Dizzy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣔⣒⣀⠒⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣽⠟⢉⣩⣤⣤⣤⣤⣤⣬⣿⣒⠂⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠯⠛⠁⠢⢴⡿⠟⠙⣓⡚⢓⠛⡛⠳⠮⢤⣀⣄⡐⠻⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡫⠂⣰⡆⢴⣆⣤⣤⣈⠛⠿⢛⠉⠁⢀⣅⢀⡀⠀⠈⠙⠻⣦⣌⠐⠕⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠏⠠⣺⠟⣡⠀⠸⠟⠉⣤⢶⣳⣬⡛⢿⠿⣶⣦⣬⣉⠒⢦⡀⠈⠻⣷⣄⠲⡡⢀
⠀⠀⠀⠀⠀⣰⣳⠋⠁⢀⡼⢁⠞⡱⠃⡨⢤⣼⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⢾⡀⡈⢿⣷⣔⡹⣷⡀
⠀⠀⠀⠀⢰⢳⡏⢀⢠⡿⢠⢃⠠⠀⠐⣁⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⢩⣍⡈⠛⠻⠎⢻⢳⡀
⠀⠀⠀⠀⣿⢿⠇⠆⡟⠁⠋⢁⡄⡸⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆⢻⡟⣦⣶⠀⠈⣏⣧
⠀⠀⠀⠀⣿⡞⢠⠁⠀⠁⠠⣿⢣⠃⣿⡇⣀⡤⡄⠀⠀⠀⠀⡠⣄⣀⣀⣃⡀⠄⠀⠀⢸⡘⡈⠗⢇⡀⣴⢻⢸⣿⡀
⠀⠀⠀⣶⣿⡇⡀⢸⣠⡆⣄⣈⠘⢢⠟⠋⠁⢀⠇⠀⠀⠀⡜⠀⢿⣿⣶⠏⡘⠀⠀⠀⠈⠓⠐⣿⢹⠆⣿⠸⢸⣿⡇
⠀⠀⠀⣿⣿⡇⣷⠈⢿⡇⡇⢻⠞⠁⠀⠀⢀⠎⠀⠀⠀⠀⣇⠀⠀⠀⣠⠜⡤⠊⠙⠀⡘⢠⠰⣿⠈⡄⡿⢀⢹⣿⡇
⠀⠀⠀⢻⢿⠀⢸⣇⠸⣟⢰⣸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⣜⣠⣄⡸⡰⢃⠃⣦⠁⠘⠄⠁⡈⣸⣻⠃
⠀⠀⠀⠸⣾⡇⠈⢏⡆⢿⣖⠗⣷⡀⠀⠀⠈⠷⢄⠀⠀⠶⠮⣥⣦⠀⠀⠈⠉⣩⡞⠁⢃⣼⣿⠃⠈⠀⢠⢡⠇⡿
⠀⠀⠀⠀⢿⣿⣆⠘⣺⡜⢿⢆⠻⡷⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⡤⠤⠤⢶⠻⠗⢀⡀⠾⠙⠁⠞⠀⠐⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢻⣎⢆⠱⡹⣎⠻⣧⣿⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⡲⢤⣴⣋⣤⠴⣪⡵⡣⢦⠄⠀⠀⠠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠛⢯⣧⡙⢿⢧⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣤⣤⠴⠾⣋⠤⠖⢁⠔⡹⢋⡴⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠪⡻⣟⠎⠳⣝⡻⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠷⠾⠋⣀⠤⣚⡵⢊⠕⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⣤⡀⠙⢳⠀⠀⠀⠀⢸⠍⢉⣳⣄⣀⣴⣛⡤⢄⣠⣶⠟⢉⣪⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡻⣶⣧⠀⠀⠀⠈⣗⣒⡲⠞⠒⠒⢒⣉⣉⣥⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡀⠀⠀⢸⣽⡿⠿⠟⠛⠛⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣄⡀⠀⢀⣥⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⢶⣚⣿⣿⣿⣿⣿⣿⣶⣤⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣭⣿⣿⣿⣿⠿⣿⣿⣿⣯⣭⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣾⣿⣿⣷⣄⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣽⣿⣿⣯⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⢀⠀⠈⠢⠄⠀⣀⣀⡀⠜
⠀⠀⠀⠀⠀⡀⠂⠩⠓⠈⠥⢀⠇⠀⠀⢻⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠂⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠃
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣴⣦⣠⣦⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠙⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠋⠛⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠈⠁⠂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠘⠻⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠈⠀⢀⠁⠀⢠⣆⣀⣀⡀⢀⣀⡀⠀⡁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢂⠀⠀⢀⠀⠆⠀⠀⠀⠙⠋⠉⠛⠛⢩⠃⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠴⣄⠴⠶⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠜⠉⠉⠙⠒⠒⠒⠒⠚⠉⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣠⡤⢶⡋⠀⠙⠲⠤⠤⠷⢦⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢉⡟⠀⠀⠀⢻⡀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋⠀⠀⠀⠀⠈⢷⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⠀⢀⣠⡼⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠎⠁⠀⠈⠉⠉⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠈⠉⠉⠁⠈⠱⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⠴⠶⠶⠶⠶⠞⠛⠛⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠶⢤⣄⡀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠈⣏⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠂⠉⠙⡛⠛⡉⣡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠒⠒⠚⠓⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠃⠀⠀⠀⠀⢀⣊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠘⣷⡀⢀⣠⣴⣾⡿⠿⠿⣶⣦⣤⡀⠀⠀⢀⡼⠋⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡟⠀⠀⠀⢠⠂⠀⠀⠀⠀⢈⣿⣿⣿⡋⠉⠀⠀⠀⠈⣿⣿⣿⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠀⠀⠀⠈⠳⠤⣤⣄⣠⣾⣿⣭⣿⣿⣷⣦⣴⡾⠿⠛⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃⠀⠀⠀⠀⡀⢀⠀⢠⣿⡟⣩⣤⣤⠀⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠠⠀⠀⢠⣿⠯⠁⣩⣥⣄⠀⠀⣺⣶⣿⡿⣿⣽⣿⣿⣿⣿⣟⠉⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⡟⠀⢸⠋⠻⢟⡧⠐⡿⢻⣿⠤⠼⣿⣟⣿⣿⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠀⠀⣛⠛⠛⠀⠀⠻⡌⠁⠀⣀⡼⣩⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⢾⣿⣶⣶⣶⡎⢹⣿⠋⠈⠁⠘⣿⣿⣿⣿⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⣀⣤⣽⣿⣯⣭⣀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣾⣿⠟⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡦⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠀⠀⠀⢀⣾⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⡙⣆⠀⠈⣿⣿⣿⣿⣿⣿⠉⠉⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⡆⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢹⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣸⡏⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠙⢿⣄⡀⢀⣼⣿⣿⣿⣿⣿⣿⡟⣴⡟⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠉⠛⠿⠿⡿⠿⠿⠿⠛⡫⠶⠛⢁⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠹⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⣼⣿⡿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣀⢀⣀⣤⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⡾⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⢴⡷⠋⠀⢐⣀
⠀⠀⠀⠀⠤⣴⠎⢀⣤⢴⠁⠀⠄⠀⡰⠋⠀⢸⠄
⠀⠀⠀⠀⢰⠃⡘⠻⡁⠘⣄⡞⠉⠙⡟⠃⠀⡼
⠀⠀⠀⠀⠈⠛⠛⠶⢼⡀⠸⡄⠀⠐⡯⣑⣊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⢒⠟⢀⡀⠀⠈⠉⠉⠀⠒⠂⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⠇⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠸⡁⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠉⠠⢀⠀⠀⠀⢠⠀⢠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡽⠍⡉⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⡆⢡⠀⠀⠀⣉⠆⠀⠘⢄⠜⢿⣿⡟⡲⡄
⠀⠀⠀⠀⠀⢀⠊⠀⠈⠱⣼⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠁⠈⡆⠀⢞⠁⠀⠀⠀⠎⠀⠈⢿⣉⣩⣿
⠀⠀⠀⠀⠀⠘⠒⠒⠒⣦⠈⠢⣀⠀⠀⠀⠀⠀⡏⣢⠔⠀⣠⠛⡄⠀⠈⡲⠀⠀⠈⢢⣴⠿⣤⠛⡏
⠀⠀⠀⠀⠀⠀⡠⢀⣴⠃⠀⠀⠀⠑⢤⡀⠀⣀⡈⢁⣠⣔⠁⠀⠃⢠⠚⠆⠀⠀⠀⠀⣼⠀⣼⢋⡨⠆⢢
⠀⠀⠀⠀⢠⠐⠥⠋⠉⠉⠀⠀⠐⠐⠤⣔⢦⡀⠀⡟⢢⣌⠳⡴⣳⠁⠐⡄⠀⢄⡠⠊⠀⣀⠈⡏⠇⠐⠊
⠀⠀⠀⠀⠀⠑⠢⣀⠀⠐⢄⠀⠀⠀⠀⠀⠙⢵⣌⠁⢸⠈⢠⡰⠃⠀⠀⠘⢆⠀⠀⠀⠀⠘⢸⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠢⡀⠀⠀⠀⠀⠀⠙⠇⢄⢀⡾⠁⠀⠀⠀⠀⠈⣓⢦⠔⠒⠉⠉⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠱⣄⠀⠸⡦⣀⠀⠀⠀⠀⠈⠙⣻⡿⣄⣀⠔⡗⢮⡿⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢤⠳⡒⠚⠁⠈⠳⣔⢾⠤⠭⢽⢧⣘⡿⢌⠑⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⢱⡀⠀⢀⣠⠞⠀⠀⠀⠈⠉⢣⠀⠈⠒⠊⠀⠀⠀⠱⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠏⠉⠀⠀⠀⠈⣆⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⢀⡀⠀⠀⠀⠘⠀⠀⣰⢷⣀⡤⢤⠔⠒⢀⡟⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⠒⠒⠀⠐⠒⠒⠒⠒⠉⠻⢭⣁⠀⢀⡤⠞⠃⠀⣼⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠯⡙⣉⣀⣠⣤⣔⠗⢠⠀⠀⠀⠀⠀⠀⠓⠤⢀⣀⣀⣀⣈⠭⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠽⠟⠛⠛⠒⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣄⠀⠀⡄⠀⠀⠀⠀⢀⡠⠤⠴⣶⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢶⣶⣦⠀⠀⠀⠈⣇⠀⠀⢰⡟⢛⡄⢰⠁⠀⠀⠐⠉⠁⠀⠀⣰⠇⢀⡤⣴⠾⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣟⡆⣴⢋⣉⣤⣼⡙⡅⠀⣰⣿⡀⠸⡄⠀⠸⡄⣠⣷⢸⠀⠀⠀⠀⠀⣠⣤⡀⠉⠀⠘⠋⠁⢀⣼⠗
⠀⠀⠀⠀⠀⣀⣠⣽⣾⣯⠥⢈⡟⠻⠿⠟⠛⣿⣃⠞⢁⠅⢧⠀⢡⠰⣶⣧⠙⠛⠀⠀⣠⣶⣿⣿⠿⠿⠟⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠛⠋⢹⡘⢇⣠⠞⠁⠀⠀⠀⠀⡧⠋⠀⢸⣀⣘⡆⠈⣆⠀⠁⠀⠀⣠⣤⠟⠉⠁⠀⠀⣰⠞⠂⠀⢀⡟⠋
⠀⠀⠀⠀⣀⠀⢀⠎⠉⠁⡽⠀⢠⡀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣷⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⡶⠛⡀⠹⣄⠖⠱⡄⠙⠛⠛⠲⣄
⠀⠀⠀⠀⠉⢁⣾⣶⣿⣿⠇⠀⠀⠙⠂⢀⢞⣭⠖⣀⠀⢀⡞⢉⡠⡀⠀⠀⠀⠀⠀⡹⢿⣇⠀⠉⠁⢠⡌⣣⠀⠘⡄⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⡌⠁⠸⠫⠆⠀⠳⡀⠀⠙⣆⠀⠀⠀⠀⢳⡀⠉⠀⠀⡄⠈⠉⣹⡄⡀⣷⠀⠠⠞
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⢈⡿⣄⠀⠀⠀⠀⠒⡊⢐⡆⢹⡀⠀⠀⠀⠀⠉⠛⢧⡀⢣⡀⢿⡿⠂⣧⣯⣤⣤⣤⣤⣤
⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠘⠒⠁⠀⠘⢦⡒⠀⠀⡇⠲⢯⣶⣿⡏⢸⡇⠀⢀⡀⠀⢀⣴⣶⣏⠒⠬⣤⠶⠾⠋⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣤⠀⠀⠀⠀⠀⠀⠈⠛⠳⣜⠦⣀⠉⢉⠀⣼⣃⣴⣿⣿⣶⣿⣿⣿⣿⣆⢠⣌⣢⣘⣴⣿⣸⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣦⡀⠀⠀⣹⡉⠓⠶⠶⠛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⠛⠿⣯⣿⡿⠿⣟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⡄⢳⣦⣼⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠿⠃⠀⠠⣎⠹⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡵⠶⠶⢦⡀⢹⣿⠿⣿⣿⣿⣿⣿⡄⢻⡟⠉⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣤⠈⠩⢭⣉⠓⡆
⠀⠀⠀⠀⢠⣦⠀⠀⣾⡡⠔⠀⠀⠀⣡⡿⠋⠉⠲⣍⢻⣿⣿⣿⡄⠁⢰⣢⠀⠻⠉⣩⡍⠉⠻⠿⢿⣿⠟⠻⠂⠀⠁⠀⠀⠀⠀⣱⠇
⠀⠀⠀⠀⠀⠁⢠⡞⠉⠀⠀⠀⠀⠘⠁⢀⡀⢠⠖⠚⣦⡻⣿⣿⣿⣆⠀⠀⠀⣀⣀⣉⣁⣐⣟⣀⣀⣀⣀⣀⠀⢠⡤⠀⣀⡴⢋⠏
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢠⣾⡫⠀⠀⠀⣾⠟⠤⣈⠹⣿⣿⡆⠀⠀⠙⠣⣟⠒⣦⠐⠒⢒⣶⠟⠉⠀⠀⣠⠞⣉⠔⢁⡀
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣰⠯⢟⣀⡀⠀⠀⠉⠀⠀⠚⠁⠙⠛⠛⠀⠀⠀⠀⠈⠑⢯⡠⠶⣯⠀⠀⠀⠀⣾⡣⠊⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠀⡏⠙⠻⠀⠀⠀⠀⢠⣞⣃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣗⡴⡼⠀⠀⠀⣸⣏⠀⠀⠲⢤⣄⡀
⠀⠀⠀⠀⢠⡀⠀⠻⣿⣄⠀⠀⠀⠀⠙⢦⣤⡴⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⢀⠎⣠⠿⠿⠀⠀⢀⠀⢿⡘⠳⣤⡀⠀⠀⠉
⠀⠀⠀⠀⠈⢿⣶⣄⠹⣿⣦⠀⠀⠀⠀⠀⠀⠲⣤⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⢀⠎⡼⠁⠁⢘⡀⠀⠛⠁⠀⠙⠢⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠈⢧⠙⠇⠙⣿⣷⡀⠀⠀⡄⠀⠀⢸⣷⢸⠁⠀⠀⢀⣠⠴⠒⠂⠀⡎⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠓⢤⡑⠄⣿
⠀⠀⠀⠀⠠⣶⣾⣧⠀⢀⣘⣿⣷⡄⠀⣧⣄⣠⣼⣿⠀⠆⠀⡴⠋⠀⠀⠀⠀⠀⢷⠀⠀⡟⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⢻⣿⡀
⠀⠀⠀⠀⠀⠈⠛⠋⣴⡿⠃⠈⢿⣿⣆⢷⡈⠉⢉⣿⠀⠈⣾⠁⢀⡤⠚⠉⠙⢷⣌⣇⠀⢣⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣠⡿⡋⠙
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⣠⣤⠀⠻⢿⠦⠙⠾⠿⠋⠀⠀⢹⣾⠋⠀⠀⠀⠀⠀⠙⣿⡀⠀⠂⠀⠀⠠⣶⠀⠀⠙⣤⣾⠏⡰⠁
⠀⠀⠀⠀⠀⣠⡄⠘⣿⣧⣿⢸⡆⣰⣶⣆⣀⣿⡆⠀⠀⢀⣿⠷⣄⠀⠀⢄⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣀⣤⠶⢋⢳⡎
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠛⠻⣿⣿⣿⠋⠉⠛⠉⠀⠀⠀⢸⣿⠀⠈⠛⠶⢤⣵⣄⣀⣀⣀⣁⣤⡤⠴⠖⢛⡉⠄⠊⠁⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⠻⠶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡴⣾⠁⠀⠀⠀⠀⠸⠁⢄⠙⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⠊⠁⠠⡏⠀⠀⠀⠀⠀⠀⡇⠘⠀⠘⠀⠈⢁⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⢰⡇⠀⠀⠀⢛⡀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⢘⠀⢿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣷⠀⢸⡇⠀⠀⠠⠈⣇⣨⡄⠀⠀⠀⠀⢸⠀⠀⢀⠀⢸⢀⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡀⠸⣧⢰⣿⣦⡾⠋⢉⣡⣄⠀⠀⣠⠆⠀⠾⢋⣴⣃⣀⠀⣰⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡄⢸⣯⡳⠌⠋⠀⢹⡀⢰⠋⠀⠀⠀⠀⠁⠀⠀⠀⠈⠈⠛⠁⠘⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠘⢧⡉⠉⠀⠀⠀⠀⢳⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⠀⠀⠀⠀⣀⡀⠀⠉⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢈⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠞⠋⠙⠻⣿⡄⠀⠀⠀⢀⣿⠿⠛⠃⠐⠤⠀⠀⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠤⢒⣀⣀⠀⠁⠀⠀⠀⠈⠁⣀⣈⣙⠃⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⠘⠉⠉⠉⠳⣀⡀⠀⠀⢤⠛⠉⠉⠉⠳⡄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⠀⠈⠀⠀⡶⠀⠀⡧⣾⠀⢰⡆⠀⠴⠆⠀⠀⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠳⣮⡇⢰⡀⠀⠀⠀⠀⣴⡿⠀⠸⣷⡀⠀⠀⠀⣠⡁⠀⣸⠀⡤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⣤⣄⣼⠁⠀⣽⣒⣀⠐⠊⠜⠳⠀⠀⠘⢟⠒⠒⠊⠁⠉⠀⠹⡏⡠⠤⠄⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢠⣸⡏⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡶⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣧⠀⠀⠀⠀⠀⠴⠃⢈⡀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⢀⡟⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⢸⠀⢰⣄⠉⠀⠀⣠⣶⡆⠘⠀⠀⠀⠀⢸⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⠛⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢸⣏⣉⣉⣉⡛⣛⣛⣒⣉⣉⣉⣉⣉⣩⡏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢨⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⣦⡀⠀⣤⣤⣤⣤⣤⡄⢀⣴⣶⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠂⠤⠶⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⠀⠀⠿⢀⡴⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠈⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠙⠲⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢦⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣯⡿⠙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠐⢎⠂
⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠁⠀⡁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠂⠢
⠈⠐⡆⠀⠀⠀⠁⠀⠀⠀⠦⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⢠⡄⠀⢅⡂⠀⠀⠀⠐⢠⠀⢀⡀⠐⠠⠀⠚⠲⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⣼⢢⡄⢀⠀⠀⠀⠂⠜⢦⠀⣤⠤⠇⠰⠃⣅⠀⢠⠀⠀⡦⠍⠈⢀⡖⠀⠀⠀⠀⢹
⠀⠀⡎⠀⠀⠀⡀⠄⢁⠰⠉⠉⠀⠨⠂⠂⠀⠘⠋⠐⠃⠀⠀⠀⠈⠄⠐⠖⠊⠑⠀⡇⠀⠂⢀⣿⠧⢄⠀⠀⠘
⠀⠌⠀⠀⠀⠀⠡⢄⢀⠀⠀⠀⠀⡦⣂⠍⠂⢀⠀⢀⠀⠀⠀⠀⢀⠔⠀⠁⡔⠁⡎⠑⢄⠀⠋⢛⠟⠛⠉⠦⡀
⠀⢐⠀⠀⠀⠀⡄⠈⢢⠑⠀⢀⠈⡠⠊⠀⢉⡟⢀⠿⠀⠠⠒⠀⠀⣠⡆⠀⢀⠌⠁⠀⠈⡀⠀⠀⠀⠀⡀⠀⢃
⠁⠈⠉⢲⠀⠀⠎⠢⠀⠂⠔⠡⠊⠅⠀⡐⠀⠀⠀⠀⢆⡀⠈⠐⠂⠁⠩⡀⣀⣀⣀⠀⢰⣾⠀⡀⣀⠀⡀⠀⢸
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠂⠐⡁⠀⢀⠎⠀⠐⠤⠤⠔⠃⠀⠤⠀⠀⠀⠀⠑⠒⠳⠤⠤⠤⠋⠙⠃⢀⠀⠓
⠀⠀⠘⠀⠀⠀⠂⠐⠂⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣉⣀⡀⠀⠀⠀⢀⣀⣤⣠⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡼⠃⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣧⠉⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⣸⣿⡿⠟⠋⠁⣴⣿⣿⣿⣿⣿⠿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠟⠀⠀⠛⠉⠀⠀⠀⣰⡿⠿⠟⠋⠁⠀⠀⣰⣿⡿⠿⠟⠛⠉⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣭⡀⠀⠀⠀⠀⠀⠐⠋⠁⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⢈⡃⠀⠀⠀⠀⠀⠀⢰⡟⠛⢷⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠘⠲⢤⡄⠀⠀⠀⠀⠀⠀⠘⠶⠛⠁⠀⡾⠃⠀⠀⠀⠸⣄⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⠋⠁⠀⠉⢿⣄⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠘⠋⢠⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠁⠀⠀⣠⡴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢠⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⡶⠶⣾⠋⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣤⡾⢿⣇⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠀⠀⠀⢳⠀⠋⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣟⠉⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣀⣘⢿⣿⡿⠋⠀⠀⣀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⣰⣁⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣻⠛⠛⠻⠿⠿⢿⣿⣿⣯⡉⠀⠀⠀⠀⠀⠀⢀⣼⠋⠉⠉⠒⠲⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠘⡆⠀⠀⢀⣠⣴⠞⠉⠀⠀⠙⠲⠤⠴⠶⠶⠛⠉⠀⠈⠛⡷⡀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠘⠀⠚⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⣄⣼⠏⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣶⠾⠿⠛⠋⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠛⠻⠾⠟⠁

⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⠁⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠞


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣥⠂⠀⠀⠀⠀⠀⠀⢚⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣀
⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠈⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡖⠀⠀⠀⠀⠀⠀⠋⣿⠃
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠊⠉⡇⠁⢐⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡴⠋⠘⡆⠀⢉⢶⠲⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣧⠔⠁⠈⣇⢀⠔⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠋⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡰⠁⢀⣀⣰⢇⠞⠉⠳⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠋⡜
⠀⠀⠀⠀⢸⠁⠀⡸⠉⣀⡜⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠤⡄⠈⠙⢦⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⡼
⠀⠀⠀⠀⡌⠀⠀⡇⠸⣅⡀⡗⢤⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠳⡀⠀⠀⠈⣧⣀⣠⠴⡆⠘⢣
⠀⠀⠀⠀⣇⠀⠀⢳⠀⢸⡇⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠙⢆⠀⢠⣄⡀⣀⡔⢰⠚⠁
⠀⠀⠀⠀⠘⢦⡀⠀⠉⠚⠀⠀⠀⠉⠀⠧⢄⠀⠀⠀⠀⢰⢢⠀⠀⠀⠀⠈⠢⡀⠀⠉⠋⠀⠉⠁⠀⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡞⠘⡄⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠃⠀⠸⠀⠀⣀⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⡜⢆⠀⢳
⠀⠀⠀⢀⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠠⠤⣄⣀⡰⠃⠀⢇⠀⡸⠀⢸⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⠇
⠀⠀⠀⢸⠀⠀⣽⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⠒⠋⠁
⠶⠶⠖⠚⠂⠀⠻⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠈⠒⠒⠂⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠶⠶⠶⠶





⣿⣿⠟⢿⢛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠛⠉⠉⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠒⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⡖⠦⢤⣄
⠀⠀⠀⠀⠀⢀⡴⠛⠁⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣦⠀⠈⠻⣦⡀
⠀⠀⠀⢀⣴⠋⠀⣠⢴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠀⠈⢳⡄
⠀⠀⢠⡾⠁⢀⡾⠁⡾⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠛⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹⠘⢧⠀⠀⢿⡀
⠀⢀⡿⠁⢠⡟⢀⢾⠇⠀⠀⠀⡼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⡇⠈⣧⠀⠘⡇
⠀⣼⠇⠀⡞⢀⡏⢸⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠰⠺⠆⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠙⠳⡄⠀⠀⠀⡇⠀⢸⡆⠀⣿
⠀⣿⠀⠀⠁⣸⠀⢸⠀⠀⠀⡟⠀⠀⠀⠀⢀⣀⡤⠤⣤⡀⠀⠀⠀⠀⣠⠤⠤⣄⡀⠀⠀⠀⠀⢷⠀⠀⠀⡇⠀⠀⡇⠀⢸
⢰⣿⠀⠀⠀⡏⠀⢸⡀⠀⢸⠃⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⡆⠀⡠⠋⠀⠀⠀⠀⠱⡀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⡇⠀⢸⡆
⠈⣿⠀⠀⠀⠁⠀⠈⣇⠀⣾⠀⠀⠀⠜⠀⠀⣠⣤⣄⠀⠀⠑⠐⠁⢀⣤⣶⠤⠀⠀⠃⠀⠀⠀⠸⡇⠀⡼⠀⠀⠀⡇⠀⢸⠇
⠀⢿⡇⠀⠀⠀⠀⠀⣘⣆⡏⠀⠀⠀⠀⠀⢼⡿⠛⢦⡇⠀⠀⠀⠀⢾⡟⠙⢶⡇⠀⠀⠀⠀⠀⠀⣇⣰⣃⠀⠀⠀⠀⠀⣼
⠀⠸⣿⡀⠀⠀⢀⡞⢁⠈⡇⠀⠀⠀⢂⠀⠈⠳⠶⠞⠁⣀⣠⡀⠄⠈⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⡇⢀⠈⡇⠀⠀⠀⠀⡟
⠀⠀⠹⣷⠀⠀⢸⡇⠈⢹⡇⠀⠀⠀⠀⠐⢤⣀⡤⠴⣾⠁⠀⠈⠁⠑⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⣿⠉⢃⠇⠀⠀⠀⢸⠇
⠀⠀⠀⢻⣧⠀⠀⠳⣤⣨⡇⠀⠀⠀⠀⠠⢈⠠⠀⠀⢷⡀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠀⠀⣿⡠⠊⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠈⣿⠀⠀⢰⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠃
⠀⢀⡀⠀⠀⡿⠀⠀⢸⣷⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡄⠀⢰⠀⠀⢻⡇
⢰⡏⠀⢀⣼⠃⠀⠀⢸⡏⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠀⢸⡇⠀⠸⣦⡀⠘⢿⣦⣀⣠
⢹⣿⣶⣛⣡⣴⠃⢀⡿⠁⠀⣼⢡⠉⠳⣄⠀⠀⠀⠀⠀⠀⢦⡠⠃⠀⠀⠀⠀⠀⣀⡴⠋⠀⢿⣆⠸⣷⡄⠀⠉⠻⠷⠶⣿⠛⠁
⣀⣩⣭⣿⡿⢃⣠⠟⠁⣀⡼⠟⠁⠀⠀⠈⠓⢦⣀⠀⠀⠘⠦⠄⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠈⠻⢷⣬⣿⣷⣶⣤⡴⠖⠁
⣈⣉⣉⣉⣉⣉⣁⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣲⣦⣤⣤⣴⣶⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢀⣴⣿⣿⠿⠛⠉
⠀⢀⣴⡿⠛⠉
⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢻⣷⣶⣦⣼⣿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⡟⠛⠿⠇⠀⠙⠛⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣀⣀⡀⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⠀⠤⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠢⢄⡀⠀⠀⣠⣾⡯⠔⠒⠒⠂⠈⠁⠒⠒⠒⠠⢤⣙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣟⣥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢶⣤⣤⣤⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠖⠛⠓⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢠⣶⣿⣿⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣾⠁⢀⣶⣿⣿⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣰⡟⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⣿⣿⣿⣷⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⢾⣿⣿⣿⣷⣶⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄
⣿⡁⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠘⠿⣿⣿⠿⠋⠀⠀⢰⡇⠀⠀⠀⠀⠀⠘⣧⠀⠈⠿⣿⣿⠿⠃⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹
⢹⣇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢤⣄⣀⣀⡀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⣀⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠙⠷⢼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡸
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠃
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠁⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠲⠶⠶⠶⠶⠒⠚⠛⠛⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⢻⡿⠶⣶⣦⣤⣤⣄⣠⣤⣤⣤⣤⣤⣴⣶⣶⠟⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣼⡿⣆⠀⠀⠀⢉⣹⡿⣛⠿⣯⡉⠀⢀⡟⣿⣆⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣸⣿⣇⠘⢧⡤⠞⠛⠍⣶⣿⣧⡌⠛⠲⣬⠁⢿⡿⣧⡀⠀⠘⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢠⡿⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣧⠘⣷⡄⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢠⡟⠁⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠈⣿⠀⠈⢿⣄⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⢀⣾⠁⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⢹⡁⠀⠀⠛⣦⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣾⠃⠀⠀⢸⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣷⠀⠀⠀⠹⣷⡀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⣼⠏⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢷⡄⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣰⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⢻⣄⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⢰⣷⣤⣿⠇⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠰⣿⣤⣾⡆⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⢸⣿⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡟⠉⠉⠙⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣟⢀⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⣿⠀⠀⢠⡏⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⣠⡾⠋⠀⠀⠘⢷⣄⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠂⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣤⡏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⡞⠁⠌⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠦⣄⡀
⠀⠀⣠⣴⣶⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠠⣿⠀⠀⠀⣠⡶⠶⢦⡀⠀⠀⠀⠀⠹⣏⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⠘⢧⡈⣷
⠀⠠⣟⢠⡟⠰⢈⡷⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⢸⡏⡛⠲⢮⣷⠀⠀⠀⠀⢠⡟⠉⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠻⣤⣀⣀⣹⡿⠶
⠀⢀⣽⢿⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣀⣴⣿⣤⣤⡤⢤⣼⣇⡶⢶⢹⡆⠀⠀⣀⣤⣴⣿⡆⢹⡯⣭⡿⠂⠀⢀⡾⠋⣉⣻⡄
⠐⠛⠁⣾⡁⠙⢦⡀⠀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣼⠇⣠⡗⢀⡼⢿⣿⡇⢸⡀⣿⠀⠀⠙⣿⣺⣟⠁⠿⣿⡿⣄⠀⠀⣿⢡⡾⠋⣿⡇
⠀⠀⢰⡟⢷⡀⢀⣹⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢃⡶⢮⣴⢏⣴⠿⣅⢸⡍⣧⠸⣧⣼⣦⠴⠞⠋⣿⠟⠃⢀⡏⠙⢮⡳⠦⠿⠛⠀⣸⠟⡇
⠀⠀⢸⣇⣠⣵⣿⡷⠛⢷⡄⠀⠀⣀⡀⠀⠀⢾⣟⣉⣭⡭⠴⠟⣱⣿⡵⢿⡅⠀⢹⣿⡇⠙⢦⣄⣠⣤⣤⣤⠴⣿⠀⠀⠙⠃⠀⠀⠙⢲⡦⠤⠾⠋⢠⡇
⠀⠀⠈⠛⠛⠛⢷⠀⠀⠀⢻⡶⠛⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⡿⠀⢀⣿⡇⠀⠀⠀⢰⠀⠀⠀⢠⣿⠄⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠷⢦⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣁⠀⣼⣿⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣽⠂⢠⡟⠀⣾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣰⠟⣿⢹⡏⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡟⢠⡟⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⣏⣸⣧⡿⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡞⠀⢰⠀⡇⢸⡇
⠀⠀⠀⠀⠀⢀⣾⢡⡟⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣉⣿⣿⠁⠀⢸⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⢁⡀⢸⠀⡇⢸⠁
⠀⠀⠀⠀⠀⣼⢃⡾⠁⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢷⡄⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡏⢸⠀⣸⢠⡇⣿
⠀⠀⠀⠀⣰⠏⣼⠁⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⡇⢻⡄⢹⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⣀⣸⣀⣿⣸⣇⣿
⠀⠀⠀⢠⡟⣼⠃⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⢿⡏⢸⣷⠟⠁⠸⢦⣤⣤⣬⣿⣦⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣽⣟⣋⣉⣿
⠀⠀⠀⣾⣱⠇⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢸⣿⣿⡾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠛⢍⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣤⣀⣸⠂⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⡆⢠⢀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣼⣀⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠁⢸⡶⠛⠉⠉⠉⠓⢦⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠙⣦⠀⢹⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⣼⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⢈⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡇⠀⠹⣄⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠙⢦⣀⣀⣀⣠⠜⠁⠀⢸⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣧⠀⠀⠘⠉⢳⢻⡏⠁⣀⣠⣤⣤⣤⡀⠀⣸⠉⡯⠇⠀⠀⠀⣸⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠾⢸⠀⠀⢿⠀⠀⠀⢀⡇⠀⠀⠀⠷⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠀⠀⢸⡆⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠓⠒⠒⠛⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢦⡈⠉⠛⠒⠒⠒⠚⠋⠉⣨⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⡄⠘⡇⠀⠀⠀⠀⠀⢰⠇⢀⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⣀⡇⠀⠀⠀⠀⠀⢸⢀⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣽⣿⡿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⢟⣯⣿⣿⣿⣆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣒⠶⡀⠀⢈⣙⣷⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠜⠛⠛⢧⠀⠀⠀⠀⠀⠀⠀⡠⠎⢁⣀⣀⣀⡟⡷⢷⣿⠿⣛⡟⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠒⠛⠛⠀⠀⠀⡀⠀⠀⠀⡀⣤⣾⣿⣿⣿⣿⡟⠀⠈⠋⠛⠝⠚⠛⠿⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢫⣟⡋⠿⠂⠀⠀⠀⠀⠀⠀⢦⠐⢛⢁⣀⣠⠿⠷⠿⢛⣟⣘⣿⠿⠗⠛⠑⠄⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠘⢬⣅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢉⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣶⠾⠓⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠉⢻⣦⣀⣀⠀⠀⢀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠋⠀⠀⠀⠁⠋⠉⠉⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⢣⡀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣁⠀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠻⣴⣷⡶⠂⠀⠀⠀⠙⡖⠲⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡄⢰⠇⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⠣⣀⠀⠀⠀⠈⠉⠉⢛⣶⣄⢳⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠾⡀⠸⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⢸⣧⡙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢀⠤⠒⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠈⠻⢦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⡈⠓⠶⠶⠶⣻
⠀⠀⠀⡔⠁⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⢬⡊⠕⠢⠤⠖⠃
⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣁⠀⠀⠀⠀⠀⢀⣴⣊⣥⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣉⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢠⡾⠿⠷⠶⠤⠤⠤⠤⠴⠒⠛⠓⠒⠒⠒⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠐⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⣀⡉⠙⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠷⠆⠀⠀⢀⠀⠀⣤⣶⡧⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠿⠋⠀⠀⠀⢠⣄⣄⣀⡏
⠉⠀⠃⠀⠀⠀⠀⠀⠀⣼⣟⡀⠀⠰⡄⠁⠀⣿⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣄⠿⠛⠻⣿⡅⠀⠘⠀⣄⡀⠀⣤⠄⠛⠁⠀⠀⠀⠀⠀⠀⣿⠟⠋⣉⣀
⠀⣀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠃⠀⠀⠀⠀⢰⠁⠸⠀⠀⠀⠀⠃⠀⠀⠀⢸⣧⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⣾⠿⠿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⡄⢸⣶⣶⣄⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣀⡀⣠⣮⠀⠀⠀⠀⠀⠸⠇⢠⣴⠶⠀⠀⠀⠀⢠⣿⢻⡿⠁⠈
⠐⠋⠀⠀⠀⠀⠀⢀⣰⣾⣿⠿⠟⠋⠉⢓⣟⣋⣉⣽⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⢻⣷⣿⠁⠀⠀⠀⠀⠀⠀⢿⠼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⠇⠀⠀⠀⠀⠘⠿⠛⣿⡿⠋⠉⠀⠀⠹⡆⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠘⣿⡄⠀⢀⣀⣀⣀⣀⣀⠀⠀⣤⠀⠀⠰
⠀⠀⠀⠀⠀⠐⡶⢶⣀⣤⠀⠀⠀⠀⠀⢠⣤⣤⣿⡇⠀⠀⠀⠀⠀⣷⠀⠀⢀⣄⣠⣘⡀⠀⠀⠀⠀⠀⠀⠀⡎⠰⣾⣿⣿⣿⣿⣿⣿⣿⣶⣽⡆
⠀⠷⠀⠀⠀⠠⠷⠾⠏⠉⠀⠀⠀⠀⠀⠸⠿⠿⠏⠇⠀⠀⠀⠀⠀⠷⠶⠶⠾⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠸⠘⠁⠈⠙⠻⠿⠿⠿⠿⠿⠿⠿⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⣿⡇
⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⣈⠀⡅⠀⠀⠀⠀⡈⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⡇⠀⠀⠀⠀⠲⣲⣾⡆⠀⠀⠀⠶⣿⣿⡟⠃⠀⠀⠀⠀⠀⡁⣹⠃
⠀⠀⡄⣀⣤⣤⡼⠀⠀⠀⢂⠲⠇⠀⠀⠀⠀⠃⠻⣿⣇⠀⠀⠀⠀⠈⢛⠇⣀⠀⠀⠀⠀⣠⠀⠀⣦⡀
⠀⠀⡇⠙⠛⣻⡷⠀⣀⣀⣿⣿⣯⣉⡅⠀⠀⣄⣀⣻⣿⡇⠀⠈⠠⠖⠀⠀⠈⠇⠀⢀⣀⣤⣤⡠⢀⢤⣄
⠀⠀⡇⣿⣿⣿⡇⠀⣿⡏⣿⠟⠹⠿⡇⠀⠀⠈⢉⠉⢹⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⡎⣿⣿⡟⠁⠀⠀⢿⡆
⠀⠀⠗⠶⠿⣿⡇⠀⣿⣇⡟⡞⢟⣻⡇⠀⠀⣶⣶⣶⣾⡇⠀⠀⢀⠀⠘⣿⠇⡀⠀⣧⣿⣟⡃⠀⠀⣄⣼⡇
⠀⠀⠁⠀⢉⠉⢹⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⠻⣿⣿⠀⠀⠀⠀⠉⠉⠉⠁⠄⡆⠹⣝⢿⣉⣠⣤⣽⣿⠁
⠀⠀⡤⢇⡀⠳⡼⠀⢠⣤⣿⣯⣭⣭⡅⠀⣤⣾⣾⣿⣿⡆⠀⠀⡀⠀⠀⣀⢐⣰⠇⠀⠹⣷⢎⢻⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣖⡲⢿⣶⣶⣶⣶⣷⣆
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠃
⠀⠀⠀⠀⣰⣿⣿⣯⣿⣭⣯⣛⣿⣟⠋⢹⠍⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣤⣄⡀⠠
⠀⠀⢀⣷⣮⣝⣉⣛⣋⣛⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠌⠟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⢀⣀⡒⠒⠒⠒⠀⠀⢀⡀⢀⠐⣄⠒⣄
⠀⠀⠀⠘⣿⣿⣿⣿⠁⠁⠦⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠐⠀⠀⠀⢈⠉⠀⠀⠀⠀⠀⠀⠀⢂⢨⣭⣄⡁⡝⡤⠆
⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠶⠋⠐⠒⠂⢄⡸⡿⠁⣠⣤⣤⣤⡄⠬⠉⢤⣀⣸⡣⣿⣗
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠲⠶⠼⠭⠿⢿⣿⣄⢠⣛⣉⣙⢃⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠘⢧⣄⢀⣀⣀⡀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀⠲⡶⠶⠒⠶⠋⠉⠳⠆
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⢸⠃⠀⠁
⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀
⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀
⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣀⢻⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⠦⡀⠀⠀⠀⠀⢹⠀⢻⢢⡀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠓⠦⣀⣀⡈⢇⠘⣷⣭⡳⢤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠙⢮⣣⣈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⣶⣶⢶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣸⠉⢹⠇⠀⠀⠀⠀⠀⣠⠛⢲⠆⠀⠀⠀⠀⠀⢀⡼⠉⣩⠟⠀⠀⠀⡤⠦⠤⣤⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢀⡇⠀⣯⠀⠀⠀⠀⠀⡴⢁⡴⠃⠀⠀⠀⠀⠀⣰⠋⠀⣾⠆⠀⠀⠀⣰⠃⠀⣰⠃⠀⠀⠀⠀⡰⠁⢹⠏⠀⠀⠀⠀⠀⠀⡴⢻⣁⠀⠀⠀⠀⢸⢇⣼⠃
⠀⠘⡦⢠⡎⠀⠀⠀⠀⡾⠤⠬⢿⢯⠟⠁⠀⠀⣠⠋⢐⡿⠂⠀⠀⠀⢠⠇⠀⢘⡷⠀⠀⠀⠀⠐⡿⠱⡷⠀⠀⠀⠀⠀⠀⢾⢀⡴⠁⠀⠀⠀⠀⢠⣌⢹
⠀⠠⡇⡸⠃⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⠟⢰⡿⠀⠀⠀⠀⠀⢈⠇⡴⠋⠀⠀⠀⠀⠀⠘⡟⣾⠁⠀⠀⠀⠀⠀⢠⣯⠞⠀⠀⠀⠀⠀⠀⣸⠙⡇
⠀⢠⡟⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣿⠶⠋⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⠃⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣸⢰⠃⣀⠀⠀⠀⠀⢰⠃⠀⡴⠃⠀⠀⠀⠀⠀⣰⠃⠸⢥⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢰⢒⠆
⠀⢀⡞⠹⢦⠀⠀⠀⠀⠀⡏⠘⠋⡟⠀⠀⠀⢀⣏⣀⠘⢛⡗⠀⠀⠀⠀⠘⣳⢖⣴⡋⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⣼⠁⡷⠀⠀⠀⠀⠀⢀⠇⠸⡄
⠀⠘⡗⢠⡏⠀⠀⠀⠀⢰⣃⢴⠀⢹⠃⠀⠀⠀⢁⡏⣀⠘⣲⠂⠀⠀⠀⠾⣥⠄⡽⠃⠀⠀⠀⠀⣴⡃⢩⠇⠀⠀⠀⠀⠀⡟⠐⣷⠀⠀⠀⠀⠀⠘⢳⢰⠁
⠀⠸⣥⣠⠋⠀⠀⠀⠀⠈⠁⠾⢶⡜⠀⠀⠀⠀⠈⠉⡝⡰⠁⠀⠀⠀⢀⣔⠁⢾⠄⠀⠀⠀⠀⠀⢹⡷⠃⠀⠀⠀⠀⠀⠀⡽⠰⡇⠀⠀⠀⠀⠀⠀⣧⡽⠂
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢉⣟⡴⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠚⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⠀⠀⣠⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢃⡴⠷⠊⢀⣤⣞⣴⠏⠀⠈⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠛⠉⢀⡠⠖⠉⠚⣡⠎⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⡠⠖⠉⠀⠀⢀⠜⠁⠀⢀⢜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡅⢀⠴⡇⣠⡆⠈⠀⠀⠀⠀⣰⣫⡄⠀⣴⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⡽⠖⠁⠐⠛⡹⠀⠀⠀⠀⠀⠼⠋⡼⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠀⠀⠀⢀⡼⠁⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡤⡿⠀⠀⣠⠎⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠔⢋⡼⡁⠀⢀⡿⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠊⠀⠀⠚⢫⢇⡤⡾⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠟⢋⣞⣥⠋
⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠟⠁
⠀⠀⠀⠀⡠⣴⡿⠚⠛⢋⡉⠙⠛⠉⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⡞⢾⡉⣀⠀⠀⠙⠁⠀⠞⢦⡀⠀⡷⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠘⡇⣨⠇⢧⠀⠀⠀⠀⢀⡀⠀⡷⢰⡇⠀⠀⠀⢀⠔⠋
⠀⠀⡴⠁⢿⠀⠈⢳⠂⠀⠀⢉⠙⢦⠀⠀⣽⠀⣠⠖⠁
⠀⠈⢻⠀⢸⠀⠀⠘⠶⠦⠀⠈⡀⣈⠇⢀⡇⣼⠁
⠀⠀⠈⢻⠈⠛⠒⢦⡀⢀⣀⡀⠉⣡⠴⢫⠖⠋
⠀⠀⢀⣮⠔⣆⡠⡄⠙⣉⠉⣛⠛⡡⠒⠃
⠀⠀⠀⠀⠀⠉⠀⠛⠊⠈⠖⠉⠋

⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡜⠀⠀⠠⠖⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠠⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠀⠀⠀⠠⣶⠠⡀⠀⠀⠀⠈⠀⠹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⠀⡀⠀⠀⠠⠋⠓⠷⠂⠉⠐⠀⢠⠀⠀⢈⠆⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠅⠀⠀⠊⢆⠀⠑⠀⠀⡤⢈⡀⠠⠀⠀⠠⠀⠀⠘⠂⠀⠀⠀⠀⠼⠆⢀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠀⠀⠁⠀⡠⠀⠀⢠⠀⠀⠀⠈⡀⠒⠀⠀⠸⢲⠀⠎⠐⠚⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣀⠁⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠠⢺⠀⠀⠒⠚⠀⠁⠁⠀⠀⠀⠀⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢌⡀⠐⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠈⠁⠂⠉⠲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠘⠀⠀⢀⠀⠀⠀⠉⡀⠀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⡀⠀⠈⠆⠀⠀⠀⠈⣤⣤⠀⠀⣀⣠⠀⠀⠆⠀⠀⠀⠀⠠⡁⠀⠈⣀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⢣⠀⠀⠂⠀⠆⠀⢠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠠⡤⠶⠂⠄⠁⠀⠄⠁⠡⠇
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣾⣶⣶⣾⣶⣶⣿⣶⣿⣿⣶⣶⣶⣶⣿⣶⣶⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⣾⣿⣿⡗
⠀⠀⠀⠀⢰⣿⣿⠏⠀⣀⡀⠀⢀⣾⣿⣁⣠⣤⣤⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⣾⣿⣿⣾⣿⠏⠀⠀⡼⠿⠟⠛⣻⡿⠁⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣷⡞⠀⠀⠀⣸⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠸⠛⠋⣹⣿⠏⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⠖⠀⠀⠘⢉⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⣾⣃⣤⡤⣤⠀⠀⠛⠋⢹⣿⣿⠋⠀⠀⠀⠀⣾⣿⣿⡤⠀⠀⠀⢸⡿⠟⢩⡿⠃
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⠁⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠘⠉⣿⠟⠁⠀⠀⠀⠈⠀⢠⡟⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠐⢿⡒⠶⠦⠤⣤⣀⣀⠀⠀⠰⢶⡶⠤⢤⣄⣀
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⠉⠓⠒⠾⠷⠶⣤⣈⠉⠙⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠙⢦⡀⠈⢯⡙⠓⠲⠶⠤⢤⣄⣀⡻⣿⣒⠶⠤⢤⣀⣀⠉⠙⠒⠶⠤⣄⣘⣷⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠦⡀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠆⠀⠈⠉⠛⠲⠦⢤⣄⣀⠉⠙⣳⣌⡙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠢⡀⠀⠘⢯⡓⠒⠶⠤⠤⣔⣿⣗⡲⠦⠤⣄⣀⡀⠈⠉⠓⠺⠿⢯⣓⠦⣍⡓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠢⡄⠀⠙⢦⡀⠀⠀⠀⠀⠈⠉⠑⠂⠀⠀⠉⠙⠒⠶⠤⣄⣰⣦⣄⣀⠙⠲⣬⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣦⡀⠈⠳⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠞⠛⠛⠻⠷⢾⣹⣍⡳⢭⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡙⢦⡀⢾⣷⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡉⣱⠞⠁⠀⠀⠀⡀⠀⠀⠀⠈⠉⠟⢛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠉⠂⠳⣍⠳⣄⠐⣿⢦⡀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⢀⠴⠋⠙⢦⠀⠀⠀⠀⠀⠀⠙⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠀⠈⠳⡈⠁⠈⠳⣝⠢⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣧⡀⠠⣾⣦⡀⠀⢈⣧⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠙⢦⡘⢮⡻⣄⠈⠳⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⠂⠱⣌⠑⢄⠈⣇⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⠳⡄⠀⢹⠐⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣸⠄⠀⠈⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢢⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⢰⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠋⠀⠸⡄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣏⠉⠹⣄⠀⠀⣸⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡈⠙⢲⠮⢤⣀⡀⠀⠀⠀⠀⡔⣀⡴⢶⠋⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢾⣠⠞⠙⠻⠿⣿⡀⣰⣛⣁⣹⣾⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⠬⡷⠯⠵⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠛⠒⠦⣄⠀⢀⠜⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠒⠂⠠⡀⢀⡞⠁⠠⣶⣶⣿⣿⡆⠀⠈⢳⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢷⣾⣶⠶⠄⢠⣄⣄⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⣀⣠⣌⠭⠖⢋⡁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣓⠭⢟⣋⣭⣴⠶⠿⠟⠛⢆⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣒⣫⣭⠭⡄⠰⠖⠒⠒⢚⠉⠀⠀⠀⣈⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠋⠳⡀⢀⠀⠀⠀⠠⠞⠘⢇⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⣀⣀⣀⠌⠀⠀⠙⡏⠀⠄⠀⠀⠰⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢀⠔⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⢹⠏⠀⠀⠀⣸⣤⣤⠋⠀⣀⣤⣷⡀⠀⠀⠉⠒⠤⢄⠀⠀⠀⠀⢐⠫⠔⣊⡤⠖⠛⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⣿⣿⡀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢣⡅⡜⢁⡴⠊⠁⠄⢐⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠀⠈⠀⠀⠀⠀⣧⣿⣿⣗⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠆⠀⠂⠀⠀⢀⠟⣿⡗⠀⠙⠦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⢃⡆⠎⠙⣼⡄⠀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣾⣶⣶⣾⣾⡇⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡇⣼⡇⢸⣿⣿⣿⡄⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⡴⢧⢹⡇⢸⠿⠿⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠶⠾⠶⠶⠶⠾⠶⢶⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠿⠓⠛⢛⠇⠘⢆⠒⢂⡛⢻⣛⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣀⠀⠁⠀⠀⢸⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣠⡔⠋⠀⣀⣈⣓⠦⠄⣴⠁⠈⢆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⡇⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠒⠂⠂⠛⠒⠐⠓⡰⠒⠋⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⢷⣶⣷⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠙⠃⠉⠉⠉⠁⠀⠍⠉⠉⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⠀⣀⣀⠀⠀⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⡾⠉⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⣿⣿⣆⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⡇⠀⠀⠀⠀⣀⠀⠀⠀⣿⠀⠀⢛⣿⣿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⡇⠀⠀⠀⣀⣿⠀⠀⠀⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⣸⣿⡿⠁⠀⠀⣾⠉⠀⠀⠀⠀⠹⣦⣀⣿⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⠓⠒⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢶⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣤⣤⣤⣾⣦⣤⣤⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠘⢿⣿⠋⠉⠈⠉⠉⠈⠉⢻⣿⠟⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⣤⡤⠴⠖⠛⠁⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⡢⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⠀⠀⠀⠀⢶⡦⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⢌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠙⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠨
⠀⠀⡕⠀⡀⠀⠀⢰⣶⠀⣦⠀⢀⣇⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢠⠀⠀⢸⡿⠀⠘⢦⣘⣟⣠⠞⠀⢰⡄⠀⠰⠀⠀⢰⠄⠀⣆⠀⠀⡆⠀⠀⠀⠀⡄⠀⢀⠦⠀⢰⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⡄
⠀⠘⠀⠀⠾⡿⢀⡼⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⢳⣄⢻⣟⣠⠟⠀⠀⠹⣄⡺⡿⣀⣀⠀⠀⠻⣄⡘⣿⣂⡾⠀⠀⠐⢿⣇⣀⡼⠃⠈⠳⣤⣀⣻⠿⣀⡾⠁
⠀⠀⢰⡟⣿⣿⡏⠀⠀⠀⠀⢠⡿⠻⣧⠀⠀⠀⠀⠉⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⡏⠉⠻⠆⠀⠈⢹⣿⣯⠀⠀⣠⠞⣿⣿⡅⠀⠀⠀⠀⠀⢹⣿⣿⠉
⠀⠀⠀⠳⣿⣿⡇⠀⠀⠀⣰⡟⠀⠀⠘⢷⡀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠘⠇⢸⡿⣧⠀⠀⠀⠀⢀⣿⣿⣿⡆⠈⠃
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠘⠋⠀⠀⠀⠀⠈⠛⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠉⣿⢻⡏⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠈⢻⢻⡟⠁
⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠿⠀⠀⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠾⠃⠼⠃⠀⠀⠀⠀⠸⠸⠇


⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠘⡒
⠀⠀⠀⠀⠀⣠⣬⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣃⠀⠀⠀⣠⡄⠈⠁
⠀⠀⠀⠀⠀⢿⣿⡿⢿⡿⢶⡤⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣷⢶⣯⠅⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⢻⡥⣮⡁⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣀⠀⠀⠘⠉⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⡄⠀⠑⠀⠀⠁⠀⠁⠀⣷⡆
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⠈⢹⣧⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⢧⡜⠀⣴⣿⣷⣤⣤⣠⣿⠁⢀⣀⡄
⠀⠀⠀⠀⠀⢀⣿⠁⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣼⠇⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠾⡆⠀⠀⠌⣧⣄⢙⡿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠉⠀⠀⠀⠀⠀⣿⢿⣇⠀⠀⣿⣿⣷⠀⠀⢰⡄⠙⣿⣷⣻⣿⣿⣧⣦⡄⠈⠓⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⣷⣶⣮⣩⣥⣴⠇⣝⣝⣄⣿⣿⣧⣿⠙⢿⣿⠟⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡻⠿⢿⣿⣿⣿⠀⠿⣿⠟⣿⣿⣾⡻⢷⣶⣶⣾⡿⠛⠉
⠀⠀⠀⣰⣷⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⣹⠾⣿⣾⣿⣿⡟⣿⡇⢸⣿⣿⣿⣷
⠀⠀⠀⢈⣭⡁⠀⠀⠈⠀⠀⠀⡀⠀⠀⢈⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢸⣿⣼⣿⣦⣿⣿⣿⣿⡇⣿⡇⣸⣿⣿⣿⣿⣇
⠀⠀⠀⢸⣿⡟⠶⠶⠛⠂⠀⠀⠛⠲⠶⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢸⣿⡇⣻⣯⣿⣿⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⣿⣿⣿⠸⣿⡇⣿⡆⣿⡏⣿⡟⠃⠙⠋⠉⢹⣿⢹⣿
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠿⠗⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⢸⣿⠁⠈⢸⣿⡇⣿⡇⣿⡇⣿⡇⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢸⣿⠀⠀⠀⠉⠀⠉⠀⣿⡇⣿⡇⠀⠀⠀⠀⠸⠟⠘⠿
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠿⠇⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠒⠚⢿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠒⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣸⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢴⡄⢠⡤⠀⣤⡀⢀⣤⠀⣤⡀⣠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⣟⠁⠀⠘⣿⣿⠁⠀⠘⣿⣏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠋⢻⡆⠀⣾⠟⢻⣦⠀⣼⠏⠹⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣿⣿⣷⣶⣶⣾⣿⣿⣿⠏⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣈⣛⣿⣿⣿⣿⣿⣿⣿⣭⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠉⠰⣤⡀
⠀⠀⢌⠻⠳⠆
⠀⠀⠘⡆⠒⠄⡇⠄⡀⠀⠀⠀⠀⠒⠊⠝⠢⠀⣒⠤⣀
⠀⠀⠀⠐⠁⠰⠛⡄⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⢤⠑⠔⢄
⠀⠀⠀⠀⠘⠐⠡⠀⠁⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⠂⢂⡰⡑⠣⠀⠀⠀⠀⡀⡀⠀⢄⢀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⠘⡁⢡⡀⠀⠀⠀⠀⠀⠀⡘⠆⢀⠁⡈⠢⡣⢆⠀⠀⠀⠓⣎⠁⢘⠩⡂
⠀⠀⠀⢀⡀⠀⢈⠆⢂⠀⠈⢄⠀⡙⢷⡆⠲⠴⠦⠞⢋⠀⠀⠀⠀⠁⢄⠢⠑⣀⠀⠠⡬⠀⢀⠢⡀⠁⢀
⠀⠀⠀⠀⢷⡐⠀⣶⡂⠀⠀⠀⠢⣑⠄⠈⠀⠐⠀⠣⠈⠀⠀⠠⡀⢢⠒⠳⢤⣤⣀⢔⡨⢀⠐⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠐⢍⠩⡃⢄⣀⠠⡀⡠⠠⣀⠀⠠⣀⠔⠂⠀⠠⠘⡀⠀⠀⢁⠌⣷⠈⡀⡀⠑⠢⠄⣼⡦⡀⢄⡢⠀⠐⡉⠒⣠⢄⡀
⠀⠀⠀⠀⠀⢀⠹⣊⡐⠂⠥⠀⠘⠚⠒⠄⠂⢄⣠⠀⣇⡀⠂⠙⠅⠊⡀⡛⡀⠀⡨⠈⠸⠦⣔⠌⠉⢊⡐⡤⠄⠀⢀⠀⠤⡬⣇⢡⡂⠄
⠀⠀⠀⠀⠀⠀⠱⣈⠒⢵⣶⠐⠁⠀⠁⠀⠐⠀⠐⠶⡂⠑⠀⣄⠀⢀⠨⠈⢀⢠⣿⣀⠢⣀⣌⢀⣈⣉⣡⠭⠤⠒⡓⠚⠒⠶⠤⡀⠋⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠈⠤⠬⣤⡀⠑⠦⠀⠂⢄⠅⣦⡑⢆⠷⠥⣘⠟⢉⠀⣀⡈⢚⡩⠟⠫⣠⠰⠵⣌⡀⠀⠉⠀⠀⠀⠈⠒⠄⡊⢆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⡒⢍⠂⠈⠊⠹⢦⡄⠠⣹⡆⢈⠃⠀⣢⠞⠁⣀⠞⢁⢀⠀⠟⠀⠰⠀⠀⠀⢀⣥⣦⠔⡈⠂⠘⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡅⠈⠞⢄⠂⠁⡐⡍⠠⠈⠂⣼⠉⠀⠁⠇⠀⠉⠉⠀⠀⠀⢀⠀⠀⠀⢸⡄⠈⠁⠘⣄⠑⡈⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢅⠀⢲⡠⡈⠂⠁⡌⡼⠁⠀⡁⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠁⠐⢿⡆⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢢⡈⡂⣻⣬⣤⡻⢰⢂⣰⡆⢁⠐⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⠖⢊⠄⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⡀⠀⢝⣌⠘⢏⢻⡎⢻⠁⠌⢀⢰⣆⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⠊⠀⢸⣇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠻⡕⢝⠶⡇⠊⠀⠓⠀⡌⠀⠁⠀⠀⠀⠀⠀⠀⠲⠁⠃⠀⢈⡑⠔⠀⠀⢸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢕⠋⣐⠅⠁⠢⠀⠜⠑⠈⠘⠄⠂⠀⠀⠀⡀⠀⠀⢹⠂⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⡀⠆⢡⠈⠎⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠳⠆⠁⡊⠀⠀⣰⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⢝⡓⠾⡤⠐⠄⢸⡾⠟⠝⠒⠦⢀⡀⠀⠀⠀⠀⠈⠀⠀⡸⢁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣅⠘⠄⠘⠢⢄⣔⠙⢶⡄⠀⠀⠠⠞⠀⠀⠠⢈⢀⠤⡪⠜⡡⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢖⠢⠄⡐⠙⠳⠶⠬⠤⣀⣉⠀⠫⠔⢊⣩⠕⠀⠔⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢢⣄⢄⠉⠤⠤⠄⠀⠀⠠⠄⢚⠁⠠⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢷⣶⠾⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠻⣷⡀
⠀⠀⠀⠀⢠⣾⠟⠋⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠋⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⡶⠟⠉⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢠⡿⠉⠙⠻⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⠀⠀⣀⣠⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣶⣶⣾⠛⠁⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣷⣦⣄⣀⣀⣸⡟⠀⠀⠀⠀⣰⣿⣇⡀⠀⠀⠀⠀⣀⣠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⠁⠀⠀⠀⣰⣿⠛⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢷⣆⠀⠀⣼⠇⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢻⣆⢰⡿⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⣦⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠻⣿⡇⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠋⠁⠀⠀⣿⠀⠀⠀⠾⠃⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣶⣶⣤⡀⠉⠀⠀⣰⣿⠃⠀⠀⢀⣶⠟⠛⠛⠛⢿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣰⣿⠟⠿⣿⣷⡀⠀⣰⣿⠃⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣵⣿⠃⠀⠀⠹⣿⣿⣶⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠈⣿⣄⣠⣤⣤⣼⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠙⣿⠈⠉⠉⠛⠻⠶⣦⣄⡀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⡔⣂⠀⠀⠀⠀⣶⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠙⠟⠀⠀⠀⣠⠛⠦⠀⠀⠀⠀⠉⠻⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢀⠼⠑⠆⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠛⠙⠂⠀⠀⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠁⠀⠀⠀⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⣦⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣤⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⡀⠀⠁⠀⡀⠀⠀⡠⠀⢓⠴⠊⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠈⣠⠀⠈⠑⢤⢠⣩⠚⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡖⠰⠿⠁⠛⠁
⠀⠀⠀⠀⠀⢠⣿⠋⠀⢀⣾⣿⣷⡤⠀⠶⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠲⠏⠀⠉
⠀⠀⠀⠀⢀⣾⠃⢀⣴⣿⣿⠿⠋⠀⠈⣠⣄⡈
⠀⠀⠀⠀⣼⠃⣠⣾⡿⠛⠁⣀⣤⣶⣿⠿⠛⠋⠁
⠀⠀⠀⣸⢃⣾⠿⢋⣠⣴⠾⠟⠋⠁
⠀⠀⢰⣷⣿⠵⠞⠋⠉
⠀⠀⠛⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⠶⠾⠿⢿⠿⣛⣛⣛⣿⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⢋⣉⣭⡷⣮⣵⡶⢛⣻⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⣿⣋⣳⣦⣭⠿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡝⢻⣶⣤⡀
⠀⠀⠀⠀⠀⣠⡶⠋⢀⡠⣾⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⡲⢏⡻⣦⡀
⠀⠀⠀⢀⡼⠋⠀⣰⣮⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣿⣭⣷⣶⡿⣦⡀
⠀⠀⣠⠟⠀⣠⡾⣭⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣼⢿⣄
⠀⣰⠏⠀⣰⣻⣶⡿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣋⣷⣥⢻⣆
⢰⡏⠀⣴⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⢻⣿⡿⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣆
⣾⠀⢠⣌⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⡟⣿⣿⣿⣿⣿⡏⣿⣿⢿⣿⣿⣿⣿⣿⣷⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡄⣿⡆
⣷⠀⢸⣯⣶⣻⠿⣿⠿⣿⣿⢿⣿⣿⣿⣽⠿⡇⣿⣧⠿⣿⣿⡻⢻⡇⣿⣿⢻⣿⣿⣻⣿⣿⡌⣿⡧⢟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⡶⡙⣿
⢿⠀⢸⣧⡿⣿⣄⡛⠻⢿⣞⣻⣯⣽⣿⣿⣿⣿⡟⣿⣼⣏⢾⣱⣯⣿⣼⣿⣯⢿⡙⢷⣛⣿⢧⣿⣿⡻⣿⡟⢷⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⡇
⠘⣇⠈⣯⣿⣟⣿⠟⠀⠀⠈⠙⠓⠾⢿⡿⢿⣟⣷⢞⣿⣦⡉⠛⠛⠿⢯⢟⡿⣷⣥⠀⠉⠑⠛⠴⣝⢿⣦⣄⠀⠈⠛⢿⣿⢿⢿⣧⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⢀
⠀⠙⣦⣭⣼⠝⢁⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠉⠉⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠑⠦⣽⡽⢿⣯⣿⣽⡿⢟⣿⢸⡇
⠀⠀⠈⣠⠁⠀⠈⠉⠁⠀⣀⣀⣀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢤⣄⠀⠈⢟⣫⣷⣿⣮⣽⡋⡹⣾⡇
⠀⠀⢰⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⠀⠀⠀⠀⠉⠀⠀⠸⠋⠀⠀⠈⠙⢿⣴⡟
⠀⠀⣾⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠠
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡁
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢻⣎⠻⣦
⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⠁⠀⠀⠀⢻⣆⣹⢷⡄
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣬⣯⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⣭⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠉⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣏⣧⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣧⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⢿⣼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣾⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢐⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠾⢟⡿⠋⢁⣀⣤⣤⣤⣤⣴⡶⣶⣶⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠁⣠⣶⣫⣴⡾⠟⠋⠉⢉⣉⣥⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⡀⠀⣠⡾⠟⠋⡉⠀⠀⣀⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⣠⠞⠀⢀⣡⣤⣶⠏⢀⣠⣾⣫⣤⣤⣤⣤⣤⣴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⠟⣋⣴⡿⠋⣠⠞⢋⡵⠋⣠⠴⠟⠛⠛⣉⣭⣭⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣡⠞⢋⣼⡡⠚⢁⡴⠋⠀⠀⠀⠀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣧⣾⠟⠁⢠⠟⠋⢀⣴⣫⣴⣶⠶⢃⣴⠟⠋
⠀⠀⠀⠀⠀⣾⢋⣠⠥⠤⠤⣅⠀⠰⠛⣩⠽⣋⣥⡶⠟⠁
⠀⠀⠀⠀⢸⡿⣩⠘⠁⠀⠀⠈⣧⢠⣞⣵⠟⠉⠁
⠀⠀⠀⠀⠈⣷⠉⣙⠀⠀⢀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠷⣭⣀⣀⣭⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠻⠏⠁⠀⠀⠀⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠸⣧⣄⡀⢠⣾⡇⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣧⠀⠀⠈⡀⠀⠇⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣰⣦⣾⡷
⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⡆⠀⣾⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠛⠿⠀⠀⠀⢀⠫⠄⢂⠁
⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠈⠛⠁⠀⣴⣦⣶⣆⠀⠀⠀⣿⣿⣇⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣦
⠀⠀⠀⠀⠻⢿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⣿⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢻⣿
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡤⠀⠀⠀⡀⠀⣿⠃⠈⣿⡁⠀⠸⠝⢛⣛⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⢀⡀⢀⡀⠀⢸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠿⡿⠟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣴⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠉⠹⣦⣤⠁⠀⠘⡟⠙⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣤⣄⣴⣿⡏
⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⣁⠀⠀⠀⠀⠀⠀⢀⠏⠘⠀⠀⠀⠀⣻⣦⡄⠀⠀⠀⠉⢻⠟⠋
⣰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣾⣼⡁⠀⠤⠄⠀⠸⠛⠇
⠿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣿⣦⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶
⠀⠀⣰⣷⡀⠀⠀⠀⠀⢀⡀⠉⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⣦⠀⠀⠷⠀⠀⠀⠀⠹⣿⣿⡇⠀⣠⣤⣴⣶⣦⣤⣼⣿⣿⣿
⠀⠀⣿⣿⣿⡄⠀⠀⠀⡚⠉⣷⡆⠀⠀⠀⠀⠄⠀⢠⡤⡀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿
⢀⣴⣿⣿⣿⡇⠀⠀⢠⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣼⣿⣿⣿⣿⠏⠁⠀⠈⠙⣇
⣿⣿⣿⣿⡿⠁⠀⠀⠀⡀⢸⣏⣿⣿⠀⠀⠀⠀⠀⠐⠀⠀⣠⣄⡀⣰⠒⢲⣾⣆⠀⠀⠀⣀⣿⣿⣇⡈⠃⠀⠀⠐⠀⠘⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡿⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣶⡶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣞⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⣷⣦
⠀⠀⠀⠀⠀⢀⣠⠴⣦⡴⠶⠿⢿⣿⣿⣫⣵⣿⡟⠀⠀⠀⠀⠀⠀⡠⢀⣴⣿⣿⠋⠁⠀⠀⠀⠀⠀⣠⠄⣷⣻⣿⣿⣿⣷
⠀⠀⠀⠀⣴⡿⠀⢶⣾⣻⣿⡆⠛⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣠⣼⣷⡿⣫⣿⠋⠀⠀⠀⠀⠀⠀⡴⠀⠐⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⢾⠁⠶⣶⠈⣿⣿⣿⣿⡞⢛⡵⠋⠀⠀⠀⠀⠀⢰⣿⣧⠉⠛⣰⣿⠕⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣆⣷⣤⣾⣿⣿⣿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠷⠠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⣿⣿⣿⣿⣿⣿⣾⠿⠁
⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⢄⠀⠀⠀⠀⠙⢸⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⢀⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⠀⠀⢰⣮⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡟⠀⠘⠋⠀⠀⠀⠀⠀⠀⣴⠒⠀⠀⢀⡛⢿⣦⡀
⠀⠀⠀⠀⠀⢀⡤⢾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣤⢾⡿⢿⡄
⠀⠀⢠⠞⠁⠋⠀⣬⣿⣿⣿⣿⣿⠀⠀⠀⠙⢛⣆⢢⠀⢀⠠⠄⠀⣽⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠰⠟⣠⣿⣾⣿⡟⠁
⠀⠀⠠⣦⠀⢸⠀⣹⣿⣻⣿⠛⠁⠀⠀⠀⠀⠀⡨⢿⠿⠧⠤⢀⣴⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣧⣿⣿⠃
⠀⠀⠀⠸⡇⢸⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢸⣷⣾⣿⡻⣿⣶⡢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠯⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠊⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣾⢿⣶⣶⡆⠀⠈⠀⠀⠀⠀⠀⢀⡷⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣽⣭⣤⣤⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿
⠈⠻⣌⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠀⠀⡀
⠀⠀⠈⢿⣽⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⣞⠋
⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠿⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠒⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠓⠂⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠂⠈⠛⠛⠛⠛


⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣙⠃
⠀⠀⠀⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⣴⣿⣿⣷⡄⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⡇⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⣩⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⡇⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢻⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⣀⣀
⠀⠀⣿⣿⣿⣿⡇⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⣿⣿⣿⣿⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣒⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣀⢸⣷⢀⡀⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣾⣧⣿⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣾⣿⣦⣤
⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢟⣭⣽⣾⣵⣶⣤
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⣿⣽⡿⢿⣭⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣡⣿⣿⣿⣿⣇⠀⢹
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠛⠛⠛⣵⣿⣿⣎⡟⠛⠛⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⡾
⠀⠀⣿⣿⣿⣿⢀⣠⣤⣶⡷⠀⠀⠀⠀⠻⣷⣶⣶⣦⡸⢿⡿⢏⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠾⠷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠛⠛⠛⠋⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠙⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠙⣿⣿⣧⡀⠀⠀⠀⠀⠉⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⡇⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⡞⠁⠀⠀⠀⠈⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⢳⡄⠀⠀⠀⠀⣿⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⠟⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⡰⠋⠀⠀⠀⣠⣾⠟⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⡄⠀⠙⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣿⠀⡠⠋⠀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⢠⠶⠋⠀⠀⠀⠉⠉⠉⣷⣄
⠀⠀⠀⣠⣤⣤⣤⣤⣶⠞⠋⠁⠀⢸⡇⠀⠀⠔⠊⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣇⣀⣀
⠀⠀⣼⠋⠀⠀⣠⡾⠃⠀⠀⠀⠰⣏⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⣰⣿⡿⠁⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠈⠉⠙⣇
⠀⢀⡏⠀⠀⢸⡏⠀⠀⠀⠀⣠⠶⠾⡷⠶⠶⠟⠛⠋⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⢹⣿⠶⠶⠶⠿⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠔⠋⠀⠀⠀⢈⡇⠀⠀⢠⠊⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣿⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⡠⠃⠀⢀⡤⠶⢶⡟⠛⠋⣿⠏⠁⠀⠘⣿⣿⡄⠀⠀⠀⠈⠙⠛⢶⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣰⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿⣿⠛⠿⢷⣶⣶⡒⠚⠛⢫⡉⠉⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠉⠛⣷⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⣤⣤⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡚⠛⠋⢩⡞⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⢻⣆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⢋⣽⣿⠟⠁⠀⣿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⢀⣴⣿⢿⡅⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⣴⣿⠟⠁⠘⣿⡀⠀⠀⠀⠹⣦⡴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠁⠀⢠⣾⡟⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠲⠁⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⢧⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⠁⠀⠀⠀⠙⡆⠀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⣧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁

⠀⠀⠀⠀⠀⠄⠀⠠⠠⠀⡀⠀⠀⢀⡀⢀⣠⣤⡟⠀⠀⠀⢀⣀⣤⢀⠀⠀⠀⠀⠀⠀⢠⡤⡄⠀⠀⡀⡐⢀⠤⢤
⢠⢀⠀⠂⠃⠀⢃⠀⢀⣐⠁⠈⠁⠀⠛⢹⠖⠚⠃⠀⠀⠀⠀⠉⣀⠀⠁⠀⣤⠶⠓⠂⠈⠀⢠⠀⠀⠀⠠⢈⠀⠐⠂⠀⠀⠁⠆⠁⠀⠉⠀⠀⢀⣐
⠘⠀⠀⠀⠁⠀⠀⠘⠁⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⡈⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣔⠲⢤⣤⣀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠚⠣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠟⢋⣉⣉⣙⢛⣻⣿⣷⣾⣿⡿⠿⢮⡉⠻⣦⡀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣯⡴⣫⢞⡡⠖⢈⣉⣿⣿⣿⣿⣽⣶⣤⣄⠀⢹⡀⠀⠹⣆⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⣴⣿⣿⡿⠞⡡⠊⢠⠾⠛⠉⠉⠁⠉⠉⠙⠛⠻⢿⣧⠀⠳⡄⠀⠈⢷⡀⠀⠈⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡼⠛⠉⠀⡠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⣹⣆⠑⡄⢻⡄⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢠⡟⢀⠀⡔⠌⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠸⡎⣿⡤⠤⠔⠒⠣⣀⠀⠀⠀⠀⠀⡠⣄⣀⣀⣀⣀⣤⡾⠛
⠀⠀⢸⡃⢸⣦⣥⣾⣟⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣀⠹⣿⣿⣼⣿⡿⣿⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠸⣇⠘⣿⣿⣿⠃⠞⠉⠉⠉⠛⠻⠆⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠈⠀⣿⣿⣿⡿⠃⣿⡇
⠀⠀⠀⠘⣷⣌⣻⣿⡆⣠⣤⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡶⠾⠶⣶⣶⠀⣿⣿⡟⢡⠂⣼⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣇⠈⠛⢧⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠿⡿⠴⠋⠀⠠⣿⣿⣰⡗⣴⠃
⠀⠀⠀⢀⡞⣿⢹⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⢠⣿⡿⢣⡿⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⣿⣿⡵⢋⣴⣏⣻⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⡇
⠀⠀⣿⣿⡀⢸⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠠⠔⠀⠐⠄⠀⠀⠀⠀⠀⠀⣾⣿⣾⢹⣿⡇
⠀⠀⠘⣟⢷⣄⠻⣿⣿⡜⠃⠀⢀⡄⠄⠀⠰⢶⣿⣷⠶⠀⠀⠀⠀⠀⣸⣿⣿⠏⡼⣸⠃
⠀⠀⠀⠈⠛⢿⣿⣾⣿⠀⠀⠀⡌⠀⠀⠀⠐⠄⢀⡀⠄⠀⠀⠀⢀⣼⣿⣿⠏⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠸⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡤⢻⡟⣾⣿⠃⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⢰⠏⢠⣦⣄⣀⣀⡤⠖⠋⠀⠸⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣡⣾⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⡀⠀⠀⢠⣾⡿⢋⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⣿⣿⡏⠻⡄
⠀⠀⠀⠀⣾⡿⡑⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⣿⣿⠀⣷
⠀⠀⠀⣀⣿⡇⠃⣼⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢻⣻⣿⡎⣯⢀
⠶⡶⠻⢤⡹⣿⣦⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣚⣣⢼⣯⣿⡇⣼⣠⠟⠓⠶⢤⣤⣀
⢄⠀⠀⠀⠙⢮⡙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣮⣛⣋⡴⠋⠀⠀⠀⢠⡞⠁⠀⠀⠑⢀
⠘⡇⠀⠀⠀⠀⠉⢶⠁⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠝⠋⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢷⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡐⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣼⠛⠿⠏
⠀⠀⠀⠀⠀⣴⠾⠿⠛⡙⣩⣥⣾⣿⡟⢿⡄
⠀⠀⠀⠀⢀⣥⣀⣶⣿⣿⣿⣿⣿⣿⣿⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⢤⣤⡄⠄⠀⠀⠀⣄⣴⣷⠶⣏⣠⠼⠗
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠋⠀⠀⢵⠴⢿⡇⠀⠂⠐⠀⠈⠹⣟⣾⣿⣷⡆⣠⡆
⠀⠀⠈⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡔⢂⣀⠀⢈⣭⣿⣿⣿⣿⠇⣿⠔
⠀⠀⠀⢹⣿⣿⣿⡇⠞⠳⠈⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡾⠿⠻⠀⠸⠭⠒⠀⠉⠑⠆⠀⢀
⠀⠀⠀⠈⣿⣿⣿⣷⡀⠴⠶⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠊⣩⠥⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣶⣄⠀⠀⡸⠂
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠃⠛⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠉⠛⠶⣶⠶⡄⢂⡀⠐⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠐⠒⠉⠁⠀⠀⠳⠶⠤⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠒⠒⠶⠤⠤⠤⠤⠶⠤⠤⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⢀⠴⠒⠋⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠶⠶⠶⠶⠤⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⣤⡄⠀⡀⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠿⠟⠁
⢠⠀⠀⠀⠀⠀⠒⠒⠃⠀⠉⠀⠀⠀⠈⠑⣿⠊⢠⣦⣼⠃⠀⠀⢺⠟⡀⢿⢋⠁⠘⣕⡏⣱⣶⡦⠀⠀⠀⠁⠀⢰⡿⠃⠀⠀⠀⠀⣤
⢚⣶⣦⠀⠀⠀⠰⣦⠘⣧⠀⠀⠀⠄⠀⠀⣀⡀⠿⠃⠋⠀⠀⠀⠀⠀⢱⡏⠀⠀⢸⣟⠀⠸⠟⠁⠸⠇⢀⣀⢀⡜⠁⠀⠀⢤⠔⣆⣿
⢀⣽⣿⣆⠀⢀⡋⠁⠷⠈⠀⠀⠀⠀⠀⣿⣿⣿⠆⡘⠃⠀⠀⠀⢸⣶⠀⠈⠡⠄⢀⠈⠀⠶⠀⢠⣖⠀⠀⠹⡿⠷⠾⠀⠀⠉⠠⢿⡶
⠛⠛⠋⠁⠀⠈⣇⣤⣶⣷⠀⠀⠀⣀⣀⠻⠿⠛⢸⡃⠀⠀⠀⠀⠘⡥⢂⡀⠀⣰⠀⠀⢠⣾⣷⡌⢿⡦⠀⠐⠀⠀⣴⣦⡀⠀⠀⢀⡀
⠀⠀⠀⣴⠃⠀⣿⣿⣿⡿⣀⣶⣾⣿⣿⣿⣶⣶⡿⠋⠀⠀⠀⢀⡀⣴⠛⠛⢠⠟⠲⣶⣾⠟⠋⠀⠀⠀⠀⢰⡶⠿⠚⢿⣿⠉⠻⣭⡀
⠀⠀⠀⠻⢿⣴⠟⠻⣿⣄⣿⣿⠟⠿⠿⠿⠟⣻⠁⠀⠀⠀⠀⠀⠀⠛⠶⡖⠻⠂⠀⠀⠁⠃⠀⠀⠀⠀⠀⠸⡶⢰⣄⠈⠁
⣀⣀⣀⣀⣠⣗⣀⣀⣹⣿⣿⣃⣀⣠⣿⣶⣾⣃⣀⣀⣀⣀⣀⣴⣗⣈⣀⣀⣀⣠⣆⣀⣀⣿⣦⣀⣀⣀⣀⣢⣀⣀⣀⣿⣀⣀⣀⣀⣾
⣿⣿⢛⠻⠟⡛⢻⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⣀⢀⣜⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠂⠙⢼⡉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣅⠀⠶⢀⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⣀⠀⠀⠀⣠⣾⣿⣿⣇⢚⣿⣿⣿⣿⡦⢀⣀⣴⣠⠄⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠒⠒⠛⠻⡧⣁⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡿⠻⣿⣿⣿⣷⠀⠉⠲⣤⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣾⣿⣿⣻⡆⠀⣴⡔⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⣿⣹⣿⣿⣿⣿⢧⣻⣿⣧⣷⣿⣷⣷⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡠⢻⣿⣿⣿⣿⣿⡿⡀⣿⢿⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠙⠻⠿⣿⣷⣿⠟⠺⡏⠻⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡤⠴⠞⠛⠁⠀⠀⠀⠀⢽⣦⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠲⣄⠀⠀⣠⠖⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠏⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠹⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⡗⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⡀
⠀⠠⣶⣦⣀⡀⠀⢠⡟⠀⠀⠀⡀⠀⠀⠀⣠⢒⣐⢦⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⢃⣐⡄⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠉⢻⢾⣧⣄⠀⣠⠃⠀⠀⠀⢇⠸⠿⠽⠀⠀⠀⠀⢻⡟⠀⠀⠘⡂⠿⠯⠏⠀⠀⠀⠀⠘⣄⠀⠀⠘⠳⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠔⠁⠀⠀⠀⠀⠈⠓⠠⠔⠊⠀⠀⠀⢸⡇⠀⠀⠀⠈⠂⠤⠐⠃⠀⠀⠐⠀⠈⠢⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⣤⣤⣼⣧⣤⣤⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠐⡀⢄⢀⡔⠋⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠙⢦⡀⡠⢀⠂⠀⠈⠛⣿⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠘⠟⠛⠉⠉⠉⢹⡏⠉⠉⠉⠛⠻⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦
⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠄⠤⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠤⠠⢀⠀⢀⠔⠀⠀⠀⠀⠀⣀⡾⠁⠀⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣠⡞⢳⣄⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢐⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⠀⠀⣿⠖⠒⠛⠉⠀⠀⠉⠛⠒⠲⣿⠀⠀⠀⠀⠀⢠⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠴⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠤⣤⣤⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⡴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⣀⠑⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠂⠘⠋⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⣿⠟⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠠⣾⠂⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠁⠺⠁⠀⠀⠀⠈⠄⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⣀⠋⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠀⣠⠀⠀⠀⠨⠀⠀⢠⣶⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⣥⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠀⢰⣋⠀⠀⠀⠀⠀⠂⠀⢀⠁⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠤⠊⠕⠀⠀⠀⢠⠂⠀⣀⠠⡀⠀⠀⢀⣤⠀⡠⠀⠒⣾⠇⠀⠀⠀⠀⠀⠁⠰⠂⠀⠁⣀⠀⠀⠀⠀⠀⠀⠒⠀⢭⡓⠶⣄
⠀⠀⠠⠂⠀⠀⠀⠀⡠⠢⠄⠜⠀⣨⠀⠀⠀⡇⠀⡘⠃⠀⡀⠀⠀⠀⠀⠀⠑⡄⠀⠘⠄⠠⣤⡢⠀⣠⣵⠤⡂⢀⡢⢄⠙⢦⡀⠀⠀⠀⠻⠃⠀⣿⠄⠘⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠙⠃⡀⠀⢀⠠⠂⠀⠀⡇⢰⡁⠙⠬⠁⡆⠀⠀⠀⠀⠀⠘⢄⠀⣠⡀⠻⠐⠀⢹⠚⢀⡁⡠⠀⡘⠀⢆⢣⠀⠠⢇⠄⠀⠀⢀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⠘⠦⠘⠀⠂⢤⠀⢣⡐⠎⠑⠢⠞⠀⢠⡐⢢⠀⠀⠀⠈⢆⠀⠀⢀⠀⠂⣁⠎⠀⠭⣤⠜⠀⢀⡸⠙⠀⠀⠂⣠⠤⣊⠎⣡⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠖⠁⠀⠀⠈⢇⠀⠈⢰⣒⠀⢰⠀⡎⠀⠈⡄⠀⠀⠃⠀⢮⠀⠀⠗⠤⣀⠀⠀⠀⠊⡰⠋⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡀⠉⠨⡄⠈⠑⢤⡁⠀⠀⠀⡐⠀⢹⠺⡀⠀⠀⠉⠁⠠⠭⠤⠤⠐⠉⠀⠀⠀⢀⣠⠴⠂
⠀⠀⡘⠀⡘⢃⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣄⠘⢇⠀⡇⠀⠀⠁⠀⠀⠀⡹⠀⠀⠠⠀⣤⠀⠣⡈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠃
⠀⢀⠃⢀⠃⠸⡀⠀⠀⠀⠀⠤⣭⡭⠰⡍⠙⠂⠀⠁⠷⢀⣀⣀⠔⠉⠉⡄⠀⠀⠐⠊⠉⠀⠀⠱⡄⠀⠪⡙⠒⠤⣤⠔⠚⠉⠀⠀⠀⠀⠀⡠⠁
⠀⠘⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠙⠛⠒⠀⢄⠀⠒⢂⠐⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡁⠘⢆⣴⡡⢄⢀⠎⠀⠀⠀⠀⠀⠀⢀⠴⢤⡤
⠀⠘⡄⠐⠓⠤⠵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⡀⢃⠀⠀⠀⠀⠀⡤⡠⠙⣄⠀⠀⠀⠀⢧⡄⠀⣼⡦⠀⠏⠀⠀⠀⠀⠀⠀⡴⠓⠊⠁
⠀⠀⠈⠢⡀⢠⠾⠕⢰⠆⠀⠀⠀⠀⠀⠀⡴⠾⠃⠀⠘⠄⢆⡴⣿⡟⠚⠹⠟⠂⠈⢦⠀⠀⠀⡇⠀⠀⠈⠉⢙⠀⠀⠀⠀⠀⠀⠜⠀⠀⠈⠢⡀
⣀⣀⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣎⣴⣋⣀⣀⣀⣒⣃⣀⣘⣆⣀⣀⣇⣀⣀⣀⣀⣸⣀⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣘⣦⣀⣀⣀⣀⣀
⣿⣿⡟⡛⠿⠛⡻⠻⣟⢟⣿⢟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡶⣶⣷⣾⡏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣔⢌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡻⢷⡕⢜⠦⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢽⣮⡓⢌⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠠⡝⣿⣧⠳⣮⣊⡳⣔⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢻⡌⣿⡇⢤⠳⡙⠚⠞⢦⡈⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⡘⠻⣮⣈⣉⠢⣝⢶⡝⢌⡑⢿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⣓⡂⠹⣿⡳⣌⢡⣌⢢⡙⠧⢻⣄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠹⣿⣆⢪⡉⡻⢄⢉⠳⠙⢿⡈⢿⣦⡝⣦⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣿⣯⡳⡌⡄⠀⠠⠈⡀⠓⢆⠉⡳⠌⢿⣶⣦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠙⢷⣤⣝⢦⣄⠧⠠⠀⢂⠀⢁⠀⠐⠉⠂⠲⠫⢿⣏⢿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣙⢿⣷⣮⢱⡌⠀⠌⠘⠂⣄⠁⡀⠈⠁⠉⠄⢐⡘⢧⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠹⣿⣿⡿⢷⣦⢀⡂⠀⠠⣀⠀⠆⠐⠀⠐⢈⡌⠀⢀⠁⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣌⣙⣻⣑⡲⢀⠈⠀⡄⡀⠀⠠⠄⢠⢈⠀⡀⠈⠈⡀⠈⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣝⢢⠀⢀⢀⠐⠀⠀⠀⢀⠈⠀⠀⠉⠐⢠⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣃⣝⠑⢀⠀⠀⡀⠀⠀⠀⠀⡀⠆⣀⠠⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⣷⠶⠀⡀⠁⢀⠀⠀⠆⠀⠤⠀⢦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⣠⣤⣈⣎⡼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠂⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⡀⢀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠈⠀⠊
⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⠿⢿⢸⠿⠿⠿⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠻⢿⡿⠃⠘⠿⣿⠿⠛⠛⠛⠛⣻⠋
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡼⠸⡀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣠⡴⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⠞⠀⠀⠙⢆⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣧⡴⠁⠀⠀⠀⠀⠈⠳⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠐⠺⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠷⠦⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤
⢀⣠⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⡦⢶⣶⣾⣿⣿⣿⣿⣿⣿
⣼⡏⠉⢉⡿⠀⠀⠀⠀⠀⠀⠀⣤⢢⡄⠀⠀⠀⠀⠀⠀⠸⣏⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⣿⠁⠘⣯⠉⠉⠉⠉⠉⠉⢹⡆
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡟⠸⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣸⡇⠀⠀⢹⡆⠀⠀⠀⠀⠀⢸⡇
⢿⣆⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⣷
⠀⠙⢻⡿⠀⠀⠀⠄⠀⠀⠀⣾⠁⠀⢹⡄⠀⠀⠠⠁⠀⠀⢸⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣧⣄⣀⣀⡀⠀⠀⢀⡿⠀⠀⢸⣇⠀⠀⠀⣀⣀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢀⣽
⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣶⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡊⠉⢹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠛⠛⢻
⡠⠒⠓⠒⠶⣄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠚⠋⠉⠉⠀⣀⣠⡤⠖⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠣⠤⠔⠈⠀⢀⣠⣴⠀⢀⡔⣋⣁⣀⣀⣀⣠⡤⠴⠒⠋⠉⠀⠠⢤⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⡰⠋⢠⡇⠀⠈⢿⡉⠉⠉⠉⢀⣤⡖⠀⢀⣀⣠⣄⠀⠉⢻⣦⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣇
⠀⠀⠀⠰⣇⢀⡾⠀⠀⠀⠈⢻⣆⣴⡾⠛⠁⣙⣲⣏⣥⡤⠾⢓⣫⡁⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠹⣿⠙⠋⢉⣭⠿⠷⠶⣤⢸⠉⠀⢻⡄⢸⡆⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠋⠇⠀⠀⠀⢸⠇⢀⣐⡊⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠒⢒⡞⣰⠋⠉⢈⣡⢾⡇⠀⠀⠀⢀⡼⠁⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⡤⠶⠶⠛⢉⣠⠜⠃⠀⠀⠀⡾⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡴⠚⠉⠙⠛⠒⠒⠦⠴⠒⠀⠀⠈⠉⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡤⣤⡴⠖⠚⠶⣄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡇⠀⣀⣀⣠⠴⠋⠁⠀⢹⠀⠀⠀⡴⠛⠳⠞⠀⢻⢀⡤⠤⡄⠘⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣴⡆⡇⢠⡇⠀⠀⠀⠀⠀⠀⠈⠳⠖⠊⠀⠀⠀⠀⠀⢸⠎⠀⠀⢸⠀⢸⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢉⡔⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⡸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠊⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠃⣠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢠⡀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⠔⠁⡴⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡙⠫⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡠⠊⠀⠀⠀⠀⠀⠈⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠖⠚⠉
⠀⠀⠈⣀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠐⠿⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼
⠀⠀⠀⠀⢻⣄⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⢶⣦⣤⣤⣤⣶⠶⠶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠶⠦⢤⣤⣤⣄⣀⣀⣀⣤⣤⣤⡤⠴⠶⠖⠋
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣦⣄⣤⣶⣿⣿⣿⣷⣦⣤⣴⣶⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⢻⡆⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣾⡇⠀⠈⣉⡙⣿
⠀⠀⠀⠀⠀⢀⣼⣿⠿⠿⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠉⠉⢸
⠀⠀⠀⠀⢀⣾⠋⢠⣤⣤⣀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡿
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⣿⣄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠻⠷⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢿⣏⠀⠘⣯⠉⠙⣿⡄⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⠀⠀⣿⡀⠀⢻⡇⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⡆⠀⠙⣧⠀⢸⣿⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠁⠀⠀⠀⠀⢨⣿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠉⠛⠻⠿⢷⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣷⡀⠀⠀⠀⠀⢰⣿⠃⠀⢀⣠⣶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠉⠀⠹⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠈⠻⣶⣾⠟⠋⠁⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⢻⣧⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⣀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⣿⣇
⢀⣤⣶⣶⣶⣶⣶⣶⡶⠿⠛⠀⠀⠀⠀⠀⣠⣾⠟⢁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠻⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⣀⣠⣴⠟
⠀⠉⣿⡟⠛⠿⠛⢻⣿⣷⡀⠀⠀⠀⢿⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⠉
⠀⠀⠙⢿⣶⣶⣶⠟⠋⠻⣷⣄⠀⣀⣼⡿⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠻⠿⠟⠋⠀⠀⠀⠀⠈⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡄⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡟⠻⣿⠙⢿⣶⣷⣶⣶⣾⣿⡇⠀⣿⣿⢶⣷⣶⣶⣾⡟⢉⡿⠟⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣷⢆⠿⠷⠶⠁⠀⠀⠀⠀⢹⡇⠀⣿⠇⠀⠀⠀⠀⠸⠶⠾⢧⣴⣻⠟⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⢀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠙⣿⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣼⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣿⡟⠛⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣀⣀⣀⣀⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢰⡇⣿⡿⠟⢸⣿⢸⣿⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⣿⡇⠀⠈⣿⣾⡿
⠿⠿⠟⠟⢀⡦⠀⢀⡄⠒⠶⠶⣴⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣾⣦⣸⡇⣾⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠛⠀⢠⠏⣀⠀⣀⡀⠈⢣⡈⢃⠀⠀⠀⠀⠀⢹⠀⠀⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣷⡿⢻⣿⠋⠒⠶⠱⡌⠀⠀⠀⠀⠀⠈⠀⢰⣿⠛⣿⣇⣿⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⡏⣿⣿⡯⡿⣿⣿⢿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠛⠁⣀⠀⠀⠀⠀⣴⢷⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⣉⣉⣉⣀⣉⣁⣈⡠⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠿⠛⠃⠚⣿⣿⣴⣿⢿⣷⡿⡏
⠀⠀⠀⠪⡀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠐⢶⡤⠉⠁⠤⠤⠤⠄⠀⠀⠀⠀⠒⠲⠟⠁⠀⠀⠀⠀⠀⢐⠀⠉⣀⣠⣤⣄⣙⡿⠛⠛⠓⠚⠗⠺
⠀⠀⠀⠤⠇⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣴⣋⣽⠽⠷⠾⣿⣿⡧⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠠⡏⠬⠉⡁⠀⠀⠀⠀⠀⡼⢀⡏⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠉⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⢂⠠⠀⡐⠑⢦⣤⠄⣆⢟⠀⠐⠌⢀⡦⠝⣢⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⢠⡤⠋⠀⠀⠌⠠⠤⠤⠠⠬⡄⠃⠋⢀⠈⢁⡄⣁⠀⠀⠀⠀⠀⢠⡴⣿⠯⠽⠿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⣀
⠀⠀⠐⣯⠴⠲⠜⠀⣇⠀⠀⠀⠀⠀⡄⢂⠠⠀⣠⣾⣿⣧⠀⠀⠀⢰⣿⣿⣷⣿⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⢉⠥⢘⡶⠉⣿⣿⣿⠀⠀⠀⠀⠙⠻⢿⣿⡶⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⣠⠀⢆⠀⠀⠀⡀⠀⠠⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠄⠀⣑⠀⠀⠠⠤⠤⣮⣦⣼⣶⣶⣾⣿⣿⣤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⢹⡋⠿⠈⣶⠈⠳⣴⣆⠀⣴⠞⠁
⠀⠀⠀⠀⠀⠘⠉⠂⠀⠀⢀⠠⣤⣤⣦⡄⠦⠠⢿⣶⣴⣧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣀⣴⠋⣁⠀⠆⢀⡀⠃⠘⣿⠛⣧⢠⢤⡀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠄⠀⢀⡀⣠⠤⢤⡀⣁⠀⠒⠩⠀⣰⢬⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠟⠉⠉⡄⠉⢀⣀⠈⡅⠀⠁⣼⡟⡇⠀⠀⠁⠄⠤⠁
⠀⠀⠀⠀⠀⢸⣄⠁⢀⣢⣄⡓⠙⣘⣀⠀⠄⠀⠀⣘⠬⠻⣈⡁⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⢰⠄⠀⢤⠀⠂⠈⠉⣀⠀⠀⠓⠈⣿⡇⠀⠀⡀⠀⡀⠴
⠠⠀⡶⠆⠠⠦⠾⠀⠠⠥⣼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶⠀⠀⣀⠀⠄⠰⣷⠀⢀⡀⠃⢘⠁⢻⡇⠀⠀⠈⠀⠁
⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⢀⣤⣤⣍⣀⠴⠀⢀⣄⠠⠁⠄⠉⠀⠈⠁
⠀⠀⠀⠑⠀⠐⠀⠀⠀⠐⠆⠀⠀⠀⠶⠀⠀⠀⠦⠀⠠⡆⠀⠐⣤⠀⠀⠀⠀⠀⠀⠈⣷⠟⠻⣿⣿⣿⣿⣦⢀⡄⠴⠛⢆⠀⠄⢰⠴⠆
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀⢀⡁⠀⠀⠀⠶⠂⠀⠀⠀⢀⣘⢁⡀⠀⠀⠙⢿⣿⣿⠈⢀⡂⠀⢈⡀⠀⢀⠀⢴⠀⠀⠀⠀⢠
⠤⠄⠂⠈⠁⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⣤⡔⠀⠀⢬⡁⠀⠀⠀⠈⠁⢸⢿⠏⣾⣿⣿⣿⣿⣿⣿⣿⡀⡼⠀⠀⠀⢀⣾⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠃⠀⠀⠐⣿⡟⠁⠀⠀⠘⡧⢀⡀⠀⠀⢀⣤⠄⠀⣻⡟⠛⠉⠉⠛⠀⣿⠀⠠⠤⠤⠤⢤⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⡄⢰⣿⣧⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀⣿⣇⠀⠀⠀⠀⠀⠀⢠⠤⠞⠛⢢⣤⡾⣧⣿⠁⠀⠀⠀⠹⣿⣿⣀⠀⠀⠀⡜⣸⣿⣇
⡄⠀⠀⢀⣴⣶⣶⠀⠀⠀⣴⣶⣶⣦⠀⠀⠀⣴⣶⣶⡂⠀⠉⠀⢸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢻⣿⣿⣀⠀⣀⣀⣴⣿⠋⠛⠳⢄⠠⣷⣿⣿⣿
⠉⣀⡀⢈⣉⣉⠉⣁⣀⣀⣉⣉⡉⠉⡁⢀⡘⡛⢉⡉⠃⢀⣾⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠻⣿⣏⠉⠉⠉⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠘⠅⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⠀⠀⠀⢨⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⡄⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣧⠀⢠⠦⠀⣴⢖⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠆⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠋⠿⠷⣨⢻⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡟⡸⡘⡼⡏⣯⣦⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢲⣾⣯⣶⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠓⠛⠛⠁⠉⠀⠀⠀⠀⢠⠱⢠⢣⢡⢳⢰⣿⠸⠸⣷⠢⠄⠀⠀⠀⠀⠀⠀⢰⣶⠚⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣀⡀⣀⣀⣀⣒⡀⠒⠂⠂⠘⢘⠀⡀⠂⠐⠰⠀⠀⠐⠒⢀⣀⣨⡇⡆⡜⡄⡈⡆⣼⢸⡇⠟⠘⡀⠀⠀⠀⠀⠀⠀⠀⢫⣥⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠃⠀⠸⣿⣿⣿⠋⠈⢿⢡⡇⣿⣿⠋⣿⢸⣇⢿⣿⡿⠃⠙⣿⣿⢰⢀⢣⠀⢷⢀⡇⣇⢷⡸⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠹⡿⠁⠀⠀⠀⣼⢀⡿⠃⠀⠘⡼⣿⡸⡿⠁⠀⠀⠸⣿⢘⢸⡜⢸⡈⢸⢰⣹⣸⣇⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣅⠛⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠴⠒⠚⠛⠛⠓⠒⣒⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠉⠀⠚⠉⡀⣀⣤⠤⠤⠴⠒⠋⢁⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⡡⠊⠁⠀⠀⠀⠀⠀⢀⡴⠒⠉⠀⠀⠀⠀⣠⠖⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠞⠁⠐⠉⠀⠀⣀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⣀⡤⠞⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠚⠁⠀⠀⠀⠀⡠⠊⢀⣠⣔⣩⠤⠤⠔⠒⠚⠉⠛⠉⠀⠀⢂⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⢀⠞⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣐⣁⡠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⡴⠊⠁⠒⠲⠀⠐⠢⣕⢢⡀⠔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠖⠋⠉
⣼⠁⠁⢴⣦⢾⠆⢺⠅⠈⠆⠃⠀⠀⠀⠀⣀⣠⠤⣤⠶⠒⠛⠉⠉
⡇⠀⢀⠐⠃⢀⣠⣈⣼⠀⡜⠀⢀⣀⣤⠤⠤⠖⠋⠁
⠹⡄⠀⡲⠤⡀⠀⠑⠮⠔⣁⡴⠋⠁
⠀⠈⠢⣌⣑⡒⢂⣀⡤⠞⠁
⣶⡶⠲⠖⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡟⠀⢀⣀⣠⣤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢼⠴⠚⠉⠁⣀⣤⠴⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡭⠽⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢓⡶⠂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⡆⠀⠀⠀⢀⡠⠤⠴⠲⡋⢁⡀⠄⠀⠀⠀⣀⣉⡀
⠀⠀⠀⠀⠀⢠⠏⠉⠉⠁⠀⠀⠰⣖⠋⠀⠀⢀⡼⠒⠒⢦⡀⡴⠋⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⢞⡇⠀⠀⠀⠀⠀⠻⡅⠀⠀⡟⠀⠀⠀⠀⣹⡁⣤⠀⠀⠀⢸⠀⠀⡀⣀⢠⣤⣤⣤⣤⣼⣿⡶⠆
⠀⠀⠀⠀⠀⠀⢸⢀⡄⠀⠀⠀⠀⠀⠘⣆⠀⣧⠀⠀⠀⠐⠛⣟⠛⠁⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠾⣷⠀⠀⣀⠤⠤⢄⡼⠀⠘⣆⠀⠀⢀⡼⠁⠙⠒⠚⣇⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡁⠚⠛⠳⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠁⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠭⣝⣉⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⠋⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣠⡯⠧⠄⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠓⠄⠀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠀⠀⢣⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠛⡖⣶⠛⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠤⡇⢸⣀⣀⣀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠉⠛⠃⠀⠀⡆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠑⠀⠀⠲⢶⠿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠒⠒⢶⣶⠖⢉⣹⣆⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠯⠭⠭⠒⠉⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⠛⢉⣱⠆⠀⢻⠋⠀⠀⠀⠀⠀⠀⠈⢹⠃
⠀⠀⢠⣀⣀⣀⣠⣤⠀⠀⢀⣠⡀⠀⠀⢤⡀⠀⠀⠀⠸⠀⠀⠀⣿⠀⠀⠀⠳⣬⡭⠥⠬⢗⣸⡿⢃⣀⠀⠀⣀⣀
⠀⠀⣸⠩⡉⢉⣧⣿⠀⢰⣿⡟⠓⠲⠖⠒⢷⠀⠀⠀⢠⡀⠀⢠⡏⠀⠀⠀⠀⠙⢧⣄⡀⢀⡽⢁⠋⠉⠉⠉⠉⠻⡆
⠀⠀⢻⡅⠀⠀⠁⣿⠀⣸⣯⣀⣀⣀⡀⠀⢸⡀⠀⠀⢸⡧⣧⣼⡇⠀⠀⠀⠀⠀⠀⠙⡤⠏⠀⡌⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⣧⠀⠀⣰⠇⠀⣿⠋⠉⠉⠙⡇⠀⢸⡇⠀⠀⠘⠇⠠⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠙⢻⣿⠏⠀⠀⢿⠀⠀⠀⠀⠃⠀⣸⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⣀⣀⣿⣿⡿
⠀⠀⠀⠀⠁⠈⠀⠀⠀⢸⡀⠀⠀⠀⢦⣤⡿⠀⣰⣫⣭⣅⣀⣁⣀⣨⣆⠀⠀⠀⣀⡀⠴⢤⠄⡀⠈⠉⣙⣛⠛⠉⢰⡆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⠇⠀⣿⠋⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠁⠒⠚⠓⠐⠁⠀⠀⠀⡊⠀⠀⢸⣷⡀⠀⠀⠀⠤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢨⣿⠀⠀⣿⣷⣦⣤⣴⣶⡄⢰⣿⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣧⡀⢀⣀⣀⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⠀⠀⡏⠀⠀⠈⠉⣻⠀⢸⣿⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⡟⢿⣿⡷⣾⣿⣿
⠀⠀⠀⣀⡤⣤⣀⡀⠀⠀⢸⠀⠀⠀⣸⡇⠀⠀⡇⠀⠀⠀⠀⢹⠀⢸⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠚⠷⠾⠟⠧⠀⡇⢸⡏⠀⢻⡏⡿
⠀⠀⠂⠘⠓⠚⠒⠊⠁⠀⢸⣷⠶⠾⢿⣧⠀⢠⡇⠀⠀⠀⠀⣸⠀⢸⣿⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠋⠀⢸⣇⡀⠀⠀⣠⣿⢀⣸⣿⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⣇⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⠿⠿⣿⣿⣿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⠿⠶⢿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠿⠥⣉⣹⣿⡤⠴⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠋⠁

⠀⠀⠀⣀⣀⣤⣤⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠀⠀⠀⠀⠀⠀⣰⠙⠛⠉⢰⠂⠀⠀⠀⠀⠀⠀⢀⡤⡶
⠀⠀⠀⡏⢠⠴⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡴⠛⣱⠁
⠀⠀⢀⡇⠁⠁⠐⢻⠇⠀⠀⠀⠀⠀⠀⡰⠃⢠⡏⠀⣀⣀⡀⠀⠀⠀⢸⠁⠀⠀⢸⠁⠀⠀⠀⠀⣠⠊⠀⠀⠓⠲⠤⣤
⠀⠀⢸⡇⠀⠀⠀⢸⣀⣀⡀⠀⠀⠀⡼⠁⠀⠈⠉⠉⠉⣹⠃⠀⠀⠀⡾⠀⠀⠀⠈⢡⡟⠀⠀⢰⣁⣤⣤⡀⠀⢀⡞⠁
⠀⠀⢸⠀⠀⠀⠀⠈⢀⡜⠀⠀⠀⣸⢃⣀⡤⢤⠀⠀⣴⠃⠀⠀⠀⠀⠓⢲⠆⠀⠀⡿⠀⠀⠀⠀⠀⣠⠏⠀⣠⠏
⠀⠀⠘⠛⢙⡇⠀⢀⡞⠀⠀⠀⠀⠿⠛⠉⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠁⠀⠀⠀⠀⣴⠁⠀⠰⠷⣶
⠀⠀⠀⠀⢸⡁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠃⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⡸⠁⠀⠀⠘⠟⠒⡆⠀⡼⠁
⠀⠀⠀⠀⢸⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠀⣰⠃⠀⠀⠀⠀⠀⢸⢣⡞⠁
⠀⠀⠀⠀⢸⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⠃⠀⠀⠀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⣶⠺⠛⢛⣳⠀⠀⠀⠀⠀⢸⢆⠀⠀⠀⠀⠀⠀⢀⠤⠤⠶⠖⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⠇⠀⠀⢘⡏⠀⠀⠀⠀⠀⢸⠘⣆⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⡾⠋⠋⠉⠉⣴⠟
⠀⠀⠀⠀⢸⡇⠀⠀⢠⡴⠃⠀⠀⠀⢀⡴⠊⠀⠈⢆⠀⠀⠀⠀⠇⠀⠀⢠⠇⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⡾⠁
⠀⠀⠀⣶⠚⠀⠀⠀⣸⠃⠀⠀⢀⡴⠋⠀⠀⢐⡴⠋⠀⠀⠀⢸⠀⠀⢀⡟⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣰⣏⣀⣠⡀
⠀⠀⠀⢻⡄⠀⠀⡶⠚⠀⠀⠀⠙⡄⠀⠀⠐⡏⠀⠀⠀⠀⠀⣘⠀⠀⢸⠇⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⣨⠇⠀⠘⣇⠀⠀⠀⠀⠀⠘⣄⠀⠀⠱⣄⠀⠀⠀⠀⡏⠀⠀⠈⢛⠟⠀⠀⢰⣏⣤⣤⡄⠀⠀⣠⠞
⠀⠀⠀⠹⡄⠀⢾⠉⠀⠀⠀⠀⠀⠀⠈⠦⢤⢂⢸⠀⠀⠀⠀⠓⠒⡿⣠⠋⠀⠀⠀⠈⠀⠀⢰⠃⠀⡴⠃
⠀⠀⠀⠀⠻⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡄⠀⠀⠀⠀⣸⣱⠁⠀⠀⠀⠀⠀⠀⠀⣿⢀⡞⠁
⠀⠀⠀⠀⠀⠹⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠉
⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⣀⡴⢚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⡤⠶⡷⠚⠁⢐⡭⠖⠋
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⡀⢀⡤⠚⠉
⠀⠀⠀⠀⠙⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠶⠋⠠⠂⠀⠁⢀⡼⠃⠀⠊⠀⠀⠀⣜⣲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠵⢶⠴⠊⠀⠀⢀⠀⢀⢄⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠀⠀⠀⠀⠀⣠⢛⣯⣥⣤⣤⣀⠁⠀⠀⠉⠀⣴⡋⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⣴⢣⡟⢃⣀⡂⣶⡍⣷⠀⠀⢄⡴⠋⠀⡀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⡇⢸⡇⠘⠶⠋⠰⠶⣼⠇⢲⡏⢀⠔⠊⠀⠀⠀⠀⠰⡥⠼⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠙⢮⠻⣆⣛⣛⣤⣴⣿⠴⠋
⠀⠀⠀⢠⡄⠀⠀⠸⠶⢬⣉⣉⡖⠋⠁
⠀⠀⢲⣉⣹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢴⣓⡆
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣀⣠⣤⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠉⠉⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠟⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⣶⡶⠶⠞⠛⣻⠁⠀⠀⠀⠰⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣀⡀⠀⣷
⠀⣄⠀⠀⢀⡶⠛⣿⡄⠀⠀⠀⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠉⠙⠛⠻⠿
⠀⠘⣦⡼⠋⠀⠀⠘⣷⡀⢀⣴⡿⠋⢿⡆⠀⠀⠀⣀⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⢠⣄⣀⡀⠀⠀⣿⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠹⣿⠟⠁⠀⠀⠈⢿⡀⣠⡾⠋⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⣄⣇
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⣾⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠰⣶⣶⡄⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠄⣀
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠀⠆⠀⠀⠀⠀⠀⠀⠻⠀⢸⡀⠀⢰⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⢴⣿⣿⣿⣿⡗⠀⢀⣠⣶⣀⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠖⠘⠓⣂⡀⠰⠐⡶⢴⠶⠀⣿⣿⣹⣶⣦⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠁⣾⣿⢿⣿⣇⢻⡿⠀⠙⠁⠀⠀⢀⣤⣦⣄⠀⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⡀⠀⠀⢻⣿⣿⡿⠛⣿⠿⣧⣶⣦⡄⠀⠛
⢀⣾⣿⡿⣿⠃⢀⣬⣵⣿⣿⣿⠈⠀⡀⠀⠀⠀⠀⡾⣿⣿⡏⣴⣴⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣈⡇⠾⠇⢰⡆⠀⠙⠛⠁⠀⣷⣦⠛⣿⣿⣧⡀⢀⡀
⢾⣧⣾⣷⣿⣦⣾⣿⣿⣼⣿⠋⠀⣠⣿⠄⠀⠀⠀⠀⠿⠏⠁⣿⣿⡁⠀⢻⣷⠄⢼⠛⠟⠻⣿⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠃⠈⢿⣿⣿⠈⣿⢿⣷⣦
⠀⣿⠉⠈⢿⣿⣿⣿⠏⠁⠀⠀⠀⠙⠋⠀⣀⡾⠁⠐⠛⠀⠀⠈⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⡟⠁⢸⡧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⣠⠀⠀⠸⡇
⢸⣿⠀⣿⡄⠛⠉⠉⠀⠀⠀⠗⠀⠀⠀⠀⡿⢰⣶⠀⢀⡀⠀⢶⣿⣷⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣶⡀⠹⣯⠀⠀⣀⣀⣤
⠘⠁⠰⠿⠇⠀⠀⠀⣤⣶⣿⣾⡂⣠⡆⣸⠃⠨⣧⠀⢘⣷⡆⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠉⣀⣴⣿⣿⣿
⠀⠀⠀⠀⣰⣦⣄⣸⣟⣿⣿⡟⠉⡋⠠⡿⠀⠀⠀⢠⣾⡟⠀⣿⡇⣀⠃⠀⠀⢀⠦⠄⠀⠖⠀⠀⣹⢿⠿⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠙⠁⢴⡅⠀⠉⢸⣿⣿⣿
⢰⣶⣤⡄⠲⣿⣿⣿⠏⠁⠀⠀⠈⠁⠒⠁⠛⠀⠀⢠⣉⡀⠀⠉⣠⣵⣄⡀⠀⣠⣶⣶⣤⡀⠄⠀⠹⠴⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⢀⣀⣈⢃⢀⣀⣾⠛⢿⣿
⠈⠙⠛⠀⠈⠿⢿⣿⠀⠀⠀⠀⣀⣠⣄⣠⡄⠀⠀⠈⢿⣧⠀⣾⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡄⠀⠀⠀⠀⠙⠿⣯⣙⢿⣿⡄⠈⠻
⢰⣾⠟⠃⠀⠐⣃⣀⣀⡀⠀⠈⠙⠿⣿⣿⠓⠀⠀⠀⠘⣷⣄⠛⢻⣿⣿⣿⣼⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢿⣆⠀⣾⣿⠒⠀⠀⠉⠹⢿⣿
⠈⠁⠀⠀⠀⠐⣿⣿⣿⣇⠀⠀⠀⠀⠈⠛⠃⠰⠦⣠⣾⠻⢿⣦⠀⣿⠿⠿⠿⣿⣯⡤⠀⣴⣷⣧⠀⢴⠟⢻⠿⣃⣀⣀⣀⣼⡿⠷⠍⢀⠀⣀⣀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⢸⣿⢺⡿⠀⠀⠀⠙⠋⢁⡄⣩⠕⠤⠀⢠⣶⡟⠠⣿⣿⠟⠈⣁⠀⠀⠀⠀⠙⠿⠛⢀⡀⠀⠀⠀⢰
⠀⣀⣤⣴⠀⠀⠀⢀⣻⡷⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣤⣀⠹⠃⣀⡀⠀⠀⠀⠀⠀⠀⠸⢏⡀⠀⠀⢸⣟⡁⠀⠻⠃⠀⠀⠛⠀⢠⣄⡀⠀⠀⢀⣾⣿⢐⡖⢸⡤
⢀⣨⣿⣿⣄⠀⠀⣘⠁⠀⠷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⡀⠛⣁⠀⠀⠀⠀⣿⣾⠀⠈⠁⠤⠐⣀⠀⠀⠸⠧⠀⢠⣾⠃⠀⠀⠹⡏⠀⠀⠿⠉⠁⠈⠁⠬⣿
⠸⠿⠟⠛⠉⠀⠀⢻⢀⣠⣴⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠠⡟⠘⣿⠁⠀⠀⠀⢛⡁⠄⠀⠀⠀⡐⠁⠀⠀⣴⣶⢦⡘⣿⣷⡄⣀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⣠⡆⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠘⠻⣷⣦⣀⣀⣤⠙⠂⠿⠀⠀⠀⠀⠻⣿⠇⠀⢀⢸⡇⢀⣤⣸⣿⠿⠇⠈⠀⠉⠁⠛⠃⠀⠀⠀⠀⢹⣷⠂⠀⠀⠚
⠀⠀⠀⠀⡿⠤⠀⣸⡟⠁⢀⡄⠀⡀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⢈⠃⠤⠒⠀⣿⠀⠀⠀⠉⠀⠈⠙⠿⢻⣤⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀
⠀⢀⣠⣶⣶⣤⡘⠋⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⢁⡟⠀⠀⠀⠀⠀⠀⠀⢠⣬⠈⣹⡀⠃⠀⠀⠀⠀⠀⢀⡽⢀⣸⣄⠀⠸⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢻⣿⣿⣿⣿⡏⠀⢠⣶⣿⣤⣶⣿⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠘⢿⣥⣈⣉⣩⣤⣄⠀⣿⠟⠓⠀⠀⢾⡟⣹⣿⣷⣿⣿⡿
⠀⢸⣿⣿⣿⡟⠀⠀⠘⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣷⣦⡀⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣉⣀⣠⣤⠀⠈⠛⠛⠋⠉⠉⠙⣷⣿⣷⡆
⢠⣿⣿⣿⡿⠁⠀⠀⠀⣽⣿⣿⣿⠟⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣾⡇⠀⢸⣆⢸⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⢰⣶⠀⠀⠀⠀⠀⠉⢻⣿⣷⣴⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⡀⠐⢄⠚⠨⠄⠄⢀⠤⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⠎⠀⢠⢞⠀⣀⠉⣀⠦⡈⢁⡁⠌⠀⠠⠤⠂⡤⠀⡁⠁⠉⠁⠀⠢⠀⠔
⠀⠀⠀⠀⠄⠀⢀⢀⡄⠀⠨⠚⢰⠴⠛⡳⢈⡀⣀⣘⣠⠟⢉⡥⢞⣤⣥⡃⠠⠖⣶⣶⣴⠿⠶⠐⣊⡼⣲⡬⢀⠀⠀⠤⠀⠄
⠀⠀⠀⠀⠀⡆⠈⠀⠄⡨⢣⣴⠇⠴⠟⠐⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠉⠁⠉⠥⠌⠀⠊⠻⠔⠐⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⢠⠈⣠⠴⠊⠉⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡀⠀⢀⠀⠂⢠⡄⠀⠀⠀⠀⢐⠀⡉⠀⠘
⠀⠀⠀⢀⠴⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢂⠀⢐⢨⣴⢪⡔⠢⠕⠛⠉⠉⠐⠡⠶⠀⠀⠁⠃
⠀⠀⠰⠃⠀⠀⠁⠀⡀⠀⡀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡤⢎⣉⡫⡈⠉⠁⠘⠐⠂⠰⣀⣄
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠴⢺⠅⡀⠉⡠⠀⠠⠐
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠔⠒⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠫⡥⠄⠑⠂⠁
⠃⢐⡈⠸⠃⠀⠀⠘⠀⠒⠠⢣⡞⠢⠈⣴⡋⡀⠀⠀⠀⣀⠔⢁⡀⢩⠂⡀⢀⡤
⠀⡀⢤⠀⠀⢰⡶⣀⠀⠀⠜⠀⠉⣤⣠⡔⡴⠃⠀⠀⠀⣁⠔⣁⣰⠦⠤⠍⠉
⠀⠡⡈⢿⣠⢮⣖⡵⠿⢉⣵⡷⢮⡀⡼⣷⢏⠀⣠⡡⠈⣼⡷⠋⠃
⠀⠀⠐⠀⠁⢌⣽⢦⠀⢥⡻⣄⣬⣰⡻⠋⢀⠔⠁⡀⠟⡉
⠀⠀⠀⠀⠐⠀⠈⠉⠉⠉⠓⠋⠉⣁⠨⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢯⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣃⠇⠀⠀⠀⠀⢿⣂⣀⡀
⠀⠀⠀⢀⣀⣀⣀⣤⢾⡉⠘⢿⣔⣢⡴⢖⠗⠛⠉⠉⠉⠉⠙⣓⣆⣀
⠀⡴⢋⡭⠄⠀⡒⢝⣷⣝⣠⣼⠟⠋⠀⠀⠀⠀⠀⠀⠤⠬⠽⢿⣿⣿⣷⣦⣄
⠸⣇⠏⠀⠀⠀⠀⠀⣼⢻⡏⡉⠀⢀⡀⢶⡀⠈⠲⣄⣀⣀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠉⠳⡄⠀⠀⢀⡞⠃⠌⢷⣿⣦⣧⣿⣾⣷⣷⣦⣄⣨⣭⣥⣤⣀⣀⠀⠀⠀⠈⠘⣆
⠀⠀⠀⠻⡷⣤⡿⡃⠹⢻⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠿⣿⣟⢦⡀⠀⠀⠸⡆
⠀⠀⠀⠀⢀⡿⠇⠹⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⣿⠀⠀⡀⡇
⠀⠀⠀⠀⢸⣧⡘⠺⣿⢀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⠀⢇⡏
⠀⠀⠀⠀⠘⣏⠻⣾⡇⠈⠀⠀⠀⠉⠛⠗⠀⠀⠀⠀⢰⠶⠖⠛⠛⠛⢆⣷⣿⣇⢠⡞⠁
⠀⠀⠀⠀⠀⠈⡳⢾⠀⢲⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⡀⢹⣿⣯⠏
⠀⠀⠀⠀⠀⠀⢡⠸⠁⠀⠙⠦⠾⠷⠀⠀⠀⠀⠀⠀⠀⢨⣿⡍⣹⠿⠃⢸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡜⠠⠉⠃⠀⠀⠀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡆⠀⠀⠀⣀⣤⣤⣼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠤⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠉⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣉⠇⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⡠⠀⣀⣀⠀⢀⠀⠀⡀⠀⠩⣠⡇⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡀⠀⠀⠀⢈⠛⠿⠿⠟⠂⠀⡈⠇⡀⠀⠉⠇⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⢠⢿⡿⣿⣄⠀⠀⠀⠁⠀⠀⠐⠁⠀⠂⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣠⠟⡾⣸⣿⡏⠑⢄⠀⠀⠀⠀⠀⠠⠛⡆⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⢀⡿⠸⡇⢻⠹⡇⠀⠀⠈⠓⠒⠒⠒⠛⠀⠙⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⠉⠁⠙⠢⢄⣀⣀⣀⣀⡤⠓⠦⣤⣤⣴⣶⠾⠛⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠑⠀⠓⠼⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠘⣿⣦⡑⢤⣄⣀⠈⣳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⠨⢿⠉⠀⠈⠙⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀
⢀⡠⠤⠒⡉⠀⠀⠈⣷⣄⠀⠀⢐⣬⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠒⠂⠄⣀
⠀⡅⠂⠀⢃⢉⠒⠴⣿⡿⠃⣠⣶⡟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠊⠩⠀⠂⠌⠠⠁⠂⠡⠀⡐⠠
⠐⠀⡘⠀⡀⠤⠐⢰⡏⢀⣾⡿⠛⠲⠒⠒⠒⠒⠒⠀⡀⠀⠀⠀⠀⠀⠘⢁⡂⡐⠠⠐⠂⡈⠐⡀⠂⠐⠠⠀⡰⡀
⠨⢠⠁⠀⠄⢃⠈⡈⠳⣤⣿⡐⢀⠀⡌⠠⠈⡌⢄⠱⠀⡠⠀⠀⠀⠀⠀⠈⢤⢁⡘⠈⠐⠀⡁⠡⢨⠈⡀⣂⠐⠰
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⡏⠉⠉⠉⠉⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⡏⠙⢻⠋⠉⡏⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⡀⠠⠤⠤⠔⠚⡣⠖⠉⠑⠶⡓⠒⠀⠠⠤⠤⠤⣅⠀⠀⠃
⡇⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠈⠑⠢⠤⢤⠄⠀⢸
⡇⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⣠⡴⠒⠦⣄⡀⠀⠀⡇⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⢀⠃⠀⢸⠁⠀⠀⠀⠀⠀⢀⣀⡤⣜⠀⢀⠙⠁⠀⣏⠀⢀⠛⡄⠀⢹⠀⠀⢃⠀⠀⠀⠃⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⡘⠀⠀⡎⠀⠀⠀⢀⡴⠊⠉⠀⠀⠈⠉⠁⠙⠢⠴⠫⣑⡆⠀⣻⠀⠈⡄⠀⠘⡀⠀⠀⢠⠀⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢰⠁⠀⢰⠒⠋⡀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡟⠀⠀⢱⠀⠀⠁⠀⠀⠘
⡇⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⢀⡇⠀⠀⢮⡀⣀⠠⠐⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢻⠀⡤⡀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⢸
⡇⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠆⠀⡸⠀⠀⠀⠀⢳⠤⠤⠤⣄⣀⠤⠴⠦⠖⢚⡇⠀⠀⠀⠀⢸⡐⠁⢲⠀⠸⠀⠀⠆⠀⠀⢀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⢠⠃⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠀⣼⡁⠀⡈⠀⠀⢣⠀⠸⠀⠀⠈⠀⠀⠈
⡇⠀⠀⠀⢀⠀⠀⠀⠀⢀⠁⢀⠆⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠀⠀⠘⠀⠀⢸⡀⡴⠚⢣⣾⠕⠀⡐⠀⠀⠀⠀⢆⠀⢁⠀⠀⠀⠀⢀
⡇⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠠⠄⠀⠀⠀⠀⠀⢉⣠⡶⠻⠋⠀⡰⠁⠀⠀⠀⠀⠘⡄⠀⡀⠀⠆⠀⢸
⡇⠀⠀⠀⢸⠀⢰⠀⢀⠁⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢤⣤⠤⢲⣶⡈⠭⠡⠐⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠱⡀⢡⠀⢰⠀⢸
⡇⠀⠀⠀⠀⠀⡄⠀⠂⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⡏⢰⠀⡸⠐⠀⠀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠀⡄
⡇⠀⠀⠀⡆⢠⠀⡈⡰⠁⣀⠀⣒⢀⢔⠈⠀⠀⠀⠀⠀⡀⠠⡇⠈⠉⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⡀⢠⠀⡆
⡇⠀⠀⠀⠁⠄⠰⡰⠁⠀⠈⠀⠈⢻⠛⡎⠁⠈⠛⠀⠀⠀⠀⠇⠀⠀⠀⠀⢰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⠈⠄⠁
⡇⠀⠀⠸⢀⢀⡞⠁⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠤⠤⠒⠁⠀⠈⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢀⠸
⡇⠀⢀⢂⣼⠯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢿⣄⣀
⡷⡏⠉⢹⣳⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠒⠢⠤⢤⣤⢤⠦⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⣻⠉⠈⡷
⡇⠑⠲⢾⠁⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⣠⠔⠚⢹⠉⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢹⠶⠒⠁
⡇⠀⠸⢸⠀⣇⠱⠀⠀⠀⠀⠀⠀⣀⡴⢚⣧⢀⣠⢒⣉⡀⡀⢀⢸⡄⠀⣇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡇⢸⢸
⡇⠀⡆⢸⠀⢹⣀⡇⠀⠀⠀⢀⠚⣃⣀⠤⠐⢛⠿⠂⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡉⠀⠀⠀⠀⠀⠀⣇⣸⠀⢸⠀⡆
⡇⢸⠀⢸⣤⣸⠉⢣⠀⠀⠀⢠⣀⡠⠔⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠴⠀⠀⠀⢠⠋⢹⣠⣼⠀⢰
⡇⡇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡴⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡇
⣿⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⠶⠀⠈⠀⠀⠀⠀⠀⣟⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢹
⡇⠀⢠⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⢠
⣗⢒⣂⣤⣒⣒⣐⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣈⣑⣒⣐⣒⣠⠤⣒⣒
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠤⠾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣃⡸⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠁⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡌⡁⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠃⠀⠀⡂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻⣿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣽⣿⣿⣿⣶⣶⣦⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣏⠁⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣴⠟⣿⠟⣿⣿⣿⣿⠇⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⡾⠁⣼⠃⢸⣿⣿⣿⡈⠀⢀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡀⣸⠃⠀⣿⣿⣿⣿⡇⠀⣼⣿⡿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠑⠃⠀⠀⠁⣼⠏⢸⡇⢠⣿⡿⠃⢻⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣷⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⣼⡏⢸⣿⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡀⠀⠀⠀⠀⠀⢿⠀⠸⡏⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠸⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠛⠻⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠅
⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⡆
⠀⠀⠀⠀⠠⣿⠓⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⢀⣐⣻⣴⣓⣀
⠀⠀⠀⢀⣀⡟⠚⠃⢤⡀⠀⠀⠀⠀⣰⣿⣿⣛⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠉⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⢠⣼
⠀⠀⠀⠛⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⠋⠁⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠑⠟⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠛⠁⠀⠀⡆⣰⣀⡀⠀⢿⡇⠀⠀⠀⡎⢢⣄⣀⣀⡠⠀⢸⡆⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⠀⠀⢈
⠀⠀⠀⣇⠀⠄⠠⠀⢸⠃⣿⡀⡀⠀⢸⣧⣤⣤⠀⡇⠀⠉⠳⠉⠀⠀⠸⠧⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣸⠶⢂⣀⠲⢰⠈⣿⡉⡅⠚
⠀⠀⠀⠘⠛⠛⠛⠈⠋⠀⠛⠉⠁⠀⠈⠛⠚⠛⠛⠃⠀⠐⠛⠂⠀⠀⠈⠛⠁⠀⠀⠀⠈⠀⠛⠀⠀⠀⠛⠛⠛⠛⠛⠓⠀⠛⠓⠒⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣦⣴⣯⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣿⠀⠀⠀⠈⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⣞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣿⡏⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⡴⠋⠀⠀⠀⣠⣴⡿⠃⠀⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠀⠀⡜⠁⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠙⣿⣿⣧⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⠛⠛⢻⣆
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡶⠟⠋⠀⣼⡇⠀⠈⠀⡤⠚⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢳⡄
⠀⠀⢀⣾⠛⠛⠛⠛⣿⡟⠁⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⣰⣿⣿⠏⢹⣿⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠓⠒⠲⡆
⠀⠀⢸⠇⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠙⣧⡀⣀⣀⣀⣤⣴⠿⠟⠉⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⢿⣿⡀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸
⠀⣠⠎⠀⠀⠀⢿⠀⠀⠀⠀⢀⡴⠛⠛⣻⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⢘⣿⠛⠛⠛⠋⠉⠉⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠞⠁⠀⠀⠀⠀⢨⠂⠀⠀⠠⡎⠀⢀⠴⠋⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⢿⣿⣿⡀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⢀⡴⠒⢺⡿⠋⠉⢉⣿⠃⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠉⠉⠛⣶⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣰⠟⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢿⣿⣿⠿⣿⣶⣤⣤⡤⠶⠶⠶⢞⠛⠒⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠙⠛⠿⣶⣄⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠈⢿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠞⣻⠄⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠙⠿⠶⠶⣶⣦⣤⣤⣤⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠁⠀⣾⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡄⠙⣷⡀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⣩⣾⣿⠟⠁⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠁⣠⣾⣿⣿⠁⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⢀⣼⣿⠟⠁⠸⣧⠀⠀⠀⠀⠙⢧⣀⣤⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣽⡇⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡾⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠻⡆⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠻⣄⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⢿⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠁⠀⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠙⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠀⣀⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠉⠓⠀⣀⠀⠀⠀⠤⠤⠘⠂⠀⡏⠀⣀⣀⡀⠀⣀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⢀⢀⣟⠃⠀⠀⠀⠀⠐⠒⠂⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠈⠀⠀⢀⡀⠀⠀⠀⢸⡁⠀⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣸⠀⠀⣤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⣄⠘⢄⠀⠋⠄⠂⠀⣀⣠⣼⣤⠤⠄⠀⢀⡀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠀⠀⠙⢦⠀⠀⡾⠣⠤⠤⠀⠀⠤⠤⠬⠤⠄⠐⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠃⠀⢀⣠⠄⡠⠖⠂⠀⠀⠐⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡄⠀⠀⠀⠀⠀⠀⠂⠀⠁⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⠀⠀⢀⠜⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠇⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢹⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣸⠀⠀⠀⠀⠈⠀⢀⠆⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⢤⣤⡤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣐⣸⣿⠀⠀⠀⠀⠐⠀⠈⠀⠀⠘⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠐⡄⠀⢀⡸⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠠⡇⠈⢉⠙⠸⠠⠘⡍⠉⠁⠁⠀⢀⡠⠀⠀⢸⢿⠀⠀⠀⢉⠐⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⡜⠂⠀⠀⠀⡘⠀⠀⠈⡄⠸⡄⠀⢳⠒⠒⠒⠉⠉⠀⠀⠀⢸⠀⢆⠀⠀⠀⠀⠀⠀⠀⢠⠁⡘
⠀⠀⠀⠀⠀⠀⢠⡝⠀⠀⠀⡜⠁⠀⠀⠀⡇⠀⠁⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠚⠉⠓⠒⠶⠶⠒⠚⠁⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠣⣤⡐⠈⠀⣀⣀⠄⠀⠉⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣤⣤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⣄
⢸⣿⡿⠟⠛⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠳⣷⠆⠀⠹⠃⠀⢀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠙⠁⠀⠀⠀⠀⠀⣴⣴⣿⠇⠀⠀⠀⠀⠘⠊⢷⡀⠀⠀⠀⠃⠀⠀⠀⠸⠷⣿⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠀⠄⠠⣤⠀⠀⢸⣌⣷⠀⠀⠀⠀⠀⠀⡀⢴⡄⠈⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⣂⡀
⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⣴⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠐⠀⠀⠀⢁⣄⠆⠀⠀⠀⠀⠀⠌⢍
⢸⣿⡆⠀⠀⢰⣶⣶⣦⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀
⠈⢉⡁⡀⠀⠀⠠⣿⣯⣀⠀⢀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠘⠛⠿⡿⠀⠀⠀⠈⠉⠛⠛⢿⡀⠀⠀⠈⠀⠀⠀⠰⣷⣦⡀⢀⡀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⡀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡀
⠸⣷⣤⡀⢰⡀⠀⠀⠀⠀⠀⠈⠀⢹⡇⠴⠴⠒⠈⠀⢀⣼⠟⠛⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣤⠀⠳⠈⠁⠀⣿⡇⠀⠀⠀⠀⠀⠘⢛
⠀⠀⠻⡟⢻⣇⠀⠀⠀⠀⢀⣤⡀⠀⠁⠀⠉⣠⣤⡀⢸⡟⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢛⡤⠀⠋⠀⢀⠾⠧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢛⣛⣢⣤⠴⣦⣤⡈⠹⠃⠀⡆⠈⡇⢀⣈⢶⢻⡎⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠠⣶⡅⠀⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⢴
⢠⣤⣀⡀⠀⠉⠛⠁⠀⠀⠹⣿⡶⠄⠈⠀⠀⠛⠛⠻⣶⠀⢠⡆⠀⠀⠀⠴⣷⣶⡛⢿⠁⠈⠃⠀⣽⣷⡂⠈⠛⠿⠛⢿⡆⠀⠀⠀⠀⠀⠈
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⢀⠀⠀⠀⠀⠀⠟⠀⠀⢙⡀⠀⠀⠀⠀⠹⣷⣸⣷⣶⠀⠀⠉⠀⣼⡗⣧⡀⠀⢠⠅
⢸⣿⣿⡇⠀⠠⢦⠀⠀⠀⢴⣿⣷⣔⣿⣶⣷⣀⣀⣠⠀⠀⠀⠈⣷⠀⠀⢸⣾⣻⣽⣿⣿⣧⣤⡶⠀⣠⠘⠀⠿⢱⡆⠰⢴⡀
⢸⣿⣿⡇⠀⠀⣰⣷⣶⠾⠿⣝⠻⢿⣿⣿⠿⢿⣷⡏⠀⠀⠀⣾⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠠⣿⣧⣀⡸⡣⠀⠀⢀⣴⡿
⢸⡿⡿⠇⠀⠀⠈⠸⠇⠀⠀⠀⣶⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⢰⣿⣿⣿⠟⠉⠉⠛⢿⣿⣿⡇⠀⠀⠀⠐⢶⣎⠁⠀⠀⠀⢸⣿⡆
⢀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣥⣤⠄⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⢻⠋⠀⠀⠠⣄⠀⠀⠀⠘⡿⢠⣷⡀⠀⠀⢸⡷⠀⢰⡆⠘⣿⣷
⢸⣿⣿⣿⣿⣿⣦⣀⠘⢷⣶⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢀⡴⠶⣧⡀⠀⠀⠂⠀⠁⣾⣿⣿⣆⡀⠸⣿⠀⠘⠁⠀⠉⢿
⢸⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⣀⣤⣀⡀⠀⠀⠛⠑⠿⠟⣿⡆⠀⣀⠀⣸⣿⣿⠿⠿⢿⣦⣹⠀⠀⠀⠀⠀⠈
⠈⠻⢿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⠿⠿⠿⢿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠴⢋⣀⣤⣤⣤⠈⠙⠁⠾⠆⣴⣿⣿⣷⡿⢢⠀⠀⣤
⠀⠀⠀⠉⠀⢙⣿⣿⡟⠀⠀⠈⠟⠻⠗⠖⠀⣾⣿⣿⣿⣿⢃⡿⠟⠃⠘⠃⠀⠀⠀⠀⠈⠙⠋⠀⠉⠀⠀⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠀⣀
⠀⢠⢠⣤⡄⠈⠙⠋⠁⠀⠀⣀⡀⠀⡀⠀⢨⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⢀⣠⣀⠙⢻⣿⣿⣿⣿⣿⡀
⢠⣷⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⢠⣿⣶⣤⣿⠿⠿⠤⠆⠀⠀⠁⠀⣠⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⠟⢠⣿⡀⢀⣀⣀⣨⣿⣶⣶⣦
⢸⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⡇⠀⠀⠀⣰⠖⢶⣿⡾⢉⣷⡟⠛⢦⡆⠀⠀⣀⠀⠀⠁⠀⠀⣿⠿⣷⠀⢻⣿⣿⣿⡏⠁
⠀⠙⠻⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⠀⢹⡆⠀⠀⠀⣿⡀⠀⠿⠀⠛⠛⠷⠀⠈⣀⣰⣾⣿⠉⢿⠀⠀⣠⣿⣷⣷⠀⢘⣡⣼⣿⣷⣾⣿
⠐⢃⣀⣨⣿⠀⠀⠀⠀⠀⠀⠀⢩⣿⡇⠀⠉⠁⢀⣀⣠⣄⡀⠀⠀⣀⠀⣶⣤⣶⣾⠟⣿⣿⣏⣠⣾⠤⠛⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣧⣀⣤⣤⣄⣀⣀⣸⣿⣥⣀⣴⣶⣾⣏⡉⠛⢿⣶⣿⣿⡀⢹⣿⠛⠁⣴⣿⣿⣿⡿⠿⠀⠰⣟⠛⠛⢣⠛⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠀⠀⠀⡷⣠⣆⠠⠔⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠐⣒⠀⠀⠘⠻⣿⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⡭⠶⣷⢀⡘⠓⣀⠘⠁⠈⢠⠰⠘⠀⢠⡀
⠀⠀⠀⠀⠀⢙⠀⣿⠀⠀⠀⣸⣿⡆⠡⠅⣎⠀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠚⢀⡈⠡⡄⠉⠶⡀⠴⠦⠰⢆⢁⡀⠖⣈⠠⠌⣁⠈
⠀⠀⠀⠀⠒⡰⣴⣿⣧⣤⣴⣿⣿⡞⠈⠁⠀⠀⠀⠉⠒⠀⠀⠠⠄⢤⡤⠶⢀⡐⢃⡀⠛⠄⡤⠘⠀⠀⠀⠙⠳⠴⠮⠰⠦⠁
⠀⠈⠁⠀⡰⠉⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠘⠓⢠⠐⢢⠀⠅⢀⠐⡦⡀
⠀⠀⠐⠈⠀⠀⠀⠀⠹⣿⠿⢿⣻⠅⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠠⣌⣀⢚⡀⠈⣀⠘⠠⡆
⠀⠀⠀⠀⠀⠀⠠⠅⠀⡜⠀⡀⣶⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣶⣶⣾⣶⠂
⠀⠀⠀⠀⠀⠀⠁⠄⠅⣁⠀⠠⠀⠀⠘⠀⡃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⢀⠒⠠⠟⠀⠁⠒⠩⢀⡀⠀⠒⠉⢈⡁⢀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣶⠉⡀⢤⣄⡓⢠⡅⠨⠒⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠉⠁⠀⠑⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣠⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⢿⢿⡿⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢸⢁⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢈⣾⣿⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⣼⣿⣧⣤⣴⣶⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠘⠿⠿⠛⠛⠋⠉⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣴⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡟⠛⠛⢻⣿⡿⠁⠀⢠⣾⣿⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⣿⡿⠁⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⣴⣿⣿⣶⣶⣶⣿⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⠏⠀⠀⠀⠛⠛⠋⢉⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⡦⢀⣴⣿⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣽⣿⣧⣿⡿⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸


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

⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠉⢿⣲⣦⣌⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠠⣈⡳⢿⣿⣷⣦⣐⠂⢔⡀⣀
⠀⠀⠀⠀⠀⠠⡀⠈⣿⣶⣿⣿⣿⣿⣿⣦⣌⠪⣕⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡈⡙⢿⣿⣿⣿⣿⣿⣿⢷⣌⡳⠄⠙⡲⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠻⢿⢿⣿⣿⣿⣲⣭⡻⣷⣦⣝⡢⠁⢬⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡉⠻⣿⣿⣿⡝⢿⣯⣿⣿⣷⣦⣙⠶⣤⣄⠈⠳⠄⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠹⣄⠀⠙⢶⡌⢿⣿⣿⣬⡙⠿⣿⣟⢿⣿⠷⣦⣌⠓⠐⣄⠈⠳⣄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢷⡄⠀⠙⢧⠹⣿⣿⣿⣦⡙⢿⣿⣿⣷⣄⡙⢿⣦⡘⠷⢦⡈⢳⡄⡑⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠻⣤⡄⢦⡀⠈⠻⣿⣿⢷⣀⡙⠻⢿⣿⣿⣦⠙⠿⣷⣶⣌⠂⣑⠘⢦⠠⣌⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠹⢮⠳⣌⠻⣿⣿⢿⡷⣤⡙⠿⣿⣿⣶⣦⡈⠻⣧⡙⠳⣌⠓⢦⡳⣌⢧⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⢀⠀⠁⠘⢷⣄⡙⠻⢿⣦⡝⢷⡄⠙⢷⣿⣿⠠⣝⠿⣷⣬⡙⣦⠹⣦⣈⠳⠄⢢⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠱⣄⠀⠀⠰⠙⣷⣮⠻⢿⣦⡹⣦⠀⠙⢿⣧⣙⣇⠈⢻⣿⡌⠳⣬⣙⡛⠦⣌⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠱⣄⠀⠘⢿⣷⣦⣬⡛⣿⠀⠀⣀⣈⠉⠻⣦⡀⠹⣷⣄⠙⠻⠿⣷⣜⣧⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠈⣂⠘⣦⣍⣉⡙⠻⣼⣇⠀⠉⠉⠀⠀⠈⣧⡀⠙⠛⠻⢿⣦⡈⠙⢿⣧⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⠓⠙⠧⠈⠻⣿⣿⣇⠈⠙⠿⣷⣚⠀⠀⠀⠈⠙⠳⠆⠀⠀⠹⣧⡀⠀⢻⡎⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⢤⣬⣙⣿⣆⠀⢠⠌⠉⠀⠀⠴⠷⠀⠀⠀⠛⠀⣤⠙⠻⣦⠛⢿⣦⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠈⠳⢦⡙⢿⣇⠉⠛⢃⡀⠐⠾⠀⠴⠂⠀⠀⠀⢀⡄⠀⠀⠀⠘⣧⠀⣿⡇⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢦⡀⠈⠈⠻⣷⣄⢈⡁⢴⠄⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠛⠋⠀⣼⡇⠸⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠳⠀⢦⣄⣈⠻⣦⢥⠀⠀⠶⠀⠀⡁⠀⠀⡄⠀⠀⣤⠆⠀⣀⠀⠀⠁⠘⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠙⢿⣿⡛⠀⠀⠀⠴⠆⠀⠉⠀⠀⠀⠴⠂⠀⠀⠀⠉⠀⠠⡄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠶⢀⣀⠀⢠⡖⠀⢠⡄⠀⢀⡀⠀⡤⠀⠀⠀⠁⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⣮⡍⠀⢠⣤⢠⠀⠤⠀⠀⠀⣀⠀⠀⠁⠀⠠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⠀⠀⠀⠘⠃⠘⠀⡬⠀⠀⡠⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⣠⣀⣀⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⢀⠈⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠔⠀⡄⠀⢠⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⠀⠐⠀⠢⢀⠠⠖⠤⣄⡐⢀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠠⢀⠀⣁⣌⠢⢨⡈⠄⡀⠕⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠣⠀⠀⡇⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠔⠉⠀⣦⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠉⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠣

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿