Jolly Emojis & Symbols

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

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

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

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠒⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡶⣶⡶⣤⡀⠀⠀⠀⣾⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡤⣬⣷⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠘⣿⡌⢻⡄⠀⢸⣿⣿⣿⣴⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠶⠾⡿⠿⢥⡤⣤⣤⣤⣤⣤⣤⣬⣙⣳⣄⡴⠋⣿⣿⣿⣿⠟⠁⠀⠀⢹⣧⠀⣷⠀⠀⣿⣿⡍⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣸⠇⡤⡄⠰⢒⠆⢠⠴⠆⡎⠉⠉⠱⣋⡀⢠⣀⡈⣿⠁⠀⠀⠀⠀⢸⣿⠀⢻⠀⢀⠛⠛⢷⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⡈⠀⠉⢁⡀⠸⣌⢸⡁⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣼⣠⣾⣷⣤⣬⣧⣶⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢿⢈⡟⣶⣒⠉⠑⠛⣋⢠⡇⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⢀⣠⣤⠞⡵⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⣤⡟⠘⡅⠙⣿⣿⣿⡿⣱⠟⠀⢀⡴⠛⠁⠀⠀⠀⠀⣨⣽⡶⠿⠛⠁⠊⣸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠙⣷⣤⡳⠀⠙⠛⠛⢱⠏⠀⠀⣸⠀⠀⠀⠀⠠⠰⠛⠛⡉⡠⢀⢀⡴⢲⣿⠸⣏⠛⠻⣿⢿⣿⢿⠟⠛⢁⡌
⠀⠀⠀⠀⠀⣀⡴⠞⠋⠘⢷⣾⣟⣮⠙⠷⠦⣤⣤⠞⠳⣄⠀⢿⣦⣀⠀⠀⠀⠈⠀⠚⠁⠁⠁⠉⠀⣿⠛⠛⢻⣃⠀⠀⠀⢁⣀⣀⡴⠞
⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠹⢿⣭⣀⣀⣠⠾⠁⠀⠀⠈⠓⢈⣳⣥⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠞⡷⠶⠟⠓⠛⠛⠛⠛⣿⣿⣍
⠀⠀⢰⠟⠳⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣷⡀⠀⠀⠀⣠⡴⢿⡀⠀⠉⠛⠒⠒⡶⠶⠿⠿⠯⠽⠿⠾⠿⠿⠿⠿⠿⠷⠿⠿⠿⡝⢷⡀
⠀⠀⡟⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣴⠏⠀⠀⢻⠄⠀⠀⠀⠀⡇⡞⠉⢉⣉⢩⣭⡍⢩⣭⡍⠉⣩⡏⠉⠙⡆⡇⠈⢳
⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠁⠀⠈⠉⠀⠈⠁⠀⠉⠀⠈⠉⠁

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⢀⢀⣑⣧⣴⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡜⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡼⠁⡸⢠⡄⠤⠤⠀⠀⠀⠄⠐⢂⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⡰⠃⠺⠷⠴⠀⠿⠄⠘⡖⠀⠋⢻⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠂⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡴⠻⡟⠃⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⣼⡿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡠⠿⡇⠀⢶⣄⠀⠈⠀⠐⠂⠉⠉⠁⠉⠀⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠁⠀⠀⠹⢄⡀⠀⠀⠀⠀⠒⠒⠂⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⢨⡍⢳⣤⣄⣂⠀⢀⣠⣼⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢂⣤⠀⢀⠄⠻⣿⣤⠀⠈⡅⠀⠀⠀⠉⠹⢉⡟⠋⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢘⡿⠆⢈⠀⢰⠿⠃⠀⣄⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠂⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢲⠀⠀⠀⠀⢠⣿⠳⠀⣠⠀⠀⠈⠐⡇⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠐⠀⠀⡜⠏⠀⠀⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢡⣤⡤⢀⣀⠀⠀⣼⠌⠀⢠⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠘⠟⠁⠀⠈⡰⣻⠟⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⡾⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⢉⣶⣫⣥⣬⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣫⣭⣿⣮⢄⢻⣿⣿⢣⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣻⣷⢿⣣⠬⠙⣸⡇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠤⢽⡗⠁⠛⣅⠈⠰⡆⠠⠈⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠀⠀⢣⣀⠰⢠⠷⠤⠈⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢛⠃⠀⠀⠊⠙⠛⣫⠄⢀⠀⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⣌⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿⡿⠿⣙⣶⣒⣶⣦⣄⣀⣀⣀⣀⣀⣀⣠⡤⣦⣤⠞⠉⡴⣧⠄⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣵⣇⣃⡾⡜⡜⣩⢏⠫⡹⡙⢉⠃⣄⣦⡾⠛⢶⣒⣲⡾⣏⠉⠉⠉⠉⠁⠙⣯⣧⣿⡀⠈⠀⠾⡻⠊
⠀⠀⠀⠀⠀⠀⣠⢾⠟⣡⣾⢋⣿⢿⠛⠛⠻⠾⠶⠷⢷⡞⠻⣯⡈⢧⡀⠈⠻⣶⠾⣿⢀⡠⠔⠋⠀⢀⣿⠕⣿⠁⠉⠉
⠀⠀⠀⠀⠀⢀⣧⡎⣸⢁⠂⡞⡟⢸⡇⠀⠀⠀⠀⠀⠀⢻⡌⠛⣣⡀⢳⡀⠀⠘⢿⡟⠛⠚⠛⠛⠋⠉⠛⠛⠋
⠀⠀⠀⠀⠀⢸⣿⣴⣿⣏⣼⡶⢡⣻⠃⠀⠀⠀⠀⠀⠀⠀⢻⡘⠿⠟⠀⢳⡀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⡞⠁⠀⠈⠙⢯⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⣼⣿⠄⢳⣀⣀⣨⣿⣦⣤⡀
⠀⠀⠀⠀⢰⡇⠀⠀⣀⣤⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⡬⠿⠒⠛⢻⡏⢹⡿⡄⢸⣧
⠀⠀⠀⠀⠀⢳⡄⠀⠉⠀⠊⠙⡿⡳⣄⠀⣀⢀⣠⡤⠖⢿⣯⠁⠹⣆⢠⣿⣷⠈⢷⢸⡇⠹⣼⢹⡆
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣠⢾⢓⣹⠟⠩⣽⠋⡟⣦⠀⢸⠈⢳⡀⢹⡄⠉⠉⣀⣸⣾⣷⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣼⢿⢡⡴⡝⣶⢻⡇⠈⠳⣼⣄⣤⣿⣾⣿⣾⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢹⠙⢾⠋⣿⠋⢙⣾⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣄⣈⣷⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⢰⠖⠺⢿⣿⣿⡿⠋⠸⣿⣿⡿⠟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠉⢻⡄⠀⢠⡞⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣦⣄⣀⡾⠁⠀⠈⣇⣠⣤⣶⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣼⣿⣿⡟⠛⠃⠀⠀⠀⠉⢹⣿⣿⣿⣦⣤⣤⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣴⡿⠋⠀⠀⠈⢧⡻⡄⠀⠀⣤⡆⠀⠀⢀⣴⠟⠁⠀⠀⣠⣴⠿⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡾⠋⠀⠀⠀⠀⠀⠈⢧⢱⡀⠀⠛⠃⠀⣠⡾⠋⠀⢀⣴⡾⠋⠁⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠞⣡⣤⣤⣤⣄⡀⠀⠀⠈⣧⢧⠀⢀⣠⣾⣿⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠛⠋⠉⠉⠛⠻⣷⣄⠀⠈⢿⣷⢟⣛⣩⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠇
⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠟⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⠁⠿⠛⢛⣻⡷⠀⠀⣰⣶⣶⣤⣤⣴⣶⣶⣶⣶⣾⠿⣦⣄⣀
⠀⠀⠀⣿⠛⠻⠿⠿⠷⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣾⣿⣿⠟⣿⠛⠁⠀⠿⠿⠿⠛⠻⢿⣿⣍⣩⣭⣭⣥⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢉⣽⠿⠋⣡⡿⢫⡿⠋⠁⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣼⠋⢠⡟⠀⠀⠀⠀⠀⣶⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠘⠻⠿⣿⠿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⢸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⣶⠀⢰⡇⠀⠀⠀⠀⠈⣹⠏⢀⣠⡆⠀⠀⠀⣸⢷⣶⣶⣦⣤⣄⣀⡀⢸⡇⢀⣠⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢠⣇⣴⠿⠋⠁⠀⠀⢀⡏⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⠿⠟⠛⠉⢹⠀⠀⢠⣿⣦⠀⠀⡸⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⢀⡟⠀⠘⣇⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⡾⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣤⣄⣀⣀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣶⣾⣧⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⢀⣐⡍⠑⣠⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠛⠶⢤⠷⢸⡷⠞⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣀⣾⢦⣷⠠⠂⢉⡳
⠀⠀⠀⠀⠀⠀⠓⡐⢲⡎⠋⣍⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸⡧⠓⢁⠋⢉⠉⠉⠉⠉
⠀⠀⠀⡀⠀⠀⠠⠀⢨⣇⢠⠈⡑⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⡂⡒⢄⢀⡠⠐⠂⣻⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢰⣁⡀⢸⢀⡀⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⠈⠁⠀⠈⠃⠀⡁⠀⠀⠀⠀⠐⠀⠀⠊⢉⡹⢢⣍⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠀⠈⠀⠈⣡⣬⠈⠁⠉⠈⢃⣀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢁⡈⢈⠉⡇⠀⡦⠉⢀⡲⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⠈⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠀⠀⢡⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠂⠚⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⡁⠀⠀⣀⡸⠀⢸⣀⣀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣄⠀⠦⡀⠀⢀⠀⠟⡀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣀⠀⢀⣀⣀
⠀⠐⢄⠂⠈⠒⠁⠀⠀⣻⡆⠀⠞⠤⠊⠠⠖⠡⠂⠴⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠦⠶⣛⢉⡩⢒⡛⠛⠻⣟⡓⠁⡀⠀⠑
⠴⠂⠀⠒⠒⠰⠷⠾⠲⠿⠿⠿⠒⠒⠢⠔⢲⠄⠈⠰⠐⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⢆⠀⠀⡄⢠⠊⠉⠀⠁⠈⢛⠂⠀⠀⢀⡀
⠐⠀⠂⠉⠀⡀⠀⠀⠀⠰⠀⡎⠐⠂⠀⠀⠀⠂⠲⠘⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⠈⠉⣷⡋⠀⠻⠈⠁⠉⠈⠾⡅⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⣠⣄⡀⠀⠀⡀⠄⠀⠘⡆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡧⠸⣷⠟⠃⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠉⡻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠩⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣠⡆⠐⠃⠇⠤⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⢤⠤⠤⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣑⣦⡋⣐⡠⠂
⠀⠀⠀⠀⡈⠉⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⢀⠀⡀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⡿⠞⡋⠀⠀⠀⢀
⠀⠀⠀⠠⠇⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠲⠶⠶⢧⡀⣅⡞⢁⠴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠖⠠⠄⣀⠀⢰⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠂⢤⠀⠀⠉⠁⠀⠀⢠⠄⠙⣀⠄⠀⠀⠈⠉⠈⠁⠓⠀⠤⠆⣀⠀⠇⠀⡨⠦⠛⠉⠁
⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⠀⠈⠀⠐⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰
⠀⠀⢠⠹⣧⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢲⠖⠊⠀⡇⠀⣀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠈⠐⠁⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠸⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢠⢀⠀⠴
⠀⠈⠀⠂⠀⠄⢻⣷⡀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⠀⢻⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⢻⣷⡶⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⡙⣿⣧⣿⣷⣶⣶⣤⣤⡀⠀⢀⣨⣼⡿⠟⠻⠿⣶⣦⣤⣀⡀
⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⡿⠿⠟⠛⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣦⣤⣀
⣶⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣦⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣁⡈⠉⠛⠿⣷⣦⣄⡀
⣿⣯⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠉⠉⢛⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⣯⣀⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣿⣦⡀⠀⠀⣉⣹⣿
⣿⣿⠀⠀⠈⠙⢿⣿⣶⣶⣶⠿⠻⣿⣉⡁⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⣹⣿⢿⣷⣶⣶⣾⣿⣿⡿⠛⠋⠉⠛⠻⣿⣦⡀⠀⠀⠀⠈⣿⣿⣾⠿⠛⠋⢹
⢿⣿⠀⠀⠀⠀⠀⢽⡿⣿⣶⣄⢸⣿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣴⠟⠁⠀⣠⣿⣏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡷⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣿⠃⠈⠳⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠏⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣶⣶⣿⠟⢋⣁⣼⣿⠀⠀⠀⢸⣿⡟⠛⠉⠀⢸⣿⠻⣿⣦⣄⡀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠟⠛⠋⠙⢿⣿⡿⠿⣿⣿⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⣿⠀⠉⠻⠿⣿⣿⢿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⡀⠀⠀⠀⠹⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣧⡀⢀⣀⣀⣬⣿⣦⡄⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢿⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠻⣿⣄⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢸⡇⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⢸⣿⡏⠀⠀⣀⣠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣷⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠁
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠤⠤⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠛⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡆⠈⢳⡀⠀⢹⡤⠴⠖⠒⠒⠒⠂⠂⢄⡜⠁⠀⠀⣀⣠⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠱⡚⠉⠙⣄⠀⢇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⣴⠋⠁⠀⠘⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠹⣦⡱⣄⠀⠘⣆⠘⡄⠀⠀⠀⠀⡴⠁⣠⠔⠃⣀⡽⢦⣀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄⠘⡆⢹⣦⡀⢀⠞⣡⣞⡥⠖⠋⠁⠀⢀⣩⣷⣄⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⡞⠉⢩⡿⠛⠉⢁⣀⡤⠴⠚⣋⣉⣀⣀⣼⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣿⡀⠀⢸⠉⠒⠚⠛⠛⠛⠛⣉⣉⣩⡭⠥⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠉⠢⣄⣀⡤⠴⠖⠋⠉⠀⠀⢀⣨⠽⠻⡏⠁⠉⠉⠙⠯⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⢢⡀⠀⠀⠀⠈⠳⢦⡀⠀⢀⣀⠤⠒⠋⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⣀⠀⢸⡇⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣀⡤⠖⢻⡟⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠼⣇⠀⠀⣿⠈⠑⢦⡀⠀⠀⠀⠀⠀⢀⠠⠄⠊⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠻⣤⡇⠀⠀⠀⠈⠢⢀⠠⠐⠈⠀⠀⠀⠀⢸⠄⠀⢀⣸⡧⠤⠶⠚⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢈⡗⢄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣀⡼⠓⠛⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠀⠀⣼⠁⠀⠈⠒⢦⡀⣿⠀⣀⣠⠴⠚⠉⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠉⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⣇⠀⢀⣀⣠⣤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡤⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁
⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡃
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠀⣇⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⡀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⠄⢠⡀⠄⠀⠀⢀⠙⠀⡋⡀⠀⠀⠀⢀⡀⠠⣚⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⠻⢀⠂⡀⠀⠀⡇⠀⠐⠀⢚⣩⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⡀⠀⠉⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠉⠠⠀⠀⠀⣠⠒⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢁⠀⡀⠀⠀⡀⠇
⠀⣀⠀⠀⠀⠉⠁⠠⠔⠁⠀⠀⠄⠀⠀⠈⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠖⣀⠠⠄⠀⢠⠀⠁⠓⠀⠄⡐⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠃⡀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡄⠱⠞⠋⠀⠁⠀⠀⠀⠅⠖⠉⠀⠹⠒⠦⣀⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢾⠀⡷⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⢰⠭⠶⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⢈⡄⠈⠁⠀⠈⠉⢠⡔⠀⠂⠀⢀⡀
⠀⠀⠀⠬⠳⣤⠀⠀⠀⠀⠠⡄⣧⠀⠀⠀⠀⢠⣀⡨⠆⠉⠀⠀⠀⠀⠀⠀⠆⠀⠶⢭⡚⠒⠃⠀⠀⠀⢼⠀⠀⠀⠈⠀⡈⠰⢗⡊⠵⠂⠠
⠀⠀⠀⠀⠀⠁⠀⢃⡶⢄⡀⠄⢃⡀⠰⣂⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠀⢄⡀⠀⠐⢿⠀⣰⡶⠞⠉⠀⠀⢤⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡌⠁⠈⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠄⠀⠙⢷⠤⠌⠁⢫⣀⣄⣀⡀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠄⡰⢖⠁⠐⠄⠰⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣠⡞⠘⣧⠀⠉⠳⠄⠀⢠
⠀⠀⠀⣀⡠⢷⠈⠁⠈⠀⢠⣦⢠⠀⠈⠉⠉⢠⣦⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢆⡀⠠⠒⠋⠉⠀⠀⡟⠀⠀⠀⠀⠀⡳⢆⡀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠡⠞⠉⠈⠃⢠⡆⠀⠀⠀⠇⠀⢠⡄⠜⠈⠛⠰⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠚⣠⢰⡖⠒⢀⠂⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠁⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡤⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⣠⣾⣻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠀⠀⠀⠀⠀⣀⣿⡿⠋⢀⣠⣴⣾⠃⣠⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡔⠛⢳⣤⣴⣾⣿⣿⣿⠏⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣭⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⣤⣀⠀⠀⠀⠀⢀⣤⣿⣷⣦⣤⣾⣿⣿⣿⠄⠀⠀⢻⣿⣿⣿⣿⣧⢻⣿⡿⠋⠁⠉⠛⠿⣿⣿⣿⣤⣀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠁⢀⣲⣾⣤⡈⣿⣿⣿⣶⣀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⢛⣿⣿⣿⠟⠃⠛⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣸⣿⡿⠋⣠⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠃⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠙⠛⠿⠃
⠀⠀⠀⠀⠔⢛⣿⠟⠁⣴⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⡀⣤⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠚⠛⠛⠁⠀⢺⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠂⣈⣯⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠰⡆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⡀⠀⠛⠉⠀⠀⠀⣠⣾⠃
⠀⠠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡟⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⣴⣾⡿⠃
⠀⠀⡀⠀⠀⠀⠀⠂⣀⣀⡤⣶⢿⣿⣿⣿⠿⠇⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡏⠀⢀
⠀⠘⠿⢆⣄⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⢀⣀⣀⣀⣀⣹⡿⢿⡿⠃⠀⣠⡞
⠀⠀⣿⣛⣷⣶⣤⣠⣴⣾⣿⡿⠿⢿⣿⣷⣶⠶⡶⢂⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠀⣠⣾⠋
⠀⠀⢿⠯⠿⠟⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡟⠀⠙⠻⠋⠁⠀⠀⠀⠀⢀⠀⠤⠄⠠⠤⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠋⠉⠀⢀⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠹⠁⠀⠀⠀⠀⠀⠀⠟⠂⠀⠀⠀⠀⣀⣠⡤⠶⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⡐⣤⣤⠀⠀⠀⠀⠀⣰⡟⠀⠀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣶⠛⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡋⠀⠀⠀⠀⢰⣿⣷⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⠿⠿⠛⠛⠿⠿⠿⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣠⣴⣾⣿⣿⡿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⢦⣤⣀⣀⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠱⠶⣤⣬⣉⣿⡿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢹⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⢀⠊⣄⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⠐⣂⠀⠄⡰⠂⠒⠊⠈⠿⠿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠈⠀⠤⢀⠱⠁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣷⣶⡶⡐⠀⠉⠂⡤⠀⡶⣄⠀⠀⠀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠛⠿⢛⡴⡁⠀⠀⠀⠧⣀⠀⠈⠉⠀⠘⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠐⠀⠋⠀⠀⠋⠀⡜⢀⠠⠃⠀⠈⢈⠗⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠤⠤⠚⡀⠁⠒⠀⠤⠀⠂⠀⢀⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⠀⠀⠀⢆⠉⠐⠀⠀⠔⠁⠀⠀⠠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠀⠀⠚⠒⠄⠈⠢⣀⡠⠀⠀⠀⠀⠀⢠⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠢⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⡠⠘⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠐⠉⠀⠀⠀⢀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⢀⠊⠀⠀⠀⠀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⢀⠤⠄⡀⠈⠐⠤⣀⣀⠠⠤⠐⠀⠀⠀⠁⠀⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠁⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠄⠀⠀⠀⠀⠀⠀⢐⠑⠁⢀⠄⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠀⠈⠙⠄⠀⠀⠀⠒⠒⠒⠁⢀⣔⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡶⠋⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣏⠔⠀⠀⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠁⠉⠀⠉⣽⠏⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⡠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣾⠃⢰⡿⠀⠀⠀⠀⣀⠀⢀⡄⠉⠉⠁⠰⠊⢄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢀⠄⠸⠁⠀⠀⠀⠀⣸⡏⠀⣼⢇⡔⠀⠀⣼⠃⢣⡟⠐⠀⠀⠀⠠⣀⡸⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢫⡀⠀⣀⣀⠀⠁⠀⣿⢀⣠⡿⠉⠀⠀⢠⡟⠁⣼⠇⠀⣤⠀⣴⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠁⢀⣡⠄⢺⡏⠀⣼⢣⡶⠉⢢⣾⠁⢠⡟⠀⣰⡏⢠⡟⠀⣠⡶⠟⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣴⠋⠸⠀⣿⠁⢠⡿⠘⢧⠀⠂⠛⠄⠈⠓⠂⠈⢁⣽⡷⠟⠋⢀⣀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⢀⠇⣸⡟⠀⢸⡇⠀⠀⠲⡄⠀⢠⡟⠂⣴⠞⢻⠏⣴⠋⢠⡞⠠⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢠⡿⠁⠀⠘⣧⡀⠀⠠⢁⣠⣾⠋⠸⠁⠠⠋⢸⡯⠀⣼⠇⠀⡄⢠⡆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠀⣀⠤⠬⣉⣤⠾⠋⢱⡏⢀⡴⠄⢀⡤⣿⠃⢠⡟⠀⣸⠃⣸⠇⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣴⠋⠁⠀⠀⣾⠃⠸⣇⠀⠜⠀⠿⠄⠘⠓⠀⠈⠀⣿⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢧⣀⠀⠀⢠⡿⢀⡀⢀⡴⠒⠒⣤⣀⠠⠤⠤⠖⣾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣼⠇⠀⠀⣿⠄⢠⠎⠉⠆⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⡆⣰⡏⠀⠀⠀⢿⡀⡈⠢⠈⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠨⣷⠋⠀⠀⠀⠀⠀⢹⠷⠄⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠤⠐⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣶⡿⠃⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠀⠉⠁⠈⠀⠑⢋
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠜⢻⡄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠠⠀⠄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠉⠁⠘⠓⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣤⣴⠗⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣈
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠁⠀⢤⣤⣠⣤⣤⣤⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⣿⣿⣿⡿⠟⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠏⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣣⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣆⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⣿
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢦⣠⡄⠀⢼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠣⡾⠀⣿⣿⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣯⣶⣶⠏⠀⠀⠀⣠⠤⠀⠀⠘⣷⣤⡀⠀⠀⠿⢿⣄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠷⢙⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⠀⣈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣼⡿⠁⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⡰⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢃⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⠛⠀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⣌⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠀⠠⡀⠐⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣿⣿⡇⠀⠀⠙⡄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢹⣿⣆⠀⠀⠀⢱⠀⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⢁⣴⡿⠇⠀⠀⠀⣰⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡏⣿⣿⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣷⣿⣿⣤⠆⠀⠀⠀⣻⠃⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⣷⣿⣧⠀⠈⠻⣷⡀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣶⣮⣤⡼⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⣿⠀⠀⠀⠈⢷⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⠀⠀⠀⠀⠈⠁⢰⡿⠿⣿⣿⢿⡏⣿⡟⢉⣤⠔⠀⠛⠚⠀⠠⠆⠐⠋⡰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⣷⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢰⠀⠀⠀⠀⠀⠀⣠⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⡇⢀⣿⠀⠀⠀⠀⠀⠸⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠛⠀⠀⠀⢀⡀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⠀⠀⣴⣿⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡆⠀⠀⠀⠀⢹⣿⣾⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠛⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣿⣿⡄⠀⠀⠀⠀⢨⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡅⠙⢻⣿⡄⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣿⢷⠀⠀⠀⠀⢹⡟⠉⠀⠀⠀⠀⠀⠙⠓⠂⠀⠀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢶⡄⠀⠀⠱⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠤⣤⣙⣦⣄⠀⠙⣆⢣⠀⠀⠀⠀⠀⠀⡠⠃⢀⠔⠉⠀⠀⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡈⠙⠻⠷⣤⣈⢙⣠⣤⣤⡀⡠⠜⡡⠔⢁⣀⣀⣀⣔⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠀⠉⠛⠒⠒⠒⠊⢹⣀⠀⢡⡏⣐⠛⣉⣉⣁⣀⣀⣀⠀⠉⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠠⢤⣤⡄⠁⠒⠒⠒⢒⠶⠾⣿⣆⣸⡇⠀⠙⣦⠨⢭⣉⣀⠉⠀⠁⣰⣧⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠒⢶⡎⣧⢀⣈⣉⣉⣁⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⡤⠖⠇⢸⣦⡀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⢀⠁⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⡸⠀⠸⡇⠉⠛⠲⠒⠋
⠀⠀⠀⠀⠈⠓⠒⠛⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⢸⡃⠀⢠⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠉⠢⣸⠀⠀⢨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠷⠒⠛⢆⢠⠇⠀⠀⢹⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠀⡆⠀⠀⢸⠀⠀⠀⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣠⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⡀⠀⡇⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣏⢹⣟⠻⣿⣟⠟⡻⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⡀
⠀⠀⠀⡔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠔⠒⡄
⠀⠀⠀⠳⠤⠤⠴⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠴⠃
⠀⡄⠂⢀⠀⠀⠀⠀⠀⠀⠴⠛⠻⠟⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⣿⠷⢦⡀⠀⠀⠀⠀⡀⠀⣀⠀⠂⡄
⠀⠈⠉⣠⣴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉
⠀⠀⠞⠉⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠤⠤⠤⣄⣀⣀⣀⣈⣉⡉⣉⣉⣉⣁⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⠔⠋⠉⠉⠙⠻⠍⠀⠀⢙⡷⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢤⣀⣀⣀⣀⣤⡤⠶⠚⠋⠀⠘⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⡀⠀⠈⠪⡉⠐⠂⠔⠁⠀⠀⡠⠊⡹
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⡲⠏⢉⡒⠈⢐⣄⣀⡈⣦⠎⣀⣀⡔⠪⠔⣊⠉⠲⢄⣀
⠀⠀⠀⡤⣒⠊⠉⠀⠀⠀⠀⠄⢀⣀⡀⠁⣂⣬⠭⡿⣤⠼⡯⣥⣄⠈⢁⣀⡀⠀⠜⠀⠉⠉⠒⡢⡄
⠀⠀⠀⡆⠀⠉⠑⠢⠤⣠⡄⣒⡩⠥⠒⠉⠁⠂⠉⡠⠋⠢⣀⠁⠀⠁⠐⠢⠍⡒⢤⡤⠔⠊⠁⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢀⠗⠋⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⠣⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⣀⢀⡴⠁⠀⠀⠀⠀⢀⠔⠉⠉⠒⠢⢄⣀⣀⡤⠐⠊⠉⠢⡀⠀⠀⠀⠈⠂⠀⣀⡤⠂
⠀⠀⠀⠀⡇⠀⢋⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⡗⢄⠀⠀⢠⠛⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⢀⠜⠓⡧⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⡷⠊⠢⡴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠣⡤⠃⠀⠀⡇⠀⠈⠉⠒⠤⢄⣀⣃⡠⠔⠊⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠧⢄⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⢸⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⢤⡤⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠨⣘⡯⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⢇⡀⠁⣀⠔⠛⠭⠀⠀⠀⠀⠀⠀⠀⡲⢤⣄⡄⠀⡀⠇⢠⠀⣸⣴⠾⡓⢄⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠓⣄⣾⠕⣫⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠊⠔⠋⢑⣚⣿⣷⣊⣀⠀⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠐⢻⠟⠉⠉⠩⣭⢛⠟⣍⠓⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⢹⠙⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⡜⠀⠘⠁⠀⠀⠉⠲⣶⡄⠀⠀⠀⠀⠀⠀⠤⣐⣒⠾⠻⠀⠈⠀⢋⢷⣒⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠄⠀⠀⠀⠀⠈⡌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣙⣇⠄⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⡀⠡⠀⣤⠀⣂⠗⠠⠀⠀⠀⠀⠀⠀⢴⡰⡀⠠⡀⡀⠀⣇⢘⠀⠀⢠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠽⠓⠤⠬⣙⣬⣠⡧⠊⣩⡅⠀⠠⠤⠀⠀⠀⠀⠩⠭⠛⠢⢔⡳⣄⡿⢁⠤⠒⠛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠓⣂⢟⠿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡻⡿⣓⠒⠣⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠠⠎⠰⠈⠖⠈⢹⠳⡶⠀⠀⠀⠀⠀⠀⠀⠉⠀⢁⡔⢡⢇⠈⠑⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠙⠀⠀⡀⠀⠀⣃⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠿⠀⠀⠄⠀⠀⠸⢱⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠐⡹⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡆⣸⢶⠀⠀⣴⠋⠂⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⣷⠰⢡⡟⠀⡆⡘⣺⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⣇⢸⣼⠙⡅⣧⠇⢿⠀⠀⢸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⣰⠁⠀⠀⡇⠸⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢶⡄⢰⡀⠀⡇⠀⠀⢸⡇⠂⣾⠸⢀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡰⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠈⢷⡀⠀⠀⠀⢸⡇⠀⣿⠁⢸⡇⠀⡿⢀⣤⡀⡀⠀⠀⡄⠀⢀⠠⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠑⠀⠀⠀⠀⡇⠀⢹⠀⠸⡇⠀⡇⠈⢛⠁⣥⡄⠀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⢄⠄⠀⠀⠀⠰⠆⠀⠀⠀⢉⡏⠀⠀⢸⠀⠇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠻⠀⣹⠸⠏⣁⡄⡀⠀⠰⡆⠀⣇⠀⢸⠃⠀⠀⣾⡐⢸⣇⠃⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢝⡑⢾⣿⡿⢊⡫⠁⠀⠀⢰⡇⠀⠘⣆⠀⠀⠀⠀⣿⠁⢸⡏⠀⣿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠋⣡⡎⣩⢹⣄⠋⠃⠂⠀⢸⡇⠀⠀⠈⠃⠀⠀⠀⢸⠀⠀⠇⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠄⠋⠁⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⡀⢠⣤⣄⡀⠀⣀⠀⢈⣤⣤⡀⣤⣤⣤⡄⣀⠀⠀⣀⠀⠀⣠⠀⢈⣡⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⢸⡇⠀⣿⢸⠁⠀⠘⡆⣿⢠⡟⠁⠀⣿⠀⣿⠀⠀⣿⡆⢠⣿⠀⢠⣿⡇⣾⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣅⣀⡏⢸⠀⠀⢠⡇⣿⢸⡇⠀⠀⠃⠀⣿⠀⠀⣿⢻⡌⣿⠀⣾⢹⠇⣿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⢈⡇⢸⣀⣰⠟⠀⣿⠈⠳⢤⡀⠀⠀⣿⠀⠀⣿⠈⠁⣿⢰⣏⣸⡀⠻⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⢸⡇⢸⡻⣄⠀⠀⣿⠀⣶⠀⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⢸⠇⢸⡇⣴⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠃⠸⠀⠘⠇⠘⠃⠈⠃⠀⠷⠀⠻⠤⠞⠁⠀⠛⠀⠀⠛⠀⠀⠛⠙⠀⠘⠃⠙⠆⠞⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣀⠀⣀⡀
⠀⢴⣴⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⣦⣄⢰⣶⢸⣷⡆⢠⣶⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⢀⣆⣿⣷⣀⡀
⠀⠀⣴⣿⣿⣿⣿⡿⣿⣿⣭⣿⣿⡇⢸⣿⣷⣶⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠛⢛⣿⣿⡟⠀⢹⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣦⣶⣶⣶⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢈⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣴⣴⠂⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠘⣿⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠆⠀⢀⣻⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⠀⠀⠘⠿⠿⠛⢛⣛⣛⠋⠀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⠛
⠀⠀⠈⠑⠒⠒⠀⢹⣿⠻⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠉⠁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠻⣶⣤⣄⠀⠀⢋⠉⠉⠉⠉⠙⠛⠛⠛⠃⠈⠁⠀⠀⢀⠜⣿⢛⣿⣿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡠⠀⣀⡞⠀⠈⠛⠛⠀⠀⠀⡏⡏⠁⠓⠢⠔⣠⢤⣤⣤⡤⢀⠤⣠⠞⠁⠸⠋
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⡀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠈⠉⢙⣷⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⢼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⢔⡾
⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣠⠴⣪⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠗⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀⠎⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⠊⡹⠋⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⠚⠉⢲⡤⢎⣩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡴⠛⠺⢅⣀⡤⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠋⠉⠉⠉⢉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡀⠀⢠⠀⠀⠀⠀⡇⢸⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⡌⣡⠄⠀⠀⠇⠀⣿⠈⡆⣞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠛⠙⠜⠀⠀⠀⠃⠚⠈⠀⠣⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡤⠀⠀⢀⡤⢤⠀⠀⡠⡤⢄⠀⠀⡠⢠⢄⠀⠀⡠⣠⢄⠀⣤⡤⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠺⡶⡃⢰⢋⡔⣁⡇⠘⣾⡭⢯⡆⢸⡴⢻⢞⠀⣌⠔⣡⠊⡁⢋⡔⠑⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⢲⣡⣴⢈⡃⡽⢃⡬⠊⡔⣡⠆⢀⠁⠁⣎⡥⣋⠔⢡⢊⡝⢋⡄⠀⣋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡹⢤⠜⡝⠰⣊⠟⣱⠇⢣⢊⠔⡡⠗⣏⠧⡤⢊⠇⢩⡔⡡⢫⠂⣏⡆⠀⣇⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠉⠉⠉⣄⠈⠉⠉⠁⠀⠀⠈⠉⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠒⠀⢈⣈⣉⣉⡁⢁⣉⢈⢈⠉⠈⠈⠉⣫⠉⡏⠈⠈⠉⢁⠉⠉⢈⠁⢉⣉⣈⣉⠀⠉⠈⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⢠⣧⡇⠀⠀⠀⠀⠀⠀⠀⡁⡿⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠰⢀⣀⣀⠇⠀⡤⣴⣾⠛⠻⣗⣦⠀⢰⣀⣀⣛⠄⢿⣀⣀⣐⡀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠾⡹⠛⠀⠐⠒⠓⠬⠷⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⣀⣠⣤⡀⠀⠀⠀⣀⣀⣠⣤⡄⠀⠀⠀⣀⣀⣠⣤⣄⠀⢀⣀⣀⣀⡀⠀⢠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠂⠀⣠⡾⠻⠋⠻⣿⡄⠀⠀⢹⣿⣿⢣⠃⠀⠀⠀⠹⣿⣿⢣⠂⠀⠸⣿⣿⡏⠁⠀⠀⢻⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣿⡇⠀⠀⠀⢹⣿⠀⠀⠀⣿⣿⠘⠀⠀⠀⠀⠀⣿⣿⠈⠀⠀⠀⢻⣿⣇⡀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⢾⣿⡇⡀⢰⣿⣿⡄⠀⢰⣿⣿⡆⠀⢀⣿⣿⡄⠀⠀⠀⠀⢀⣿⣿⡅⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠫⠎
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣿⡗⠁⠘⢻⣿⠔⠀⠀⣿⣿⠃⠀⠘⣿⡿⠣⠃⠀⠀⠀⠘⣿⡿⠃⠂⠀⠀⠀⠀⠀⠻⣿⡟⡈
⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⣠⣿⠁⠀⠀⠈⣿⣦⣀⣰⣿⡟⠄⠀⠀⣿⣇⡀⢀⣠⣾⠁⠀⣾⣿⡀⢀⣠⣾⠰⠀⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠘⠻⠿⠿⠟⠈⠀⠀⠀⠻⠿⠿⠿⠿⠿⠸⠀⠻⠿⠿⠿⠿⠿⠰⠀⠠⠾⠿⠿⠧⠠

⢠⢀⡲⠒⠄⠒⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⢐⠐⠀⠰⠄⠢⠀⢒⡄⣤⡄⠀⠀⠐⠀⠀⢦⢀
⡨⢞⡀⠀⠀⠀⢡⠁⠈⡆⠀⠀⠀⣆⠐⢂⡀⠩⠀⠠⡘⠆⢉⡈⠁⢠⠆⢈⠣⠈⠉⠀⢀⠃⣆
⠐⠘⣇⠑⠦⠄⢀⠀⠐⠅⠀⣀⢀⠈⠓⠔⡀⠤⠰⡠⠒⢡⠀⠀⠀⢈⠀⢢⢕⠁⠀⠀⠀⢢⠡
⠸⡀⠈⠻⡀⠤⣾⢶⠈⠷⠶⣿⣟⠋⡀⠀⠢⢉⡅⠀⠡⠀⠀⠀⢈⢠⡏⠸⠛⢈⠀⠈⠉⠁⠠⣀
⠀⠈⠢⡀⠴⣦⡻⡘⡂⠘⠁⢀⠳⢧⡑⡤⠁⣆⣠⠒⠄⠀⠈⠕⢖⠒⠀⠀⠀⠩⢭⠠⠄⠀⡀⣀⡀
⠀⠀⠀⠈⠐⢄⠙⢄⡇⠂⠀⣼⠗⠂⠐⠯⠇⠀⠀⠹⠀⠒⠀⠀⠨⣄⠀⠀⠀⠀⠀⠃⡀⣢⠘⡦⠧⡤⣄
⠀⠀⠀⠀⠀⠀⢆⠀⢹⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⢀⡤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠘⡀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠧⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⣎⡀⠀⢀⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⢤⠀⠀⠀⠀⢀⠴⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⢲⣄⢳⠀⠀⣰⠃⢀⠴⠒⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢎⢧⣠⠇⢰⣋⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠒⠚⠛⠒⢺⠆⠀⠀⢯⠤⠤⠶⠥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⠛⢦⠀⠀⠉⠀⠀⡴⠉⠉⠳⣆⠀⢸⠁⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀⠀⠀⠀⡼⢁⡔⡂⠀⡧⠀⠄⠀⠀⡇⠠⠙⣦⡘⣆⡼⠀⡴⠋⠀⣀⣀⠙⡆
⠀⠀⠀⠀⠀⡜⢁⣠⣄⡀⠉⢦⡀⠀⡇⣾⠃⠀⡰⠃⣀⣄⣀⣀⢱⢄⡀⠘⢧⢳⣇⡜⢀⡴⢋⡠⠊⢀⡇
⠀⠀⠀⠀⠀⢣⡈⠑⠤⣉⣳⡀⠱⠶⢣⣡⠴⠞⠓⠉⠁⠈⣇⠈⠉⠉⠙⠓⠲⠆⠍⠁⣜⣉⣁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⠤⢼⠂⠈⠈⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⢰⠒⠉⠁⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠆⠂⠈⠀⠀⠀⢩⠀⠀⠀⠙⠀⠀⠀⠗⠀⠠⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠀⠀⠈⠈⠄⠀⠀⠰⡀⠀⠀⠀⠈⠙⡆⠀⠀⡄⠀⠀⢀⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠠⠀⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⡈⠀⠀⠁⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⣀⣀⠴⠴⠒⠉⠉⠒⠒⠲⠤⠤⠴⣄⣀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⢴⡧⢊⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⣸⣇⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⢸⡇⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⡇⣬⡀⠆⠸⠀⣷⠀⠀⠓⢦⢣⣄⠰⣿⣺⣷⣿⠆⣠⡜⡴⠞⠀⠀⣾⠀⠗⠰⢀⣭⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠝⠿⣶⣦⡁⣤⣤⣄⢸⡯⣟⣷⡖⣡⣌⢲⣾⣻⢽⡇⢀⣤⣄⢈⣤⣴⠿⠫⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⡤⠉⣛⣿⣾⠯⠼⠻⡟⠉⡅⢿⣿⢸⠉⢻⠿⠧⢽⢷⣷⣛⠉⢤⡀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠾⠛⡆⢿⣷⣮⣓⣼⡼⢧⣷⣒⣵⣾⡷⢠⠻⠷⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠴⠾⣟⣿⠂⠪⢭⣿⠉⢄⡠⠉⣿⡭⠕⠐⣿⣻⠷⠦⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⠻⢿⡿⢂⣚⣭⣿⣄⡈⢉⣠⣿⣭⣓⡐⢿⡿⠟⠋⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⣴⣿⣶⠃⠿⠿⢋⠔⡻⢻⡟⡟⠢⡙⠿⠿⠘⣴⡿⣦⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⣉⣤⣵⢿⣿⣯⢱⡾⣧⣠⠇⣿⣿⢸⣀⣼⢷⡎⣽⣾⡿⣮⣤⡉⠡⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⣱⣾⠛⠋⣄⠙⠋⠁⢼⢟⣯⡿⠳⣌⣡⠞⢿⣽⡻⡇⠈⠙⠋⣠⠛⠛⣶⣆⡁⢄
⠀⠀⠀⠀⠀⠀⢀⡴⠦⠩⠁⠂⠼⠀⠟⠀⠀⠶⠋⡎⠁⠰⣿⢿⡿⣿⠆⠈⢱⠗⠦⠀⠀⠿⠀⡦⠐⠨⢍⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣈⡉⣹⡏⢉⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡁⢹⠏⢈⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢹⡏⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡾⠛⠛⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠐⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢱⡀⠸⠆⣀⠀⠀⠀⠀⠀⠀⠀⠶⡁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢸⡇⠀⠀⢹⠀⠀⠀⠀⣴⡿⠷⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⡀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠃⠈⠉⠀⠶⠄⣀⠀⠀⢀⣤⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠘⢶⣤⡀⠀⣿⣿⡞⠀⣿⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣿⠀⠀⠀⠀⠈⠙⠺⣧⡈⠛⠛⠹⣿⡟⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⡀⢻⣿⣿⡄⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠸⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣟⣢⠀⣨⣿⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠚⠛⣶⡄⠀⠘⢿⣿⣿⣿⣿⣾⣿⣿⣄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⢸⣿⣿⡿⠋⠉⠙⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣿⡏⠀⠀⠀⠀⠀⠘⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⣀⣠⣶⡶⠶⢶⣄⡀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⢠⣿⠁⠀⠀⠀⠚⢷⣄⡘⡟⠀⠀⠀⠀⠀⠀⢀⡔⢻⡦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡆⠀⠀⢸⣿⣶⣶⣄⡀⠀⠀⠙⠻⠷⠀⢀⡀⠀⢀⡴⠃⣰⡿⠁⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣥⡀⣀⣸⣿⡇⠀⠘⢿⣄⠀⠀⠀⠀⠀⠈⣹⣶⣿⣁⡾⠋⠀⠀⣼⠏⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⣴⣶⣤⡴⠶⠾⢿⠿⠋⠉⠉⣠⠏⠀⠀⠀⠀⠀⡸⣿⡄⠀⠀⠀⠀⠀⢹⣿⡟⠛⠀⠀⠀⣸⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⠀⣠⣾⠇⠈⠹⠆⠀⡀⠐⡞⠁⢈⣤⠀⠀⠀⣠⣿⣿⣧⣾⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠠⠈⣰⡿⠃⠀⠀⠀⠀⠀⠁⡰⠋⠀⠛⠉⠀⠀⢰⣿⡿⢸⣿⠋⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⣤⠎⠀⠀⠀⠀⠀⠀⢰⣿⠟⣠⠾⣿⣿⣏⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⢈⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⢸⣤⣿⠘⠿⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⢠⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠃⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⣸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠿⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣿⣿⠏⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣾⣿⠏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣼⣿⠛⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⡏⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡄⠀⠀⢠⣿⡏⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢀⣿⡟⢁⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⣼⣿⢃⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⢰⣿⠟⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⡇⢀⣾⡿⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡄⠀⣼⠿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠿⠀⣿⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⠀⣻⡤⠤⠾⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⠀⣿⣧⣄⣀⣀⡀⠈⠙⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠈⣸⡟⠩⡟⠉⠿⠋⠻⠆⠀⢀⣈⠙⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡿⠏⠋⠻⣏⡀⠀⠀⠀⠀⠀⠀⢘⠿⢙⡙⢿⣁⠀⠀⠀⠀⠀⠀⠲⣾⣶⣿⣿⣮⠴⠀⠀⠀⠀⠀⠀⠑⠄⣒⠤⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡄⡉⢠⡟⠉⠀⠀⠀⠀⠀⠈⢻⣧⢙⠃⣼⠋⠁⠀⠀⠀⠀⠀⠤⢽⣿⣿⣿⡿⡤⠀⠀⠀⠀⠀⠉⠀⡡⢛⠪⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠝⠟⠈⠙⠀⠀⠀⠀⠀⠀⠀⠘⠩⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠎⠋⠮⠑⠀⠀⠀⠀⠀⠀⠀⠚⠀⠰⠀⠈


⠀⠀⠀⠀⠀⠀⢀⡄⢐⢂⣔⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⡴⡴⡠⠀⠀⠀⠀⠀⠀⠀⣄⠦⣶⢖⡄
⠀⠀⠀⠀⠀⠀⣄⠐⣾⣿⣷⠢⠆⠀⠀⠀⠀⠀⢀⡀⠐⢬⡌⠄⠤⠀⠀⠀⠀⠀⠀⠈⣟⢲⣿⣷⢺⢒⠀⠀⠀⠀⠀⢘⡊⢡⣷⡎⠬⠅
⠀⠀⠀⠀⠀⠀⠀⣬⠹⠛⠏⡅⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⡹⣤⣉⣧⢟⠂⠀⠀⠀⠀⠀⠐⢢⢄⢸⢴⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠖⠞⢶⣀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠬⢤⡔⠄⠀⠀⠀⠀⠀⠀⠀⠢⣴⠛⠶⠶⠂⠀⠀⠀⠀⠀⠐⢶⣶⣿⣶⣗⡄
⠀⠀⠀⠀⠀⠀⠂⣿⡸⠽⢀⡧⡂⠀⠀⠀⠀⠀⠐⠂⣗⡺⢇⡿⡘⠀⠀⠀⠀⠀⠀⠰⠈⣿⠶⡤⠤⠠⠀⠀⠀⠀⠀⠾⢿⠿⢿⠿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠎⠳⡓⠋⠒⠀⠀⠀⠀⠀⠀⠀⠠⠊⠍⡍⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⠋⢢⡖⠲⠀⠀⠀⠀⠀⠀⠀⠴⠰⢰⠦⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠙

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠡⢰⣾⣿⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⣹⣿⣿⣿⣿⣿⢿⣹⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⣴⣾⣿⠟⠷⢮⡙⢿⣿⢿⢿⣻⠍⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠉⠋⠀⠀⠀⠀⠐⠂⠁⠀⠈⠉⠀⠀⠀⠀⠃⣱⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸⣦⢾⡿⣿⠁⢠⡄⢀⣀⣀⠝⠛⠀⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠐⠟⠆⠁⠀⠀⠞⢋⣿⣬⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠿⣻⠛⢛⡯⠙⠿⠗⠀⠀⠀⢾⣦⣤⡾⣴⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠈⠀⣀⠀⣶⡄⢀⠀⠀⠀⠉⠙⠿⢣⣿⣿⣿⡏⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⣾⡂⠀⠀⠀⠀⠀⠈⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⠙⢦⡀⠀⠀⠁⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄










⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣽⣽⣮⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣾⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⢃⠀⠀⢰⡟⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣾⠷⠖⠃⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⢀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣴⡏⠀⢸⡿⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠉⠙⠛⠒⡶⠧⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢾⠀⠀⠀⠀⢀⠔⢰⣿⠀⠀⣿⠃⢀⠆⠀⠀⣴⠋⡅⣴⠇⡔⠀⠀⠀⠀⡇⠀⢸⠄⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⠀⢀⠔⠁⠀⣾⡇⠀⢸⣿⠖⠁⠀⠀⢰⡯⠊⢰⣿⠐⠀⠀⠀⢀⡀⠈⠛⠉⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣤⠆⠒⢂⠀⠀⣸⣿⠴⠒⣿⠇⣀⡀⣀⠀⣿⠁⠀⣾⠏⠀⢠⡗⠀⣼⠇⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⡖⠉⣿⡇⠀⢸⡿⣸⡏⠁⠸⢺⡟⠀⣰⣿⠀⡠⣾⢃⣰⡟⢀⣠⣶⠟⠋⠀⠀⠀⠀⠂⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣴⠋⢠⡇⢰⣿⠁⠀⣿⠇⠹⢇⣀⠜⠈⠓⠊⠀⠙⠈⠀⠈⣀⣾⡿⠟⠋⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⡼⠀⣾⡏⠀⢸⣿⠀⠀⠠⠤⡄⠀⠀⢰⠞⡂⢀⣴⠟⢻⡏⣰⡟⠁⢠⡟⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠑⠊⠁⣸⡿⠁⠀⠘⣿⠀⠀⠀⠀⡜⠀⢀⣿⠞⢰⡟⠁⢠⠟⢀⣿⢀⠂⣾⡥⠋⡀⠀⢀⡀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣀⡴⠟⠁⠀⠀⠀⠙⠳⠤⠄⣊⣤⠾⢻⡇⠀⢀⠁⣀⡁⠀⣼⡟⠀⢰⡟⠀⢀⡿⠀⣼⠃⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⢒⣶⡞⠋⠀⠀⣿⠃⢠⡿⠙⠀⡽⠈⣿⠃⠀⣾⠇⠀⢼⡇⣠⣿⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢰⡟⠁⠃⠀⠀⢰⡿⠀⠸⣧⡀⠴⠁⠀⠛⠀⠊⠉⠃⠀⠀⠁⣼⣇⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢸⡄⢀⠄⠀⢀⣾⡇⡀⠀⢀⣴⠖⠛⠒⣄⣀⣠⡤⠤⠴⠖⢲⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡠⠬⢄⠀⠀⢰⡿⠀⠀⠈⣿⣇⠀⣠⠞⠉⡇⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢸⠀⠀⣿⠃⠀⠀⠀⣿⡈⠆⢿⣀⡴⠁⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⢸⡇⣼⠏⠀⠀⠀⠀⠘⢷⣼⠀⠀⠀⠀⠀⢀⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⢀⠏⠛⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡀⠉⢠⣾⣿⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣟⢁⣴⣄⣹⣿⣤⣤⣀⣠⣾⠛⠿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⣉⠔⡀⢀⠀⡦⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢁⢄⠀⠂⠠⣴⠆⣤⠀⠐⡀⠀⠀⠀⢰⢟⡟⠛⠛⠛⠛⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⢠⢐⢸⠈⠀⡄⠉⢁⡞⠈⢄⣀⠴⡄⠀⠀⢸⠋⠀⠀⢀⠃⠐⡀⢰⠇⠀⡴⠦
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⣷⠀⠖⠀⠉⣀⡴⠋⠀⠀⢸⣅⣤⣼⣀⠀⠈⣆⠀⠀⠀⠀⠈⠉⠙⢻⣦⡗⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢿⣇⠀⡀⣸⠃⠀⠀⠀⠀⠀⠈⣿⣿⡏⣻⠖⢏⡀⠀⣠⠆⠐⠇⠤⠚⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣍⡤⠋⠁⠀⠀⠀⢠⠄⠒⠲⠿⣿⡃⣿⢠⠊⢰⣆⠀⢄⠀⠀⠀⢀⡦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣿⣧⢹⠀⢰⣯⡼⠦⢄⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⢀⡄⠂⠉⠑⡀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠘⢻⣿⣿⣧⣯⣷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢈⠕⠶⠀⠀⠀⠈⠳⣇⠂⠀⠀⢁⡹⣿⣷⣿⣿⣿⣿⣿⠁
⠀⢀⡤⢀⣤⠤⠄⠠⠤⠄⠠⠗⠀⠀⠀⠀⠀⠀⠀⣹⡌⠁⠀⠀⠐⢻⣾⣿⣟⠛⣿⠏
⢀⠇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣴⣿⣷⡄⠢⠀⠀⠘⠋⠀⣠⡧⠋
⠘⠥⠄⣀⠐⠂⠀⠀⠀⠀⢀⡴⠋⠙⢽⣿⣿⣿⣿⣿⣿⣿⣦⡀⣠⣴⣶⣿⠏⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠒⠀⠻⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠛⢀⡎⠻⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣦⠀⠀⢹⣿⣿⠿⢿⣿⣿⡇⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡟⠰⠆⠀⠈⢌⣿⣿⣿⣿⣿⠿⠋⢰⣿⣿⣿⣧⠀⠀⣿⣿⡄⢀⣿⡟⢰⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠷⣦⠀⠸⠂⠀⠛⣉⠛⠯⣄⠀⠀⠀⢹⣿⣿⣿⣆⠀⢸⣿⣿⣿⣏⡀⢾⣿⢻⣿⣿⢿⣿⣿⠃
⠀⠀⢀⡴⣶⣋⠙⣯⠝⢯⡐⠋⣿⣤⡘⠗⠀⠙⠀⠀⠙⠓⠦⣄⢸⣿⡏⢿⣿⡄⠀⣿⣿⠉⠻⣿⡆⠀⢸⣿⡏⠈⠛⠃
⠀⢀⣏⠀⡟⠂⡇⣿⠆⢠⠌⢁⢰⡟⠁⢴⣶⠄⡛⠀⠺⠃⣠⡏⠈⣿⣿⣿⣿⣷⠀⢸⣿⡆⠀⢻⡇⠀⣸⣿⡇
⠀⢸⠃⢺⠁⠀⡇⣿⠘⠀⣤⢉⡾⠀⠀⡝⠰⡆⠁⠘⠃⣀⣙⡇⠀⢿⣿⡁⠹⣿⡇⠀⣿⡇⠀⢸⡇⠀⣿⣿⠁
⠀⢸⠀⣼⠂⠰⠟⣧⠿⠒⠈⠊⠁⠀⠈⣷⠐⢲⠿⣤⠼⣟⣻⠦⡄⣸⣿⣧⠀⣿⣷⡄⣽⣷⣄⣸⣇⢀⣿⣿⡀
⠀⣸⣦⣽⣷⣠⣾⣷⣶⣶⣶⣶⣶⣶⣾⣇⠀⠉⠓⠒⠒⠛⣿⣴⣧⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣧
⠈⠛⠛⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⢿⣿⣿⣾⣷⣤⣦⣤⣤⠽⣾⠿⠛⠛⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠋⠽⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⢀⣸⣇⡀⣄⣤⠆⠀⠀⠠⣼⡆⢹⡟⢁⣿⡤⠀⠀⠰⣤⣆⢶⣼⣇⡶⢠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠋⠙⣿⡍⠀⠀⠀⠀⣛⡋⣙⣿⣿⣟⢉⣛⡀⠀⠀⠨⣭⣿⣿⣿⣿⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠻⠷⡾⠟⡶⣤⡀⠀⠀⠸⢿⡏⢹⣇⠉⣿⠏⠀⠀⢀⣴⡶⣻⡿⡿⣟⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠏⠓⠈⠀⠀⠀⠀⠀⠀⠉⢼⡷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣀⠦⣽⠴⣀⣴⡶⠀⠀⠲⣤⣀⠀⣹⡁⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣝⣳⣾⣞⣛⠌⠀⠀⠀⠀⠀⠉⣛⣾⣶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⡇⣿⢬⡛⣶⣦⠀⠀⣠⡶⠞⠉⢺⠒⠉⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠀⠘⡏⠀⣠⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⣷⡚⠉⠁
⠀⠚⠒⠂⠒⠂⠒⠒⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠉⡇⠉⢻⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠓⢦⡀⠀⠀⠀⠀⢀⣀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⣠⣾⠏⠀⠀⠀⠀⠀⢈⡿⠀⠀⢀⣴⡟⠉⠀⠀⠀⢹⡆⠀⠀⠀⠀⡞⠁⠀⠀⠙⢷
⢀⡤⠒⠀⠒⠲⠤⣄⣾⡿⠁⣰⣿⠃⠀⠀⢀⡀⢀⣀⡾⠁⠀⢀⣾⡟⠀⠀⠀⣀⡤⣼⠧⣀⠀⠀⠀⢳⣄⣀⣰⡆⠈
⢿⣤⡄⠀⠀⠀⠀⢸⣿⡟⢲⣿⣧⣀⣠⣾⡡⠴⠚⠙⡆⠀⠀⣾⣿⠁⠀⢀⣾⣏⠔⠃⠀⠈⣷⠀⠀⠀⠉⠉⠉
⠀⠁⠀⠀⠀⠀⠀⠸⣿⣁⣿⣿⠀⠀⢿⣁⡤⠀⠀⢀⡇⠀⢰⣿⡿⠀⠀⢺⣿⠁⠀⠀⠀⢀⡟⢀⡀⠀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿⣁⠀⠀⡼⠁⣠⢾⣿⡇⠀⠀⢿⣿⠤⠤⠤⠖⠋⢠⣿⠃⢸⣿⡄⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⢿⣿⠈⢉⡞⠁⠉⠀⢸⣿⡇⠀⠀⣸⣿⡆⠀⠀⠀⢀⣿⡟⢀⡟⢻⣷⡴⠋
⠀⠀⣠⡶⠚⢶⣦⡀⠀⠀⠀⣀⠤⣀⠀⠙⠋⠁⢀⣤⡶⠒⢦⡙⠷⠤⠞⠁⠙⠿⣦⣠⠴⠋⠈⠉⠁⠀⣨⣿⣇
⠀⠀⠛⠀⠀⠈⣿⣇⠀⠀⢾⣁⣤⠄⢳⡀⠀⠀⣾⡟⠀⠀⣼⠃⣠⡞⠉⠉⢶⡀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣸⣿⣧⣀⣸⣇⣀⣼⣿⡧⠄⠚⠁⣸⡿⠀⠀⠀⣸⠃⣀⡤⠀⠀⢠⣤⢰⡇⠀⢸⣿
⠀⠀⠀⠀⢀⡀⣿⣿⠒⠉⠘⣿⡇⠀⡿⠀⠀⢸⣿⡇⠀⠀⢠⣿⡇⠀⢀⡴⠃⣴⣿⠁⠀⢀⣿⣿⡄⠻⠤⠼⠋
⠀⢀⠤⠊⠁⢀⣿⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡗⠊⠉⠀⢠⣿⡇⠀⢀⡞⢹⣿⣇
⣠⠋⠀⠀⠀⣸⡿⠀⢀⣤⡴⠶⢦⣄⡀⠀⠀⠘⣿⣧⠀⠀⣸⣿⣇⠀⠀⢀⡞⠙⠓⠒⠋⠀⠈⣿⣿
⡏⠀⠀⠀⣴⠟⠁⢰⣿⠏⠀⠀⠀⠈⢻⣆⠀⠀⠘⠿⣦⡴⠃⠙⠿⣦⠴⠋⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀
⠙⠢⠴⠚⠁⠀⠀⠘⣿⣄⡾⠃⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⢀⣿⡏⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢺⡟⠛⠓⠲⠶⢤⣄⣀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⢛⡾⢧⣄⣀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⠛⠲⣶⡿⠥⠤⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⢀⣠⠞⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠤⠶⠶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠈⠑⠦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⢱⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠜⠁⠀⠀⠀⠀⠠⠤⠂⣡⠶⣄⠀⠀⠀⠀⠀⠀⡖⠲⡄⢀⢀⡀⠈⠧⢤⡆
⠀⠀⠀⠀⠀⠀⠈⠦⣄⣀⠀⠀⠀⡤⠠⠀⠔⠁⠀⠇⠀⠘⠀⠀⡏⠉⡏⠈⠀⠀⠓⢸⡡⠎⠀⢠⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢎⣁⣀⡀⠀⢦⣀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⣵⠀⠀⠀⠀⠀⢱⠀⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢒⡶⢀⣀⣀⡱⢄⡀⠀⠀⠀⠀⠀⢰⠈⣿⠀⠀⠀⠀⠀⢰⠗⠒⠒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⣩⠗⠦⣄⡀⠀⠀⠑⠂⠀⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠉⠀⠈⠉⠉⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣳⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣏⢘⠄⠀⠀⠀⠀⢌⢙⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣀⣤⣄⣀⣠⣄⣠⣄⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠇⠀⠀⠀⠀⠀⠀⠧⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⣿⡄⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⢻⣧⠀⠀⠀⠀⣴⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣾⣿⡆⠀⠀⣾⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠁⢠⣦⠸⣧⠈⠛⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣠⣿⠟⠉⠈⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⢋⠀⠀⠀⠀⠀⠀⠛⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣠⠀⠀⠀⠀⠀⠀⠀⠛⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⡇⠀⠀⠀⠀⡀⠀⡀⣀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⡟⠁⠀⣼⣠⡿⠀⣄⠀⢰⣤
⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠷⠀⠐⠟⠋⣀⠀⠘⣧⣠⣿⠁
⣾⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⡇⠁⢰⡄⠙⢿⠏⣴⠀⠀⠀⠀⢀⣀⣀⡀
⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢱⠀⣤⠂⣤⠀⠀⠀⠀⠿⠿⠿⣧
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⡿⢻⣿⣿⣆⢿⣷⣼⢦⣿⣦
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢀⣴⣿⣿⣿⣿⣎⢿⣿⣧⣿⣿⣤
⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⡿⠏⠉⢿⣿⣿⣿⣞⣿⣿⣿⣿⡀⠀⢰⡄
⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣴⣧⣸⡇⢉⣿⣿⣿⣿⣿⣿⢿⠀⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣾⠛⠋⠀⡿⢿⡿⣿⣿⣿⣿⣾⣶⢿⣇
⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⡛⠿⢿⣶⣶⣶⡿⠿⣿⣿⣿⣿⠏⠙⠛⣿⡇⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣧⣘⣧⠉⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⢻⠃
⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠈
⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣤⠈⠃⠀⠀⠀⠀⠀⢀⡾⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢿⡿⠁⠀⠀⠀⠀⢀⣠⡾⠿⡿⠟⠀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣷⣤⣿⠛⡦⣬⣿⠂⠀⠀⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⣀⠀⠛⠛⣷⡌⠁⠀⠀⢠⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⣅⣠⣴⣾⣿⡇⠀⠀⢠⡎⣤⡤⠁⠀⡄⠀⠀⠀⠀⠀⢀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠇⠀⠀⣾⡇⠀⠀⠀⠸⠃
⣿⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⣿⡇⠀⠀⡘⠻⠀⠀⠀⠀⠀⠀⠤⠤
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠛⠂⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣿⣿⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣊⣠⣬⣿⡏
⣀⠀⠀⠀⣠⣤⣴⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⢠⡟⠃
⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⠶⣤⣤⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⢿⣿⡿⣿⣿⣤⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⣷⣾⣿⠟⠋⣤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⠙⠛⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢀⣨⣿⣿⣿⡷⠖
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣾⣿⣿⣿⣿⣾⣷⣶⣾⣿⣶⣶⣿⣿⣶⣶
⣿⣿⣯⣭⣿⣭⣭⣏⠉⠠⣤⣵⣦⡄⠈⢻⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⡏⠠⠀⠸⣿⣿⣿⣮⡿⠋⡤⠀⠀⢀⠀⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣥⣼⣏⣉⣿⡟⣿⣿⣿⣿⣿⠇⠀⠤⠀⢸⣿⠿⣿⠟⠀⠀⠀⠈⠙⠃⠀⠀⠐⠖⠀⢤⣤⣼⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣶⣦⣤⣄⣾⣗⠀⠁⠀⠀⠈⠁⠀⠀⠆⠀⠀⠀⠀⠀⣀⠀⠁⣠⣝⢛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⢤⣽⣿⣿⣿⣿⣿⣿⡀⢀⣤⠈⠙⠛⠻⠿⢿⣿⣿⣿⣆⣀⡀⠀⠴⠆⠀⠀⠀⠀⠀⣀⠀⠉⠀⠚⠛⠛⠃⠰⠂
⠀⠀⠀⠀⢠⣤⠈⠉⠀⠀⠁⠄⠒⠤⠄⠀⠉⠁⠀⠀⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠑⢀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠈⠀⣠⡶⠆
⠀⠄⠀⠀⠈⠉⠴⠶⢂⡠⠄⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠠⢀⣠⣀⣀⣀⣉⣀⣀
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⠄⠀⣀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠿⢿⣿⠿⠋⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⡴⠀⣀⡀⠉⠀⠀⣀⡀⠀⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠉⠻⠉⠀⠙⠁⠀⢀⣾⠏⡇⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣰⠃⠀⣾⠇⢸⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⢰⣿⠀⠀⣸⣿⠇⠀⢠⡶⢤⡀⠀⢠⣿⡿⠁⠀⢸⣿⣠⠏⣠⣤⡀⢰⣶⠈⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⣼⡏⠀⠀⣿⣿⠀⠀⠈⠳⢤⣽⡤⢺⣿⠀⠀⠀⣾⡟⠁⠀⢸⣿⠀⣸⡏⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠐⣿⣏⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣴⡿⠂⠈⡇⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⡏⢠⣿⣇⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢠⣿⡷⠶⠻⣿⡇⠀⠀⣿⡇⠀⢠⡇⢸⣿⡄⠀⣸⢻⣷⣀⣼⣿⣧⡾⣸⣿⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣴⣄⣀⣤⡶⢻⣿⡇⠀⠀⣿⡇⠀⢸⣿⣧⣀⣼⠁⠀⠻⣿⣶⠏⠀⠻⠟⠁⠈⠉⡴⢻⣿⠀⠀⠀⠠⣶⣦
⠀⠀⠀⠠⣦⠀⠙⠛⠛⠁⠀⢸⣿⣷⠀⠀⢻⣿⣦⠟⠈⠛⠛⠁⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⣿⠀⠀⠀⠀⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠈⠁⠀⠀⠀⠀⠀⣴⡏⢈⡇⢠⡞⢹⡄⠀⠀⠀⢿⣴⣿⡟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢰⣿⠁⡼⢡⣿⠁⣸⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠐⣶⣷⡞⠀⠺⠆⠀⠀⢀⣠⣴⡟⠁⣠⠤⣄⠀⠀⣸⣿⠞⠁⢸⣿⠜⠁⢸⣿⡆⢸⣿⠀⠀⠀⠀⢙⣿⣿⣅
⠀⠀⠀⠀⠐⠉⠻⠉⠀⠀⢀⣤⡾⠟⠁⢸⡇⠀⠘⣲⠤⢷⡔⣿⣿⠀⠀⣾⣿⠀⠀⣼⣿⠀⣼⣿⠀⠀⠀⠀⠉⢹⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢸⣇⢀⣾⡿⠁⠈⡇⢿⣿⠀⠀⣿⣿⡀⢠⣿⣇⡴⢹⣿⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣤⣦⡀⠈⠛⠗⠀⠀⠀⢸⣿⢸⣿⡇⠀⢀⡇⠸⣿⣄⡼⠁⠻⠷⠏⠙⠛⣡⠾⣿⡏⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣤⣶⡄⠀⢸⣿⠘⣿⣇⢀⡼⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⣰⠃⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣾⠏⠁⠀⠀⣾⡿⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣿⠃⠀⣠⣤⡈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣰⡿⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠻⠾⠟⠁⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠋⠀⠀⠶⠀⠸⣿⠇⠀⠀⠀⠠⣤⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠴⠛⡿⠓⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⠀⠀⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣿⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⡈⢻⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⣸⣇⠀⣶⠀⠀⠀⠀⠀⣤⣘⣿⣁⣤⠀⠀⠀⠀⠀⣴⠆⣰⡟⢀⣠⣴⠾⠛
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣾⠿⢿⣦⣿⡀⢴⡀⢠⣤⡈⢻⡿⠋⣡⣄⠀⣠⡀⢰⡟⣰⣿⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠶⠿⠛⢷⣾⣇⠘⠻⣇⢸⡇⢀⡿⠛⢀⣿⣤⡿⣿⣉⡉⠛⠛⠳⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣉⠙⠳⣦⣽⣾⣷⣟⣠⡶⠟⠛⠿⢦⠄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠗⠲⢶⣿⣿⣿⣿⠷⠶⠶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠐⠻⣶⣤⣴⠞⢋⣽⢿⡿⣟⠻⢦⣄⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠷⠶⢦⣤⣤⣈⣉⣿⣾⠛⣿⠁⣴⣾⠁⢸⡇⢹⣦⡄⢹⡿⢷⣤⣶⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣿⠏⣼⠇⠈⠋⠀⠉⢉⣠⣾⣧⡈⠛⠃⠸⠷⠨⣿⠻⣷⣶⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣤⡶⠟⠋⠁⣼⠏⠰⠟⠀⠀⠀⠀⠀⠛⢉⣿⡍⠛⠀⠀⠀⠀⠀⠶⠀⢻⡏⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣧⡈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣿⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿⠀⠀⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠲⡄
⠀⠀⠀⠀⠐⠀⣁⠃⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠉⠡⡀⡄⠠⠐⢀⠀⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿
⠀⠀⠨⠀⠁⠀⠀⠄⠈⠃⠋⠀⠀⠀⠀⠐⠀⠀⢒⠀⢢⣦⣄⠱⣤⣠⠊⢁⣨⠴⠠⠀⠂⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠋⠁⠉⢉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠙⡿⠛⠓⢾⣃⣒⣚⣃⣀⣀⣀⠀⠀⠌⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠀⠈⠉⠉⠀⠀⠀⠀⠸⠁⠀⠀⠄⠀⠀⠿⠠⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠈⠀⠀⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⡄⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠰⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⢹⠂⡀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠜⠐⡇⠂⠀⠀⢰⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠒⠓⠀⠘⠒⠂⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠘⢾⢀⡠⠞⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⡁⠂⠀⣳⡤⠨⡭⠖⠐⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠅⠀⠄
⠀⠀⠀⢀⡠⠄⢲⠁⠀⣀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠖⠁⣀⠀⢄⣀⠠⣀⠀⠀⢠⣠
⠀⠀⠀⠀⠤⠠⡰⡀⡠⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠲⡈⠳⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡁⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠙⠀⠘
⠀⠀⠀⢀⣀⠀⢀⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠈⠠⡀⣜⠄⢸⣀⣄⢬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠽⡶⢃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡄⠀⠤⠄⠀⠢⢼⠄⠸⠴⠤⠴⠶⠦⠤⠤⠄⠀⠀⠀⠀⡀⠐⠰⠁⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢂⣾⣿⣿⡿⢆⡄⠉⠹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉⠉⠉⠉⠀⠀⠠⢤⢤⠤⠂⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾⣶⣷⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⣀⠀⠀⠀⠀⠋⠉⠙⡄⠀⠀⠀⠀⠰⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⣠⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⡀⢀⡀⠀⠀⠀⢙⣥⡀⠀⠀⠀⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⣧⠈⠉⠉⠁⠀⠀⠴⣀⣿⠇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠂⠀⠀⠀⠀⢅⣻⡁⠀⠀⠀⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡏⠀⠠⠀⠐⣾⣿⠧⠀⠠⠀⠸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢦⠀⡀⠀⠈⣷⣆⡀⠄⠀⣹⣿⡇⠀⠀⠀⡐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢛⠙⢦⢀⠻⣻⣿⣿⡄⠀⠀⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⡓⣻⠿⣿⣿⣟⠃⣠⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⣠⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣀⡀⣀⣀⢀⣠⠄⣠⡀
⣀⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣭⣅⣽⣏⣭⣮⣹⣿⣿⣶
⣉⣭⣯⣍⣩⣭⣉⣩⣭⣭⣥⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿
⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠿⠟⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠴⠄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠉⠁⠉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⡀⠀⠀⠀⠀⠀⢀⡸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠒⠐⢶⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢂⣴⠀⠀⠀⠀⠀⣀⣠⢄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⠀⠘⠣⠀⠀⠀⠀⡶⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠙⠃⠀⠀⠀⠀⠀⢴⣆⠠⠦⠄⣀⡀⠈⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠆⠀⠀⠀⠀⢸⠟⢿⣶⡄⢠⠤⣶⠄⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠇⠘⣄⠀⠀⠀⠀⠳⡄⢸⡀⠀⣠⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠰⡶⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⡛⠛⠛⠻⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣀⣀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡗⠲⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠠⢤⠄⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠚⠁
⠀⠀⠀⢀⠜⠀⠙⢦⡀⣀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡰⠋⠀⢀⠴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡜⠁⠀⣴⠎⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⢀⣠⣼⣇⢀⣠⡤⠀⠀⠀⠀⡆
⡆⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠠⠈⠀⠉⠙⣿⣿⣷⠆⠀⠀⠀⠁
⠷⠠⠤⠠⠤⠤⠤⠀⠀⠰⡎⠉⠉⠉⠀⠀⠈⠁⠀⠼⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠠⠤⢤⡤⠤⠁⠸⠤⣤⠤⠤⠤⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣾⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣿⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡌⠓⣤⡀⠀⠀⠀⠀⠀⢰⣦⣀⣠⣾⣿⣿⠃⠀⢸⠓⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡔⠉⠙⢠⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢶⣶⣎⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠄⠀⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⣻⣤⡀⠀⡀⢀⢺⣗⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠄⡄⠢⡝⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢝⡳⣦⢰⠟⠎⣀⣲⠵⣛⠫⠁⠀⠀⠀⠀⡀⡀⠠⣦⡀⢫⢛⠌⣠⡞⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣰⢒⢿⠐⣏⠂⠁⠀⠀⠀⠀⠀⠀⢐⠭⡻⠓⠂⠨⣷⣿⡯⠤⠄⢶⢶⠆
⠀⠀⠀⠀⠀⠀⢀⣠⡵⢿⠛⠸⠹⡾⠅⠋⠛⢥⡢⠄⠀⠀⠀⠀⠀⠀⠀⠐⠛⣡⢂⣝⡌⠱⠶⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⢭⠏⡂⠀⠀⠀⠸⠱⠆⠀⠀⠀⠀⠀⠠⢐⣸⡆⠀⠃⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣟⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠹⡅⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢤⠀⠀⠁⢀⠀⣨⠾⠡
⠀⠀⠀⠀⠀⠀⠠⣁⡀⠀⠀⠙⡇⢀⠀⣆⢠⠶⡀⠀⠀⠀⠀⠈⠟⡺⠤⢀⡀⠀⠣⣸⣀⡪⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠊⠔⠩⠛⠒⢴⣶⣅⣊⠉⠑⠉⠂⠁⠀⠀⠀⠐⠋⠀⠀⢀⣀⠉⢶⡉⢹⢐⠊⠛⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠔⢹⠱⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠔⡱⢻⠙⢆⠈⠹⡳⡣
⠀⠀⠀⠀⠀⠀⠀⠤⢄⡶⠉⠀⠀⠐⡙⣦⠴⠤⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠘⢁⡌⠀⠀⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠁⠀⠀⠀⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢱⢯⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠠⣤⣀⠀⠀⢀⣤⣀⣀⣄⣀⣠⣀⢂⡴⠀⠠
⠀⠀⢀⣠⡤⠴⢿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⡆⢸⡟⣿⡿⠽⣀⣻⠏⠈⠉⠁
⠀⠀⠛⣏⣠⡠⠆⠀⠀⠀⠀⠀⢠⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣋⠛⠞⠑
⠀⢀⡠⠊⠀⢠⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄
⠀⠀⠑⠒⠢⢄⡀⠀⣀⠀⠀⢐⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣈⢣⠠
⠀⠀⠀⣀⡀⠀⠉⠛⠿⣆⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠉⡀
⠀⠀⣾⣿⣿⣷⣮⣴⣔⣘⣾⣿⣿⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦⠋
⠀⠀⠺⢿⣿⣿⡿⠿⢿⡝⢚⢁⣬⣿⣦⡀⢀⣴⣿⣤⣼⣴⣄⡀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⢀⠀⣀⣠⠝⠋
⠀⠀⠀⠀⠈⠉⢁⣹⣿⣿⣿⣿⣿⣿⡏⣩⡿⣿⣶⣾⣿⣿⢿⣿⣶⣷⠶⠖⠞⠛⠟⢳⠀⣠⣟⡉⠻⢤⣄⡀
⠀⠀⠀⠀⠀⡀⠘⣻⣿⣿⣿⣿⣿⣿⣏⣿⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣥⣀⣈⢠⣏⣿⣿⣷⣤⣈⠛⢶⣤⡀
⠀⠀⠀⠀⣘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⣮⣽⢻⢽⣿⢿⡿⢿⣭⣭⡁⠄
⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠙⠛⠋⠙⠛⠟⠛⠛⠛⠅⠙⠛⠛⠚⠛⠛⠋⠉⠙⠛⠛⠣⠈⠀⠀⠘⠓⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣾⡇⣴⣶⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣿⣿⠿⠓⣠⣤⣤⠀⢸⣿⠇⣼⣿⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠉⢸⣿⣿⠀⣸⣿⠟⢙⡇⣾⣿⠀⣿⡇⠀⠛⠿⠤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⢸⣿⣿⠀⠸⣿⣿⠀⣿⡟⠀⣼⠁⣿⡇⢸⣿⠇⠀⠀⢀⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣀⣠⣬⣥⠀⠙⠻⠛⠁⠈⠿⠁⢸⣿⠀⠀⣰⡟⠁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⡄⢡⡄⠀⣿⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠘⣿⣿⠀⠀⣠⣤⡄⠀⣼⣿⢀⣿⠇⠀⣬⡍⠋⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣾⣿⠁⢹⢠⣿⡇⣾⣿⠀⣸⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⣿⣿⢸⣿⡇⠀⡿⢸⣿⢃⣿⡇⠀⣿⣿⣠⡴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠏⠀⠻⠧⠞⠁⢸⣿⢸⣿⡇⠀⠈⢉⣩⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠶⠟⠋⣀⣠⣤⣤⣤⣤⣤⣍⣀⠛⠃⠀⢠⡾⠋⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠉⠉⣉⣩⣭⣍⣙⣂⠀⠸⢧⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡏⠀⢰⡆⠀⣴⣿⠟⢛⠛⠛⠛⠛⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠞⠁⠀⢿⣷⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⠟⠛⠻⠟⠋⠁⠻⠟⠩⣽⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⠀⢀⣠⣤⣀⠀⠀⣠⡤⣤⣄⠀⠈⢙⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣯⣾⣿⣿⣦⣼⣷⣿⣿⣶⡇⣷⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⡁⠻⢿⡿⠽⠃⠈⠳⠯⠵⠞⠁⢈⡟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡟⠦⣤⡤⢤⣶⣦⠤⠤⠤⣤⣖⠋⠙⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢰⡑⡐⠀⠩⢀⡔⡆⠂⠀⠈⣧⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣢⣿⣿⣷⣴⣾⣷⣶⠚⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢿⣷⣜⣿⣿⣧⣼⣿⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⢽⣿⣿⣿⣿⣿⣿⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⣸⣿⣿⣿⣿⣿⣿⠀⢡⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠿⠿⠿⠿⠿⠿⠿⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠉⢉⡓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣎⠉⠙⢻⠀⠙⣇⠀⠀⠑⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠳⣄⠈⢻⠙⠋⠀⠀⢸⡄⠀⢀⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠘⠰⣄⣸⠖⠛⠀⠀⢀⡔⠋⠀⣤⣈⠫⠑⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠓⢦⡀⢀⣠⡤⠶⠶⠯⣅⠀⢠⠀⠀⡜⢀⠞⠙⠿⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀⠀⢸⡉⠑⠒⠢⣄⣿⣋⡁⠀⠀⠀⠀⢸⠀⠀⢀⡾⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠺⡍⠉⢉⣙⣓⣦⡴⠋⠀⠀⣸⣓⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠙⠚⠁⠀⢀⡈⠉⠀⠀⠀⡿⠛⢉⣁⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠠⢀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⡼⢭⡟⠂⠀⠀⠀⠞⠋⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⣠⡎⠑⣾⡇⠀⠀⠀⣶⠠⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⢈⣿⣿⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠫⣄⣸⠋⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠠⠚⠉⢳⡠⠞⠹⣀⠟⡿⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠏⡼⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠁⠀⠀⠀⠀⠀⠈⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⡤⠴⠺⡇⠀⠀⠀⠀⠀⠀⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡰⣎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⣖⡛⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠖⠚⠋⠁⢀⡠⠴⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⠜⠻⡄⣠⠞⡉⠉⠉⠉⠁⠀⠀⢀⣀⡤⠖⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⢡⡖⠒⠺⢥⡀⢰⡀⢀⣀⡤⠔⠚⠉⣁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠖⠛⠫⠽⠆⢹⠆⠉⠉⠀⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢈⡿⠥⢛⣀⠀⠘⠠⠤⠤⠔⠒⠚⣿⣦⡀⠀⠀⠀⢠⠆⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⠤⣀⡘⠦⠖⠋⠈⣇⣀⠤⠤⠤⠤⡤⣤⡸⠿⠿⠦⣄⣠⠋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢆⠀⡞⢩⠟⠒⠲⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣁⣀⡤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠉⠀⠀⠈⢻⡁⡎⣀⣀⣠⣤⣤⠤⠤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⢻⢳⠒⠤⣤⣤⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠘⠋⠉⠈⠀⠀⢀⠠⢽⡦⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠴⠴⠄⠤⠄⠛⠛⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⣸⣿⡟⠉⠀⠼⣷⡄⠀⡔⠒⠰⢄
⠀⠀⠀⠀⢀⣤⢀⡀⠀⢸⣿⠀⣿⣿⣼⣧⣼⣿⣿⣿⡇⠀⠀⠻⠿⢁⠀⠀⢐⣂⠸⣇
⠀⠀⢀⣶⣾⣿⣿⣇⣠⣼⣿⠀⣿⣿⡿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⡈
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⢿⡿⢄⠀⠈⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠀⠀⠀⠀⣠⡴⢊⠥⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⣤⣤⣦⣴⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⣦⣤⣀⣉⡔⣶⣾⡄⠱⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣶⠇⠀⠀⠀⠀⠀⠀⣀⡤⠠⢰⣾⣿⣿⡿⠛⣯⣵⡟⠋⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢿⣀⢀⣠⣤⡶⠊⢉⣡⣴⣶⣿⣿⣿⣿⡾⣿⣿⣿⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠖⠋⠹⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⠀⠉⠋⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⢠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠫⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⣟⠻⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⡄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠉⠈⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠲⠦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠤⣄⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠞⠳⢦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⡞⠉⠀⠀⠀⠀⠈⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠉⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀
⠀⠀⠀⡐⠒⣲⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣧⣧⢜⡆
⠀⠀⠀⣘⣬⡼⢦⠾⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠉⡝⢉⠟⡇⠀⠀⠀⣀⣴⢶⠚⢒⣊⣳
⠀⠀⢀⠀⠀⡄⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠚⡀⡎⠀⡇⠀⠀⠈⡷⢧⡬⢶⣶⣮⣅⣄
⠀⠀⢸⡄⣠⠇⠸⠤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠉⠉⠉⠋⠉⠉⣉⠉⠉⠍⡇⠀⠀⠀⡄⠀⣠⠃⡇⡴⡇⠀⠀⠀⣠⠊⢠⡎⢇⡱⣘⣾
⠀⠀⢸⠤⠤⡀⠀⡤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠀⠁⠀⠈⢀⠀⠀⠀⠄⠀⢁⠀⠀⠀⡗⠊⠁⡠⢻⠀⡇⠀⠀⢸⣁⠜⠁⡳⠚⢀⡴⢻
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠐⠂⠀⡄⠀⠐⠀⠀⠆⣸⠀⠀⠀⡗⠒⠉⠀⡼⢀⡇⠀⠀⠀⡁⡠⠊⠀⡴⠋⡐⡻
⠀⠀⠈⠉⠀⠓⠚⠓⠒⠒⠒⠈⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⣀
⠀⠀⣀⣫⣄⣦⣠⣯⣖⡀⠀⠀⠀⠀⠠⡀⠐⡆⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠤⢤⣤⣿⣧⣤⣤⠜⠀⠀⠀⠰⣒⢞⣛⣛⣯⡭⢀⣁⣱⠀⠀⠀⠀⠀⠀⠠⣈⣇⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢄
⠀⠀⠀⠘⡀⢸⡇⢀⠇⠀⠀⠀⠀⢠⣡⠁⠀⠀⡀⠀⠗⠾⠂⠀⠀⠸⡁⠩⠍⠤⠤⠭⠯⠯⠭⠭⠭⠭⢽⠀⠀⠀⡏⠉⠉⠉⠉⠉⠷⡀⠀⠡⡀
⠀⠀⠀⠀⣇⢸⡇⣸⠀⠀⠀⢠⣔⠟⢹⢠⠀⠀⠇⠀⡄⠀⡅⠀⠀⠀⡄⢒⢐⡒⡂⢒⢖⠒⡒⣒⣒⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⣤⠊
⠀⠀⠀⠀⠘⠾⠷⠃⠀⠀⠀⠀⠈⠀⠸⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠃⠠⠥⠬⠬⠤⠥⠤⠤⠤⠥⠤⠞⠀⠀⠀⠣⠠⠤⠤⠤⠤⠤⠦⠼

⠀⠀⠀⠀⢀⢆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡠⠉⢢⡀
⠀⠀⠀⢠⢸⠴⣟⢰⣦⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣋⣄⣊⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠄⠰⠛⠛⠂⠀⠤⠂⠀⠀⠀⡆⠀⠙⠛⠛⠛⠛⠋⠀⡂⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠄⠀⡃
⠀⠀⠀⠸⣀⢀⢘⠀⡸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣃⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⡀⣈⣈⣀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠐⢦⡀⠀⣿⣿⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⣴⣿⠇⠀⠀⢹⡀⣿⣿⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⡏⠀⠀⠀⢸⣷⣿⣿⠁⣴⣶⡆⠀⠰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣿⣿⠤⠤⠐⠂⡾⢹⣿⣿⡜⣼⣿⠃⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠂⠹⣿⡆⠀⣀⠞⠁⠈⣿⡟⢰⣿⡏⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⢉⣩⠥⣄⡀⠀⣀⠀⣿⣿⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⡟⣰⡶⠀⢻⣤⠟⠀⢿⡇⠀⣠⠞⢁⣤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⢀⡠⢼⣿⡇⠀⢸⠇⠀⠀⠈⠛⠛⠁⣴⡿⠁⠀⠀⢸⠂⠀⠀⠀⠀⢀⣴⡆⠀⢀⡀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⡁⠀⢸⣿⡇⣠⠏⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣿⠁⠀⣿⣿⠀⠀⠀⢀⡟
⠀⠀⠀⠀⡠⠖⠉⠀⠀⣿⡇⠀⠀⠉⠉⠁⣠⢖⣪⡐⠲⣄⠀⣿⣿⣀⡤⠞⣡⡾⠛⠉⠛⡆⠀⣿⡏⠀⣠⣿⣿⠀⠀⢠⠞
⠀⠀⢀⡞⠀⠀⠀⠀⠀⣸⡇⠀⠉⣿⣿⡀⢿⣿⡟⠀⣀⣿⢺⣿⡟⠀⠀⣰⡿⠀⠀⠀⡠⠃⢰⠿⠧⠴⠃⣿⣿⢀⠔⠁
⠀⠀⠘⡄⠀⠀⠀⠀⣠⠟⠀⠀⠀⣼⣿⡇⢿⣿⠋⠁⣴⠃⠀⣿⡇⠀⣰⣿⡟⠒⠚⠉⠀⢀⡏⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠈⠀⠀⠐⠊⠁⠀⠀⠀⡠⣿⣿⠀⠈⠛⠗⠚⠁⠀⠀⠈⠛⠚⢩⣿⡇⠀⠀⠀⢀⡞⠀⠀⣠⠖⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣠⠋⠀⢀⡾⠁⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠊⠁⠀⠀⢾⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣂⠀⣾⠛⢭⣬⣭⠉⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⢉⣼⣿⢫⣼⣿⣦⣄⣠⡴⠞⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣳⡄⠈⢿⣃⣼⣿⣿⣿⣿⣾⣯⣯⣁⣤⣤⣇⡀⠱⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⡴⡽⣦⣄⣯⠁⢉⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡇
⠀⠀⠀⠀⢠⣾⣿⡿⣯⣿⣿⣿⣽⠟⠀⠀⠾⣶⣷⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⢠⣿⣿⣿⣽⣿⣿⣏⡿⠃⠀⠀⠀⠀⣿⠿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣥⣤⣄⣀⣀⠀⠘⣧⠾⣿⣿⣽⣿⣥⠎⢸⡏
⠀⠀⠸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⡁⠤⢸⣿⣦
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣭⠉⠄⣀⡴⠛⣿⣼⣧⣄
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡯⣽⣿⣿⣿⣿⡷⠈⣡⠶⠛⠁⠀⢰⣿⣿⣿⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢿⣾⣿⣿⣿⡿⢫⡾⠋⠁⠀⠀⠀⠀⣾⣿⣯⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢇⣿⣾⣿⣿⢋⡶⠋⠀⠀⠀⠀⠀⠀⣰⢻⣿⣿⣽⣽⣿⣿⣿⣿⣿⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣿⣿⢿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⡟⣾⣿⣿⣿⡟⢿⣿⡿⠿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣿⣿⡟⢃⠀⡿⢠⡀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣴⣿⣿⠠⢸⣿⣿⡙⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣾⣿⣯⠀⠁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣟⢐⣻⣿⣿⠷⠆
⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⢁⠄⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⣿⠿⣟⣹⣿⣿⣛⣻⣿⣯⠠
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡳⠂⠰⡖⠂⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⠏⠻⣿⣤⡀
⠀⠀⠀⠀⠀⠀⡷⣿⣿⣿⠅⠄⣾⠀⠀⠀⠀⣀⣤⣶⣿⠿⠻⣟⣋⣹⢻⣿⢟⣁⡀⢹⣦⢻⣦⣥⣀⡙⠁
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣣⡐⠐⡷⠡⠤⠶⠚⠛⠻⢛⣴⡾⢿⣿⣿⣿⣿⣿⡛⢙⡇⣸⢿⣬⣿⣿⣿⣻⡄
⠀⠀⠀⠀⠀⢸⢻⣿⣟⢿⢆⠘⡷⢀⢆⠀⠀⡰⠀⡜⠉⠀⠩⢾⣿⣿⣿⢿⣏⣼⣷⣿⣾⣿⣿⣿⣽⣿⠆
⠀⠀⠀⠀⠀⢠⡽⣿⣿⢯⣿⢂⣿⠰⠘⡡⠎⢴⣜⢷⣤⣤⣶⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⡉⠛⠋
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣳⣿⣿⣾⣔⢁⡀⠘⢧⣭⣧⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠃
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣽⡿⡯⡋⡩⣻⡙⠛⡙⠀⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣷⣿⣿⣿⡁
⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⣗⠟⡠⡪⠂⠀⠘⣿⣿⢿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣕⠁⠔⡔⣅⠈⠉⢻⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠰⣧⡟⣿⣻⡿⣿⣿⣿⣿⣿⣿⣾⡠⠢⠢⠀⠊⠶⠇⢹⣿⣿⣿⣿⣿⣷⢺⣿⡧⡟⢾
⠀⠀⠀⠈⠐⠷⢯⣯⣍⣟⡿⣿⣿⣿⣿⣿⣿⣷⠀⠔⠀⠁⠀⠀⢺⣿⣵⣾⣿⡧⣹⢾⣇⠁⠆⠘
⠀⠀⠀⠀⠀⠀⠠⠀⠉⣿⢿⡿⢿⡿⠟⠿⠿⢿⡆⠀⠀⠀⠀⠀⠘⣿⡻⠟⢻⡱⢻⣆⠛⣀⠂
⠀⠀⠀⠀⢀⠄⠀⠀⠀⡯⠸⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠐⢠⠂⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠒⠬⢘⢭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠄⠐⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠄⠀⠀⠀⠀⠁⢀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠠⠀⠌⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡤⠓⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠰⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⣖⡶⢔⡠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠌⢘⣦⡀⠄⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠔⠚⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠈⠀⠈⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⢿⡿⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⠀⠙⢿⡿⠋⠀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢻⣿⣿⡄⣿⢸⡇⣿⢠⣿⣿⡟⠀⠀⠀⠀⢠⡄
⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⣬⣭⣿⣿⣸⣇⣿⣿⣭⣥⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⢀⣀
⠈⢿⣿⣿⣦⢹⡇⠀⠹⣿⣿⢠⡦⢸⣷⣄⡀⠀⠈⢻⡟⠁⠀⢀⣠⣶⡇⢴⡀⣿⣿⠏⠀⢸⡇⣴⣿⣿⡿
⠀⠀⠙⣛⣛⣿⣿⣜⡻⢾⣿⡿⠁⠘⣿⣿⣇⢠⡆⢸⡇⢰⡄⣸⣿⣿⠃⠈⢿⣿⡷⢟⣣⣿⣿⣛⣛⠉
⠀⠀⠀⠉⠉⠀⢶⣭⣛⢶⣽⣇⠀⠠⣬⣝⣿⣿⠁⢸⡇⠈⣿⣿⣫⣥⠄⠀⣸⣯⡶⣛⣭⠶⠀⠉⠉
⠀⠀⠀⣠⣾⣿⣶⡾⣿⠟⠛⠙⠻⣦⣄⡈⠉⢿⡄⢸⡇⢠⡿⠉⢉⣠⣴⠞⠋⠙⠻⣿⢷⣶⣿⣷⣄
⠀⠀⠉⠛⠛⠛⠋⢼⠏⢀⣀⠐⢶⣄⠉⠛⢶⣤⣻⣾⣷⣟⣤⡶⠛⠉⣠⡶⠂⣀⡀⠹⡧⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡷⣿⡷⠶⠶⢾⣿⣿⣿⣿⡷⠶⠶⢾⣿⢾⣿⣿⣿⡷⠆
⠀⠀⣀⣤⣤⣤⣄⢺⣆⠈⠉⠡⠾⠋⣀⣤⡶⠛⣿⢿⡿⣯⠛⢶⣤⣀⠙⢷⠌⠉⠁⣰⡗⣠⣤⣤⣄⣀
⠀⠀⠈⠙⢿⣿⡿⢷⣿⣦⣄⣠⣴⠞⠋⢁⣀⣾⠃⢸⡇⠘⣷⣀⡈⠙⠳⣦⣄⣠⣴⣿⡾⢿⣿⡿⠋
⠀⠀⠀⣀⣀⠀⠶⢛⣭⡾⣿⡏⠀⠐⠛⣫⣿⣿⡀⢸⡇⢀⣿⣿⣝⠛⠂⠀⢹⣿⢷⣭⡛⠶⠀⣀⣀
⠀⠀⣀⣭⣽⣿⣿⢋⣵⢾⣿⣷⡀⢰⣿⣿⡟⠸⠇⢸⡇⠸⠇⢻⣿⣿⡄⢀⣾⣿⡷⣮⡙⣿⣿⣯⣭⣀
⢀⣼⣿⣿⠟⣹⡇⠀⣠⣿⣿⠘⠗⢸⡿⠋⠁⠀⢀⣼⣧⡀⠀⠈⠙⢿⡇⠺⠃⣿⣿⣄⠀⢸⣏⠻⣿⣿⣧
⠉⠉⠉⠁⠀⠈⠀⠀⢹⣿⠟⠁⠀⠀⠀⢛⣛⣿⣿⢹⡏⣿⣿⣛⡛⠀⠀⠀⠈⠻⣿⡏⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣼⣿⣿⠇⣿⢸⡇⣿⠸⣿⣿⣷⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠀⣠⣾⣷⣄⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣾⣷⡙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣼⡇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣿⠀⣠⡀⠀⠀⢠⡄⡆⢀⡄
⠸⡇⢿⡄⢸⣿⡄⠀⠀⠀⠀⠀⠸⡟⣿⢸⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⡏⢠⣿⡇⠀⠀⢸⢷⡇⢸⣿
⠰⣿⢾⣇⢸⡟⣿⡀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡿⠀⡾⢻⡇⠀⠀⠈⠘⠃⠟⠘
⠀⠙⠘⠟⠀⠃⠀⠀⣀⢠⡄⢀⠀⠀⠀⠀⠀⠐⣀⠤⠔⠗⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⠁⣼⡇⠀⠀⠀⠀⠊⠁⠄⠀⠀⣀⠠⠄⠒⠠⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣧⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠇⡿⢠⡟⡧⠀⠀⠀⢀⡀⠀⣠⠐⣉⠀⠀⠉⠀⠂⠀⣀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠀⠈⡏⢿⠀⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢅⠚⠰⠈⠀⠲⡄⠀⠀⠈⠒⠀⡀⠀⠀⠀⢀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠈⠛⣆⠟⠀⠀⠀⢀⣀⡀⠙⠠⡀⠈⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⠄⠁⢀⠀⠀⠀⢉⣭⣍⡉⠀⠈⠢⠀⠐⡄⠈⢁⠛⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠑⠄⣀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠱⡀⠀⡀⠀⢣⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢠⠯⡁⠈⣄⡈⠨⠀⡇⠀⠀⠀⢠⣿⣷⣦⣤⣉⣉⣩⡽⠀⠀⠀⢀⢠⠔⡠⠸⠀⠇⠀⠀⡇⠀⢸⠄
⠀⠀⠀⠀⠀⠋⠐⣈⠔⠈⠀⣠⠀⢻⡀⠀⠀⣿⣿⠿⢿⡿⣿⣿⣿⡇⠀⠀⠀⠈⠚⠠⠁⠨⡇⡆⠀⠀⡇⠀⠋⠂
⠀⠀⠀⠀⠀⠑⢅⠁⠀⠀⢴⡃⠀⠀⠳⡀⠀⠻⣿⣤⣀⣀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡠⠒⠢⠈⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠣⠐⠈⠙⢄⠀⠀⠈⠢⡀⠈⠉⠙⠉⠉⠀⠀⠀⠀⣀⣀⡴⢄⣀⣀⣀⡤⠓⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⢳⠒⠒⠈⠉⠒⠲⠤⠀⠀⣀⠖⠚⠛⠩⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠂⠈⠉⡋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠤⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⠉⠉⠀⠉⢴⠂⠉⠀⠈⠀⠅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⠀⣀⠀⠀⢀⣀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣀⣀⣀⣀⣀⣀⣀⣉⣀⡸

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠟⠁⠶⠀⣈⠉⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢉⠀⢀⠀⠐⢛⡀⣲⡞⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⣠⡀⠔⢃⠀⠀⠉⠉⠀⠀⠀⢙⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠠⠃⣠⡿⠂⠈⠁⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠂⣀⠜⠁⠀⢀⡀⠀⠀⠔⠋⠀⠀⠀⠀⠀⣰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⡁⠴⠁⠀⢀⠄⠀⠀⠀⠀⠄⠀⠀⠀⣠⢠⠄⢡⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⢐⡐⢿⡃⢀⠀⠘⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⣿⡦⠉⠐⠀⢈⢻⣆⠀⢀⣉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠈⠀⠀⠀⠙⠢⠤⢛⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡢⡀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠶⣤⣙⡷⣔⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⣮⠡⡀
⠀⠀⠀⠀⠀⠀⠂⠘⠱⠄⠀⠈⠁⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠀⢯⠁⢡⡈
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⡰⠂⡄⢀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠄⠀⠀⠛⢷⣥⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠁⠠⣤⣶⠟⠂
⠀⠀⠀⠀⠀⠐⠄⢠⠀⠀⠀⠀⠉⠀⠁⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠈⢆⠁⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⠑⠠⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠌⠁⠀⠀⠀⢀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠈⠡⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⡀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠁⠀⠀⠀⠀⠈⠃⠀⠀⠈⢆⠀⠀⠀⠸⡀⠀⠙⠁
⠀⠀⠀⠂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠶⢶⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠘⠇⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⣀⣠⠄⠀⢀⡤⠀⠀⠀⠃⠸⠁⠀⠀⠀⠀⠀⠀⠙⠃⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠈⠣⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠦⠘⢯⠷⠄⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿⠁⠀⡆⡄⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣶⡆⠑⠀⠀⢠⡄⠀⠈⠀⠀⠀⠀⠂⠀⠀⢈⠁⣾⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡀⠀⢩⣽⠌⣒⣶⠄⠀⠀⠀⠚⠿⠁⠐⠿⠃⠀⢈⣅⠋⠀⢘⠀⠀⠀⠀⠀⣀⣀⣀⢀⠲⢀⠶⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠷⠈⠉⠙⠋⠀⠸⣻⠃⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂⠀⠁⠀⠀⠀⠀⠀⠀⠰⠟⢸⣷⡾⠋⢠⣤⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠢⠈⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠄⠍⠥⠖⠈⠙⠛⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⣀⡠⠔⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠡⠀⠒⠒⠒⠊⠉⠀⠀⠀⠀⢀⣤⣤⣤⣀⣄⣀⠀⠀⠀⣀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣠⣤⣶⠶⠲⠶⢄⠐⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘⡀⠀⠈⢀
⠀⠀⠀⠀⢀⡼⠀⢀⣠⣶⣧⣽⣿⣗⣿⣶⡄⠀⠑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢦⡄⠈⢂
⠀⠀⠀⢀⡞⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⣿⣦⡈⢆
⠀⠀⠀⡌⣰⣿⣿⣿⣿⣿⣿⠿⠀⣿⣿⢿⣿⣷⠀⢀⡀⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢻⡛⠛⠈⢣
⠀⠀⠠⢱⣿⣿⣿⣿⠿⠿⠏⢶⣀⣸⣿⣾⣿⣿⡇⠰⣿⣧⣴⣦⣌⣂⣄⣲⣦⣤⣤⡀⠀⠀⠛⣶⣄⠀⢣
⠀⠀⠀⢨⠿⠋⠁⣀⣤⠀⠀⢿⡿⠿⠿⠿⠛⠉⠀⠘⠛⠛⣉⣝⣿⠿⠛⠛⠛⠻⠶⢮⣄⠀⠣⡈⢿⣿⡄⠃
⠀⠀⠄⠁⠀⠐⢛⣻⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢁⣴⣶⣶⣶⣶⣶⣤⣤⣤⣁⡀⠳⡀⢿⣿⢸
⠀⠀⠀⠀⠀⣀⡾⠿⢀⠄⢀⠠⢘⣃⠐⣄⡀⣤⡀⠤⡞⠂⢟⢻⣿⢟⣷⣿⣿⣿⣿⣻⣿⣿⣿⡆⠱⠈⣿⢸
⠀⠸⢠⢎⣡⣵⣾⣿⣿⡆⢰⠀⠀⠏⠀⢿⠁⠸⠇⠀⣿⢲⣌⢸⣿⢸⣿⣿⣿⣿⣿⠃⣠⣿⣿⠻⠿⣠⣿⢸
⠀⢀⢡⠞⢿⢿⣿⣿⠁⣅⣈⣤⣴⣶⣶⣾⣷⣶⣿⣶⣦⣤⠛⣯⡻⣮⣻⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⡀⡀
⠀⠘⡸⣷⣌⡤⢿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⣀⠗⢸⣿⣿⣷⣶⣿⢿⠻⣿⣭⣿⣿⣿⠃⠇
⠀⠀⠑⢌⣿⣿⣿⣿⣷⠾⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣿⣿⣿⣿⣧⣿⣶⣿⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠉⣽⠋⣀⣿⣏⣹⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠃
⠀⠀⠀⠀⠀⠑⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣛⣿⣿⣿⡿⠿⠿⠿⠿⡿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠘⣿⣿⣟⠉⠁⠈⠉⠛⠛⠿⠿⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⣿⡿⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣄⣾⣿⡿⢱⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⣿⣿⡗⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠐⢶⣽⣿⣿⠇⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣥⣀⣹⡆⠀⠀⠀⠀⣀⠄⡤⠾⡁⠰⡄⣸⣯⣿⡟⣼⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⡄⢰⠙⡓⢲⠀⢸⠁⢁⠀⣶⠀⣸⣶⣯⣿⣿⣏⣰⡏⣽⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⣾⣄⠷⡈⣄⢸⣂⣸⡧⣿⣝⠙⠋⠻⣿⣿⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠋⠈⠸⠟⢹⡇⢹⡍⢰⣿⠉⠓⣠⢜⣡⣮⣿⣿⣿⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠐⠆⢠⣈⣷⣈⢳⣘⣿⣾⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⣿⣟⡭⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡤⠤⠴⠦⠄⠀⠈⠅⠒⠁

⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣷⠀⠀⢠⣾⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣿⢫⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⡏⠀⠀⣾⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⡇⠀⠀⣿⣿⡄⠀⠀⢀⣤⣤⡀⠀⢸⣿⣿⡇⠀⠀⠸⣿⣿⣷⠀⠀⣾⣿⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⣿⣿⠀⢠⢿⣿⣇⠀⣰⣿⠟⠛⠟⢀⣨⣿⣿⣷⠀⠀⠀⣿⣿⣿⣄⡜⢿⣿⣄⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡶⠃⢸⣿⣿⡀⣿⣿⡀⠀⠐⡏⠀⠈⢿⣿⣆⣤⠞⠁⠈⠉⠁⠀⠈⠉⠁⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⠈⣿⣿⣇⠙⠿⣷⣤⠞⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⣿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⡇⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⡗⠀⠀⠈⠛⠛⢠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢀⣿⠃
⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⡶⠃⠀⣾⣿⣿⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠸⣷⣄⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣤⠀⢸⣿⣿⡿⠀⠀⢀⣾⣿⡇⠀⣴⡆⠀⠀⠀⠉⠉
⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⡟⠀⠀⢿⣿⡇⠀⠀⠈⣿⣿⡷⣠⣿⡿
⠀⠀⠀⠈⣿⣇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣼⣿⣿⡀⢀⡀⠉⠛⢃⠠⢶⣶⠛⠋⠁
⠀⠀⠀⠀⣿⣿⡀⢠⣿⡟⠀⣤⠀⠀⠀⡼⢿⣷⣀⣴⠏⠈⠛⠛⠉⠀⠀⡰⠋⠀⢻⣿⠃
⠀⠀⠀⠀⢸⣿⡇⣿⣿⣇⠀⢹⣧⡤⠞⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⣷⠘⢿⣿⡶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣼⣿
⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠅⢀⠔⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠠⡀⠀⢐⢐⠀⠀⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢠⠂
⠀⠀⠐⡖⢢⠀⠀⠀⠰⡀⠀⠈⠀⠀⠀⠀⠊⠀⠀⠃⠀⢀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⠀⢀⡀⠉⡃⠁⣀⠀⠀⠀⡐⠀⠀⠄
⠀⠀⠀⣀⣀⡉⠤⡠⠀⠁⠀⠀⠀⠨⠀⠀⠀⠀⢘⠀⠤⠁⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠬⡀⠀⢁⠐⠒⢈⠀⠀⢀⠀⢅⢋⣀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠒⠄⠀⡀⠸⠐⠀⣄⠠⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠗⢄⡐⢖⣀⠀⠳⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠂⢨⢣⠣⣉⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⢰⠆⠀⢀⠐⠍⢀⠄⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⡠⠌⠂⡠⠈⠁⢈⠑⠶⡀
⠀⠀⠀⢀⣀⠀⠀⢀⡀⠘⠉⠀⠈⠀⢨⠀⠂⠀⠀⠀⢠⣦⠄⠤⠄⠀⠀⠀⠀⠀⠀⣀⠄⠘⢲⠃⠀⠀⠀⢘⡄⠄⡘⠀⠠⠂⠤⠁
⠀⠀⠀⠀⢀⣠⠄⠉⠁⢠⠀⠀⠀⠀⠰⠀⠀⠀⠀⢆⠀⠠⠈⢃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⢔⡢⡀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠔⠀⠁⢠⠃⠀⠂⢠⠔⠁⠀⢠⠀⠐⢄⠀⠘⠀⠀⠳⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠁⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢾⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠙⠀⠰⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠐⠀⠐⣦⡀⠀⢹⠀⠀⠀⣐⠂⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠠⡀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢦⣀⠀⢠⠈⠲⡟⢄⡀⢀⠃⠀⢀⣈⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠈⢆⠈⠐⢀⠴⠁⠀⠈⢀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠙⡛⠈⠁⠀⠁⠘⠀⢚⠶⡾⠋⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢀⡆⠙⠰⠁⡈⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠖⡊⠉⠄⠈⠠⠂⠄⠀⡠⠒⠀⢀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡈⠉⠀⠀⠈⠁⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠀⠀⠀⣠⣭⡄⢂⠀⠀⠁⠀⠀⠃⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡀⠀⠏⠀⢀⠀⠫⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠉⠁⠎⢈⠇⠀⢠⠁⠂⠞⢁⣄⠀⠀⠤⠅
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠔⠉⣈⠀⢤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠀⡀⠀⠀⡸⠂⠁⠠⣦⠀⠀⢬⠀⠀⠀⠃⠤
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠆⠀⢁⡀⠀⠸⠀⡀⠀⠰⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠁⠀⠘⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠞⠀⠼⠈⢆⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣷⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠸⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣂⠀⠖⠂⢠⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢶⣿⡄⢂⠉⡃⠀⠠⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣟⢉⣽⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠉⣄⠟⣿⣿⣿⡆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡛⠶⣤⠭⠿⣿⡿⠛⣦⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠗⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡴⠂⠀⠀⠙⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠰⠶⠆⠀⢿⣿⣿⣿⣿⣿⡿⠩⣌⣦⢠⠄⠠⠂⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀⠀⣼⡇⠀⡘⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣹⣧⣼⡇⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣶⣿⠷⠦⠤⠄⠀⠀⢹⣿⣿⣿⡟⣿⣿⣆⠈⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣂⠀⠀⠀⠀⢸⣿⣿⣿⣀⣿⣿⠅⠈⣀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣋⣭⠌⠛⢿⠿⣼⣿⣿⡏⠉⢿⣿⡟⠉⠉⠉⠛⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⢿⢿⣷⣶⣶⠿⠿⣧⣤⣼⡉⠁⠀⠀⠀⠀⠀⠈⢉⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠉⠉⠉⠛⠒⠀⠀⠉⠙⠛⠛⠀⠁⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠶⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⡟⠋⠉⢿⣴⠶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢣⠀⠀⡼⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡦⣤⡷⢶⣷⠲⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⡴⠚⢉⣴⠞⢹⡇⢸⠈⠳⣄⠈⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣁⠀⣰⠟⠁⠀⢸⡇⢸⠀⠀⠘⢧⡀⣽⡇⠃
⠀⠀⠀⠀⠀⢀⣀⣐⠒⣠⠤⠤⠶⠶⣮⡭⠿⣞⣹⣄⠀⠀⢸⡇⢸⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⣠⠟⠡⡍⢻⡇⠀⠀⠀⠀⡿⣸⠃⠈⠛⣷⡖⠒⢾⠀⢸⠲⠶⠶⢶⡟⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠐⣇⠀⠀⠙⣾⣇⠀⠀⠂⢸⣿⠏⠀⠀⣰⣿⠀⠀⢸⠀⢸⠀⢀⠀⢸⠇⠀⣤⠶⠶⠛⠛⠛⢷
⠀⠀⠀⠀⠈⢧⣤⠖⠚⠿⢤⣴⠤⠟⠛⢶⣡⣾⣋⣿⠀⠂⢸⡇⣿⠠⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⢸⣷⠞⡛⠳⣦⡀
⠀⠀⣠⡤⠶⣾⠃⠀⠀⠀⣴⠛⣆⠀⠀⠈⢿⡉⠛⠛⢻⡆⢸⠃⢻⣠⡴⠾⠋⢰⡂⣇⠀⠀⠀⠛⠀⢸⡇⢠⠇⠀⠈⠙⣷
⠀⠀⡇⠀⢰⣏⣴⣦⠀⣼⠃⠀⢸⡆⠀⠂⠈⣧⠀⠀⢸⡗⢺⠀⢺⣹⡇⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⢀⡿⢰
⠀⡆⢿⣠⣼⣯⣤⣽⣴⡇⠀⠀⠀⢿⠀⠀⠀⢸⡶⢶⡾⠛⣽⠀⢸⠀⢳⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⡋⠀⠀⠀⣠⡾⠁⠁
⠀⠘⡄⣿⠀⠀⠁⠀⠉⣧⠀⠀⠀⢸⡇⢰⠷⣬⡇⢸⣇⡀⢸⠀⣸⣀⣀⣿⣦⠔⠒⠒⠚⣳⡤⡖⠋⠉⠉⠳⣶⠾⠿⠶⠶⠶⠶⣦⡈
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⡏⠀⠘⠁⢸⡇⠁⢸⣼⡏⠉⠁⢠⠏⠀⠀⠀⡼⠁⠀⠈⢦⡀⠀⠀⠘⢷⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠁⢸⡏⠀⠀⠀⠀⣸⠄⠀⢸⡝⣷⡤⢤⣿⡴⢶⡄⢸⠗⠖⠛⠛⠻⢷⡀⢠⣄⡈⢿⠉⠉⢉⡏⠁
⠀⠀⠀⢸⡇⠀⠈⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠠⠀⣾⠠⠀⢸⡇⣿⢹⡄⠁⠀⠀⠙⠟⠀⠀⠃⠀⠀⠈⣧⠟⠈⠙⠻⠆⠀⣾⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡋⠀⢀⣀⣿⣤⣤⣼⣇⣿⣄⣱⣤⣤⣤⣤⣤⡤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠶⠶⠶⠾⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠈⠁⠉

⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠸⣿⣧⠀⠀⢠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣿⣿⠀⠀⢠⣾⡿⠟⠛⠻⢿⣿⣄⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢹⣿⣧⣠⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⠀⢠⣿⡟⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⡄⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⣿⣿⠀⣾⣿⠀⠀⠀⠀⠀⢠⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⣿⠀⢻⣿⣄⠀⠀⠀⣠⣿⣿⠃⠀⣿⣿⣇⣀⣤⣄⡀⢸⣿⣿⠀⢀⣀⡀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠻⠿⠀⠀⠻⣿⣿⣶⣿⣿⠟⠁⠀⠀⢿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⡄⢹⣿


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





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠿⠿⠟⠛⠛⠻⠿⠿⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠉⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣶⣶⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⢀⣠⡾⣿⡄⠀⡼⠟⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠙⠻⢧⠀⠀⣤⣄⠀⠀⠀⠀⠀⠸
⣿⠀⠀⢀⣴⠟⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⢷⣦⡀
⣿⡆⢠⣿⠁⠀⠀⠀⠸⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡏⠀⠀⠈⠻⣦⡀⢰
⢹⡇⠘⣿⡀⠀⠀⠀⣰⡿⠀⢸⣟⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⣿⠇⠀⣾⡇⠀⠀⠀⠀⢹⡇⢸
⠸⣷⠀⠈⠻⠶⣶⠾⠛⠁⠀⠈⣿⡀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣤⣤⣤⣴⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⢰⡿⠀⠀⠘⣷⣄⣀⣀⣠⡿⠃⣾
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠉⠛⠛⠉⠀⢰⡟
⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟⠛⠻⢶⣦⡶⠟⠛⠻⣦⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢶⣶⣤⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⢛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠋⠉⠙⠛⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠍⠡⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⢤⡀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⠀⠀⡀⢀⡽⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢠⠏⠀⡔⠈⠀⠀⢀⣰⠏⢠⠏⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣿⠀⠎⠀⠀⠀⢠⠁⣿⢠⡏⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⣀⣀⣁⣢⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⢿⡘⠀⠀⠀⠀⠸⠀⠻⣾⠀⡠⠔⠁⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢼⡆
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡄⠀⣿⠊⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⣀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⣒⡛⠛⠛⠓⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇⠀⢻⡀⡠⠐⠉⠀⢸⠇⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠈⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡟
⠀⣀⡀⠀⠀⠀⢸⠇⠀⠀⢠⠁⠀⠀⡿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣟⡼⠁
⡼⢻⡃⠀⠀⣠⠎⠀⠀⢀⠆⠀⠀⢸⠇⠘⡶⠖⠒⠶⢧⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠋⠁⠀⠀⢻⡁
⣧⠀⠉⠛⠋⠁⠀⡀⠔⠁⠀⠀⢠⡟⠀⢸⠁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠘⢷⡀⠀⠀⠉⠁⠀⠀⠀⢀⡴⠋⠀⠀⠸⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⡇
⠀⠀⠙⠓⠶⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⢿⡟⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠥⠤⠤⠒⠉⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣾⠛⠖⣶⠤⠤⠤⣤⣤⡤⠔⠒⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠉⠛⠒⠒⠚⠛⠛⠛⠉⢉⡀⠀⠀⢹⡄⠀⣇⠀⠀⣀⣀⣀⣀⠀⠸⢈⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡂⠤⢀⡶⠀⠀⠀⠀⠊⠉⠒⠀⢈⠀⠀⠀⢿⠒⠋⠉⠉⠉⠉⠉⠈⠙⠢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⣶⡃⢀⡴⢳⣀⣥⠄⡀⢀⠎⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠋⠀⠈⠡⢼⡁⠈⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠛⠓⠦⣄⠲⡖⠃⠀⢀⣀⣀⣀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠠⣤⡤⠄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⢼⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠉⠉⢹⡏⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⢸⡇⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣉⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⣄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢂⠄⣀⠀⠀⠀⠀⠋⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⡅⠀⢠⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣧⣤⡄⠒⣀⠀⠀⠀⢸⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠋⠙⢫⡀⣠⠿⣧⡀⠀⢀⣿⣿⡿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠟⠋⠀⠀⠀⠙⠉⠀⠹⡟⠃⣾⣿⣏⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⡿⠛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠹⣿⣿⣿⣻⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⡿⢿⣿⢦
⠀⠀⠀⢀⡄⢾⢿⣿⣯⣛⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣠⣼⣹⣿⣷⣿⣁⣼⠃
⠀⠀⠀⣻⣕⣿⣿⣿⠟⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣶⣾⣿⡏
⠀⠀⢰⣿⣿⣿⣿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣶⡿⠁
⠀⠀⠸⢩⠄⠘⠋⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠐⠇⠱⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⣁
⠀⠀⠀⠀⠀⢻⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⡵⢈⣄
⠀⠀⠀⠀⠀⢘⢧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢁⡏⣸
⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠛⠛⠿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢷⣌⣑⡧⠂
⠀⠀⠀⠀⠀⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡁⠙⠷⢨⣌⠉⠙⢿⣿⡷⠸⠀⠃⢻⣄⠁
⠀⠀⠀⠀⢀⣿⣿⡇⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠘⠳⣦⡔⣩⢳⠞⣳⠃⠀⠃⠀⢿⠟⡀⢦⢀⡌⠙
⠀⠀⠀⠀⠸⣿⣿⣿⣶⢳⡆⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⢶⣆⡑⠪⣍⣡⢼⠇⢀⡈⠀⠠⡏⠞⣽⣧⡨⠇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢶⣀⠘⡃⠀⠀⠹⣆⠀⢤⠀⠀⠀⠀⡀⠈⣁⠀⡉⠛⣑⡿⢁⡀⠀⠀⠀⠀⠀⢻⢰⣟⣿⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣷⣀⠀⠀⠘⣧⡀⠂⠤⣄⠢⣉⢢⣄⣙⢛⡟⡹⣣⠎⣰⠁⠀⠀⠀⠀⢸⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠙⠟⣄⠀⠈⠙⢻⢿⢿⣿⡟⠷⢿⡿⢰⠷⠀⠠⠒⠰⠷⠿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠰⡄⠈⠢⣐⣤⡄⣸⣾⠀⣀⣀⣠⡤⢀⣀⡀⣀⠀⠀⢀⣠⡄⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠐⠀⠀⠈⠙⢟⣿⡟⢩⠟⠛⠛⠋⡟⠋⠑⢯⡵⠜⢿⣽⣿⡘⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⢄⡙⠻⠿⠟⢻⠇⣿⣿⠂⠖⠃⠐⠒⠒⠟⠛⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⢀⠀⠻⣵⣄⣤⣤⠄⢱⣉⣀⡀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠁⠀⣤⠀⠁⢄⡀⠉⠉⠁⣆⠀⢿⣻⣿⣿⣿⣶⣦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠂⠀⠲⣄⢠⣌⣰⣦⠀⠻⣃⠈⠻⣿⣿⣿⣿⣿⣿⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢜⢀⡁⢬⢿⢽⡛⠁⢀⣼⣿⣧⡀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⣈⠣⣶⣷⣶⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⠾⣢⣤⡶⡤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟⠀⢿⣿⣿⣿⣿⣿⡄⢂⣠⠄⣥⠐⠢⣁⠲⠖⢂⣈⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠇⠀⠀⠀⢸⣿⣼⣿⣿⣿⠉⠉⠀⠁⠉⠁⡋⣀⣠⠘⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⣄⢤⣾⣽⣿⡿⢿⣿⠀⠀⠀⣀⠶⠄⠡⠍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁⢠⣼⣇⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠁

⠀⠀⢀⡀⠀⡀⠄
⠀⠀⣀⠀⠀⠂⢠⡔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠀⡀⢤
⠀⠈⢹⠀⠸⡇⠘⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢲⡘⠀⠞
⠀⠐⠀⠀⠀⠤⠄⣠⡁⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⡄⠈⠋⠠⠖⠐⢀⠖
⠀⠀⢣⠀⠀⠀⠀⠀⠁⡰⠶⠾⠀⠀⠀⠀⠀⠰⣶⠀⣀⠀⠐⣀⠀⠶⠄⠠⢆⠀⠀⠀⠀⠈⠓⢦⠀⢥⠀⠀⠀⢀⠻⢦⡀
⠀⠀⠀⠓⣤⡀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⠀⠁⠀⠀⠞⠁⠈⣃⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⡜⠁⠒⠛
⠀⠀⠀⠀⠀⠹⠂⠸⡄⠀⠀⠀⠀⠀⠀⢀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣐⠆⠀⠀⠀⢱⠄⢀⡠⠴⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⠀⣘⡷⠄⠚⠓⠂⠘⠛⠒⠤⣀⠀⠀⠀⠀⢞⠁⠀⠀⠀⡰⠃⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⡄⠘⣧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⢄⠀⠀⠈⢓⠀⢀⡞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢃⡀⢘⣄⣠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠆⠀⠀⣸⣄⠎⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡞⠉⠉⢹⠇⠀⠀⠀⠀⠀⠊⠁⠙⠀⠀⣀⡤⠚⡁⠀⠀⠸⠀⠀⣿⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⣤⣤⠤⠤⠤⠔⠒⠋⠁⣠⣼⡇⠀⠀⠐⠓⢶⣟⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⠀⢻⣿⣄⣀⣤⣤⠤⠚⠛⢻⡟⠀⠀⠀⢦⡤⠎⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠀⠀⠀⠈⢿⣿⡿⠿⠇⠀⠀⣸⠋⠀⠀⠀⢠⠟⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⠀⠙⠓⠀⠴⠖⠊⠀⠀⠀⢀⣴⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠢⠤⠤⣄⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠰⠒⠒⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢣⡀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⠤⠒⣇⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄⠀⠀⠉⠉⢹⠉⠉⠚⠟⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠀⠀⠀⣧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣳⠀⠀⠀⠀⣠⠜⠁⠀⢠⡄
⠀⠀⠀⣀⠀⠀⢀⣤⠔⠊⠁⠀⠀⠀⢻⣀⠠⠔⠊⠁⠀⠀⠀⢸
⠀⠀⢘⠁⠄⢀⣈⣃⣀⣀⠀⢀⡠⠛⠊⠉⠣⠀⠀⠀⠀⡀⠁⢰
⠀⠀⠈⠁⠀⠤⠤⠤⠤⠤⠀⠿⣈⠀⠄⠀⠀⠰⠀⠤⠄⣛⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠛⠛⠛⠛⠓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⣿⠀⢹⡆⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⣿⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠛⠉⠛⠲⣤⡀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣷
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⠀⠀⣠⠏⠳⣄⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡧⣤⣤⠎⠀⠀⠈⠈⠉⠁⠀⠀⠈⠙⠓⢋⠀⠈⠳⠤⣤⣤⣤⠴⣞⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⠴⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⡤⠒⠋⠉⢶⡀⣿⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠛⠛⠉⠉⠁
⣿⠀⠀⠰⣄⡸⠂⠀⠀⠀⠐⢆⣠⠀⠀⢰⣄⣀⡴⠁⠀⢀⡇⠀⢀⣤⣀⠀⠱⠀⠀⠀⠀⠀⠀⢹⡼⠀⣠⣶⣦⠀⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠸⡇⠀⠺⣿⡿⠀⢀⡇⠀⠀⠀⠀⠀⠈⣧⠀⠙⠿⠛⠀⣸⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠲⣆⠀⢀⡇⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⢹⠳⢤⣀⡀⠴⠋⢹⡄⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⣀
⠸⣇⠀⠀⠢⠀⠀⢰⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠉⠉⠉⢹
⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠠⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⡾
⠀⠀⠹⣦⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣼⠁
⠀⠀⠀⠈⠳⣤⡀⠀⠈⠛⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠇
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣲⠶⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣀⣤⠶⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⡾⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢀⡾⢄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣸⠇⠀⠀⢠⡞⠁
⠀⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⢷⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⣯⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣴⠏⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠛⠶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠁⠀⠀⠀⠀⠀⠉⠛⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢈⢉⡁⠉⠉⠉⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⣤⣄⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠁⠠⣄⡀⠙⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⠤⠄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠩⡢⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠇⠐⢒⡨⠒⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠙⢛⣳⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠾⢂⠜⡠⠒⠁⠀⢀⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠒⠞⠛⠢⢴⠋⠀⠀⠀⠀⠀⡄⠁⠁⠀⡸⠀⠢⣤⠊⡡⠊⢀⢠⣤⡞⣯⣿⣯⣿⣿⣧⣀
⠀⠀⠀⠀⠀⢀⠤⠤⢌⠀⠙⠲⡂⠁⢙⠋⠉⠁⠁⢀⢄⠀⠁⠱⠀⠀⠀⠀⠈⠒⠒⠊⠀⠀⠀⣴⠬⠔⡚⢃⡾⣿⣯⣿⣿⢿⣟⣿⣿⣿⣄
⠀⠀⠀⢀⠴⡉⠒⢄⠀⠑⡄⠀⡟⠉⣉⠇⠀⠀⠀⢠⠀⠣⠀⠰⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠜⢡⡟⠉⠀⣼⠟⠏⣹⣵⢾⠿⡫⠭⣭⣿⣿⣦⡀
⠀⠀⠀⡇⠀⠈⠱⠀⢇⠀⠟⠊⢠⠊⠁⠀⠀⠀⠀⠀⠢⠀⠐⡄⠈⠉⠳⠤⣄⣀⣤⢤⠞⠁⣰⠏⠀⠀⣺⢅⣵⡾⢋⡬⣶⣇⠼⣲⡲⣶⣿⣿⣇⠀⠀⠀⠀⣠⡤
⠀⠀⠀⡇⢲⡀⠀⠀⡸⠠⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠀⠀⢀⢖⠏⠠⢠⠂⠀⢠⠁⠀⢀⢣⣿⡿⣋⡴⢿⣷⠾⣙⣛⣿⣿⣷⡿⣿⣿⡆⠀⠀⡼⠱⠚
⠀⠀⠀⠘⣄⠉⠉⠉⠁⠀⠀⠀⠣⡤⣤⣤⣀⠀⠀⠀⠀⠀⠁⡄⠈⢠⠃⠀⠇⠀⡏⠀⢀⡇⠀⠀⠼⠿⠿⢿⡿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣲⣷⣿⣺
⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣷⣶⣤⣤⣤⣴⣦⢤⣦⠀⠂⠀⡇⠀⢎⠀⠀⢀⣦⣦⣦⣼⣴⣶⣴⣴⣤⣶⣧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⢀⡰⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⣀⣱⣒⡒⠳⣆⣸⠀⠀⢸⣷⣶⣤⣶⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⣉⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁⠀⠀⠙⢤⡀⣸⣿⣿⣽⣿⣿⣷⡿⢩⢤⣍⡻⢽⣮⢽⣷⡿⣻⣿⡧⢸⣿⡏
⠠⠤⠤⠔⠒⠀⠴⠤⠤⠆⠹⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⢻⣿⣿⣿⣿⣏⠭⣒⢶⣵⣿⣯⣿⡪⡛⣿⣿⣾⣷⢿⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠉⣭⠭⠥⡉⠉⠀⠀⠀⠀⠙⣗⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣟⣿⣷⣶⣤⣽⡳⣭⡻⣟⡿⢮⣮⣿⣿⣿⠽⠃⠀⢀
⠲⠤⠤⣀⣀⣀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣍⣀⣉⣀⠐⠒⠒⠋⠑⠒⠻⣤⡀⠀⠀⠀⠀⠀⠀⢠⠇⢀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣜⣝⢎⣟⣮⣿⣿⣿⣟⡟⠀⠀⣸
⡀⠀⣠⡀⠀⠈⠉⠉⠑⠒⠢⠤⢤⣀⣈⡑⠒⠀⠩⠭⠍⠉⠉⠉⠹⠿⠿⣆⣀⡀⠀⠀⠀⠀⡞⠀⣸⣿⣦⣿⣿⣿⣿⣼⣿⣿⣿⣎⢎⢳⢿⣿⣿⣿⣿⠁⠀⢠⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠞⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠀⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⣷
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⢿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⠲⢤⣀⣀⣀⣀⣀⣤⡶⠀⠀⢀⡇⠀⠀⠀⠀⠐⠛⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠤⣀⣉⣉⣩⠵⠋⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁⠈⢻⣿⡆⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⡿⡄⠀⠀⠹⠿⠿⠿⠿⠿⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠒⠋⠁⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠠⣤⡀⠈⠙⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⠁⢀⠴⠦⣀⣀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁⠘⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡟⠁⠀⠀⠁⠀⢢⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴⢿⣿⡁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠗⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⠀⣀⣀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣌⣳⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣡⠟⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣠⣦⡀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢿⡿⠋⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢈⡓⠚⢉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⣄⣀⠉⠉⠁⢀⣠⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠠⠤⠀⠀⠴⠆⠐⠶⠒⠲⠖⠒⠒⠒⠋⠉⠙⡷⠄⢀⡼⢱⠄⡀
⠀⠀⠀⠰⣋⣉⣡⣤⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠋⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠯⣙⠚⠀⣸⠀⠒⠠⠄⣀⠀⠀⠀⡄⠀⡇⠀⢠⠂⠀⠀⠠⠤⢤⣄⣀⠀⠀⠐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠒⢧⡠⢧⣴⠡⠔⠒⠉⠉⠉⠉⠉⠉⠉⠂⠀⠈⢄⡠⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⣄⣤⣄⡀⠘⡄⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⣀⠤⠞⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣵⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢀⡤⠒⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡞⠣⣀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢀⠤⠎⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢿⣿⡀⠀⢣⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠀⠈⣆⣀⣀⣀⣀⢠⣆⡠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠛⠒⠢⠤⣸⡇⠀⠈⢿⣧⠀⢸⣁⣀⡀⢄⣤⡭⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢈⣿⡴⠋⠀⠀⠉⠑⠪⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢧⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣓⣤⣀⢳⣀⡤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⡤⠠⡄⠀⠀⣿⠀⠀⠈⠉⠙⡟⠺⡷⢮⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡀⣈⠳⣄⣠⣼⣖⣙⣤⣯⣤⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⡿⡥⢞⠻⢍⡀⠀⠘⣧⣤⣄⡜⠀⢸⠀⠀⠀⠀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠰⢯⡴⠇⠀⢳⡄⣾⠷⠶⣏⠀⠀⡇⠀⡼⢍⠙⢯⠁⣹⡇⠀⡱⠂
⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⡥⠾⡀⠀⢏⠃⠀⠀⠹⣆⠠⠇⣸⣀⣀⣙⠦⣧⣼⣴⣋⣠⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⠓⠦⣧⣀⠈⡆⠀⠀⠀⠹⣼⠀⢸⣁⣨⡭⠿⠿⣿⢿⠍⠀⠈⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣄⠀⢠⡶⠛⠛⠉⠁⠀⠀⣀⣀⣽⡎⣶⠚⠋⢻⡀
⠀⠀⠀⢰⣧⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⣠⣤⠴⠖⢛⡏⠉⠀⠼⠛⢇⠐⣂⣈⣷⣠⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢹⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡾⠀⠀⠀⡐⡗⣀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠃⠀⠀⠿⡪⡻⣗⠓⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⣧
⠀⣾⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠛⠛⡛⠛⢻⠛⠛⠉⠉⠁⠀⢠⡄⠀⠀⢹
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⣀⠀⠀⠀⠤⣶⣶⠄⠀⠀⠀⢀⣇⠀⠈⠀⠀⠀⠀⠠⡮⣶⢕⠵⠆⢸
⠀⣇⠀⠀⠀⠀⠀⢀⢄⣹⣳⣿⣼⣏⡠⡀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⣱⠔⠢⢄⠀⠀⠀⠈⠡⡇⠁⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⣽⠟⣿⡻⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣸⠤⠤⠤⢀⣀⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠑⠤⠤⠚⠁⠀⠀⢀⡀⠤⠒⠉⢧
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠉⠙⠛⠓⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⠋⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⢀⡿⠃⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠈⣿⣿⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣔⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⡻⣿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⣿⣷⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⣿⡟⠿⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣭⣿⠿⠽⠿⠿⠾⠿⠷⠤⠤⠤⠠⣤⣤⣤⣤⣤⣬⣿⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⡤⣼⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣒⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠛⠉⠀⠀⠀⠀⢀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠴⣶⣶⣤⡄⠀⠀⠀⠀⢀⠘⢷⣿⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣧⢙⠀⠀⠀⢠⣿⠀⢹⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡿⠛⠓⢦⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠡⡿⠀⠀⠀⣼⣿⠀⢸⣿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠚⢘⡀⠀⠀⠀⠀⠀⠀⢸⣤⠆⠀⠀⢀⠀⠀⠚⠃⠐⣧⣀⠀⠀⢻⣿⠀⣸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⡁⠈⠘⣷⠀⠀⠈⠀⠀⠀⠇⠀⢠⠔⠒⢁⣤⢴⡦⠀⢈⣿⡿⣿⣶⣼⣿⠀⢺⣿⡄⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠁⠈⢛⢾⣅⡀⠀⠀⠀⢀⣡⠈⠀⠄⠀⠀⢿⡿⣾⠁⢀⣾⠏⠀⣹⣿⠏⠙⣧⠘⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠠⣸⣿⣿⣿⣆⣀⡀⠁⠠⠴⣴⠂⠀⠘⠘⠟⢉⣠⠜⠁⢀⢂⠄⠉⢀⠀⠸⣷⡟⣷⠀⣼
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⣰⣆⢹⣿⣿⡿⠉⠁⡀⢠⣄⣰⡏⠀⠀⠀⠀⠈⠉⠀⠐⢂⣩⣯⣺⠄⠀⡓⠀⠰⢻⣿⣤⣿⠂
⠀⠀⠀⠀⡠⠀⠈⠀⠈⠂⠀⡀⣹⣿⣿⣿⣿⣇⠔⠀⢰⣾⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⣿⣿⣧⣀⡀⡘⠀⢻⡇⠛
⠀⠀⠀⠀⠑⡄⡀⢀⣠⣴⣿⡿⠟⠛⢿⣿⡏⠘⠻⡄⠟⠁⣼⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠈⠘⣿⣿⣿⣌⠁⠀⠀⡸⢁⣀
⠀⠀⠀⠰⣶⣶⢀⠼⣵⠛⠁⠀⠀⢈⣿⣿⠁⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⣿⣿⣿⡿⢸⡞⠁⡀⢀
⠀⠀⠀⣀⣏⠙⡞⠰⠋⢷⣤⣄⣀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣷⠈⢀⣾⠇⣦⡀
⠀⠀⠀⠸⣿⡦⡇⠀⢸⣾⣿⣿⣿⣿⠇⠄⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⣿⣿⣿⣿⡇⠀⣼⠻⣼⣿⡀
⠀⠀⠀⠀⠈⠁⠳⠀⠘⣿⣿⣿⣿⡿⠀⠸⡀⠀⠀⠀⠀⠀⠒⢿⡞⠒⠀⠀⠀⠀⠀⣈⣙⣿⣿⣿⣿⣿⣿⠁⣸⣿⣴⡿⠓⣠⡄
⠀⠀⠀⡀⢠⡞⣾⠀⠀⠸⣿⣿⣿⡇⠀⠀⣅⠀⠀⠀⠀⠀⠀⠘⡆⠈⠙⠿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⡟⢠⣿⣿⢏⢀⣴⣿⠃
⠀⠀⠀⠀⣿⢛⣿⡆⠀⠀⢻⣿⣿⠁⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢼⣿⣦⡀⠀⠀⠸⣿⣿⣿⡇⣸⣿⣷⣠⣿⡟⢁⡅
⠀⠀⠀⢸⣿⣿⢹⣿⠀⠀⢸⣿⡟⢠⠀⠀⠀⢃⠀⠀⠀⠀⣀⣀⣤⠷⠀⠀⠘⣿⡇⠘⡆⠀⠀⠙⢿⣿⡿⣿⣿⣿⣿⠏⢡⣾⡇
⠀⠀⠀⣸⠃⢼⢹⣿⠀⠀⠀⢻⠁⢸⠛⠉⣲⣾⣶⠿⠛⠛⠻⡤⣈⣡⠀⠀⠘⣿⡇⠀⠱⡀⠀⠀⠀⢿⣿⣿⣿⡿⠁⠀⣿⣹⠇
⠀⠀⠀⠈⠀⠉⢸⡿⡇⢠⠾⣿⠀⠘⣟⡛⠋⢉⣀⠁⠀⠀⠈⠀⠻⣿⡶⠶⠖⣿⡇⠀⢠⠃⠀⠀⠀⢸⣿⣿⠏⠀⠀⢰⡟⣿⡇
⠀⠀⠀⠀⠠⠤⠟⡖⢻⠀⠘⣿⡀⠈⠪⠙⢿⡟⠀⠀⣀⣀⡌⠀⠀⣸⠃⠀⠉⣿⡇⠀⠀⡇⣠⡤⠶⣿⣿⣿⣿⡷⠲⠿⣧⣾⡗
⠀⠀⠀⠀⠀⢀⡞⠀⢸⡄⠀⣿⡇⠀⠀⠀⢸⣹⡿⠿⠛⠉⠉⠉⠙⠁⠐⠀⢦⣿⡇⠀⠀⡀⠀⠀⣾⡿⠛⠛⠑⢃⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⡜⠁⠀⡟⢇⠀⢹⡇⠈⠉⠓⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⡇⠀⢰⣿⠃⠀⣠⣶⣾⣿⣿⣿⣏⡇
⠀⠀⠀⠀⠀⠘⠟⠻⢿⣾⡖⢛⣿⠠⡤⠤⠤⣤⣤⣦⣶⣶⣶⣶⡦⠤⠀⠋⠉⢻⡇⠀⠠⢣⠞⢩⡅⣠⣤⣿⣿⣿⣿⣿⡏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣿⣤⡶⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣦⣶⣿⡷⠖⠆⠀⠀⣻⣅⣹⣿⣿⣿⠿⠛⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠁⠀⠀⠘⠛⠛⠿⠿⢿⣿⠅

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣼⣦⠄
⠀⠀⠀⠀⠀⢲⢧⣆⢈⣿⠁⣶⡼⡆⠀⠀⠀⠀⠀⠀⢲⣿⣀⠈⡇⢀⣸⣶⠂⠀⠀⠀⠀⠀⠐⣶⡇⡀⣰⡄⣀⢿⡶
⠀⠀⠀⠀⠀⠀⣉⢉⣟⢉⣿⣉⣉⠀⠀⠀⠀⠀⠀⠀⣉⠀⣉⣿⣿⣏⡁⢉⡁⠀⠀⠀⠀⠀⢈⣉⠀⣹⡉⣹⣏⠀⣉
⠀⠀⠀⠀⠀⠴⡞⡏⢈⣿⡉⠽⢳⠆⠀⠀⠀⠀⠀⠀⠼⣿⠉⢀⡇⠈⢻⠿⠄⠀⠀⠀⠀⠀⠠⠿⡇⠁⠹⠃⠉⣾⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⡟⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢻⠟⠂

⠀⠀⠀⠀⠀⠀⡀⠀⣒⣿⣃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⣧⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣼⣦⠄⣀
⠀⠀⠀⠀⠀⡾⢷⣇⡈⣿⢁⣾⡾⢧⠀⠀⠀⠀⠀⠀⣳⣿⣀⣨⣇⣀⣼⣶⡂⠀⠀⠀⠀⠀⠐⣶⠇⣀⢰⡆⣀⢿⣞
⠀⠀⠀⠀⠀⠀⣍⣈⣽⣿⣯⣁⣩⠀⠀⠀⠀⠀⠀⠀⣉⡀⣸⣇⣈⣯⡀⣈⡁⠀⠀⠀⠀⠀⢈⣁⠀⣩⣿⣿⣍⠀⣉
⠀⠀⠀⠀⠀⠹⡟⠏⣀⣿⣀⠿⢻⠏⠀⠀⠀⠀⠀⠀⠾⣿⠁⢈⣏⠈⢹⠿⠄⠀⠀⠀⠀⠀⠠⠿⡇⠁⢸⡃⠈⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⡔⠶⣿⠶⢢⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡯⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠲⢿⡷⠂⣄⡀
⠀⠀⠀⠀⠀⣨⡿⢧⣤⠿⣤⡼⢿⣁⠀⠀⠀⠀⠀⠀⠛⠿⣤⣘⣇⣠⡾⠟⠁⠀⠀⠀⠀⠀⢀⣿⠦⣄⡼⢧⡤⠞⣿
⠀⠀⠀⠀⠀⠰⣄⣤⠷⣤⠿⣄⣠⠆⠀⠀⠀⠀⠀⠀⡀⣄⣤⠟⡟⢦⣄⣄⠀⠀⠀⠀⠀⠀⠠⣦⣠⡼⣤⡼⢧⣀⣴
⠀⠀⠀⠀⠀⠘⢹⡆⣀⣿⣀⢰⠏⠃⠀⠀⠀⠀⠀⠀⠛⠉⠀⢸⣯⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠛⡧⢀⣸⣇⡀⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠸⠀⠈⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⢄⠀⢣⠀⠀⢿⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀⠀⠐⢤⡑⠄⠱⣄⠈⣧⡠⠤⡒⠁⠀⠀⣀⡀⠤⠊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣌⣉⣁⡀⣉⣉⣐⡊⣑⣦⣬⡗⢻⢡⠀⠘⣍⠉⠁⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⢸⠉⠉⠁⠐⠒⢲⠐⠂⢉⠭⠽⠟⠉⣉⢁⡉⣲⠰⠀⡴⠮⢍⠉⠉⠛⠛⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣤⠄⢀⡀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⢸⠀⠀⢇⡀⢀⡀⠠⡤⠆
⠀⠀⠀⠀⢀⠜⢸⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠁⠐⠚⠒⡇⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠓⠂⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣⠔⠂⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂⠐⠧⣀⡠⠄⢀⡀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠴⠀⠠⠔⠐⠒⠂⠀⠁