣀⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡶⢶⣶⠚⠛⠻⣶⡀
⠀⠀⢀⣠⣴⣶⠿⢿⠛⠉⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇
⢀⣶⡿⠟⢹⡇⠀⢸⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⠙⠃
⣾⣿⠁⠀⢸⡇⠀⠸⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠀⠀⡸⠃
⣿⣿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⢿⡟⢿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⣿⠛⠋⠉⡏⠀⠀⢸⠃
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⢻⠁⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸
⣿⣿⠿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸
⣿⣿⠀⠀⠀⠉⠙⠛⠻⠿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⣀⣨
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢀⣥⣤⣶⡿⠟⢻
⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠀⢀⣠⣿⣴⡾⠿⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣷⣶⣤⣤⣀⡀⣸⡏⢀⣀⣤⣿⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡶⠛⠳⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢺⣶⣦⣤⣤⣶⣶⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⡄⢸⣿⣿⣿⣿⣿⣿⠀⣤⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠛⠛⠉⠉⠉⠛⠛⠂⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣤⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⠻⠿⣿⡿⠿⠛⠉⠙⠻⠿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠛⠋⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠉⠛⠻⠿⠿⠿⠂






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

⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣀⠀⠔⠉⢹⣄⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⢶⡶⣿⠶⣿⣾⡆⢰⡙⣿⠤⢤⡀⢻⣏⢤⢼⣿⠀⣀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⢸⡇⠀⠘⠷⣤⡈⠉⠉⠋⠉⠻⡍⠉⠙⠟⠛⢧⣸⡗⠒⠛⠛⠛⡀⠸⣿⣶⣶⣶⡶⡆⢐⢲⡾⣧⡤⠴⣶⢤⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⢦⣄⠈⠳⢾⣦⠰⠧⡴⠚⡿⣛⡀⠀⢠⣾⣿⢺⣿⣿⣿⣿⣷⠀⠛⢿⣋⠉⠛⠳⣿⣀⡉⠛⠦⣀
⠀⠀⠀⢸⡇⠀⠀⠀⣤⠼⠧⢄⣈⣀⡀⠈⡇⢀⣠⣤⡄⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠙⣿⠛⠋⠀⠀⣶⠉⠛⠶⠒⠚⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠈⢱⣶⣌⠛⠻⢿⡛⡟⠛⠿⢿⠏⠈⠻⠿⠟⠛⠉⣿⣿⣿⣿⣿⣿⣿⣷⢀⡀⠀⠀⠠⣶⡿⣶⠀⢸⣶⡤⣧⣴⣶⣶
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠉⢳⡶⣶⡂⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⡟⠁⠀⣉⡁⠀⣀⠀⠉⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡀⠀⠈⡇⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⣿⣿⣿⣿⡿⠉⠉⠀⠀⣷⠀⠀⡋⠉⠠⣤⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⣀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⠋⠈⠉⠿⣿⡇⠀⠀⠀⠀⣿⢤⡀⣿⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠘⠓⠠⣠⣇⣤⣤⣤⣶⣶⠂⡠⠔⣻⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣤⣄⠀⠀⣿⠀⠈⡿⠄⣀⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡦⡤⠤⠬⠤⠬⣿⣿⣿⣿⣿⠿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⣼⣿⣿⠿⠿⢥⣥⣿⡀⠀⠀⠀⠈⢟⣠⣤⣦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠁⠀⠲⣦⠀⠀⡿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠉⠉⠁⠀⠀⠀⣠⣿⣿⣿⣷⣶⣿⣿⣿⠈⠓⢶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢴⣞⣷⣤⣀⣇⣨⣿⣷⣦⠘⠛⠿⣿⣿⣿⣿⣿⣿⣷⠒⠒⠒⠂⢲⣿⠟⠛⢿⣿⣿⣿⣿⣿⣶⣶⣼⣈⣿⣿⣟⠛⠟
⠀⠀⠀⢸⡇⠀⠀⠀⠈⢻⠋⠁⠀⠏⠙⠟⡿⠇⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣇⠀⠀⠠⣾⡟⠀⠀⠀⠈⠙⠿⠟⣿⠉⠀⠨⠈⠛⢺⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⢀⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⡈⣉⣉⣙⡋⢀⣴⣇⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿⠍⣉⣋⠉⠉⡏⠉⠉⠫⣭⠭⠭⠭⠿⢋⡉⢍⡻⣿⣏⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠲⣿⡃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠈⠛⠀⣀⣇⠀⠀⠀⠀⠉⣛⣇⣊⣙⣋⣨⣿⣋⡉⣙⣦⣤⠀⠀⠀⡀⠀⠀⠀⣿⣤⠀⠀⠉⢢⣶
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠁⠀⠈⠉⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣠⣄⣶⣶⣿
⠀⠀⠀⢸⣷⠆⡀⣤⣀⣀⣠⣤⣠⣷⣿⣿⣿⣿⣿⣷⣶⣶⣦⣰⣤⣤⣷⣶⣶⣾⣿⣷⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⠛⣽⣏⣉⣯⣫⣿
⠀⠀⠀⢸⣏⠠⡌⢠⣬⠙⠿⠷⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⡔⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⢀⣀⣀⠒⠒⠒⠒⠒⠒⣶⣶⡖⠒⣒⣒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢀⣙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠋⠉⠀⠈⠉⠙⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠂⠁⠀⠀⠀⠉⠒⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣠⡄⠀⣠⣄⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠿⠏⠀⠻⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠓⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢤⠄⠀⠀⠀⠤⣤⢄⡀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠎⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣟⣀⡀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠥⡄⠀⠀⠀⠀⠲⠆⠼⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠛⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠂⠠⠤⠤⠤⡦⠀⠀⠱⠆⣀⣀⠀⠀⠀⣌⡛⠢⠠⠄⠄⢀⣀⡀⠚⢦⣀⠀⠤⠤⠤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠐⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡈⠁⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠐⠂⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠐⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⡀⠀⠀⠀⢀⣀⡀⠀⠀⠳⣀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠈⠉⠛⠓⠒⠒⠢⠤⠤⠤⠔⠒⠂⠈⠉⠉⠉⠙⠛⠛⠂⠉⠉⠛⠂⠚⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⠶⣶⠶⣦⣄⠀⢀⣤⠶⠶⠶⢶⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇⣿⠁⠀⠀⠀⠀⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣤⣿⡀⠀⠀⠀⠀⣸⡧⣿⡀⠀⠀⠀⠀⣸⣧⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡌⠋⠠⠿⠁⠈⠛⠛⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠋⠀⠹⠧⠈⠋⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣤⣤⣄⣀⢀⣀⣤⣤⣤⣤⣀⡀⣀⣠⣤⣤⣤⣄⣀⣀⣀⣤⣤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣉⣀⣀⣀⣈⣙⣛⣉⣁⣀⣀⣈⣉⣛⣋⣉⣀⣀⣀⣉⣙⣛⣉⣀⣀⣀⣈⣙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠉⢹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣿⠉⢹⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⣷⣄⠀⣀⣠⣤⣶⣶⣶⣿⡿⠿⠿⠿⠿⠿⣿⣶⣶⣶⣤⣄⣀⠀⣠⣾⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢹⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⡏⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣤⣤⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣤⣤⣤⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠈⠉⠉⠉⠉⠉⠻⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⠟⠉⠉⠉⠉⠉⠁⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⢃⠀⠀⣀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡀⢀⢀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⡘⣿⣧
⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣇
⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣹⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡏⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣶⣶⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠒⠲⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠓⠢⠤⣄⡉⠑⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠋⠉⠛⠒⠦⣄⡈⠙⠓⠦⢤⣉⡙⠒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⡳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣬⡷⠤⠤⠤⣄⣉⠙⠒⠦⢄⣉⠙⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠁⠀⠀⠀⠉⠓⠺⠵⣦⣤⣀⣀⣾⡟⠁⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠈⠉⢾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢭⣓⡦⢄⡀⠀⠀⢀⡤⠒⣉⡩⠟⣒⣦⣼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠽⣶⡛⢚⣩⠥⠒⠋⠁⠀⢹⠀⢸
⠀⠀⠀⠀⢠⢴⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⢻⠀⠀⠀⣀⡤⠖⠛⠀⢸
⠀⠀⠀⠀⡏⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢀⣸⠴⠒⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠶⢤⣀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠉⠙⠲⢤⠴⠒⠋⠁⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⠀⢈⡇⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡷⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⣀⡀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀⡀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣛⠛⡻⣛⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⢻⣟⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠄⠀⠀⢸⡀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠒⠒⠲⠤⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠒⠒⠞⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢺⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣤⠤⠤⢤⣄⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⢄⡪⠕⠊⠉⡠⠀⠠⠂⠀⡀⠀⡈⠁⠒⠸⢼⠁
⠀⠀⠀⠐⠲⡊⠔⠁⠄⠠⢂⢀⢀⣀⣀⣠⣬⠤⠤⠖⠚⠛⠛⠚⠒⠐⢆
⠀⠀⠨⠀⣴⢰⢁⡠⠞⠃⠉⠉⠀⠀⠀⠀⠈⠳⢶⡄⠀⠀⠀⠀⠀⢀⠀⠑⡄
⠀⠀⠀⠀⣿⠧⠂⢢⠐⠄⠀⠀⠀⠀⠀⣀⣠⠀⠀⣫⠀⠤⠤⠄⠘⠒⠐⠒⠓⠒⠒⠒⠤⢀⡀
⠀⠀⠀⡆⠁⠑⠀⢅⡓⠤⣅⣀⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀
⠀⠀⢰⠡⣢⢁⠀⠀⠀⠑⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⡀
⠀⠀⢨⠁⣄⡁⠫⢆⠄⡈⠀⠀⠀⠀⠀⠠⠐⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠍⣐⠠⣄
⠀⠀⠀⠑⠿⢿⡶⣄⠉⠒⢝⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠢⢄⠈⠑⠪⢔⡤⣀⠀⠀⠀⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⢀⢀⢀⣀⣀⣀⢀⣨⡀⣄⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⣤⡀⠈⠓⠥⡒⠤⡀⠀⠀⠀⠀⠀⠀⡘⡠⠂⠀⡀⠠⢪⡁⡁⡉⡉⡌⡈⣈⣘⣜⣜⣎⣾⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣦⣀⠀⠑⠊⢕⡢⢀⠀⠀⠀⠅⠀⠀⢀⠔⣐⣵⣌⠤⠤⣴⣼⣬⣬⣬⣭⢍⣥⡴⢟⣿⣿⣿⡿⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⣄⡀⠈⠑⠭⣂⢀⣒⡓⡭⢩⡭⠴⣒⠐⠫⢏⢉⣁⣀⠴⣵⠰⠶⣓⣛⣍⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢲⣤⡀⢸⡎⠉⢀⣐⣯⣮⠮⠶⠾⠝⣛⠋⠛⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢨⣐⣼⠯⠧⡔⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣯⣶⣽⢬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣯⣼⣿⣳⣿⡽⢖⣒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢏⠉⠲⣆⠒⠀⠡⠀⡀⠠⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⢾⣄⠀⠈⠙⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡶⣖⠚⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠋⠒⢁⠘⠞⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡤⠤⠤⠤⠾⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠒⠂⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⡁⠇⢀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⢀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠂⠀⠂⢸⣿⡈⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡷⠖⠐⠀⠀⠠⣤⣤⣄⣀⣀⣀⣉⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣯⣟⡤⡀⠖⠀⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣴⡶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠶⢶⣶⣦⣶⣦⣬⣿⣿⣿⣿⣶⣶⣶⣾⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠖⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠒⠶⠴⢶⣶⣿⣿⠟⠉⠉⣿⣿⡟⣻⣒⣲⣖⣤⣤⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣦⠀⠀⣬⣿⢹⣯⠀⢁⠀⡿⠀⠈⠀⠉⣋⠿⢿⡟⡿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡻⣿⣿⣿⠂⣴⣷⣿⣿⣿⣶⣿⣶⣿⣢⣈⣠⣴⣿⢶⣿⣮⡑⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡉⣁⣠⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠀⠀⠙⠿⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⡯⣿⣣⣟⠻⢿⡿⠋⢿⢿⣿⣏⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⠳⢿⣴⠋⠘⣄⢺⣿⣿⣮⣽⠟⣎⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⣁⢛⣟⠀⠀⠉⠉⣽⡉⠉⠀⠀⢸⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣆⠍⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣿⢸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⣼⠈⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠁⠀⠈⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡯⠃⠂⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⡀⠠⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠀⣠⣿⣿⣿⣇⠀⠀⣀⣠⡄⣠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠤⠤⠤⠤⠤⢤⡤⡄⠤⠤⠤⠤⠴⣁⣼⣿⣿⡇⢠⣾⣿⣷⣤⣽⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⣠⡴⠖⠶⠛⠛⠷⠶⠶⢤⡀⠀⠈⢉⣿⣿⡧⢸⣿⣿⣿⡟⢻⣷⡧⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⣿⣿⣇⣿⣿⣿⡿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⣀⣈⣻⣿⣿⡿⠘⠫⢑⣤⠬⠡⠃⢹⣧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠉⠛⠓⢢⣶⣿⣿⣰⣿⣦⣘⣿⣿⣏⣄⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣃⣀⣠⠀⠀⠀⠀⠛⠛⡟⠩⠛⠋⡙⠻⣿⡿⠿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣴⣇⡀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⢀⠀⠀⠀⠀⣀⠁⣸⣟⣷⣾⣿⣾⣿⣤⠠⠠⣠⡄⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣶⣤⣤⣴⡀⠀⢸⣿⡄⠀⢻⠄⠀⠀⠉⢻⠉⡏⠰⠿⠾⢹⠏⠙⠀⠀⠀⠉⠀⣻⠉
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⡇⠀⠀⠀⠀⠀⠀⠈⠀⠐⠄⣀⣀⠆⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⢀⣀⠀⢀⣀⣀⠀⠀⢰⣶⣨⣤⠎⣈⠄⠀⠐⠂⠈⢈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠰⣈⡀⢀⣀⢁⠄⡠⠤⢲⣟⢭⣭⡭⢿⣷⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡶⠐⠶⠖⣒⣒⣃⣻⣃⣴⣩⡟⢻⣤⣶⣶⣾⣷⣶⣲⣾⣿⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⡟⢿⣾⣿⣿⣿⡿⢿⣿⣿⡭⢍⣽⣧⠈⠉⠁⠚⣩⣿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⣹⣿⣿⣿⣿⣯⣭⠩⢯⣿⣟⠉⠻⠟⠻⠋⠈⠁⠘⠀⠀⠐⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠭⠾⠷⠍⠉⠁⠀⠅⠉⠅⠉

⠻⠻⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠿⠏
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⣀⣀⠀⠀⠀⣤⡄
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⣙⡛⠛⠻⠏⠀⣼⣿⡇
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣍⠉⠉⠒⠒⠀⣰⣿⣿⠃
⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠿⣿⣭⣂⠀⠄⠀⢠⣿⣿⣿
⠀⠀⠈⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢷⣴⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢤⡀⠀⠀⠀⠀⠀⠀⣀⢿⣏⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣌⣻⣿⣿
⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢄⢠⡀⣀⠀⢢⠀⠀⢼⣶⣤⣤⣴⣶⡄⢸⣷⣴⢂⣀⣤⣤⣀⣀⣄⣀⣠⣄⡀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠤⠤⣤⢈⢟⣽⣿⣿⣶⡶⣤⣘⣿⣿⣿⣿⣯⣭⣥⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣁⣒⣂⡀⣀⣄⣠⣀⣼⣿⣿⣿
⠀⣠⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠒⠒⠲⠦⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⠶⠒⠋⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠦⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿
⢀⣤⠴⠶⠖⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠀⠀⠀⢀⡠⠔⠁⢰
⢸⡟⢆⠀⠀⠀⠀⠀⠀⠠⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⣸
⠀⣧⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠠⠄⠀⠀⠀⢀⣠⠴⠚⠁⠀⢠⢒⠄⠀⣠⠏
⠀⢸⡆⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⡤⠖⠋⠀⢠⠤⢴⡺⠕⢁⡴⠚⠁
⠀⠀⠻⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠈⠉⢁⡤⠞⠉
⠀⠀⠀⠘⢧⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠙⠂⠉⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠀⠀⣀⣄⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢦⣀⣠⡤⠖⠚⠉⠁⠀⠀⠀⢀⣠⠴⣫⠞⠀⠀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠯⠭⠭⠒⠉⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢳⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠷⠶⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⠘⠇⠈⠛⠁⠉⠀⠉⠛⣭⡁⠉⠀⠀⢉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢶⣀⡂⠀⢴⣾⣷⣶⣆⣁⠂⠀⠁⢀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡴⠀⢰⠄⠀⠀⠸⣿⣟⠻⡿⣿⠁⠀⠀⢘⡏⢱⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⠀⢸⠀⠀⠀⠠⠘⢀⡠⠰⠃⠀⠈⠀⠈⠗⠐⢄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠘⠀⠀⠀⠂⢀⣣⡀⠀⠦⠀⣤⣄⠀⡇⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠂⠐⠓⠈⠉⣈⠩⠅⠒⠒⠀⠤⣀⠀⠈⠉⠉⠒⠀⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⡀⠠⠤⠔⠒⠂⠉⠀⠀⠀⠀⣀⣀⠀⠀⠉⠀⠐⠒⠒⠒⠀⠐⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡀⠀⠀⠀⠀⣀⡀⠠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⡠⠄⠂⠉⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠁⠢⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠔⢊⡁⠤⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠀⠀⠀⠀⠀⠁⠂⠄⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⠋⠀⠀⠀⢀⠐⢀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠁⠀⠂⠀⠁⠀⠀⠁⠀⠀⠈⠁⠈⠁⠀⠉⠉⠀⠀⠀⠉⠀⠈⠀⠀⠀⠁⠀⠁⠁⠀⠒⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡎⠁⠈⠉⠁⠠⠭⠍⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠳⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠠⠀⠐⣂⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⠀⠤⠤⠼⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡅⢉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠐⠂⠀⠀⠀⠀⠒⠀⠒⠒⠰⠶⠅⡠⠔⠛⠛⠋⠙⠉⢨⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠨⠅⠀⠀⣀⣀⡀⠀⠀⠠⠥⠄⠰⠀⠁⠈⠀⠘⠂⠀⠠⠈⠉⠈⠀⠀⠀⠀⠀⠉⠉⠙⠋⠀⠐⠒⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠉⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠀⠐⠂⠒⠒⠒⠒⠒⠀⠐⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠀⠒⠒⠒⠀⠐⠐⡆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠁⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⢖⣶⠞⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠛⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠀⠀⠀⠀⠀⠀⠀⠘⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠆⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠤⠐⠖⣒⠀⢉⡉⣉⡉⠉⠉⢉⠉⣉⠉⢀⡒⠲⠂⠤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⡆⠀⣠⠔⢒⡩⠤⠐⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠬⢉⡒⠦⣄⠀⠰⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣋⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣱⡆⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠶⢤⣤⣄⠠⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄⠀⣤⣤⠴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠒⠐⠓⠒⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠐⠒⠒⠒⠒⠺⠖⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠉⠉⠀⠐⠒⠶⠶⠦⠀⠀⠀⠀⠠⠀⠀⠀⠴⠖⠒⠊⠉⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠄⠐⠒⠒⠒⠒⠒⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠤⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠶⠚⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⢆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠐⠒⠒⣒⣒⣂⣀⣀⣀⣒⣒⠒⠒⠒⠠⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠴⠚⣉⡠⠤⠄⠐⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠤⢄⣉⠓⠦⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⣊⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡑⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⡀⣸
⠀⠀⠀⠀⠀⠈⡦⠠⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠦⠤⠤⠆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣤⣤⣤⣤⣴⣷⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣦⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿