Beauty Shop Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡆⢴⣒⣓⡲⣄
⠀⠀⠀⠀⠀⣀⢿⣿⣿⢟⣽⣶⣶⣏⡳⡄⠀⠀⠀⠀⠀⢀⡤⣤⡀⠻⣾⣹⡇⠟⣩⣭⡙⢮⡇
⠀⠀⠀⢀⣾⡋⠀⠈⠉⠈⠁⠈⠉⢿⣿⠄⠀⠀⠀⠀⠀⢜⣫⣭⣛⠓⠈⠓⠙⠀⠀⠈⢿⡆⡄
⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠠⣋⢍⡴⠂⠀⠀⠀⠀⠀⠀⠀⢸⣳⣧⠄
⠀⠀⠀⠀⢻⠉⠉⠉⡏⠹⠋⠉⠉⢹⠀⠀⠀⠀⠀⢀⣼⢏⣏⣴⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⣸⣤
⠀⠀⠀⠀⡏⠒⠒⠚⠁⠀⠑⠒⠒⢡⠆⠀⠀⠀⣠⠞⣱⠟⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡘⢯⡓⠂
⠀⠀⠀⠀⢸⣤⣠⣴⠖⠰⣦⣄⣠⡘⠀⠀⠀⠀⠀⠀⣰⣲⡿⢀⠀⠀⠀⢀⣀⡀⠀⠀⢿⣌⠳⣔⠮⡤⠤
⠀⠀⠀⠀⢸⣶⣄⠀⢠⣄⠀⢠⣶⣿⠀⠀⠀⠀⢀⣤⠴⣛⡽⢸⡆⠀⠀⠻⠿⠟⠀⠀⠀⣭⡛⡒⠂
⠀⠀⠀⠀⠈⢿⣿⣗⣾⣿⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠛⠉⣾⣸⣿⢀⠀⠀⠀⠀⠀⠀⢸⣟⢿⡌
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡏⣼⠀⠀⠀⠀⠀⠀⢷⡻⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⠀⠀⠀⠀⠀⠀⣼⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢻⡀⠀⠀⠀⠀⢠⢣⠏
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡗⣩⠃⠀⠀⢀⣀⣄⠀⡀⣿⠀⡇⢀⠀⣴⠖⠒⠦⠀⢀⡔⢳
⠀⠀⠀⠀⠀⣸⡇⢸⡇⣏⣰⣧⡏⣾⣸⢁⡟⣸⣇⡇⠀⢿⣄⢀⡠⢄⢸⡇⢪⣄⡀⣴⣄⢰⡖⣦
⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⠉⠈⠈⠉⠋⢀⣼⠃⠀⡼⢿⡼⣥⣏⢸⡇⢸⣀⡟⣇⡼⣼⠇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠓⠚⠁⠀⠁⠀⠉⠉⠁⠀⠈

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

⣿⣛⣷⠀⣿⣛⣛⠀⣰⢿⣆⠀⣿⠀⢸⡇⠙⢻⡟⠋⢻⣆⣾⠃⠀⠀⣾⣛⡛⠀⢠⡿⣇⠀⣾⡇⠀⢠⡾⠛⢳⡄⢸⣷⡄
⣿⣍⣽⠇⣿⣭⣭⢠⡿⠶⢿⡄⢿⣄⣼⠇⠀⢸⡇⠀⠀⢹⡇⠀⠀⠀⣬⣉⣿⢂⣿⠷⢿⡆⢻⣧⣤⠘⣧⣀⣼⠇⢸⡇⠻


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⠋⠉⣉⣀⣤⣤⣬⣍⣉⣛⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⠴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⡀⠀⠀⣠⠖⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣽⣦⣀⣀⢀⣀⣤⡞⠁⡴⠛⠁⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡾⠁⠀⢩⠞⡽⢻⢻⣿⡿⣏⠳⣄⠀⠀⠀⠀⠀⠀⣼⠐⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⣧⠀⠀⠀⠘⠡⠏⢼⠀⠇⠘⠆⠈⣠⣴⣄⣀⣀⣰⠏⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠋⠛⢯⡍⢻⡄⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋⠹⡄⠹⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢹⡄⠹⣆⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⢀⢀⣀⣈⡳⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠰⠏
⠀⡟⢻⠉⢉⣉⣉⠉⠉⢉⣉⡉⠉⢳⠀⣴⠶⣦⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣠⠶⠚⢛⡉⢉⠛⠓⠦⣄⠀⢠⣼⢷⣄
⠀⡇⢸⢠⡏⠁⠉⣧⢰⡏⠀⠙⡆⢸⠀⢿⣠⡇⠰⣏⣿⠀⠀⠀⠀⠰⢿⡁⣹⠿⠀⠀⡦⢸⡇⢸⠀⣶⠀⠈⠳⣄⠙⠏
⠀⡇⢸⠀⠳⠦⠴⠋⠈⠳⠤⠾⠃⢸⠀⢸⢹⡇⢰⡏⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾⠀⡇⢸⡇⢸⠀⣿⠀⡇⠀⠘⣆
⠀⡇⢸⢀⡴⠚⠲⣄⢀⡴⠒⢶⡄⢼⠀⢸⣼⡇⠸⣇⣿⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⣇⢸⡇⢸⠀⣿⠀⡇⢸⡇⢹⡄
⠀⡇⢸⠘⣧⣀⣠⡟⠸⣇⣀⣠⠇⢸⠀⢸⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⢸⠟⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⢻⡇
⠀⡇⢸⠀⣀⣭⣅⡀⠀⣈⣭⣅⠀⣸⠀⢸⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⣾⢾⡇⠐⠛⢋⠛⠂⠀⠀⠘⢛⡙⠛⠀⢸⠿⣧
⠀⡇⢸⢰⡏⠀⠈⣷⢸⡃⠀⢈⡇⢸⠀⢸⢸⡇⢸⡇⣿⠀⠀⠀⠀⢰⣇⢸⡇⠀⠀⠘⠀⠀⣼⠀⠀⠈⠃⠀⠀⢸⠁⣿⡄
⠀⡇⢸⠀⠙⠲⠞⠁⠀⠛⠶⠚⠁⢸⠀⢸⢸⡇⢸⡇⣿⠀⠀⠀⠀⣾⠙⠶⣧⠀⠀⠀⠀⠀⠻⠤⠀⠀⠀⠀⠀⣾⠞⠉⣇
⠀⡇⢸⢀⡞⠋⠛⣦⢠⡞⠉⠳⡄⢹⠀⢸⢸⡇⢸⡇⣿⠘⢦⡾⢱⡇⣶⠀⠹⣆⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣸⠃⠰⡆⢹
⠀⡇⢸⠈⢧⣄⣤⠏⠘⢧⣄⣴⠇⢼⠀⢸⢸⡇⠸⡇⣿⠀⠀⠀⣾⡀⣿⢸⡇⠙⢦⡀⠀⠀⠉⠁⠀⠀⢀⡼⠃⢸⠀⣧⢈
⣀⣧⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣞⣀⣈⣋⣁⣀⣙⣁⣀⣀⣀⣉⣉⣉⣛⣛⣛⣛⣛⡲⣦⣤⣤⣴⢞⣛⣛⣛⣛⣋⣉⣉
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⣶⡄⠀⠘⣷⠀⢠⡄⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣧⠀⣿⣿⣷⣸⠀⠸⣧⠀⡇⠐⢿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠳⠿⠛⠁⢹⣿⣿⣿⠀⠀⠛⢇⠀⡸⠀⠈⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣠⣾⡆⠰⠀⢀⠢⠀⠹⣿⣧⠀⠀⠀⠈⠆⠁⠀⠀⠀⣬⣿⣿⣿⠟⠁⣀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⣴⡿⠋⣀⣠⠴⠒⠀⠀⢠⣴⣤⠀⠐⣄⠀⠈⠀⡆⢀⣼⣿⡟⠁⠀⣒⣋⣁⣀
⠀⠀⠀⠀⠀⡾⠀⠀⣾⠟⠁⠈⠛⢁⣠⡴⠞⠃⠸⣯⠿⣧⠀⠈⠷⠀⢀⡇⣸⣿⠟⠀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣸⠇⢠⡞⠁⠀⠀⢠⣶⠿⠁⠀⠀⠀⠀⠘⣿⣽⡇⠀⠀⠀⢸⡇⠛⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⡔⢠⠰⠋⢀⡀⠀⠀⠀⠁⢀⣀⣤⣾⣿⣄⠀⠈⠙⣧⠀⠀⠀⣿⡇⠀⠀⡄⢀⡀⠘⢿⣿⣿⣿⠏
⠀⠀⠂⠐⣈⠀⢠⡿⢡⣴⣾⣿⣿⣿⣿⡿⠟⠛⠋⢠⣤⡄⠘⣧⠀⠀⣿⣿⣄⠀⡧⠈⠉⢉⠀⠀⠀⢠⣶⢂⣠
⠀⠀⠀⠀⠀⢠⡿⠁⢻⣿⣿⣿⡿⠟⠁⠀⡼⠃⠀⠀⢯⡽⠀⠈⠁⠀⠉⠙⠁⠀⣀⡤⠚⠁⢸⣷⣠⣿⣷⣿⣫⡦
⠀⠀⠀⣠⢹⣿⠃⠀⠀⠉⠙⣿⣷⡀⠀⠘⠁⠀⣤⣄⠀⢻⠀⠀⡀⢀⡀⠤⡶⠟⠁⣀⠀⠀⣿⣿⣿⣿⣿⣿⣯⡦
⠀⠀⠀⢠⣼⠏⠀⡠⠂⠀⠀⠘⣿⣷⠀⠀⢀⣼⡿⠃⠀⠈⠆⠀⠀⡮⢤⠊⠀⠀⠢⣿⡄⣼⣿⣿⣿⣿⡿⠟⠉
⠀⠀⢠⣿⠏⢀⡾⠁⠀⣠⡄⠀⠘⣿⡆⠀⣾⣿⠁⣰⣿⠂⠀⠀⠀⠂⣈⠀⠸⣷⠀⠹⣇⠈⠙⢿⡿⠋⣴⣶⢄
⠀⠀⠀⠁⣰⣿⠀⣴⣿⣿⣄⡀⠀⢈⡀⣰⠋⠀⢰⡟⠁⡦⠀⠀⠸⣦⡈⠳⠄⢻⣇⠀⢻⠀⠀⠄⢀⣾⡿⡋
⠀⠀⠀⠐⠛⠃⠀⢹⣿⡆⠀⢁⡴⠋⡰⠁⠴⣤⣈⠀⠐⠀⠀⠄⠀⠈⢿⣦⠀⠈⢿⡄⠘⠀⡆⡀⣼⡿⠅
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⠀⢀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠘⣷⡀⠀⠈⣴⠟⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⢠⣝⣷⣾⣿⣄⣾⣇⠀⠀⢱⣿⣦⠀⢄⡀⠈⠻⣦⠘⣧⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⣽⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⡄⠀⠙⢦⣄⠈⢳⣽⡂⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⠀⣼⠁⢿⣿⡀⠀⠀⢛⠷⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⢸⡟⠀⠘⠿⠗
⠀⣀⡀⠀⡀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣛⣿⣛⡳⢦⣤⣠⣤⣶⣿⣿⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡾⢱⠿⠿⣿⣿⣆⠀⠉⠉⠀⠐⠚⠲⣶⡝⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠈⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠘⠿⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⣤⣤⠶⠶⠿⣿⣿⡆⠈⠙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⢠⣦⡉⠀⠀⡄⠀⠀⠘⣿⡇⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠒⠀⠀⢸⡏⣷⣄⠀⢿⣆⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠻⣷⣮⣿⣿⣦⣄⣢⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⣴⣶⣤⠘⣆⠀⢹⣿⡝⣷⠈⠛⢷⢹⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠙⠛⠋⢀⠸⣷⣸⡟⠁⠃⣀⣤⡞⣸⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠒⠀⠀⣿⡟⢠⣠⣾⣿⠏⣰⠏⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣷⣸⣟⢿⣏⠀⢇⣼⠟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢼⣿⠈⠻⣿⣷⣝⢦⡘⢿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠂⠀⠀⠀⣴⠁⣾⣇⠀⠀⠈⣿⡏⢳⣷⠀⢙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⡄⠙⠿⣦⡀⡰⠋⠀⣸⣿⡧⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⢀⣿⠗⡠⠞⢫⣿⣧⡀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣿⠁⡞⠀⠀⠈⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡇⠈⢳⡇⠀⠀⣰⠐⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢻⡄⠀⢿⣆⠀⠘⠢⣀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⣠⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣀⣸⣷⠀⠀⠀⠀⠀⠀⠀⣼⡏⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢉⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣧⢀⣴⠟⠁⣸
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢨⣿⠇⠀⠀⠀⠀⠀⠀⠘⠿⣥⡤⠔⠁
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢹⣿⠄⢉⡆⠀⠀⠀⠀⠀⠀⢠⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠿⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠦⠄


⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⣿⣿⣷⡄⠀⠀⠀⠀⢸⠁⠙⢿⣝⢮⡹⣧⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡟⠛⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⡄⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠈⢀⣀⢨⣿⢸⣧⡟⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⡇⣿⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⣿⣿⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⣾⣫⡾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠛⠘⠉⠉⠉⠁⠙⠖⠃⠀⠀⠀⠀⠀⠀⠿⠶⠶⠇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠿⢿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠺⠋⣀⠀⠀⣀⠛⠗⠀⠀⠀⠀⠀⠀⠀⢪⢔⡵⡃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠴⣫⢞⡅
⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣕⣡⡆
⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⡐⠦⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⠧⣀⣽⠀⠀⠀⠀⠀⠀⠀⢠⠐⢦⡷⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠼⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⠦⡴⠃⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣤⠀⠀⠀⠀⣠⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⣰⣿⣷⡗⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⣰⣿⣿⣟⣿⡿⠃⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠀⠀⠀⢼⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢯⠙⠁⠂⠀⢀
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢣⠃⠀⠀⢀⠎⠝⠀⠀⠀⠀⠀⠀⣠⣈⠃⡀⠀⠈⠋
⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠞⠀⠀⡠⢋⠌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣦⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠃⠀⡔⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢢⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣴⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠠⡀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⢀⠬⠉⠉⠲⢹⢠⡀⠀⠀⠀⠀⢀⠔⠊⣁⠄⠀⠀⠐⢄
⠀⢠⠒⠚⠋⡠⠒⠄⠀⠀⠀⠀⡀⠀⠀⢰⢳⠋⠁⠁⠀⠈⢮⡀⠀⠀⠀⢠⠏⢀⠞⠡⠴⠚⠀⠈⡄⢇
⠀⢸⡋⠀⡖⢨⠆⠰⠀⠰⠀⠈⡹⠀⠀⢸⠀⠀⠐⠀⠀⢀⠐⠆⠀⠀⠀⢸⠀⡎⢀⠀⠀⠈⠁⠃⢹⢸⡀
⠀⠀⠁⢀⠘⠉⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠀⠀⢀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⢀⢀⣀
⠀⠀⡰⣭⡺⠃⢙⣛⡗⠂⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⣡⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠱⡀⠠⣭⠟⠁⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⢀⡴⢡⡎⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠢⠀⠀⠈⠀⠁⠀⠀⠀⢀⠔⠒⠋⠉⢀⡞⠀⠁⠀⠁⠈⢢⠀⠀⠀⠠⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠤⠒⠋⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢠⠁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠑⠀⡀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠘⢰⠀⡇⢠⠀⢀⢠⡇⠀⠀⠀⠀⠐⢀⠠⠀⠀⠰⢰⠀⠀⠀⠀⠀⠘⠀⢰⡉⠀⢠⠈⢄⡀
⠀⠀⠀⢀⠀⢸⠠⠁⠀⢀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⠀⠀⠰⡁
⠀⠀⠀⠀⠀⠸⠀⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⢠⡞⠉⠉⣷⣄⢰⡀⢳⠀⠢
⠀⠀⠀⠀⠀⢸⣿⣿⣗⣹⣿⠨⣃⢸⡇⣱⣠⡆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠓⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢠⡟⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢸⣿⠇⢀⢿⡇⠀⠀⠀⠀⠀⠀⡄⠀⡄⢰⣤⠄⠀⠀⣗
⠀⠀⠀⠀⣿⣽⠈⢯⣿⣆⠀⠀⠀⠑⢤⠛⠶⣓⡦⣙⡳⢤⠔⠉⠆
⠀⠀⠀⠀⢻⣾⣶⡜⡾⣿⡆⠀⠀⠀⠄⠈⢃⡈⢁⡈⣁⡀⠀⠀⠈⣄⣠⣤⠤⠶⠒⠒⣛⣉⣹⣭⣉⡅
⠀⠀⠀⠀⠘⣟⣟⣁⠷⠼⠷⠒⠒⠚⠋⠉⠉⢉⣍⠋⠉⠙⡄⡆⠀⢻⣶⣶⣶⠾⠛⠋⠉⠉⠉
⠀⠀⠀⢰⣿⣩⣤⣴⡄⢀⣀⣀⣤⣦⣾⣾⡿⠛⠛⠛⡃⢹⠀⡆⠀⣸⣿⠋
⠀⠀⠀⠘⣿⣯⣶⣿⣿⣿⣿⣿⡟⠁⠀⠀⡇⠀⣠⢠⡇⣾⣸⣷⣤⠿⠇⠠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣀
⠀⠀⠀⠀⠈⠉⣿⢻⡿⠀⢋⣿⣷⠀⠀⢠⣷⣰⣿⣾⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢱
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⢸⣿⣾⣷⣾⣟⣯⣭⣍⣠⣤⡤⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣧⣾⣦⠣⡻⣿⣿⠛⡟⠉⠉⢉⣉⣉⣩⣭⣭⣉⠉⠳⣴⠀⠀⠀⡀⢀⠀⠀⠀⣶⣸⡆⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣖⠻⣿⣎⣐⣻⣿⢻⡍⠭⠭⢁⣒⣒⡒⠲⢬⣱⡄⣿⡇⠀⠀⠀⢸⡂⣶⢰⣿⣿⡿⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⢾⢯⣉⣉⣿⢿⡉⡋⠉⠁⣹⣡⣄⡮⠟⣿⣿⡇⠀⣸⣆⣾⣿⠿⠟⠋⠉⢀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠪⠬⠽⣮⣯⣉⣀⣻⣛⣋⣉⣥⡞⣿⣿⣷⡾⠟⠛⠉⢀⠀⠀⠱⢄⠘⠁⠀⢈⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠖⠂⢾⣏⣂⣾⣽⣿⣿⣿⣍⠀⠘⢛⢿⣓⣆⠀⠀⢸⡁⠀⠦⣤⠈⠐⠁⢸⡇⠀⢀⡤⢒⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠦⠤⢤⣤⣥⣬⣼⡏⢿⣿⣇⣈⡈⠺⣿⣿⡝⣆⠀⠉⠙⢦⠀⠀⣦⣴⡟⣋⡵⠊⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣖⠻⠝⢙⠛⣽⠃⣨⣿⡞⣿⣿⣿⣿⣿⢿⣿⡄⠀⢀⣼⣶⠞⢫⡉⣻⣇⠀⠀⣶⢸⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠿⠋⠈⠁⠀⠀⠀⠀⠒⣊⣹⣿⣿⣿⣿⣿⣿⣿⠈⢹⣷⠶⠛⠁⢠⠈⢼⣇⣸⡏⢰⣼⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠁⠒⢀⡄⢀⣠⣿⠿⢿⣿⣿⠿⠿⣿⣽⢿⣿⡾⠛⠋⢉⡙⠛⣹⣿⣿⢿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠾⣿⣿⣛⣉⣛⣛⣋⣉⣠⣤⣾⣿⣷⠿⠛⡋⠁⠤⢉⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢱⠀⠀⢀⣼⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣠⡾⠛⣡⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⠉⠻⣿⣿⢷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡙⠲⠤⠴⠞⠋⣀⣼⣿⣿⣿⣿⣿⣿⢟⣵⣿⡿⠃⠀⠀⠈⢿⣧⡹⣦⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠈⠒⠦⠤⠶⣚⣯⣾⣿⣿⡿⢟⣫⣴⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⣷⡘⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣤⣤⣿⠿⣛⣭⣴⣾⠿⠟⠻⡁⠀⠀⠀⠤⠒⠒⠒⠂⣿⣿⡇⢹⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣿⣿⣿⡏⢶⣤⣤⡤⠀⠀⠀⠀⢠⣤⣤⣤⣶⣿⣿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡿⢋⣽⡟⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣴⣿⣿⢡⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣾⢻⣿⡏⢸⣿⣿⣆⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⣿⡏⣾⢡⡞⠋⠻⣦⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣾⡏⢸⣿⡇⢸⣿⣿⣿⣦⡀⠀⠱⠤⠤⠌⠀⢀⠴⣿⣶⣿⡘⣷⣦⣴⡿⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⣿⡇⢸⣿⣿⣿⣿⣿⣦⣄⣀⢀⣠⠔⠁⠀⢻⣿⣿⣷⣄⣉⠁⠀⣀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠹⣿⠘⣿⣿⣿⢿⣿⣿⣇⠉⠉⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⢟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠙⢷⡹⣿⣿⣦⡙⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⣿⣷⣄⠀⠙⢿⣿⣿⣿⣦⡈⠙⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢿⣿⣿⣶⣤⣌⡛⢿⣿⣿⣶⣄⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠈⠛⠻⢿⣿⣿⣳⢬⡻⣿⣿⣷⡀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⣀⣈⡙⠻⣷⡙⣞⣿⣿⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠀⠀⠈⣿⢸⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⣼⡾⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⡢⢤⣤⣤⠼⢛⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣽⣧
⠀⠀⠀⠀⢀⡶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡀⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠘⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⠀⢀⡤⠖⠛⣇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠻⣿⣥⡞⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⣿⣿⣷⣄⣠⣤⣾⠟
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠘⠟⢻⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡟⠁
⠀⠀⠀⠠⣾⣀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠻⣦⣄⣀⠀⠀⠀⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣤⣤⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠿⢻⣿⣿⣿⣿⠘⠿⣦⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣾⠳⣄⠀⠀⠀⠀⠀⢹⣿⣷⣶⣦⣾⣿⣿⣿⣿⣷⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣿⣠⡏⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣷⣾⡿⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣰⣦⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣿⣿⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⡿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⢾⣿⣿⡶⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣼⣿⣿⣷⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣡⠶⢚⣓⠒⠬⣙⠢⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⢿⣟⡯⠶⠿⢿⣿⣿⣿⣾⣷⣌⡳⢄⡀⠀⠠⠤⠖⠋⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣈⣉⠁⠁⠈⠉⠻⣶⣌⠻⣿⣿⣿⣿⣿⣶⣭⣙⣒⣒⣒⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣫⡅⠀⣀⠀⠀⠀⣈⢿⣷⡌⢿⣿⣿⣿⣿⣝⠻⠿⠿⠟⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣶⡾⠁⠀⠨⣷⣶⠝⢿⣦⡙⢿⣿⣿⣿⣿⣷⣾⡿⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣛⠂⠈⣉⣛⣛⣩⣵⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⣤⣀⠀⢀⣾⣿⣦⣌⡛⠻⠿⠿⣿⣿⣿⢟⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣄⠈⠛⢁⡠⠋⠠⢬⣉⣛⠻⠿⠿⠿⣛⣫⡷⠋⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠈⠁⠉⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢋⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣾⣄⣈⡀⠀⠀⠴⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣷⣴⣆⡠⠶⢶⣦⣄⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠀⠀⠀⢀⣶⣿⣿⡿⠿⠟⢹⠁⠀⠀⠉⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠘⣧⠀⠀⠀⠀⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⢀⢜⣿⣿⡇⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠟⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠄

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠟⠀⢠⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣲⣯⣿⣿⣿⣿⠿⠋⢀⣴⢿⢏⣾⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⢁⣠⣶⠟⣱⣿⣿⠟⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠛⠛⣉⣉⣤⣤⣶⠿⠟⢋⣴⣾⣿⣿⠋⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⡶⠾⠿⠟⠛⠋⠉⠁⢀⣠⣶⣿⣿⣿⣟⣵⣾⡿⠃⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣤⡾⠛⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠖⢛⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⣀⣀⣠⣾⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⡵⢞⣿⣿⣿⣿⣿⣿⠟⣫⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠙⠛⠿⠯⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢋⣴⣿⣿⣿⣿⣿⠟⢁⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣾⣿⣿⣿⠟⣿⠃⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⡿⢸⣿⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⢿⡄⠀⢿⣿⣿⣿⣀⡆⠐⠀⠀⠙⠻⠿⠟⠁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄⠈⢿⣄⠀⠻⣿⣿⣿⡇⠀⠈⠢⣀⠀⠀⠀⡠⠊⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠙⢷⣄⠈⠻⢿⣧⡀⠀⠀⠀⠉⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠑⠀⠀⠙⠿⣦⣀⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠢⣄⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣌⡙⠶⣤⣀⠀⠀⢤⣉⠛⠷⣦⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣬⣙⡲⢤⡉⠻⣶⣬⡙⠳⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡈⠙⠻⢿⣿⣿⣿⣿⣦⡙⣿⣿⣷⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢬⡻⣿⣿⣿⠇⣿⣿⣿⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡿⠟⣋⣼⣿⣿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣴⣿⣿⣿⡿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣧⣄⣤⣤⣤⣴⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠉⠻⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠙⠿⠛⠀⢸⣿⣿⠋⠀⠀⠀⠈⠙⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⡾⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⢀⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣤⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⣰⣿⣽⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠟⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⣿⡏⢻⣿⣿⣿⣿⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣀⣠⣾⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣇⠀⡿⣿⣿⡿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⢸⣿⠁⠀⢀⣿⠃⠀⠀⠀⠀⠤⠶⠶⠴⠶⠶⠶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⣼⣿⣄⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⣠⡿⠋⣀⣻⣿⣿⣿⣇⡈⠻⣷⡀⠀⠠⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢼⡿⠿⠟⠋⠉⣿⣿⡿⠉⠛⠻⠿⣿⠆⠀⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠟⠛⠛⠻⠶⢦⠀⠀⠨⡿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⡟⣿⡇
⠀⣴⣿⣿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢿⣿⣿⣿⠀⣿⠇
⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣯⠀⣿
⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⠀⠀⢀⣠⡿⢠⣿⣿⣿⣿⣄⢿⣄⣀
⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢌⣷⣤⣿⣻⣭⣶⣿⣿⣿⣿⣿⣿⣷⣭⣝⣿⣦⣦⣤
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷

⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⠿⣿⣿⠿⠿⠿⣿⣿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣤⣶⣶⠶⠶⣦⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⣤⣽⣿⣷⣄⡰⣿⣡⣤⣶⠾⡿⢏⡁⠈⠒⣶⣽⣿⣬⣼⢿⣖⣦⡀
⠀⠀⣠⡾⠃⣀⠀⠀⠙⣯⣷⣶⣔⣿⡶⠿⣿⣿⣿⡿⢟⣿⣧⡤⠖⢋⣷⣶⣭⣷⣾⣿⠟⠛⢿⣿⣿⣧⡀
⠀⠀⢿⡐⣦⠀⠚⢯⠐⣄⢀⣀⢽⣯⡾⣿⣶⣾⣿⣿⣿⣿⣹⣶⣶⣿⣿⣿⣿⡟⠉⠀⠀⡀⠀⣿⣿⣿⠃
⠀⠀⠈⠉⠁⢤⣲⣦⣷⣽⣄⣹⣦⣿⣧⣯⣽⣿⣭⣤⣾⣿⣿⣿⣿⡏⣹⢻⡏⠀⠀⠲⢾⠟⢀⡜⣯⠛⠳⣄
⠀⠀⠀⠁⠀⠀⣩⣿⣿⣿⣯⣯⣙⡲⠿⣿⣝⣛⣛⡿⠟⣻⣿⣿⣿⣿⣅⡈⠛⠀⠀⠀⠀⠀⠘⠃⠹⣦⠀⠈⠁
⠀⠀⢀⣴⣤⠘⠍⠊⠁⠋⢸⣷⣯⣝⣿⣿⡿⢟⣫⣴⣿⡿⣋⣭⣿⡿⡿⣇⠀⠀⠀⠀⠀⢤⡏⠀⠐⣾⡀⠀⡠
⠀⠀⠀⢄⣸⡇⠀⠠⣯⠤⣤⣽⡿⢿⣿⠷⠿⢟⣿⡿⢋⣼⣿⣿⡾⠁⡇⠙⣷⣦⣄⣀⠀⠛⠀⠀⠈⠀⠉
⠀⠀⠀⢮⣩⢶⣿⠃⣲⣷⣾⣤⣴⣶⣶⣾⣿⠟⣡⣾⣿⣿⣿⡿⠁⠀⡇⠀⢸⣿⠉⠉⠉⠁⠀⠒⢤⡀⠤⠄⢀
⠀⠀⠀⠠⣿⣿⣿⣟⡛⠿⣿⣯⣾⣿⡟⣽⣧⣾⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠈⣿⣀⠈⠛⠶⡰⣴⣌⣿⢠
⠀⠀⠀⠀⠿⠟⠉⠛⣙⣿⣿⣿⣿⣋⣴⣿⣿⡿⠛⠉⠉⠁⣤⠀⠀⠶⠀⢀⣼⠻⣿⣷⣄⠀⠁⠘⣿⠏⣼
⠀⠀⠠⣶⢤⡄⣤⠀⠨⢽⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠉⠛⣖⡤⢚⣡⡾⣣⠂⢀⠄
⠀⠀⠀⠁⠀⠀⠻⣤⡴⢀⡴⣹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⣥⡶⣛⣥⣚⡥⠞⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⡻⣧⣏⢹⣿⣿⣧⣤⣤⣤⣤⢾⣻⠀⠀⠀⠠⢐⣨⣵⠶⣛⣫⡵⡾⣛⠯⠅⠂⠈
⠀⠀⠀⠀⠀⠀⠈⠳⢦⠀⠛⠛⠛⠛⠛⠛⠛⠋⠉⠐⠋⣁⣀⣤⠴⠶⣛⣯⠵⠖⣛⡭⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠤⠤⠀⢒⣚⡛⠯⠭⠭⠗⠒⠋⠉⠁⠄⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⠦⣄⣀⡀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠤⣍⣉⣉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⡾⠀⠀⠀⠀⢀⡤⠖⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠃⠀⠀⢀⠔⠁⠀⠀⣠⠞⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡟⠀⠀⢰⣏⣀⣤⠴⠚⣡⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⠁⠀⠀⠀⠙⠦⠴⠶⣏⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⡼⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠹⡄
⠀⠀⠀⠀⢰⡧⢤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠙⢦
⠀⠀⠀⠀⠈⣇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠔⢊⢧
⠀⠀⠀⠀⠀⢿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⢻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠵⠒⠚⠛⠒⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣩⣭⣙⡲⡄⢀⢼⡋⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⢀⣀⡀⠀⠀⠙⣆⣷⠀⠀⠀⠀⠀⠀⠀⡼⢡⡴⠛⠉⠀⠀⠈⠙⢿⣸⠀⠙⢦⣀⠡⠠⠌⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡴⠋⠉⠙⢷⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⣠⠖⠛⠛⢳⡄⢸⠏⢧⣄⠀⠈⠙⠲⠦⣤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠁⠀⠀⠀⢸⠇⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡏⠀⠀⠀⠀⢳⠞⢠⠏⠈⠛⠶⢤⣄⣀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⡄⠀⠀⠀⠈⣷⠏⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢠⠏⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠏⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⠏⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣰⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣠⠏⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢠⠿⡄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⠏⠀⢻⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢠⠏⠀⠀⠈⣧⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⣧⠀⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢸⡇
⠀⠀⠀⠀⠀⢿⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠁


⣿⣿⠿⠿⡿⢻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⡤⠴⠶⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣀⣀⣠⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢦⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠝⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣶⡀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡧⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣇⢀⣠⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣠⣴⣿⡿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠻⠿⠛⠉⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠤⠤⠀⠠⠄

⠀⠀⡠⠒⠒⠋⠉⠉⠉⢒⠶⣤⡀
⠀⠘⠁⠀⠀⡤⠒⠒⣒⣚⣷⢦⣾⣿⣷⠂⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢰⣦⣄⡀
⠀⠀⠀⢀⡜⢀⠔⠁⠀⡰⠃⠿⠿⠿⠁⠀⣀⠰⣶⣤⠝⣟⡛⠉⠐⠠⠄⣑⠦⡈⠑⠤⡀
⠀⠀⠀⠀⠈⠳⣄⢔⡝⠁⠀⣠⠼⠆⠀⠀⠉⠹⣷⠆⠀⡀⠉⠉⠐⠒⠤⡀⠉⠺⢶⣀⠈⠢⡀
⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠉⠑⠒⢤⡀⠀⠁⠀⠀⠈⠀⠀⠈⠒⢄⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⠉⠒⠦⠤⠤⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠟⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣤⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠉⠉⠁⠠⠄⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠋⠃⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠛⠿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣟⣹⣿⣼⣿⣙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠀⠀⠀⠀⠀⣰⣿⣤⠟⠋⠉⠉⠉⠛⢧⡼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣶⣆⠀⠀⠀⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⢷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠁⠁⠀⠀⠀⣷⣾⡇⠀⠀⠀⠀⠀⡀⠀⢸⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠋⠉⠀⠀⠀⠀⠀⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣸⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⣀⣀⣀⠀⣯⣽⣯⣼⣿⣬⣿⣭⣿⣭⣿⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⠄⢠⣤⣤⡖⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠙⠻⣿⣿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⠿⠿⣶⢠⣴⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉


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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠳⠉⠁
⣿⣿⣿⡟⠛⠛⠛⠛⢋⣩⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠌⠉
⣿⣿⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣄
⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣿⡿⠟⢻⣿⣷⣾⣿⣿⣿⣏⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠣⠁
⣿⣿⣿⣿⣿⣿⣷⣤⣬⣛⣫⣥⣶⣿⣦⢿⣿⣿⣥⣭⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⢇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣭⣍⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠘⢿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣠⣾⢿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣤⣤⣶⣿⠿⣿⣿⣿⣿⡿⠿⠈⢀⣠⠶⠛⠉⠀⠀⠈⠈⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣡⣿⠿⠛⢉⣃⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⢀⡞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠀⢀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⡛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠦⠀⠀⢳⠀⢀⠴⢂⡻⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠏⠀⠀⠀⠨⡁⠂⠀⠤⠠⠤⠄⠒⠋⠁⠀⠀⠀⠀⠈⢧⠘⠚⠩⠤⢤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠌⠈⠁⠀⠀⠀⠀⠀⠀⠁⠒⠤⢀⡀⠀⢄⣀⠔⠊⣩⠟⠀⠀⠈⢳⡄⠀⠀⠀⠙⣧
⣿⣿⣿⣿⣿⣿⣿⢷⣍⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠹⡇
⣿⣿⣿⣿⣿⣿⡻⡇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⠋⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⠂⢠⠃
⣿⣿⣿⣿⡁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⢀⠎
⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢠⡾⠁⠀⠀⠀⠀⣠⡶⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠐⠶⠾⠋⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠃⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⡤⠀⠀⠀⠀⣦⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢀⣾⣿⣿⠟⠓⠒⠦⣤⣤⣴⡿⠁⢀⣼⠋⠀⣀⠀⠀⢠⡁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠀⠘⣿⣿⠃⠀⠀⠀⡄⠀⠉⢉⣴⡤⠟⠁⠚⠋⠁⠀⠀⠈⠉⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⢀⣾⠃⠀⢠⡾⢻⡇⠀⠀⠰⠶⠶⠖⠀⠈⠳⣴⡶⣯⣟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⢀⠌⢀⣠⣴⣿⢏⣠⣶⡿⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⢟⣾⡿⠟⣫⡾⣻⣿⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⢨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡏⢸⠉⠀⠀⣿⠰⣿⡇⠀⣰⠃⠀⣾⣿⣿⡦⠀⠀⠀⠈⠀⠠⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠹⣷⡘⢷⣦⣄⡘⠀⠙⣷⣼⡏⠀⠄⠈⠉⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠈⢷⡈⢻⣿⣿⣦⢂⢸⣿⡇⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠻⣷⡌⡇⠀⣿⡿⣿⣟⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡟⢀⡴⢋⣴⣿⡿⠋⢸⡿⡌⠑⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡞⠋⢠⣯⡴⢻⣿⠏⠀⠀⠈⣷⡙⢦⡀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠃⠀⣾⣿⠀⠘⣿⡀⠀⠀⣰⣿⡇⢠⣿⡆⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠙⣿⡿⣦⡀⠈⢳⣴⠞⠋⠉⢀⣾⣿⠇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⣾⣿⣷⠈⠙⢦⡙⢿⣦⡀⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠃⠀⠀⠀⢹⠀⣼⡧⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢰⡄⠀⠀⢸⡟⠋⠀⡟⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡠⠞⠁⠀⣰⣿⠁⠀⣰⣇⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢻⣷⡀⠰⠃⡏⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠈⠙⠢⠀⠸⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠒⠚⠁⠀⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢸⣿⣿⣿⣿⣿⣿⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⢸⣿⣿⣿⣿⣿⣯⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣿⣿⣥⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣽⣿⣷⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⣀⢞⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢋⣻⣿⣿⣷⣷⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣩⣉⠙⢲⠀⠐⡿⣎
⠀⠀⠀⠀⠀⠀⠀⠘⣁⣼⡟⠉⠉⠙⢿⣿⣿⣆⠀⠀⠀⠀⣿⡿⡯⣙⣉⡛⠻⣷⡌⠀⠸⡿⣔
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⣾⡿⠿⠋⠀⠀⠀⠀⠉⢶⡿⣿⣿⣿⣷⣭⣷⠀⣰⠽⣿⠂
⠀⠀⠀⠀⠀⠰⠋⠙⠉⠙⠟⠶⠶⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠋⠁⠀⢰⢏⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠛⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣪⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣋⡙⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⢀⣠⡶⠖⠀⠰⢗⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⢟⢛⡋⠙⠛⢛⣻⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣷⣷⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠶⣶⣾⣿⣶⡦⠄⣀⣤⡿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣩⣶⣦⣤⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣸⣿⢿⣿⠟⣋⣿⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡊⠀⢩⡖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⠇⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⢸⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⡎⠀⠀⣰⣿⠇⢀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢣⠀⠀⣿⣿⣦⣄⠠⣾⣵⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠧⡀⢿⣷⠸⣿⣿⣦⣀⠀⠀⡠⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢌⢻⣧⡹⡧⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠡⡙⣿⢹⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿
⣤⣤⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⡯⣿⠿⢷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠘⠿⠿⠶⠷⠀⠀⠀⠀⠀⢸⢋⣽⣛⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢰⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣦⡸⢿⣿⡟⣿
⣛⣻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣆⠀⠀⢸⣿⣿⣇⠀⡀⠘⣿⣇⠀⢸⣿⠀⢀⣴⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡹⣯⣿⣆⠀⢿⣿⣿⣿⣠⡇⠀⠹⢿⣆⠈⣿⠀⠘⢿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⢷⣹⣿⠿⠂⠸⣿⣿⣿⣿⡿⠀⠀⠻⢿⡄⠈⠀⡶⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣠⣀⠀⠀⡩⠤⣀⠀⠀⠻⣿⣿⣿⠃⠀⠀⠀⠀⠹⡀⢸⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣼⣿⡟⠀⠘⠄⠤⠤⠝⠀⠀⠸⣿⠿⠇⠀⠈⠀⠀⠀⠱⠀⠀⡄⠀⠀⢀⣶⣿⣿⠿⠿⠟⢋⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣾⣿⠋⠀⢀⣠⡴⠖⠋⠉⠀⠀⣠⣴⣦⡄⠀⠘⢦⡀⠀⠀⠡⢀⡇⠀⣠⣿⣿⣿⠏⠀⠠⠖⢛⣋⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣴⣿⡿⠁⠐⠿⡻⠋⠀⢀⣠⣴⠷⠀⢿⣶⠼⢷⡄⠀⠀⢻⣦⠀⠀⢸⡇⢀⣿⣿⡿⠃⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣠⠟⠋⠀⠀⠀⠀⣀⣤⡾⠟⠉⠁⠀⠀⠈⢿⣷⣻⣿⡂⠀⠀⠋⠁⠀⢸⠇⢸⣿⠟⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⠁⢀⡼⠋⠀⠀⠀⠀⠰⣿⡿⠛⠀⠀⠀⢀⣀⡀⠀⠀⠿⣿⣾⡧⠀⠀⠀⠀⠀⣿⡇⠘⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣠⠂⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣠⣾⣿⣮⠀⠀⠀⠉⠛⣧⠀⠀⠀⠀⢹⣿⡇⠀⠀⢰⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠠⡃⢉⡤⠛⠄⠀⢠⣿⠇⢀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⠅⠀⠀⢀⡀⠹⣷⡀⠀⠈⢸⣿⣷⡀⠀⢀⣧⠀⠛⠓⠶⠤⠉⠙⠛⠋⠁⣠⡄
⠀⠀⠀⠀⠀⠐⠄⠀⢠⣿⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢡⣄⠈⠀⠐⣿⣿⣧⠀⠘⣷⡀⠸⢻⣿⣿⣿⠂⠠⠿⠀⠀⢀⡤⠀⣤⡄⠀⢀⣾⡿⢁⣴⡷
⠀⠀⠀⠀⠀⠀⠁⢠⣿⠏⠀⢻⣿⣿⣿⣿⣿⠿⠋⠀⠀⢀⡿⠋⠀⠀⠀⠻⣶⡞⠀⠀⠈⡠⠤⢀⠀⠈⠁⠀⠀⣀⣤⠞⠋⠀⠀⣿⣿⣠⣿⣿⣷⣿⣿⣴⡶
⠀⠀⠀⠀⢀⡔⢻⣿⡟⠄⠀⠀⠙⠛⠻⣿⣿⣦⠀⠀⠀⡜⠁⠀⠀⣀⡀⠀⠘⣷⠀⠀⠀⠀⡒⠘⢁⣠⣤⣶⠟⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⠘⠀⣿⡿⠁⠀⠀⡠⠀⠄⠀⢻⣿⣿⣏⠀⠐⠀⠀⠐⣾⣿⡟⠂⠀⠸⡄⠀⢐⠃⢈⠟⢁⡠⠊⠁⠀⣄⢾⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣰⣾⡿⠁⠀⣠⠎⠀⠀⠈⠄⠀⢻⣿⣿⡄⠀⠀⢀⣾⣿⠟⠀⣀⡀⠀⢣⠀⠀⠉⡝⠀⠻⠀⠀⣄⠀⠘⢾⣿⡀⠾⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⢠⣿⡿⠁⢀⣼⠋⠀⠀⣀⣤⠀⠀⠄⠹⣿⡇⠀⠀⣾⣿⡏⠀⣰⣿⡿⠀⠈⠀⠀⠔⠃⠈⡁⠀⠈⣿⣧⠀⠈⢿⡇⠀⠈⠛⢿⣿⠏⢁⣼⣿⡶⡄
⠀⠀⠀⠈⠉⠁⣠⣿⠇⠀⣠⣾⣿⣧⡀⠀⠀⠄⠙⠁⠀⣰⠟⠁⠀⢠⣿⠟⢀⣄⠀⠀⠀⢰⣦⡈⠫⢧⡀⠘⣿⡆⠀⠈⣿⠀⠀⢀⠀⠀⢀⣾⣿⣣⠏
⠀⠀⠀⠀⠀⣰⡿⡻⠀⠾⣯⣽⡛⠙⠛⠦⠀⣠⣾⠇⣴⠋⠀⠀⠀⠿⠃⠀⠘⢋⠑⡂⠀⠈⠻⣿⣆⠀⠉⠀⢹⣿⡀⠀⢹⠀⢐⠉⣳⢀⣾⣿⡿⡡
⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⢸⣻⣿⣀⠀⠀⠾⠋⠁⡼⠃⠀⠛⠷⠦⣀⠀⠀⠱⡀⢀⢐⠀⠀⠀⠈⢻⣷⣄⠀⠀⢿⣧⠀⠀⠀⢈⣀⠆⣼⣿⢏⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⡀⠀⠀⠁⢴⣄⠀⠀⠀⠀⠀⠀⠀⢡⠈⡄⢧⠀⠀⠀⠀⠙⢿⣦⡀⠘⣿⣇⠀⠀⠀⠁⣼⡿⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢿⣶⣌⢿⣷⡄⠀⣾⣆⠀⠀⠀⠃⣤⣶⡄⠀⠀⠀⠐⠈⠻⣿⡄⠈⢿⡄⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠿⠟⠁⠻⣶⣽⣿⣿⣿⣿⣦⣿⣿⠀⠀⠀⠈⢿⣿⣿⠀⠈⠲⣤⡀⠀⠈⢿⣦⠘⣷⡀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡈⣿⣿⣧⠀⠀⠈⠻⣶⣄⠀⠙⢧⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣰⡏⠸⣿⣿⣆⠀⠑⠀⠘⣿⣷⡄⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠟⠁⢰⡿⡇⠀⢹⣿⣿⡄⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠻⠟⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠉⠀⠀⠠⣄⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣀⠀⠀⠠⠙⠶⡄⠀⠙⠢⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢄⠀⠈⠁⠈⢧⠀⠀⠀⠀⠹⡀⠀⠀⠘⢏⠉⠉⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣆⠳⣄⠀⠀⠀⠱⡀⠀⠀⠀⠱⣄⠀⠀⠘⢶⠒⠐⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣄⠀⢈⣑⢌⠳⡄⠀⠀⠘⢆⠀⠀⠀⠹⣧⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⢜⠢⣜⢿⡷⠁⠘⢆⠀⣠⣤⣠⡀⠀⠀⢳⣳⡀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⠁⠀⠀⣠⣤⠶⠞⠳⣝⣿⣿⠗⢸⣿⣦⠇⠳⡜⣿⡧
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠋⠀⠀⠀⣠⢶⠿⠋⠁⠆⠀⠀⢈⣈⡥⠄⡈⢿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⡠⡪⠒⠁⠀⠀⢀⣠⠴⠚⠉⣡⠴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠋
⠀⠀⠀⠀⠀⢠⣴⣶⠀⢀⡴⠋⠀⠀⠀⢀⡠⠚⠉⠀⢀⡴⠟⠁⠐⠀⢀⣠⡤⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠰⡿⠟⠡⢞⣽⣾⡆⠀⣠⠴⠋⠀⢀⡠⠞⠉⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣫⠴⠋⠁⠀⢀⣤⣶⣦⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠯⠵⠚⠉

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢡⠇⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⣄⠀⠀⠀⢷⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⡴
⠀⠀⠀⠸⡄⠀⠀⠸⡆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⢷⠀⠀⠀⢻⡀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠈⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⠇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡰⠋⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠉⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⣠⠖⠁⠹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇⠐⣆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⢃⠄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⢧
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡘⣆⠀⠀⠰⡷⢋⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⣧
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢷⣜⣦⠀⠀⠀⡼⢀⡶⠀⡄⢀⡀⣸⠁⠀⠀⠀⠀⢸⡆
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣄⠸⡄⠉⠀⠀⢠⡇⣸⠇⢼⠃⣾⢁⠇⠀⠀⠀⠀⠀⠀⣷
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⢻⠰⡆⢹⡆⢻⡄⢡⠀⠀⠀⠸⠠⢹⠠⢿⢰⢹⠸⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣼⠀⠀⠀⠀⠀⡴⠹⡆⠀⠈⣇⣿⡈⡿⡌⣿⣼⠀⠀⠀⠀⠃⠨⠜⠀⠌⠈⠀⠀⢰⠣⡄⠀⠀⠀⠀⣧
⠀⣿⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⢩⡇⢣⢳⢳⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢻⡀⠀⠀⠀⢹⡄
⠀⡏⠀⠀⠀⠀⢸⠇⠀⠸⡆⠀⠀⠀⠘⠞⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⡇⠀⠀⠀⠈⣇
⢰⡇⠀⠀⠀⠀⣼⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢻⠀⠀⠀⠀⢻⡀
⢸⠇⠀⠀⠀⠀⡟⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡆⠀⠀⠀⠸⡇
⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿
⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸
⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠾⠟⠛⠛⠛⠉⠉⠉⠉⢉⣛⠛⠛⠻⠷⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⠉⢛⡛⠛⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠓⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠀⠐⠂⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⠤⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⡿⠋⠀⠀⢀⠀⠚⠉⡛⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠠⠱⡱⣠⣠⠃⠀⠐⠄⠃⠀⠀⠈⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠦⣄⣣⣿⣿⣿⠃⣠⡀⠀⠀⠀⠲⢄⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠉⠉⢻⣿⡀⢹⣷⡀⠀⠀⠃⠈⠴⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠶⠟⠛⠾⠟⠋⠉⣩⡽⠟⠋
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠷⠦⠄⣀⣤⠔⠉⠒⠦⡤⠠⠄⠀⠀⠛⠿⠷⠟⠛⠁⠀⠀⠀⠀⠀⣴⣾⣿⣷⣆⡀
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⠶⣖⠀⢡⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣈⠲⣄⠀⠀⠀⠀⠀⠀⣸⣷⣦⣄⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠿⠛⣇⠈⢻⡇⠰⡀
⠀⠀⠀⠀⠀⢹⣶⣄⡀⠠⡀⠀⠁⠀⣱⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢰⣿⠀⠘⠀⠀⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣬⣲⣤⣰⡾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢿⡄⠀⠀⣸⠇
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣃⣡⣼⡿⠿⠿⠿⠿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢠⠃⢡⠀⠀⢻
⠐⠿⠿⠟⠫⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣸⠁⠀⠘
⣀⣀⣀⣀⣀⣉⣛⣉⣀⣙⣛⣛⣃⣻⣿⣿⣿⣿⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣋⣀⣀⣀⣃⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣯⣿⣯⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣻⠿⣭⢭⣍⣉⡿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⠉⠶⢻⡽⢯⣽⣿⠶⠿⣡⣿⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣛⣛⠛⠛⣋⣉⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠻⣿⣿⡹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡅⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⣻⡶⠶⣍⠻⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢠⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣍⣻⣷⡤⠜⢧⣹⣿⣛⣿⡿⠿⠿⡿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠒⣿⢿⠘⡇⢿⡿⠿⠿⠿⠶⢶⣶⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣜⠛⢁⣟⣿⡇⠀⠀⠀⢀⣾⠏⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠋⠀⠀⠀⠀⣾⣯⣦⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠉⠭⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠏⠉⠔⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣧⡀⢾⣛⢧⡌⣙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡛⢦⣅⣐⡋⠹⠆⠈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠫⠥⢶⣞⣖⣢⡍⢛⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣶⣤⣍⣙⣛⣓⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠷⣬⣉⡫⠽⠖⠗⠩⠄⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣽⡟⠉⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣭⣙⣛⣛⠛⠛⣛⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠟⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠛⣿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⢋⣡⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣷⣀⣸⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⠻⠱⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠭⣭⣭⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣷⣾⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠚

⠀⢠⡤⠀⠀⢀⠀⠀⠀⢠⠀⠀⢰⠆
⠀⠀⠀⢀⠀⠈⠀⠀⠀⠈⠀⠀⢀⠀⠀⠒⠀⠠⣤⠖
⠠⣤⠀⠉⠀⠀⠀⠘⠂⢰⡖⠀⠸⠇⠰⢤⠄⠀⢄⡀
⠀⠀⠤⣤⠆⢠⡄⠀⠀⠠⣤⠖⢀⠀⠃⠀⠀⢠⠈⠁⠀⣶⠆
⠰⢤⣤⠄⠀⠐⠃⢀⡀⠀⠀⢠⡄⠃⢀⡈⠁⢸⡧⠀⠐⢮⠆
⠀⡌⠀⠀⠀⣦⠀⠀⠀⢰⠀⠘⠇⠀⠿⠀⠀⠀⠁⠀⢄⡀⠀⠸⠃
⠘⠻⠀⠀⠀⠚⠀⠀⢀⡀⠀⠀⢰⡴⠂⠘⠋⠀⠄⠀⠀⠀⢤⠠⣄
⠀⠀⠀⡀⠀⠀⠀⣀⠈⠓⠀⢀⠈⢁⡀⠀⡀⠀⣀⠐⣶⠀⠙⠀⠀⠀⢂⣰⣴⣆⣤
⠀⠀⠀⠙⠁⠀⠀⠈⠀⠀⡄⠀⣀⣘⣋⠀⡙⠀⠋⢠⡁⠀⡶⠀⠀⠚⣨⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠶⠒⠀⠀⠈⠋⠁⠈⠓⠀⣦⠄⣷⣤⠀⢠⡄⠙⢛⢚⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⢸⡄⠀⢠⠷⢰⡅⠀⢀⡀⠸⢃⡀⠸⠂⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡴⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣈⡀⢐⣨⣭⠤⣶⣾⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠰⠃⣀⡉⠀⣀⠀⠉⣴⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⡛⠀⠀⠘⠏⢀⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡄⠀⠤⢠⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⢹⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣰⣷⡾⣿⠛⣁⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣙⢿⣿⣾⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠟⣿⣿⣃⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣤⣄⣠⣶⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠈⣸⣿⣆⢹⣿⣿⣧⣤⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣏⣿⣿⣾⡷⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢟⣻⣿⣿⣿⠟⣶⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠛⠛⠋⣉⣙⣛⣛⣉⣉⣙⣫⡿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣈⣿⣤⣬⣭⣽⣛⣛⣋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡙⢿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢺⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⡀⢤⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⢿⣿⠿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠈⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣦⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⣹⣿⣿⣼⣿⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣧⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣋⣹⣿⣿⣿⣿⣿⣿⣵⣎⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣟⠉⠉⢻⣋⣉⣻⡭⣭⣝⣻⡛⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣍⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡅⢻⣿⣿⣿⣧⣴⣾⣷⠄⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠿⡟⢢⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣶⣶⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡴⣫⣷⣿⣿⣿⣿⡿⠶⠾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢖⣵⢞⣽⣾⣻⣿⣿⡿⠋⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣫⣷⣿⣿⣿⣿⣿⠋⠀⠐⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣫⣾⡿⠁⠀⠀⠀⠾⢿⣿⡄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣳⣿⣿⣿⡿⢋⣴⣿⢏⠔⠀⠀⠀⠀⠀⠀⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣼⣿⣿⡿⠋⣠⣿⣿⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣸⣿⣿⠏⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠃⣰⢟⣿⠏⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⢠⣏⣾⠇⠀⡴⠀⢀⡄⠀⢸⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢸⢸⣿⠀⣸⡇⢠⣿⠃⠀⢸⡿⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢸⣧⠀⣿⡇⣿⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣿⡀⢿⣧⣿⣿⡀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⢷⡈⠻⢿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡻⣆⠘⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⠣⢻⡇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢴⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⡀⠀⠀⠀⢠⣶⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣠⠂
⠀⠀⠀⠘⣿⣿⣿⣿⣧⢠⣿⠀⡀⠀⣸⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡞⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣼⣿⢀⣷⢀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⢀⡴⠋⡏⠀⠀⠀⠀⠀⠀⠀⢀⠔⢩⠃⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢹
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡃⠀⣿⣿⣿⣿⣿⡟⠀⠀⠠⠚⠻⢀⣼⠀⠀⠀⠀⠀⠀⡔⠛⡏⢀⡎⠀⠀⠀⠀⠀⠀⠊⢩⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠇⠀⢸⣿⣿⣿⠟⠀⠀⠀⠐⠄⢻⠃⢨⠀⠀⠀⠀⠀⠀⠃⠐⢻⠁⡹⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢯⠀⠁⠀⠀⠀⠀⠀⠀⣴⣿⡟⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢟⠔⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠗⠅⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣺⣿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟
⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢝⢝⠏
⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠂⠀⠀⣴⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠺⠿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣠⣶⠀⠀⠀⣼⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡌⠁⠙⢷⡀⠀⠀⠻⢿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⣼⣿⠁⠀⢀⣼⡿⠃⠀⠀⣰⠟⠁⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣻⣿⣿⡇⠀⠀⠀⠉⠀⠀⠀⢠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣠⣿⠏⠀⠀⠀⠈⠁⠀⠀⠀⣾⣷⣷⡇
⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣵⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠛
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⣿⠛⡿⢿⡿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣛⡛⠛⠿⣿⣦⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠻⣫⡭⠷⠶⠾⠯⣭⣿⣻⣦⡳⡄⣍⠻⣆⠀⣿⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⢝⣡⠾⠛⠛⠛⠛⠳⠬⣭⣔⢮⡻⣿⣇⣿⣿⠄⢸⣾⡟⣼⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⢆⣼⢋⣴⣵⣿⣶⣿⡿⠿⠿⠿⣷⣷⣾⣿⣿⣧⣿⣿⣿⣯⣇⣼⣽⣿⡿⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣀⢨⡾⣡⣿⣿⠿⢛⣛⣛⣛⣿⣶⣦⡀⠤⣭⡿⠿⠛⠛⠛⠻⡿⠿⢿⣿⣽⣞⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠠⣧⣾⣷⣿⠿⣶⣿⠿⣟⡛⠿⠿⣿⡢⠈⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⣶⣶⣿⠿⢋⣵⣾⣿⠗⣋⣥⣤⣶⣶⣾⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠘⠡⠞⣡⣴⡿⢛⣭⢖⠿⢟⣛⣛⣻⣿⣿⠀⠀⣠⠴⠒⠶⢶⣤⣤⡄⠀⠀⢀⣠⢼⣿⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⢋⣥⡾⠋⠀⠀⢀⣤⣾⣿⣿⣿⡇⠀⠀⠀⢀⣤⣦⣤⡄⠈⠀⠀⢠⡟⣥⣠⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠀⠀⠀⠀⢠⣾⠿⣲⣶⣾⣿⣯⡤⠴⠶⠶⢿⣿⣻⣥⣿⠟⠀⠀⠀⠘⠿⣿⣿⣏⠙⠀⠀⠀⣾⣾⢟⡿⡿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢼⠀⠀⠀⠀⡟⠁⢼⣿⣿⠿⠒⠛⠛⣿⣶⣾⣏⠈⣝⠻⣭⠾⢦⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⣄⠙⣏⠀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠢⠤⠴⢶⢁⠀⢈⣽⣶⣶⣿⣿⣿⣟⣫⢿⣿⡄⠤⢦⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⣠⣤⣤⣤⣭⢰⠏⣹⠿⣻⣿⣿⣯⣭⠶⠋⣿⣿⠦⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⣠⢢⣞⠟⣩⣥⠶⠶⠾⠎⢸⢋⡾⠛⢉⣁⣠⡤⣀⣼⠟⢹⡆⣾⠙⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⢿⢃⡞
⠀⠀⠀⠀⠀⢠⡞⠁⣿⣤⠟⠉⣠⣶⠞⣡⣿⡇⠘⣠⡾⣻⣿⣿⣿⣯⠷⣪⣿⡇⠙⠾⠃⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡾
⠀⠀⠀⠀⠀⣾⠀⠀⣿⠁⣠⢾⡿⠁⢰⡇⢸⣇⢸⠫⠞⣋⣭⣭⣴⣶⠾⣿⣿⣷⠀⠀⠀⢠⠀⠈⠻⣿⣶⣤⣤⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⡀⠘⠦⣴⡟⣴⢣⡟⣰⢸⠘⢿⣄⣹⡎⣰⠟⠛⠉⠉⡁⢀⣼⢟⣾⣿⠀⠀⠀⠘⡆⠀⠀⢻⣿⡟⠁⠀⠉⠉⠉
⠀⠀⠀⢠⠇⠀⠀⣿⣿⣿⢻⡇⣿⠸⣆⢲⣯⣭⣽⣷⣾⣾⣿⣿⣾⣿⣷⡿⣹⡟⠀⠀⠀⠀⢻⡄⠀⠘⣿
⠀⠀⠀⢸⡄⠀⠀⢹⡻⣇⢸⣧⡻⣷⣝⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⠞⠻⠃⠀⠀⠀⠀⠀⠻⣆⠀⡏
⠀⠀⠀⠀⢧⠀⠀⠘⠳⣽⣦⣿⣷⣤⣉⣉⡉⠉⠉⠉⢙⣛⡛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣧⡀
⠀⠀⠀⠀⠈⠳⣄⡀⠂⠤⠤⠤⠤⠥⠽⠖⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠈⠉⠉⠉⠙⠛⠳⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠁⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⢀⣤⣾⣿⠟⢛⣋⠙⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣴⣿⣿⠋⣤⡾⢋⣡⣴⣶⣶⣤⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⣠⣾⣿⡟⢡⣾⠋⣰⣿⣿⠟⠉⠉⠉⠉⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣴⣿⣿⠟⢠⣿⠃⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣼⣿⣿⠏⢠⣿⠃⠀⣾⣿⡇⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⣼⣿⣿⠏⢀⣾⠇⠀⠀⣿⣿⡇⠀⠀⠀⠔⠛⠛⠛⠛⠻⠿⠦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢀⣾⣿⣿⠏⠀⣼⡏⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠋⠀⣠⣿⣿⣿⠏⠀⣼⡟⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠲⠟⠛⠃⠀⠈⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⣠⣾⣿⣿⣿⠏⠀⣰⡟⠀⠀⡀⠀⠀⡀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠐⠦⣄⣀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⡟⠁⠀⣴⡟⠀⠀⢰⡇⠀⢰⡇⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚
⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣠⠾⠋⠀⠀⢠⣿⠃⠀⣸⡇⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠚⠋⠁⠀⠀⢀⣴⣿⠏⠀⢀⣿⠁⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⠿⠃⡀⠀⣾⡟⠀⠀⠈⣿⣿⣿⣿⣿⠀⣇⠀⠀⠀⠀⠀⠀⠙⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⠿⠟⠋⠁⣠⡔⠁⣼⡿⠁⢠⠆⠀⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⢀⣠⣶⣿⡿⠿⠛⠛⠉⠉⠉⠀⠀⢀⣠⣴⡾⠋⢀⣾⡿⠁⣰⡏⠀⠀⣿⣿⣿⣿⡿⠀⣿⡇
⠀⠀⣠⣾⡿⠋⣡⡴⠚⠉⠀⣀⣤⣴⣶⣿⠿⠛⠉⢀⣴⡿⠋⣠⣾⠏⠀⠀⢀⣿⣿⣿⣿⠃⣸⣿⠃
⠀⢠⣿⠋⣰⡿⠋⠀⣀⣴⡾⠿⠛⠉⠁⠀⠀⠀⠐⠋⣁⣤⣾⡿⠋⠀⣠⠆⣼⣿⣿⡿⠃⣴⣿⡿
⠀⢸⠇⣸⡟⠁⢠⡾⠛⠁⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⡿⠋⠀⢀⡴⠃⣼⡿⠟⢋⣠⣾⣿⡟⠁
⠀⠘⢀⡿⠁⠐⠋⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⢀⣠⣶⠟⠀⠈⣁⣤⣾⣿⣿⡿⠋
⠀⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⡿⠟⠋⠉⣁⣀⣤⣤⣶⣾⠿⠋⣀⣤⣶⣿⣿⣿⡿⠟⠉⠀⡼
⠀⠀⠘⠁⠀⣾⣿⣿⡿⠛⠁⢀⣤⣶⡿⠿⠟⠛⠋⠉⣠⣴⣾⣿⣿⡿⠿⠛⠉⠀⠀⣠⡾⠁
⠀⠀⠀⠀⢸⣿⣿⡟⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠁⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠞⠁⠀⢀⡴⠂⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣴⠋⠀⠀⣰⣿⡿⠋⠀⠀⠀⢀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⣾⠇⠀⠀⠀⣿⡿⠁⠀⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⡟⠀⠀⠀⠐⣿⠇⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⣶⡄⠀⠀⢀⣀⣸⣿⠠⠄⢈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣠⠈⢛⣿⣷⣾⣿⣿⣿⣿⣦⣴⣿⣿⣷⣶⣤⡀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣼⣿⠇⠀⠉⢹⡿⣿⣿⣿⣿⣿⡄⠉⢟⣉⠙⠛⠻⣿⣼⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⡿⠿⠶⣦⠀⠙⠁⠘⢿⣿⣿⣿⣷⠀⣘⢛⡑⢿⠇⡋⢹⡟⢻⣘⣀⠁
⠀⠀⠀⠀⠀⢠⣶⣾⠿⣿⣷⣫⣤⠀⠶⠾⣷⣿⣿⠟⠈⠀⠀⠉⠉⠀⠻⣿⣽⣶⣾⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠸⣏⣠⣴⣶⣿⢉⣡⣶⣦⡄⠋⠛⠛⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⣘⠟⣅⠈⡙⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣛⣿⡦⣸⣤⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠈⢡⣾⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⡇⠀⠈⢹⠇
⠀⠀⠀⠀⠀⠀⣀⣬⣷⣾⡿⠟⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⢹⣿⣈⣥⣤⣀⣿⡇⠈⠀⠈⠁
⠀⠀⠀⠀⠀⣾⡿⠿⣿⣿⣶⠀⠀⠀⢿⣷⠀⠀⢀⣤⣤⣀⠀⠈⠄⠀⠃⠐⢿⠟⠻⠿⠿⣯⡁
⠀⠀⠀⠀⠀⠈⠀⠠⠙⢿⡇⢀⣴⣶⣿⠿⢇⣀⣾⠛⠛⠛⢶⣦⣠⣤⣤⣤⣸⠐⣀⠐⢸⠉⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁⣼⣿⣿⡇⠀⠐⠭⠇⠄⠀⠤⠸⣿⠛⠛⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣘⣛⣛⣾⣿⣶⢾⣿⡄⠀⠻⣿⠟⠃⠀⠘⠃⠀⠀⠀⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣭⡭⠭⠍⣉⡉⠿⠿⢿⣶⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⣹⡀⢀⣴⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣶⣶⣶⣶⣶⣶⣶⡾⣻⡿⠃
⠀⢀⣴⣾⣿⢿⣿⠿⣻⠝⣻⢟⣿⠟⢋⣽⣿⡽⣿⢟⣽⣾⢿⣫⢟⣯⣿⡽⣛⠝⡫⢞⠵⣿⠟⠙⣫⣾⠏
⣠⣾⡷⠿⠖⠛⠷⠚⠕⠋⠕⠋⠀⠚⠋⠘⠁⠈⠀⠉⠁⠀⠈⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⣇⠀⠀⠀⠀⠀⠆⠀⠀⠻⠀⠐⡀⠀⠉⠀⠸⠀⠰⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⣿⠀⠀⠆⠀⠀⠀⡄⠀⠀⠀⠁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠈⢠⠀⠃⠀⠀⠰⡆⢐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠘⣏
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢃⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠂⠀⢰⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢸⡇⢰⠀⠀⡄⢀⠀⠀⠃⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⣿⠀⠀⠁⠁⠈⠀⠃⠀⠀⠀⡆⠀⠀⠐⡄⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣿⠀⡄⠂⠁⠐⠆⢆⢰⠸⡄⠃⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿
⠀⠀⣿⡇⠀⢆⢲⠀⠀⡄⠈⠀⠁⠀⡄⠀⡀⡄⠃⢦⠀⠠⠀⠀⢻⠀⠀⠸⣤⢀⠀⢰⡄⠀⡃⠘⠈⢠⡆⠀⣿⡄
⠀⠀⢹⣧⠀⠈⠬⠁⠀⠐⠐⡶⠀⠃⠙⣆⣁⠀⠀⠀⠀⠠⠄⠨⠄⠀⠤⠀⢀⣈⣀⣀⣀⣀⣀⣀⣀⣀⢐⣂⡿⠛⢷⣄
⠀⠀⠘⣿⣦⡷⣾⣷⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣦⠀⠙⣧⡀
⠀⠀⠀⣿⣿⣷⣜⢿⣿⣿⣾⣟⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣤⣄⠊⠻⣦⡀
⠀⠀⠀⣿⢻⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣃⣀⠈⢻⣄
⠀⠀⠀⣿⠀⡿⣿⢿⡿⣦⡻⣿⣿⣍⣻⣷⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣿⣛⣟⣻⣛⣻⣿⣿⣿⣧⠈⠙⢷⡄
⠀⠀⢸⣿⠀⢘⣻⣯⢻⣿⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⡿⢿⣷⡄⣹⣦⡀
⠀⠀⢸⣿⠀⣌⡛⠙⢇⢻⣮⣿⣷⡝⢿⣿⣿⣿⣭⣽⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠟⠻⠛⠛⢻
⠀⠀⠈⢻⣤⣿⡉⠻⣜⣧⢻⣿⣿⡟⠀⠍⠉⣁⣀⠀⢀⣀⠀⠀⣄⡄⠀⠀⣀⣤⣤⣶⣶⣶⣶⠞⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣆⡵⣜⣷⣿⣿⡇⠀⠚⠉⠁⠀⠈⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⣿⣿⣶⣶⠌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣿⡖⡝⣾⣿⣿⣇⣴⣶⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⢿⣿⡙⣿⣷⡸⠿⢻⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠇⠀⠔⢃⣠⣴⣾⣿⣿⣿⣿⣿⡿⣿⣷⢶⣙⢷⣌⣿⣷⣾⣿⣿⣿⡿⣿⣍⣾⣿⣿⣿⡿⠟⠋⠁⠚⠉⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⢾⣍⣈⣿⣾⣿⣶⣿⣿⣷⣿⣿⣯⣼⣷⣿⣿⣿⡿⠟⢃⣤⠀⠐⠀⣞⠛⠯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣟⣭⣶⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣉⣻⣉⣝⢛⣟⢋⣿⢻⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠞⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠤⠴⠖⠒⠲⡒⠉⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢼⠉⠑⢌⠻⢆⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⣀⣨⠶⣀⡸⠀⠀⢉⣲⠖⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣟⢰⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠴⠆⠀⠀⣠⣿⠘⡀⠀⢠⣗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠤⠒⠊⠈⣿⠀⣇⢤⠖⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢾⣻⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢭⢈⣀⣉⠶⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⡀⠀⢨⠀⢈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⣰⠚⠧⡀⠸⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡏⠬
⠀⠀⠀⠀⠀⠀⠀⠳⣚⣉⣽⡙⠒⣳⠸⡿⢣⡌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠓⢤⠇
⡰⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠛⠉⠑⡖⠀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⢂⠔⠀⠰⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⣯⣄⣠⠀⢧
⠈⠢⣄⣀⣀⣠⡤⠔⢪⠋⠀⠀⠀⡸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢽⠗⡸⢁⣠⠔⠊⠉⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⡠⠞⠁⠀⠀⢠⠀⠀⠀⠀⠀⢠⡄⠤⢴⣒⠒⢡⡪⢊⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣀⣀⣀⡀⢸⢳⠊⠢⣀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠱⣤⣄⡀⠀⠀⠑⠒⠐⠒⠒⠊⠉⠉⠙⠦⠬⠥⡀⢹⠳⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⣉⣉⣉⣙⣉⣍⠁⠀⠀⠀⠈⠋⠰⠢
⠀⠀⠀⠀⠀⡅⠀⠑⠂⠄⠀⠀⠀⠀⠀⢀⣉⠁⠂⠆⢠⣀⣀⣀⣀⠀⠀⣀⣠⠇⠆⣸⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠒⠢⠤⢤⠀⣀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠘⡀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢍⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣄⡀⡰⠲⡄⠀⠀⠀⠀⠀⡆⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠎⠀⠀⠈⢡⠎⣀⠀⠀⠀⠀⠀⡇⠒⠒⠒⠂⠠⠬⢁⣀⣀⣀⡀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠑⡄⠰⢦⠀⠜⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠁⠒⠒⠂⠤⠤⠀⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚

⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⢸⣣⣷⣹⠆⠀⠀⠘⢿⣷⣤⠶⠛⠉⣿⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣼⡟⠁
⠀⠀⠈⣷⣿⣿⠀⠀⠀⠀⢸⣿⣇⣠⣄⣠⣽⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⢿⡃
⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣷⣶⠆⠀⢀⣤⣀⣤⣿⡇⠀⠀⠘⣧
⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠸⣿⠉⠙⠛⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⢀⣠⣈⣿⣧⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠒⠶⣾⣷⠶⠖⠒⠒⠀⠀⢸⡏⠙⠛⠻⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠃⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣄
⠀⠀⢀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣟
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣴⣶⠂⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⡀⠠⠴⠶⠾⠿⠿⠿⠿⠿⠟⠋⠙⢦⣀
⠀⠀⠀⠀⠹⢷⣶⣴⠶⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⣤⣤⡀⠀⠈⠛⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠤⠤⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣏⡀
⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⣾⡿⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡼⢿⣿⣿⢿⡄
⠀⠀⠀⢹⡿⢿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣼⣿⢻⣷⠀⠀⠀⠘⠛⣿⣿⣿⡞⠛
⠀⠀⠀⣿⡇⠘⠿⠇⠰⠖⠶⠶⠿⠃⠀⠀⠹⠦⠀⠀⠀⢀⣴⡿⠟⠃⠀⠙⣷⡀⠠⣶⣿⣿⡿⣿⣿⣷⠆
⠀⠀⠀⠹⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⢸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠉⠙⠛⠛⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣈⠙⢦⡀⠀⠀⢀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⡙⢄⠀⠀⠑⡺⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢹⣿⣿⣯⡻⣿⣷⣄⠀⠀⠨⡺⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣌⢻⣿⣷⣄⠀⠈⢙⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢀⣤⡶⠖⠛⠒⢸⣿⣿⣿⣿⣧⡙⣿⣿⣆⠀⠈⢑⣽⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⡿⠛⠁⠠⠴⠶⣶⠞⣿⣿⣿⣿⣿⣷⡈⢿⣿⡆⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠⣤⣤⣴⣯⠀⣿⣿⣿⣿⣿⣿⣧⠘⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⡿⠛⢉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠘⣿⠏⠀⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⠿⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⣿⠃
⠀⠀⠀⠀⠀⠘⠀⢰⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠲⣿⣿⣿⡷⠂⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⠁⠀⢳⣄⠀⠀⠈⠉⠉⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⣷⣦⣄⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠸⣿⡏⣿⣿⣿⣿⠀⠀⠀⠈⢿⡟⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⣿⣿⣿⣧⠀⠀⠀⠘⠇⠀⣰⣿⣿⣿⠋⢸⣿⣿⡇⠰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠻⣿⣿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠙⠻⠷⠄⠈⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣷⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢶⣶⣤⣀⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠉⠛⠻⢿⣶⣤⡉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠉⡿⠋⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠑⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⡂⠀⠀⠀⠀⣰⢻⣷⠀⢥⢠⠀⠀⠀⠀⢠⣀⡼⢠⡠⠄⢻
⠀⠀⠀⠀⣀⠀⠀⠀⠀⢾⠇⠀⢠⡆⠀⢠⠇⠀⠀⠀⠀⠃⢸⣿⠀⠈⠺⡄⠀⠀⠀⠀⢹⡇⠋⠇⠀⢸
⠀⠀⠀⠀⠀⢈⣧⣄⠀⢸⠀⠀⢈⡟⡰⢻⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠄⠀⠀⠀⠀⢸⠁⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣧⣸⠀⠀⣸⡟⡵⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠐⠂⠀⢻⡉⠀⣠⣼⣋⣁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⣀⡇⠀⠀⠀⠀⠀⡆⠀⣤⣌⣹⡿⠃⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠒⠒⠚⠓⠾⠛⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠎⠘⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⡦⣀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⠣
⠀⠀⠀⠀⠀⠷⢤⡀⠀⠈⠛⢠⠀⠀⠀⠇⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⠞⠳⠀⠀⠀⠀⡃⠀⠀⠀⣐⣀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⣾⠂⡐⡶⠀⠀⠀⠀⠈⠙⢦⡀⠀⠐⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣼⡄⠸⠀⠀⠀⢸⡏⠀⡴⢣⠀⠀⠀⠀⠀⠀⠀⠈⠘⡇⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣧⠀⠁⠰⢀⣿⣇⢠⠀⠈⢠⠀⠀⠀⠀⠄⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⡌⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⣿⢣⣠⠞⠁⢿⣿⡷⠀⣾⡿⠀⠀⠀⠀⠐⢄⠀⠀⣿⠃⠀⠀⠀⠀⠀⠧⠀⠀⣰⡶⠇⠀⠀⠀⠀⠀⠀⠚⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠀⠀⠀⠀⠃⠀⠀⣯⢇⡀⠀⠀⠀⠠⠄⠀⠀⢠⣴⡆
⠀⠀⠀⠀⠀⠁⠀⢀⡶⠁⠀⠀⠀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠰⢵⣃⡤⠴⠚⠑⠚⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁

⠀⠀⡠⠐⠈⠉⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠶⢶⣤⣀
⡾⠀⠀⠀⠀⣀⠤⠀⠉⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⢿⣶⣦⡄
⣇⠀⣤⡾⠋⠁⠀⢰⠁⠀⠀⠀⠀⠀⠈
⠹⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠒⠊⠉
⠀⠙⠿⣦⡀⠰⠊⡀⠀⠀⠀⢀⣤⡆⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠘⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⣠⣤⠤⣤⠀⠀⠀⠈⠑⠲⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠛⠛⠉⠁
⣼⣿⢱⡟⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠈⠳⢤⣀⣀⡀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⠒⠒⠒⠛⠛⠛⠛
⠹⣿⣌⠛⢁⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠉⠉⠁⠀⠱⣄⠀⠀⠀⠀⣀⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⢀⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣿⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣹⣿⣠⣾⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣤⣄⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⡀⠀⢀⣼⠟⣿⣿⣿⣿⢻⣧⡀⠀⣀⠀⠀⠀⠀⢀⣿⠸⣿⡿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢀⡀
⠀⠀⠀⠻⢿⣿⣥⣴⣿⣿⣿⣿⣤⣬⣿⡿⠟⠀⠀⠀⠀⠐⣿⣦⣿⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣿⣿⣷⣾⣿⡟
⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⢿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣿⣶⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣋⣽⣿⣿⡇⠀⠀⠀⠀⢸⣇⣀⠀⠀⢿⣿⣿⣶⣀⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣿⣿⣿⡿⠛⠛⠋⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣶⣦⣶⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠤⠤⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣀⣀⣀⣀⡐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⡊⠀⠠⠄⠤⢾⣷⣶⣒⡂⠄⠙⣆⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠰⢮⣝⣀⣐⢧⡽⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢄⠂⠠⠤⠤⠭⠥⠤⠤⢤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣶⠚⠋⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣤⣬⣭⣭⣥⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⠛⢺⣧⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡂⠠⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢯⡑⣲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠢⠍⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡶⠬⢭⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣁⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⠉⠹⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡂⠄⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣍⣓⢲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢁⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⠈⠽⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡁⢲⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢁⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⠤⣬⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡲⠤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⣿⣿⣿⠿⣿⡿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣞⣷⠦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡈⠉⠳⢬⡙⢾⣅⢦⣀⠀⠀⠁⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢄⠈⠛⠯⡙⠣⣝⢳⣄⠀⠀⠀⠔⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠹⢦⣍⠻⣌⢷⡀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠙⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⣦⣄⠈⠙⢦⣀⠙⠳⠈⠀⢷⡈⢩⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢠⡆⠀⠀⠀⠹⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠘⠈⠷⣄⠀⠁⠀⠀⠀⠀⠘⢷⡉⢀⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣸⣿⠃⠀⢠⠿⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⢂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⣧⠉⠯
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣿⡇⠀⢰⣃⠜⠁⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣶⡧
⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇⢀⡀⠈⢁⡟⠹⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣴⠋⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⡷⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠰⠄⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣵⣿⣿⠀⠀⠀⠀⢀⠏⢠⠓⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠙⠏⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⡾⠋⣽⠎⠀⠴⢣⣾⣿⣿⠇⡀⠀⠐⠁⠀⠠⣂⣾⣿⢿⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠐⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⢿⣥⠾⣇⠀⢐⣵⣮⣿⣿⣁⠀⠀⠀⠀⢀⠔⠀⠹⡎⠁⢹⣷⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠹⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣠⢒⣀⣭⣶⠿⠚⣧⠙⠂⠀⠀⠂⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⣻⠖⠥⠶⣣⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⠈⠒⠉⠙⣇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣄⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡉⢿⣴⣶⣤⢀⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⣿⠋⠁⠀⠀⢀⣴⣿⣦⣄⡀⠀⠀⠀⠀⠘⠁⣠⣸⣯⡛⠻⠟
⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡀⠀⠀⠠⣤⣿⣿⣿⣿⣷⣶⣶⠏⠀⣀⡀⢉⣹⣃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣿⣦⠞⠁⠀⣦⣾⢻⣯⡻⢿⣿⣿⣿⣿⠀⠘⠿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢀⠀⢹
⣿⣶⣦⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⠃⠀⠀⠀⠈⠻⣿⢿⡻⡄⢻⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣤⡀⣶⣾⡆⣾
⣿⣿⣿⣧⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣆⠓⣿⡈⣿⣿⣿⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽
⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠾⡇⠀⣠⣤⣤⣈⣿⣶⢸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⡀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⠈⠿⠞⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠨⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣞⣛⣻⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⢿⣷⣶⣶⣾⣿⣷⣤⣄⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠛⠛⣛⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⣀⢀⠀⠀⠀⠀⠀⠀⣀⠁⣈⢳⡄⠀⠀⠀⠀⠄⡀⠀⢀⠠⠀⠄⠀⠀⠀⠀⢀⠤⠐⠠⢄⣀
⠀⠀⠀⠰⠡⠄⠀⠤⢹⠀⠀⠀⠀⠘⠃⠉⠀⠉⠙⡁⠀⠀⠀⢘⠂⠁⠀⠈⠀⡹⠀⠀⠀⠀⠀⣄⢠⣀⠀⡀⣸
⠀⠀⠀⠀⡄⠀⠀⠀⠸⠀⠐⠀⠀⠐⢦⠀⠀⠀⠀⡤⠢⠤⠶⣔⠁⠀⠀⢀⡘⠀⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢰⡦⠤⠴⣁⠄⠀⠀⠀⠀⠀⠑⠤⠔⡤⠃⠒⠀⠒⠋⣀⢌⠉⢹⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠐⠀⠃⠀⠘⠁⡇⠦⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢠⠘⠉⠐⠃⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⠜⠀⠀⠀⠀⠀⠌⠒⠈⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠒⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠐⠄⢰⠄⠞⠀⠀⢀⠀⡔⠉⢃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠂⠈⠈⠀⢀⠀⠀⠀⠸⡄⠉⠀⠉⠈⠆⠀⢀⠀⠤⠀⠠⠌⢢
⠀⠀⠀⠀⠀⠀⢀⢚⡁⠈⣙⢫⠆⠀⠀⠂⠀⢄⡊⠈⢒⠢⠈⠆⠀⠀⠀⠱⡀⠀⡠⠃⠀⠀⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠈⢀⢐⠀⠀⠀⠰⠀⠁⠀⠉⠀⡀⠀⠀⠀⠠⡖⠂⠀⠑⡢⠆⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢀⡜⠘⠀⠀⠀⠀⢄⠀⠀⠀⡐⠀⠀⠄⠀⠀⠙⠁⠀⠈⠁⠀⢀⠀⠀⠈⠒⠂⠁⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠍⠁⠒⠙⠡⠀⠀⠀⠀⠀⡨⠑⠒⠘⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⠀⠀⠀⠀⣸⣛⢛⣓⣀⠀⠀⠀⠀⣴⣶⠶⣆⠀⠀⠀⢀⣶⣾⣶⣦⠀⠀⠀⠀⠠⢒⣆
⠀⠀⠀⠀⠀⢻⠿⠿⠿⣿⠀⠀⠀⠛⠋⠀⣀⠈⠛⠃⠀⠀⠀⡟⠀⠀⢻⠀⠀⠀⣾⡧⠀⠠⢿⡇⠀⠀⠀⡖⠋⠋⢲
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⢀⣠⣄⣀⣦⡄⠀⠀⠹⣷⡄⠀⣾⠃⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠀⠀⠄⠀⠀⠂⠀⠀⢀⠀⠀⠈⠗⠻⡛⡗⠃⠀⠀⠀⣀⠁⠀⢁
⠀⠀⠀⠀⠀⡀⠀⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣷⠹⠄⠀⠀⠀⠈⠂⠔⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠁⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡿⡿⠟⢿⠀⠀⠀⠀⣾⠋⠈⢿⠀⠀⠀⠀⣾⠛⠛⢻⠀⠀⠀⠀⠸⡇⠀⠉⡓⠀⠀⠀⠀⡿⠛⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣄⠀⠀⠀⠰⣤⣤⡼⠁⠀⠀⠀⠀⠁⡀⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⡟⠀⠀⠀⢀⠉⠉⠁⡀⠀⠀⠀⠀⠴⢠⣀⠦⠀⠀⠀⠀⣄⠈⠀⠡⠆
⠀⠀⠀⠀⢠⣾⣿⣷⣷⣿⣷⡄⠀⠈⠀⠂⠀⠂⠈⠃⠀⠀⡙⠄⠀⠼⡃⠀⠀⠀⠀⠳⠀⠠⠇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠠⠶⠀⠀⠰⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⡄

⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⠛⠛⢿⠀⠀⠀⢠⣿⡿⠋⣿⡆⠀⠀⠀⢐⣋⣉⣙⣓⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⣴⣿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠌⠉⠉⠡⠀⠀⠀⢸⣇⠁⠀⢠⡇⠀⠀⠀⠘⠁⠀⠀⠋⠀⠀⠀⠐⡟⢛⠛⣿⠀⠀⠀⢸⣿⠃⠐⢉⣧
⠀⠀⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠀⢸⣿⠇⠀⢿⡇⠀⠀⠀⠀⠴⢀⠴⠄⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠘⢿⡦⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠐⠃⠁⠀⠀⠀⠀⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⣤⡐⠀⠐⢁⣴⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⡮⢵⣿⣿⡇⠀⠀⠀⠘⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠛⠛⠃⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⣀⠤⠔⠛⠛⢻⣿⣿⣿⣿⣿⡻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣞⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⢬⡿⢿⣝⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠘⠶⣄⡀⠀⠀⠀⠙⠿⣿⣿⡄⠀⠀⠈⠓⠯⢟⡿⣿⣿⣿⣶⣦⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠃⠀⢄⡀⠀⠛⠀⠀⠀⠈⠉⠽⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⢀⣤⣆⡀⠀⠀⠀⠀⣀⠴⠒⠉⠉⠁⢀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀⢧⠀⠀⣠⠾⠥⠤⠤⠐⠀⣰⠋⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⢣⠀⠈⠻⣿⣿⣧⠰⠇⠀⠀⠀⠀⣄⡰⢁⡞⢠⠆⡞⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠈⠀⠀⠀⢈⣿⡿⠀⠀⡠⠀⠀⠜⠁⠁⠈⢀⠎⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀⠀⣼⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣴⣾⣿⣏⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⠿⠿⠿⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢲⣶⣴⣿⣯⣿⣿⣿⣿⡏⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⢃⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⢀⠀⠙⠛⣿⡿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠉⢹⣇⣨⣽⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠴⠵⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⠛⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠟⠁⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠔⠂⠈⢿⣦⣀⠀⠁⠀⣀⡀⠀⠀⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⠶⠶⣦⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⠀⠀⠙⢿⣦⡀⠀⠸⣷⡀⠀⠤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣄⣈⣻⣦⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠘⠉⠀⢀⡀⣀⣠⣾⣿⣿⣥⣴⡾⠿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠂⠀⠘⣿⣿⣿⣿⠛⠉⠉⠀⠀⠂⠊⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠋⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⢿⣿⢿⣿⣿⣿⢿⣯⡀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠹⢿⠘⠻⠾⣿⣻⣿⣷⣶⣿⣾⡿
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣿⣷⣦⣴⣶⣦⣬⣽⣿⣿⣿⣿⡿⠂⣠⣾⣄
⠀⠀⠀⠀⠀⢰⣿⡿⢿⣤⣀⣀⣀⣉⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢛⣿⠇⣠⡀
⠀⠀⠀⠀⢰⣄⠻⣿⣶⣬⣽⣿⣿⣿⣿⡿⠿⠛⠿⠟⠛⢿⣿⣿⣿⣷⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣤⣴⣦⣤⣈⣉⣀⣀⡈⠉⠛⢛⣉⣿⣿⡿
⠀⠀⠀⠀⠀⢿⣿⠛⣿⣿⣿⣿⣿⠟⠿⣿⣿⠿⠿⠟⠻⠿⠿⢿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⠏⢀⣠⣤⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣅⣴⣾⣿⣿⠿⠟⣋⣽⣿⣿⣿⣿⣿⣿⣿⣶⣆⣈⣉⣙⠻⣯⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⠫⠟⢁⣰⣾⣿⣿⣿⡿⠧⠄⣉⣉⣤⣤⣿⠿⠿⠛⠛⠛⠛⠋⠉⠙⠻⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠈⣿⡿⠉⢀⣴⣿⣿⡿⣯⠤⠄⣀⣤⣶⣬⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⣿⣿⣿⣿⣛⣀⣰⡿⠿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣍⣁⡀⠀⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⡟⠛⣋⣉⣴⣾⣿⠔⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⣤⣾⣿⣿⡌⢻⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠸⣿⣿⣿⣿⡟⢁⣌⠑⣜⢣⡀⠀⠀⠀⠘⠛⠿⣍⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⣻⣿⣿⣿⡇⠀⠙⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⢰⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⣷⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠻⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⣾⣿⡿⠀⠀⠀⠸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣄⣀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡿⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⢣⡤⣼⡟⠋⠛⠛⠛⠛⢋⠛⢋⡙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⢰⣷⡆⠀⠙⣧⠀⢶⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡹⣿⡄⢻⣿⣷⣾⠀⢹⣆⠸⠀⡛⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣉⠉⢙⢋⠀⠈⢿⣿⡟⠀⠀⠉⠄⠰⠁⠀⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠁⠀⢀⣼⡟⠁⠁⣀⠈⠀⠈⠟⠓⠀⡀⠀⠈⠀⠀⠀⢠⣾⡿⠛⢋⣡⠶
⠀⠀⠀⠀⢠⠇⠀⣠⣿⠋⠠⡶⠋⢁⣠⡄⢸⣟⣷⡀⠈⢦⡀⠀⡇⢠⣿⡿⠁⣠⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⣼⠀⣠⠏⠁⠀⢀⣤⡾⠋⠁⠀⠀⠻⣞⣷⠀⠈⠁⠀⡇⢾⠟⠀⠰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡈⡁⡴⠁⠀⠀⠀⠘⠉⠀⢀⣠⣶⡄⠀⠙⢻⡅⠀⠀⢰⣇⠀⠀⡀⠀⠈⢿⣿⣿⣿⡟⠁
⠀⠀⠈⠖⠁⠀⣼⢃⣤⣶⣶⣾⣿⣿⠿⠿⠗⢀⣠⡄⢻⡄⠀⣸⣿⣄⠀⡇⠙⠛⠂⠉⠉⢉⣴⠀⣀
⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⣿⡿⠋⠁⣰⠇⠀⠀⢯⡟⠀⠙⠀⠉⠙⠃⠀⣀⠴⠊⢁⣿⣀⣾⣯⣾⣧⡄
⠀⠀⠀⠄⣿⡏⠀⠀⠉⠙⣿⣧⠀⠀⠁⠀⣤⣄⠀⢧⠀⠀⠀⡠⠀⡶⠋⢁⡀⠀⢸⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⣴⡟⠀⡠⠂⠀⠀⠹⣿⡇⠀⠀⣼⡿⠁⣀⠘⠀⠀⠰⠐⠎⢀⡀⠫⣿⠠⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠸⠟⢀⡾⠁⣀⣴⠂⠀⠙⠿⠀⣸⠿⠁⣼⡟⠀⠀⠀⡈⠠⣄⠘⣷⠀⠸⡇⠀⠙⠟⢁⣾⡟⠂
⠀⠀⠀⢠⣿⠃⢼⣿⠟⠳⠄⢀⡶⣰⠃⠀⠸⠋⠐⠓⠀⠀⠻⣦⡈⠂⠹⣇⠀⠇⠀⠒⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⡐⠉⠰⢁⠈⠛⠂⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⢻⡆⠀⠘⠄⣼⠿⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⢦⣜⣷⡀⢠⡄⠀⠀⢠⣤⠀⠀⠀⠙⢷⡈⢻⡀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠁⢙⣿⣿⣿⣿⣾⣷⠀⠀⠘⣿⣇⠀⠲⣄⠈⠻⣌⣧⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣧⠀⣴⢹⣿⡄⠀⠈⢿⣦⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠟⠁⢸⡇⠀⢿⡿

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠑⠄⠀⢀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡀⠀⢠⠊⠉⠑⡄
⠀⠀⠀⠀⡠⠔⠢⣀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠐⠀⠈⠀⠀⠀⡆⣠⠄⠢⢄
⠀⠀⠀⠀⣇⠀⠀⠈⢮⢇⠀⠀⢀⢰⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⡆⡀⠀⠀⡸⡵⠁⠀⠀⣨
⠀⡠⠤⣀⠨⠳⡀⠀⠀⡈⢆⠀⠘⠘⠂⠀⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠢⡀⠐⠃⠃⠀⡰⢡⠀⠀⢀⠞⠍⣀⠤⢄
⠸⠀⠀⠀⠑⠄⠑⢄⠀⢡⠈⢆⡴⠀⡀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠁⢄⡐⠁⡌⠀⡤⠊⠠⠊
⠀⠉⠒⢄⣀⡈⠢⡀⠳⣌⠂⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠐⣣⠞⢀⠔⢁⣀⡠⠒⠉
⠀⠀⠀⠀⠀⠈⠑⠚⠂⠀⠀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠒⠂⢤⠀⠀⠐⠓⠊⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠔⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⢘⠀⠀⡃⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠊⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⣿⣿⣿⣿⡟⠿⠿⠿⠿⠿⠿⣿⢿⠿⠿⢿⠿⡿⢿⠿⡿⠿⠿⡿⠿⢿⡿⢿⠿⡿⠿⢿⠿⡿⡿⠿⠿⡿⡿⢿⠿⡿⠿⠿⠿⣿⣿
⣿⣿⣿⣿⡗⠏⣐⣒⡐⡶⠀⣿⢨⠀⠀⢸⠀⠇⠘⠀⠀⠀⠀⠁⠀⠀⠁⠸⠀⠀⠀⠸⠀⢀⠃⡆⠠⠇⡇⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⡇⠀⣿⣿⣷⠁⠀⢻⠀⠀⡄⢈⣰⣀⠤⠤⠤⠤⠧⠤⠤⠤⠴⠤⠼⠤⠤⠤⠶⣀⠀⠠⡇⠀⡇⢸⠀⠀⢀⣀⠴⢺⣿
⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⢸⠀⠤⣧⡇⠀⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢊⣢⡇⠀⡇⢸⡔⠲⠏⠀⠀⢸⣿
⣿⣿⣿⣿⡇⣄⣛⣛⣛⣀⠀⢸⠐⡗⠓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⡾⠁⠀⠁⢸⠂⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⡇⠀⠁⠈⠈⠁⠀⢠⠀⡇⢠⠇⠀⠀⠀⣀⣀⠀⣀⠀⠀⡄⢀⠀⠀⣀⡀⠀⠀⠀⠀⢡⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠇⠄⠄⣀⠌⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⢹⠀⠀⡟⡆⠀⣷⡞⠀⠈⣧⠄⠀⠀⠀⠀⢸⢀⠀⢸⣾⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠀⠂⠀⠀⠀⠀⠀⢸⠀⠁⢸⠀⠀⠀⠀⢸⠀⠠⠧⢻⠀⡯⠙⠂⠀⢧⡤⠒⠀⠀⠀⢸⢸⠈⢸⢸⡇⠀⢀⣀⣠⣼⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡷⠏⢽⢿⠿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠘⢠⣾⠀⢰⣠⣰⢰⠀⡆⢶⠀⢰⠂⠀⠐⠶⢡⠚⡄⡆⢠⠀⡆⠀⠀⠀⢸⡇⠤⣘⡨⠷⢽⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢿⡀⠀⠋⠃⠘⠉⠃⠛⠃⠘⠀⠀⠀⠘⠈⠓⠁⠳⠞⠀⡧⠀⢸⠀⢸⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣀⣴⣠⣾⣿⡿⠛⢻⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⢸⣆⠀⡇⢻⡡⠀⢸⣭⠀⠀⢺⠑⡄⠀⠀⠀⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠀⣠⣞⠻⣷⠀⠀⢸⡇⡄⢸⠀⠀⠀⢸⡌⢦⡇⢸⣀⠔⠘⣇⡤⠂⢸⡠⠏⠀⠀⠀⡇⠸⠀⡄⢸⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⡛⣛⣿⣶⣟⣛⣛⣛⡇⡇⢸⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⡀⠀⢸⠇⠐⠠⢀⠀⢸⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠁⠀⠀⠀⡇⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⡇⠀⠀⡇⠀⠇⠀⢸⠀⠀⠀⠀⠁⢻⣿
⣿⣿⣿⣿⠀⠀⠈⠀⠈⠀⠈⠁⣿⢸⢸⠤⡄⠀⠀⡄⡀⠀⠀⡧⠀⣜⠀⢠⡇⠀⠤⠀⡇⣄⡃⡇⠀⡇⡇⢸⠀⠀⠀⠀⠀⢺⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣺⣷⡁⢋⡇⠄⣐⠆⣗⠀⢸⢁⠀⠁⠀⡇⠀⡆⡇⠂⣇⠀⠇⠇⢸⠄⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢟⣛⡎⠀⡏⡇⠀⡿⠁⣞⠀⢼⢸⡄⣽⠀⡄⠈⠁⡇⢠⢹⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠸⠁⠇⡄⠏⠒⢹⠈⠢⡇⢤⢨⠥⠳⠻⠆⢰⣶⠂⠹⠁⣿⠀⠀⠀⢸⠀⢢⡀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡧⡆⢰⠀⠃⢀⠀⠸⠀⠀⠁⢰⠀⠀⠀⠀⢀⠀⠘⢸⠀⢠⠁⠀⠀⢸⢸⠀⠀⠹⡄⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣶⣶⣷⣶⣾⣶⣶⣶⣷⣾⣶⣶⣾⣶⣿⣶⣶⣶⣾⣿⣶⣶⣶⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠫⣿⣟⢿⢻⣿⣿⣿⣿⣟⡽⡟⡟⣿⣟⡻⡟⣿⢿⣿⡟⢿⣿⡟⡟⣿⡏⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣟⣿⣿⣮⣸⣟⣯⣿⣿⣿⣧⣇⣼⣻⣏⣷⣇⣿⣮⢹⣷⣽⣯⣧⣬⣻⣧⣻⣿⣟⣾⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠛⠋⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⢻⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠠⢬⣉⠾⠇⠺⡿⣿⣿⣧⠕⠀⣀⠀⠀⢴⣾⣷⣮⡙⠲⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣄⡶⠂⠀⠀⠀⠀⠀⣀⣼⣿⣶⣶⣶⣤⣤⣄⡲⠢⣿⡶⢷⡆⠐⠚⠓⠈⠉⠘⠙⠛⠐⠂⢻⣿⠟⠛⠿⣿⣿⣿⣿⣿⡛⢃⣀⣀⠈⡉⢻⠛⠛⢻⣿⣿
⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⣿⣿⣿⡿⠿⣿⣿⣦⣴⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠉⠙⠃⠠⠤⠄⠀⠁⠀⠀⠠⠿⣿⣿
⠹⠟⠛⠿⠿⠷⠦⠤⠶⠦⠬⠁⠁⠐⠀⠉⠛⠏⠀⠀⠈⠙⠛⠓⠀⠀⢠⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣴⢦⣶⠾⠶⠿⠿⠿⠿⠞⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠤⠤⠤⠤⠤⠶⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣶⣤⣤⣤⣼⣿⣇⣻⣿⣿⣿⣿⣿⣿⢸⣿⣷⣴⣶⣶⡿⢟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣾⣿⠿⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣠⣶⣾⣿⠿⠿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠀⢹⣿⣇
⠀⠀⣠⣾⣿⠟⠛⠉⠀⠀⠀⠚⠃⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠿⠟⠋⠉⠀⠀⣠⣾⣿⠇
⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠾⠛⠛⠋⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⠟⠋
⣾⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⡶⠿⠛⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⡿⠟⠛⠋⠁
⣿⣿⣟⠛⠛⠛⠻⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠿⠋⠉⠉⢹⣿
⠈⠿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⡾⠿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠈⠉⠉⠙⠋⠉⠉⠉⠉⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠁⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⢀⣴⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣾⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⢀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠛⠋⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤
⣿⣿⣿⣿⣿⠿⠛⢿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠔⣋⣩⣭⣬⣭⣉⡙⠛⠿⣷⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡾⠃⣠⣶⡿⠟⠋⠉⠉⠉⠉⠙⠓⠄⠈⠻⣷⣽⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⣴⡿⢁⣴⣶⣿⣿⣿⣷⣶⣄⠙⢦⣄⠀⠈⢿⣧⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣿⣿⣿⢰⡿⢀⣾⣿⡿⠿⠿⣿⣿⣿⣿⣷⡈⢿⣷⡰⡌⣿⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡻⣿⣿⣿⠈⣇⠸⡿⠋⠀⠀⠀⠀⠹⣿⣿⣿⡇⢸⣿⣷⣷⠈⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣉⠻⠿⣧⡈⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡿⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣙⠛⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢀⣿⣿⡿⠁⣼⡿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⢀⣠⣤⣀⣀⠀⠀⠀⠀⢠⣿⣿⡿⢡⣿⡿⠋⢠⡾⠋⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠉⠛⠿⣦⡄⢀⣿⣿⠟⢰⣿⢏⠀⢀⡡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⡇⠀⢿⡟⠛⠓⠆⠀⠈⠀⢸⣿⡟⠀⣿⠃⡆⣠⡾⠁⠀⠸⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠃⠀⢼⣿⣦⣤⣤⠄⠀⠀⠸⣿⡇⣆⢻⡀⢷⣿⣇⢠⣀⠑⠒⢛⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠻⣇⢻⣦⡑⠈⢻⣿⣦⡈⠙⠛⢛⠋⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢿⣶⣄⡙⢿⡿⢷⣦⣬⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠀⢉⠻⢿⣷⣝⣦⡈⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⣄⡀⠀⠀⠀⢻⣷⣌⠻⣿⣷⠈⣿⣶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⣿⣿⣿⣿⠏⢀⣀⣠⣾⢿⣿⡇⣹⡟⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⡵⠿⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠾⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢀⣤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⢠⡄⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⡇⢠⡄⡏⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣤⢰⠃⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⠀⠀⠀⠀⠃⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⣿⠀⠀⣠⣄⠀⠀⠀⠀⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⡙⠒⠒⠶⠦⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⡾⢠⣟⠀⠀⢻⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⢇⠈⢌⢳⠀⠘⡆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⡆⢸⢀⡆⠀⢻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀⡏⠘⠛⠁⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢇⠹⡄⠀⠀⠀⢸⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠄⣇⠀⠀⠀⠈⢆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⡀⣇⠀⠀⠀⠀⢸⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢃⠸⡄⠀⠀⠀⠈⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡄⣷⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⣁⣹⡄⠀⣀⣀⣸⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣉⣽⡷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣃⢠⣉⣉⣙⣛⡛⠙⠋⢀⣩⡇⠀⠀⠀⠀⠀⢰⡟⣰⡟⠁⠀⠳⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣿⡆⢻⣉⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⣿⠈⡇
⠀⠀⠀⢀⡠⠐⠒⠈⠉⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⣽⠇⣨⣹⣿⣿⣿⣿⣿⣿⠉⠙⠛⠛⠓⠶⠶⢾⣧⣽⣇⠀⠀⢠⡿⣠⡇
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣷⣿⠿⠋
⠀⠘⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠙⠛⠛⠛⠛⠋⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠙⢿⣭⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣠⣤⣤⣤⣤⣤⣴⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣝⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡲⠄⠀⠀⢀⣤⡤⠤⠦⠤⠀⢿⣷⡀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠹⡷⠀⠀⠀⢰⠞⣿⡶⣦⡤⢸⣿⡇⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⡇⠀⠀⢸⡿⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣠⡿⢡⠃⣽⡗⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⣾⠟⠀⢀⡾⠋⠐⣡⣾⡿⠃⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⢠⣿⠋⢀⡴⢋⠔⣡⡾⠛⠁⢀⣠⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠁⠀⣼⡏⠀⡼⡡⢃⡾⠋⣀⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡧⢡⡟⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠘⣿⣿⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣯⣭⣽⣭⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⣤⣾⣆⠀⣰⣇⠐⢆⡀⣀⠀⣀⢦⣸⣷⣿⣿⣿⣿⡿⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢤⣽⠻⣿⣶
⠀⠀⣿⣿⢧⣀⠾⣹⣿⢨⣿⢻⣷⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⢰⣾⠐⢟⠊⣤⠀⠊⢻⡆
⠀⠀⣿⣿⣿⣿⣾⣿⣿⢁⡿⣮⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡀⠀⢸⡄⠀⠀⢤⠀⠀⢠⣿⠃⠀⠸⣄⠀⠸⡷⣌⡃
⠀⠀⣿⡟⡟⠍⣻⣿⣿⠜⣵⡿⣿⣿⣿⣿⣿⠟⠉⠀⢰⣿⡥⠀⠀⠇⠀⡰⠎⠀⢠⡿⠏⠀⠀⠀⠘⣊⠀⣧⢿⡃
⠀⠀⣿⡃⡠⠴⣛⡷⢿⣿⣿⠿⣦⢘⣿⣿⠏⠀⠀⠀⢈⣧⡀⠠⡄⢠⠀⠀⠀⣠⣿⠶⠖⢏⠀⠀⠀⢈⡏⠀⢾⡇
⠀⠸⡇⢱⣡⡶⠘⣇⣀⠙⠿⣒⠇⣸⣿⣿⣆⣀⣶⣶⣿⣿⣧⡴⠸⢾⡦⠐⠀⢠⠤⠂⣀⡀⠀⠀⢀⣈⡁
⠀⠀⡄⢿⣿⡅⠶⠿⠃⠙⢶⠏⠀⠘⣿⣿⣶⣤⣬⣅⣍⠙⠉⣷⣾⡀⠀⠤⠀⢠⣾⡦⣇⠀⠀⠐⠰⠷⠀⠄
⠀⠀⣴⣿⢱⡟⣠⡶⠒⠀⠀⠐⢹⠂⢙⣿⣿⣿⣿⡿⠃⠀⢘⣻⣿⣿⣶⣶⣾⡾⢿⣷⠿⢀⡤⠀⢶⣶⣄⣾⣏⡀
⠀⢠⣿⣯⣿⣄⣏⣱⢁⡴⠒⠂⣀⣀⠈⠽⠻⠟⠛⠀⠀⠀⡛⢻⣿⣿⣿⣿⣿⣿⣿⣧⡶⣿⠋⠀⠸⣿⣿⣿⣿⠇
⠀⢠⣴⣾⣿⣿⡿⠓⠀⡰⢶⣿⣿⣿⠘⠛⠷⠛⠋⠀⡀⠀⡅⢈⢙⠻⣿⣿⣿⣿⣿⣿⡖⠋⣀⡄⠀⣨⣿⣿⣿⣅
⠀⠘⣿⣿⣿⠿⣃⣤⣶⣾⣷⣾⡿⣿⠿⣿⠟⢰⣿⣿⡇⢀⠟⢮⡏⣰⣿⣿⣿⣿⣏⣶⣦⣼⡏⠛⢺⣿⣿⣿⣿⣟
⠀⢸⣿⣿⣿⣿⣿⡿⠿⠇⣭⣅⡐⢻⣰⣿⣶⣿⣶⣿⢛⠒⠀⠉⠙⣿⡿⠛⠉⠛⢹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⡿⣿⢯⣶⠧⣠⡏⠛⢷⠀⠨⣍⠀⢻⣿⠿⠲⣿⣃⣶⣿⣌⣿⣷⣦⡔⢘⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿
⠀⢸⣿⠛⠃⠚⠻⣿⠀⣿⣿⠆⠙⠑⣦⡀⠲⣾⣩⢀⠼⢿⣿⣿⠟⠻⠿⣿⡏⠃⢠⣿⣿⣿⡁⢸⣿⣿⣿⣿⣿⡟
⠀⠈⣿⣿⣶⠶⠀⢈⢳⡌⠉⢢⣀⣘⣉⣿⣆⠹⣟⣇⣠⣾⣿⣷⣶⡶⣤⠘⠇⣠⣿⣿⣿⣿⣧⣸⡗⢩⣯⢻⢿⣧
⠀⠀⣿⣿⣏⣤⡞⢿⣏⠠⣾⣿⣿⡿⣿⣿⣿⠦⠈⠉⢁⠄⠉⢙⣿⡿⠀⠀⢘⡛⠻⢿⣿⣿⣿⣏⣾⣿⣿⣶⣿⣯
⠀⠀⡿⣿⣿⣿⣋⠜⠻⠷⠌⠉⠻⠿⠤⠶⢿⡆⠖⢋⣀⣠⡶⠎⢁⣸⣶⢖⠚⣿⣀⠀⠀⠉⠙⠿⢿⣷⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠁⠀⠀⠀⠀⠈⠋⠃⠉⠙⠉⠘⠓⠄⠉⠁⠘⠓⠚⠛⠛⠛⠓⠖⠀⠀⠛⠋⠉⠙⠉⠁

⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠙⠿⢿⣿⣧⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠒⠀⠀⠄⠊⠹⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠲⠤⠀⠀⣠⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠰⢻⠉⠛⠒⢞⠋⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡠⠾⣿⡿⠋⠸⣟⣧⡀⣆⠀⠀⠈⠀⠀⣸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢣⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⡘⡠⠛⠦⣄⣀⣀⡙⠿⣿⠛⠗⠒⠃⠀⢠⣇⡤⠜⠋⠉⠉⠉⠉⠉⠒⡴⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠄⠂⠈⢢⡿⢴⡀⠀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠃⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⢸
⣷⠀⢀⠄⠀⢼⠘⢧⡀⠀⣥⠄⠀⣠⠞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⢳⡄⡀⠀⢸
⣿⠀⣾⠀⠀⠈⢿⡄⠙⠓⠲⠖⠛⠁⠀⠀⠀⠀⣸⠀⣠⡴⠖⠲⠦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣡⣀⣠⠤⠴⠚⠀⣷⠀⢸
⣿⠀⠘⢷⣄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣣⠞⠁⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⢸
⣿⠀⠀⠀⠙⠻⢦⣤⣈⠑⠒⠢⠴⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿
⣿⣿⣿⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣿⣟⣛⣿

⠀⢀⣠⣴⣶⡾⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⢰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⢿⣷
⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣀⡀⠀⣼⡟⠀⣿⡀
⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢻⣿⣿⣿⣿⣴⡟⠀⠀⢻⣿⡆
⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣴⠀⠀⢰⣿⡅
⠀⠀⠈⠙⠻⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣯⢿⣧⢆⣤⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⣝⠋⣴⣟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠑⠜⠳⠤⠴⢼⣧⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠉⠛⠶⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢈⣠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⢹⣋⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣾⡿⠁⠀⠀⠀⠀⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⣼⡿⠁⠀⠀⢀⣠⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠋⠁⠀⢀⣾⡟⠁⠀⢀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⣠⣿⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣼⡟⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡄⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡶⠶⠶⠾⠿⣯⡟⣤⠀⢠⣿⣿⣿⣧⠀⠀⠀⣴⠛⠛⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⣈⠿⣿⠀⠈⣿⣿⣿⣿⣆⠀⠀⠻⣦⡀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡿⣳⠛⢧⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⣀⣀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠎⠇⠀⠀⠀⠉⠛⠻⠿⠿⠛⠛⠋⠁