Olympic games Emojis & Symbols

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⡨⡄⢀⡶⠀⠀⠀⠀⠈⠀⢰⣄⡄⠀⠀⠀⠀⢦⣜⠀⠀⠀⠀⠀⣠⠾⣶⠒⠒⠀⠀⠀⠀⠀⢰⣷⡆⠀⠀⠀⠀⠀⠀⡼⢸⣇
⠀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⠆⠀⠀⠀⠔⠁⠙⢵⡄⠀⠀⠀⠈⠃⠀⠺⡆⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠠⠺⡣⡦⠁⠃
⠀⠐⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠰⣀⣀⡀⠀⠀⠀⠀⠠⣤⣶⡶⠀⠀⠀⠀⢀⣈⢶⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠰⣤⣤⣄⠀⠀⠀⠀⠀⣀⣠⣶⡄
⠀⠀⠀⡹⢫⡞⢧⠀⠀⠀⠀⠞⠙⠀⠀⠀⠀⠀⠀⠀⠴⠛⠦⢀⠀⠀⠀⠀⣠⢴⣶⢻⡟⠀⠀⠀⠀⣯⠰⡋⢣⡄⠀⠀⠙⠖⠋⣩⣺
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠡⠔⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⡀⣄⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠
⠀⠀⠀⠟⢾⡇⠀⠀⠀⠀⣀⠞⠻⠛⣳⡀⠀⠀⠀⠈⠉⠹⠏⠁⠀⠀⠀⠀⠈⠛⡊⠙⠓⠀⠀⠀⣠⠚⣄⡐⢄⠀⠀⠀⠐⠚⠿⢷⠘⠁
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃⠀⠀⠁⠳⠀⠀⠀⢀⡴⠃⠘⡆⠀⠀⠀⠀⠙⠷⠝⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⠋⠁⣰⢧⣀⡀⠀⠀⢴⣄⣠⠤⣦⣄⠀⠀⠀⢘⣚⢧⡄⠀⠀⠀⠀⠤⠴⠿⣦⠔⠀⠀⠀⠀⠀⠻⢾⣦⠀⠀⠀⠀⠀⠀⠀⣷⡅
⠀⠘⠷⣾⠁⣉⡿⠁⠀⠀⠉⠉⠁⠘⠉⠉⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠃⠉⢩⠊⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠈⢡⣾
⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠋⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣿⣟⣻⡾⢿⠷⡿⠾⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⣠⣾⠟⠛⠋⠙⠛⠻⣷⣄⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠙⠛⠿⣶⣄⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠛⠛⢿⣦⡀
⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣀⣚⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣾⡇⠙⣻⡻⢿⣦⡀⠀⠀⠀⠀⣠⣾⠟⣿⠏⠙⣿⡻⢷⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣦⡀⠀⠀⠀⠀⣰⡿⠁⣴⡿⠁⠀⠘⣿⣄⠙⣿⡆⠀⠀⣼⡟⢁⣼⡟⠀⠀⠹⣷⣄⠹⣷⡄⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠙⢿⣦⣤⣄⣠⣿⢳⡿⠋⠀⠀⠀⠀⠈⠻⢷⣮⣿⣄⢸⣿⣴⠿⠋⠀⠀⠀⠀⠈⠻⣷⣽⣧⣀⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣤⣤⣀⣤⣴⣾⠟⠁⠀⠀⠀⠀⠙⢿⣶⣤⣄⣀⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⣶⡶⠶⠶⠒⣚⣻⣿⣯⡨⠙⣀⣀⢀⡀⠠⠀⣀⣾⣷⣠⣦⣀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠈⣿⣷⠓⢂⣫⢶⣛⣽⣿⣿⡿⣧⣤⣤⣀⢘⣻⣿⣿⣾⣷⣶⣿⣿⣿⣿⣿⣇⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢲⡄⢹⣿⠟⣯⡿⠿⠟⢿⡏⣿⢿⢿⠿⠯⢿⠿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢻⣆⣀⡀⠀⣃⣼⣿⣀⣀⣤⣄⣤⣤⣤⣠⣦⣤⡄⢠⣀⠀⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣬⡿⠋⢻⣾⣿⣿⣿⣅⣿⡇⣿⣿⢸⣿⡇⠸⣿⡇⢸⣿⣰⣿⣿⡿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣟⡛⠛⠋⠛⠒⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠰⢆⣨⣿⣿⣿⣟⠋⢻⣧⣿⡿⢸⣿⡁⠀⣿⠆⣾⣿⠻⣿⣿⣷⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠁⠉⢹⣿⡿⡷
⠀⠀⠀⠀⠀⠀⠠⢼⡌⠈⣻⣿⣍⢿⠃⠡⣼⣿⣿⣿⣼⣿⣥⣾⣿⣿⣿⣿⡀⠘⠦⢉⣟⣤⣱⣽⣿⣿⣿⣿⢻⣿⢿⣷⣦⡀⢸⣿⣷⠆⣀
⠀⠀⠀⠀⠀⠀⠀⣿⡒⢸⡟⢁⣿⣯⣵⣬⣯⣽⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⡀⠀⠀⠘⠋⠙⣿⡜⠙⣂⣈⠛⠝⠓⡸⠿⠋⠝⠋⠋⠉⠛⠈
⠀⠀⠀⠀⠀⠀⠈⣬⡁⠺⠟⡊⠉⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⠲⠀⣬⠚⢰⠉⣿⢿⡏⢁⠀⢀⠘⣿⣦⣄⠙⢲⠂⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡤⣝⢻⠫⣀⣹⣿⣿⡟⡸⡽⢟⣻⠿⣿⣿⣿⣿⣿⣿⠋⡅⢱⡦⠿⠿⠀⣿⡿⣿⡸⠄⠚⠆⣸⣿⠿⣗
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡧⡜⠁⠉⡿⣿⠾⠷⠾⠶⢤⢠⣾⣿⣿⣿⣿⣧⣤⢀⠀⠀⢓⡲⢾⠛⠛⠛⠟⢿⡟⠛⠉⠉⣑⣬⢿⠤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠂⢤⣀⣽⣿⣇⣂⣠⣄⣻⣤⣿⣿⣿⣿⣿⣿⠿⠀⠄⠀⣠⢠⠀⣀⠀⣀⣴⣧⠄⣀⣤⣶⡏⠻⣿⢶⣶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⠛⡀⠿⡈⠛⠛⠉⠻⢏⣉⠉⠉⠙⠀⣼⢋⠉⠀⠙⠓⠲⢤⢄⡅⠈⠀⢿⣍⣉⢿⡣⣠⣿⣿⣿⣥⣤⣄⠠⣈⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡄⣲⡀⢳⣄⡀⣠⣤⣤⣀⣶⣿⣷⣆⠈⣧⣤⣤⣌⣤⣤⣀⡈⣷⢢⣄⣤⡟⣿⣿⣿⣷⡈⢿⣮⣿⣿⣿⣿⣿⡟⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠿⢹⣿⣿⣿⣟⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⠀⠀⠀⠈⣉⢉⣉⠋⠅⠈⠁⠁⠐⠃⠚⠤⢹⡿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣿⠛⠿⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠻⠷⣧⣿⣿⣾⣿⣷⣄⢀⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠛⠛⠛⠛⠛⠒⠐⠐⠒⠐⠀⠀⠀⠐⠒⠒⠂⠀⠂⠀⠀⠀⠀⠛

⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡆⣸⠃⠀⠀⠀⠀⡤⢤⣠⠘⠀⠀⠀⠀⠀⢠⠖⠀⠂
⠀⠀⠤⡄⠹⠋⠁⠀⠀⠀⠀⠀⢠⠲⣌⡗⠢⠬⠀⠀⠀⠀⢠⡇⢚⠈⠃⠀⠀⠀⠀⠀⠀⣴⡄⠘⠶
⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠠⠏⠠⠇⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠔⠀⠀⠀⠀⠼⠙⠁

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡀
⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠠⣆⡤⠖⠀⠀⠀⠀⠀⡘⢃⡀⠀⠀⠀⠀⠀⠀⢠⣁⢞⡴⠋
⠀⠀⠠⠚⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⡉⢥⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⠒⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠓⠀⠀⠀⠙⠂⠀⠀⠀⠀⠰⠚⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢀⢀⣴⠀⠀⠀⠀⠀⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠄
⠀⠀⠀⠀⠈⠏⡾⠀⠀⠀⠀⠀⠀⠈⣙⠲⢤⡀⠀⠀⠀⠀⢬⡳⣄⣙⠆⠶⠀⠀⠀⠀⠀⢶⠳⠿⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡞⡩⠃⠀⠀⠀⠀⠀⠀⣠⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢉⡴⠃
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣘⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡨⡄⠀⠀⠀⠀⠀⠀⢆⠀⣀⠀⠀⠀⠀⠀⠀⠐⠚⠳⡴⠂
⠀⠀⠒⡦⠈⠉⡀⢀⠀⠀⠀⠀⠀⣴⡦⠙⠲⠄⠀⠀⠀⠀⠀⠈⠙⠻⠦⠄⠀⠀⠀⠀⠀⡖⣦
⠀⠀⠚⠁⠀⠀⡽⢪⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠃⠹


⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢡⠖⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠚⣙⠐⣂⣀
⠀⠀⠀⠀⠰⢾⡶⠀⠀⠀⠀⠀⣠⠞⠲⡈⠉⠉⠀⠀⠀⠘⡽⣯⠀⠙⡽⣯⠁⠀⠀⠀⢠⢇⠽⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣔⠢⣄⠀⠀⠀⠀⠀⠀⠤⣤⣴⣆⠀⠀⠀⠀⠀⠀⠰⡤⠖⠳⠀⠀⠀⠀⠀⠀⢀⢐⢆⡀
⠀⠀⠀⠀⠮⡟⠊⠀⠀⠀⠀⠀⠀⣠⣶⠄⠬⠃⠀⠀⠀⠀⠀⢀⣈⠓⠂⠀⠀⠀⠀⠀⠠⠬⡯⠠⣧⢤⡀
⠀⠀⢠⠟⢠⠏⠀⠀⠀⠀⠀⠀⢠⠞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢧⡀⠀⠀⠀⠀⠰⠃⠀⡟⠐⣆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⡀⠀⡤⢀⠄
⠀⠀⠑⡄⠀⠠⠢⡀⠀⠀⠡⡀⢈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⢄⠔⠀⠀⢀⠎⠴⠆⡠⠃
⠀⠀⠀⠀⡄⠀⠀⡘⣄⠀⠀⠑⡀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠀⢀⠊⠀⠀⡰⠁⠔⢀⠌
⠀⠀⠀⠀⠈⢄⠠⠀⠉⢄⠀⠀⠐⠄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠄⡠⠁⠀⢀⠜⠀⠊⡰⠁
⠀⠀⠀⠀⠀⠀⢂⠀⢀⠈⢆⠀⠀⠈⢆⠂⡌⢄⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠌⡒⠁⠀⡠⠂⠄⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡈⢃⠀⠀⠀⠡⠀⢈⢆⠀⠀⠀⠀⢠⠊⠀⢀⢠⠊⠀⠀⠔⠁⠂⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠡⡄⠀⠀⠐⠄⠀⢡⠀⠀⡐⠁⠀⠀⡰⠁⠀⢀⠈⠀⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠂⠘⢆⠀⠀⠈⠂⠠⢱⠊⠀⠐⣀⠜⠀⠀⡐⠁⠀⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠀⠡⠣⡀⠀⢀⡵⠁⠀⠀⡠⠋⠀⢀⠌⠀⠂⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⣬⠎⠀⠀⢀⠜⠁⠀⡠⠁⠀⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠀⡴⠃⡀⠀⡡⠂⠀⢀⠌⠀⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠶⠴⠴⠤⡼⡥⠤⠤⠮⠴⠤⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠲⠺⣥⣤⣤⠽⠶⠤⠤⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⠖⠶⠭⠩⠽⠶⢦⢤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡝⡧⡶⠀⠀⠀⠀⠀⠀⣀⢈⣽⣗⡽⢶⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⢳⣼⠟⣧⠷⠀⢀⠤⠤⠤⣄⢈⡖⣻⣿⣜⡦⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡝⣈⢡⣝⢧⡝⠁⠀⠈⢀⡀⠀⠀⣧⢀⡩⠙⠶⣛⣷⡿⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠠⣴⠗⣻⠀⠀⡄⠀⠀⡅⠀⠀⢹⠌⠁⡅⠀⢩⣽⣧⠿⡌⣧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⡣⣼⣯⠾⠀⠀⠀⠀⠀⠁⠀⢠⣿⠀⠈⠐⠀⢠⠶⣾⢼⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡆⢡⢼⡦⣷⡀⠠⠤⠀⠀⢀⢔⠞⣝⠀⠀⡔⢈⣟⣿⡷⣿⠧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⡌⣶⣻⣏⡦⠀⠀⠀⠀⢿⠖⠖⠟⠀⠈⣠⡛⣼⣟⣾⠏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠢⡡⢿⣳⣬⣍⡄⠦⠁⠌⠈⠠⡄⢳⣬⠷⢿⡽⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠥⣛⢉⡨⣽⡿⠷⠲⢾⡞⣿⣯⡿⠛⣣⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠷⠶⢥⣤⢤⣬⣤⢴⠶⠟⠋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡠⠀⢀⠀⡀⡀⢀⠀⡀⢀⢀⢄⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠙⠻⢶⣄⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⣟⠛⠛⡷⢶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⠶⡟⠛⢛⡷⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⣸⠏⠀⠀⢿⡄⠈⠻⣆⠀⠀⠀⢀⡾⠋⠀⣼⠇⠀⠈⢿⡀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⣾⠁⣠⡾⠋⠀⠀⠀⠈⠻⣦⣀⢹⣆⠀⢀⣾⢁⣠⡾⠋⠀⠀⠀⠈⠻⣦⡀⢻⡆⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⢾⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣿⠶⢾⡇⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁







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

⠀⠀⠀⣀⠴⠒⣛⣉⣉⣙⡓⠦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠶⢚⣋⣉⣉⣛⠲⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣉⣉⣓⠲⢄⡀
⠀⢠⠞⣡⠞⠉⠁⠀⠀⠀⠉⠓⢦⡙⢦⠀⠀⠀⡴⢋⡴⠚⠉⠀⠀⠀⠈⠙⠲⣌⠳⡄⠀⠀⢠⠞⣡⠖⠋⠁⠀⠀⠀⠉⠙⢦⡙⢦
⢰⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⠀⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡄⢠⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢳
⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣾⣃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⣾⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢸⢠⣧⡿⠧⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⡇⣸⣼⠿⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢯⠏⡼⠉⢭⣷⡲⣌⠳⣄⠀⠀⠀⢀⡴⢋⡼⡽⢡⠏⠩⣽⣖⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⣸
⠀⠱⣌⠳⣄⠀⠀⠀⠀⡼⢡⢎⡴⢋⡼⠁⠀⠈⢧⡙⢦⡳⢬⣦⠀⢠⠏⡴⣣⠞⣡⠏⠀⠀⠹⣌⠳⣌⡦⣽⡄⠀⠀⠀⢀⡴⢋⡴⠁
⠀⠀⠈⠳⢬⣙⡓⣲⢸⢡⣯⣯⠖⠋⠀⠀⠀⠀⠀⠙⠢⣍⣙⠒⡆⡏⣸⣽⡥⠚⠁⠀⠀⠀⠀⠈⠓⢦⣉⡓⠒⠒⠒⣛⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠉⠉⢿⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣵⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠇⢧⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⠞⠀⠘⢦⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⡒⠦⠤⣤⠤⠴⢚⣩⠔⠃⠀⠀⠀⠀⠑⢬⣙⠲⠤⢤⡤⠤⠔⣊⡤⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⢀⣴⠞⠉⠉⠀⠉⠉⠛⢦⡀⠀⠀⠀⣠⠾⠋⠉⠁⠈⠉⠙⠷⣄⠀⠀⠀⢀⡴⠟⠉⠉⠀⠉⠉⠳⣦⡀
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡴⠞⢫⡟⠉⣯⠛⠶⣄⠀⠀⠀⠀⣠⠶⠛⣿⠉⢹⡝⠳⢦⡀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠻⣄⠀⠀⠀⠀⢠⡟⠀⣠⠞⠁⠀⠘⢧⡀⠘⢷⠀⠀⡾⠃⢀⡼⠃⠀⠈⠳⣄⠀⢻⡄⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠛⠶⢤⣤⣿⠶⠚⠁⠀⠀⠀⠀⠀⠙⠳⢾⣧⣼⡷⠞⠋⠀⠀⠀⠀⠀⠈⠓⠶⣿⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣤⣤⠴⠞⠁⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⡤⠾⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣿⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡛⠐⣿⣾⣿⣿⣤⠎⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣶
⣿⠁⢩⣿⠋⠙⣿⡅⢹⣿⡄⣀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡶⠋⠀⠀⠠⣾⡿
⣉⣤⣬⣋⣁⣠⣦⣈⣛⣿⣿⣷⣶⣾⣾⣽⢿⣬⣸⣤⣤⣀⣰⢶⡄⢠⣶⡀⠈⠀⠀⢸⡀⣀⢶⡛⠀⠿⠁
⣿⣿⣿⣿⣿⠿⠿⠿⡿⢍⠿⡿⡮⡛⠿⣻⡲⡟⠛⠛⠻⠿⢿⡶⠟⠻⢿⣿⣦⢉⣿⣷⣯⠶⠒⢧⣴⣿⡋
⣹⡿⣿⣶⡇⣶⡂⠀⠀⠸⣤⡦⡄⠈⠙⢾⡀⣀⣶⣤⣀⣀⣿⣅⣠⣷⣼⣿⢿⠃⠀⠁⠆⠀⠰⣴⡄⢻⠛
⢙⡗⢻⡏⣀⣨⣽⠀⠀⠀⠀⠛⠋⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠟⢙⠟⢿⣿⣧⠛⠚⠁⠀⢀⣰⣿⣷⣏⢢
⣸⣧⠘⢻⣿⣿⣿⣇⠀⠀⠆⠀⠀⠀⠀⠀⠀⢘⣧⠭⠟⠻⢿⣦⣿⣿⠛⠟⠁⠀⠀⠀⢠⣾⢿⣏⢷⣿⠻
⠙⠚⠿⠿⠗⠚⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⢀⣀⣉⣸⡿⠒⠀⠒⠀⠒⠶⢎⡻⠿⠿⠂⠹⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣿⣤⣤⣶⣿⡏⠿⠁⠀⣠⣤⡄⢸⡄⠖⠃⠀⣿⡄⠀⣿
⠀⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣿⣿⣿⡿⠟⠃⡽⠋⠈⠉⠉⠀⣀⣽⣿⠿⠿⢄⣥⣶⣧⣄⣿⣿⠀⠹
⣴⠀⠂⠘⠛⠉⠉⠉⠉⠉⠁⠉⠉⠙⠛⠛⠛⠛⠃⠀⠀⢀⣤⣤⣾⣿⣿⣿⣾⣿⣾⡟⠉⣨⣿⡾⠟⠂
⠈⠀⠐⠳⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⠉⠁
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣖⡆⠀⢀⠤⡀⠀⠀⠀⢠⢠⠀⠀⢨⠀⡄⡀⢸⡆
⠀⠀⠀⠀⣴⠂⠵⢆⡸⢾⣇⡎⠀⠠⢀⠀⣤⠀⢘⠕⢀⡛⠙⡇⢠⡋⢀⡀⡄⣤⡁⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠻⢤⢀⠀⠃⠀⢸⣇⠄⠰⣈⣈⡫⠀⠘⡄⣬⣀⡀⠘⠀⠲⡕⠟⠰⠩⢁⡅⡈⢀⣈⠠⢤⠄⢀⠂⡈⠃⢀⢀⡠⡄
⠀⠀⠀⠀⠂⠚⢠⣶⠳⡔⡈⠀⣤⠀⠈⠁⢤⠀⢀⠰⢶⠖⠀⠲⠍⢀⠠⠈⢘⣲⣞⠀⡁⠘⣘⣶⠧⢉⢘⡒⡡⢹⣼⡷⠠
⠀⠀⠀⠀⠀⣤⢸⡇⠀⣥⠀⠀⡯⣠⢠⡅⠾⠃⠴⠒⠛⠀⠲⠆⠀⠸⠧⡄⠴⣧⣑⣨⢠⠀⠃⠀⠈⠁⠈⠉⢿⠛⠹⡇
⠀⠀⠀⠀⠀⠉⠙⠿⠶⢦⡠⠴⠟⡻⣿⠾⠀⠠⡤⠤⡤⠀⠰⣶⣤⠀⡖⠃⠀⠀⠼⣿⣆⠀⠲⠙⢿⠆⠀⠁
⠀⠀⠀⠘⢷⣶⣾⡗⠐⠂⠀⠍⡁⡗⢀⠘⠀⢸⠃⠀⠀⠸⠾⠭⠉⠀⠏⢁⡛⠁⠉⢁⠌⠀⡀⢝⠂⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠈⠠⢁⣀⡈⠀⠀⠀⠀⠀⠇⢠⡀⠀⠈⠑⠀⡐⠋⠚⠆⣈⣹⣅⠄⠀⢀
⠀⠀⠀⠀⠀⡏⡃⠀⠀⠀⠀⠀⠤⠀⢰⣿⣿⣽⣇⡀⣀⠀⠀⣀⠌⢻⣄⠈⠁⠀⠀⠀⠀⠀⠂⡈⣍⢹⣷⠈⢡
⠀⠀⠀⠀⠀⠇⢃⠀⠀⠀⠀⠀⠀⢀⠠⠙⠉⠃⠀⠀⠉⠀⠀⠀⠀⠚⠃⠰⡞⠁⠀⠀⠠⠀⠀⠀⠀⠉⠛⢷⠈⠁⠘⢁
⠀⠀⠀⠰⠆⣠⡈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⢀⡀⠐⠢⠤⣄⡀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⢀⣶⣇⠀⠒⠃
⠀⠀⡜⠉⠉⠂⢀⠴⠋⠁⠈⣆⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠶⠶⠊⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⣶⠋⢸⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠐⠁⢤⣤⣤⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣝⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠻⣄⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⣤⣀⡓⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠉⣻⡧
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠶⠶⣤⣀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⢿⣿⠀⠀⠐⣄⠀⠀⠀⢀⣠⠄⠀⠀⠚⠁⣴⢞⡫⠅
⠀⠀⠀⠀⠠⢤⡿⠛⣿⡿⠒⠒⠒⠈⠷⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⣹⠀⠀⠀⡞⠀⢀⠀⠀⠀⠀⠃⡞⠀⢀
⠀⠀⠀⠀⢀⠏⠀⢰⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣸⣿⠀⠀⣠⡿⠀⠀⠀⢧⡀⠈⠀⠀⠀⠀⠀⢧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣤⣿⣯⣤⣾⠟⠁⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⢹⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠠⠴⢃
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⠟⠁⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠠⡤⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⢈⠆
⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⣿⣿⢀⣠⡴⠋
⠀⠀⠀⠀⢀⠞⣿⣇⢀⣀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⣾⠟⢿⡏⠁
⠀⠀⠀⠀⠁⠀⠘⠿⢿⣿⣿⠿⠿⠿⠶⠒⠶⠦⠤⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠉⠙⠋⠁⠉⠛⠉⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠠⠤⠄⢀⠀⠈⠐⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡄⠊⠀⢀⡀⠤⠤⠄⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠁⠀⠀⠀⡠⠘⢂⣉⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣈⡁⠚⠡⡀⠀⠀⠀⠈⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠦⣀⠘⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⢀⠀⡇⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠀⠀⠤⠤⠤⠀⠀⠀⠐⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠔⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠄⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠄⢊⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⡐⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠢⢐⣨⣴⣶⡏⠀⠀⠀⠀⠀⠁⠒⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡎⠀⠀⠀⠀⠀⠀⠂⠉⣦⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⠀⣀⣠⠞⠈⡿⠿⠟⠛⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠏⣠⠏⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣶⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⣀⣴⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠀⠀⣈⠀⠀⣿⣿⣿⣷⣄⡠⢀⣠⡤⠾⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠂⠶⠍⡀⢠⠀⠄⠆⠐⠀⡄⣀⣿⣿⣿⣿⡿⠁⠈⡄⠀⠀⠈⠻⣿⣿⠿⠛⠉⠀⠙⠶⣄⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠂⠈⠉⠀⠀⠄⢈⠁⠁⠀⠀⠀⠀⠀⠀⠈⠐⢈⡡⢞⠟⠁⠀⠀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠐⠒⠀⠀⠲⠖⠚⣋⠥⠖⠁⠀⠀⠂⠀⠉⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡠⠀⠀⢰⣤⣴⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢡⡀⠀⠀⠀⠀⠀⣀⠀⢠⠥⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⠸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠒⠀⠉⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠙⠀⠀⠀⠀⠀⠈⠉⠉⠉



⣶⣶⣶⣶⣶⡿⣾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⢷⡶⠾⠿⠶⢶⣾⡿⠶⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠲⢶⣿⣄⠀⠀⠀⠀⠈⢓⣶⣖⠀⡀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠤⣾⡧⠤⠀⠀⠀⣿⡶⠀⠀⠀⠀⠠⠌⠓⢶⣿⠀⠀⠀⣾⡧
⠀⠀⠀⣾⣯⠀⠉⠀⠀⠀⢀⣼⣿⠉⠉⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠠⣿⡁
⠀⢀⡴⠃⠈⢳⡀⠀⠀⣠⠋⠀⠈⠳⣄⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢰⠇⠘⠇⠀⠀⠀⢰⠋⢿⡀⠀⠀⠀⠀⢻⠏⠁⠀⠀⣀⡿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠈⠀⠈⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⢈⢀
⠀⠤⠤⢤⣾⣇⡀⠀⠀⢠⣿⠖⠒⠂⢀⡴⣿⡶⠒⠲⠀⠄⠴⣾⡶⠊⠁⠀⣼⣾⡉⠀⠀⠀⠲⣶⣿⡄⠀⠀⣾⣿⡋⠁⠐⠳⣾⡦⡤
⠀⠀⠀⣼⣏⠀⠉⠀⠀⣹⣿⣄⠀⠀⠈⠀⣹⡟⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⣹⣿⡇⠀⠀⠀⢸⣿⠋⠉⠀⠀⢰⣿⠁⠀⠀⠘⣿⡇
⠀⠀⡠⠞⠘⡇⠀⠀⢀⡏⠀⠸⡂⠀⠀⠀⠙⡦⠀⠀⠀⠀⢐⠃⡇⠀⠀⢀⡏⠈⠻⡄⠀⠀⡟⡿⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠏⡇
⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⢄⠀⠁⠀⠀⠀⠀⠁⢀⣀⣀⣀⠈⠁⠁⣀⠀⠈⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠈⠓⢲⣾⡿⡆⠀⢰⡿⠻⣫⠋⠀⢀⣼⡃⠀⢢⣠⡶⡶⣶⠆⠀⠀⠀⡄⣀⠤⠀⠀⣐⠄⠀⠀⠐⢤⣤⣤
⠀⠀⠹⣤⡏⣀⠀⣀⣠⠀⠀⠀⢸⠃⠀⠁⠀⠀⠀⠀⠋⠀⠐⣿⣻⡃⠀⠘⠟⠉⠻⠫⠀⠀⠀⢀⡿⠋⠀⠀⠀⢻⣧⡀⢸⠛⣿⠉⠈
⠀⠀⠀⠙⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⠻⡇⠀⠀⠀⠀⠀⡟⡇⠐⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣤⡶⠀⠀⠀⠀⣠⣆⠀⠼⠀⠙⠢⠀⢠⡄⠀⠀⢻
⠀⠀⢀⣶⡟⢀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⡠⢾⣿⠢⠀⠀⠀⠈⢼⣿⣭⠂⠀⠀⢠⣿⠿⢦⡄⠀⠀⠀⢰⣿⠃⠀⠀⠈⠀⠐⢺⣿⠶
⠠⢄⣸⡟⠛⠳⣄⡄⠀⢊⣿⠀⠀⠀⠀⢀⣑⣋⢿⡄⠀⠀⠀⠶⠒⠼⢧⠄⠀⠐⠛⠁⠀⠀⠀⠀⠀⠉⠛⢫⠭⠖⠞⠀⠤⢴⡿⠍⠄
⠀⠀⠘⠛⠳⠆⠀⠀⢀⡟⠟⠉⠁⠀⠀⢀⠀⠀⠠⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠁
⠀⢀⡀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠀⠀⠀⠈⠃⠒⠒⠉⠀⠀⠢⠧⠖⠚⠉⠀⠀⠐⠒⠒⠀⠉⠁⠀⠀⠠⠄⠺⠒⠂⠀⠀⠐⠀⠀⠀⠂
⠀⣧⡄⠀⠀⠀⡜⣠⡀⠀⢀⡄⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⢰⣿⠳⡀⠀⠀⠘⣿⡖⠂⠀⢹⣷⠃⠀⠀⡼⣿⡿⠀⠀⣰⣿⡀⠀⠀⢿⣿⠓⠄⠀⠀⠀⢀⣀⣀⠀⢻⠿⣶⡄⠀⠀⠀⠀⣠⣄⡀
⢸⡏⠷⠈⠁⣀⠼⣿⠇⠀⠀⢸⡿⢷⠀⠀⠀⣿⣿⠀⠈⣰⢿⡆⠀⠀⣾⢿⡈⠀⢀⢀⣾⣟⡿⠉⠀⠈⠀⢾⣧⡀⠀⠀⠀⣯⣿⣿⡀
⠈⡇⠀⠀⠀⠈⠀⠙⠋⠁⠀⠛⠃⢸⡀⠀⠀⢹⠁⠀⠀⣻⠈⠀⠀⠀⠈⠂⠙⣄⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠀⠀⠋⠉
⠀⠈⣀⣴⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠀⠀⠀⠀⡄⠀⠀⠀⣤⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠎⣽⣿⠂⠀⠀⡴⣿⠾⠀⠀⠀⢴⣿⡆⠀⠀⣾⣦⠀⠀⡔⣿⣷⠀⠀⠀⣿⠄⠀⠀⣾⡿⡆⠀⢞⣿⠦⠀⠀⢶⣾⡇⠀⠡⣿⡶⠄
⠀⢠⣿⠃⠀⠀⠀⣹⠟⢶⡄⠀⠀⢸⢿⡆⠀⠀⡿⣧⠀⠀⢸⡿⠀⠀⠀⢰⡿⣄⠀⠀⣿⡟⠁⠀⠀⣿⠁⠀⠀⢸⣿⡁⠀⡴⠾⣷⠃
⠀⠰⠁⠀⠠⠒⠉⠁⠀⠀⠦⠀⠀⠉⠘⡃⠀⠀⢱⠹⠀⠀⠸⠀⠀⠀⣠⠻⠞⠁⠀⠀⠉⡇⠀⠀⠰⠹⡀⠰⠚⠉⠀⡃⠀⡵⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⢤⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⡀⢀⣤⡀⢀⡀⠀⣠⣄
⠀⠀⠈⠓⢶⣿⣦⣀⣠⠄⠶⣤⣀⣻⣗⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢻⣷⢤⣤⡀⠀⠀⠀⠀⢨⣿⣿⣭⡾⠋⠉⠙⢷⣭⣿⣿⣏
⠀⠀⠀⠀⣸⣿⣇⡀⠀⠀⠀⠀⠀⠉⣿⣯⠉⠙⠀⠀⠀⠀⠀⠰⣿⡉⠙⠛⠋⢹⣿⠆⠀⠀⠀⠉⠉⠙⡿⠀⠀⠀⠀⠀⢻⡏⠉⠉
⠀⠀⠀⢸⡏⠀⠘⣷⡀⠀⠀⠀⠀⣿⠉⠙⠷⣤⡀⠀⠀⠀⠀⣼⠏⣿⠀⠀⢠⡟⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣈⠛⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣦⠀⠀⢲
⠀⢸⡗⠂⢲⣦⣿⣷⡆⠒⠺⡇⠀⠀⠀⠀⠀⣠⣶⣞⠛⠀⠀⠀⠈⠓⠦⣽⣏⣠⡴⠃⠀⠤⣤⣤⣼⡛⠀⣠⡀⠀⢘⠋⠺⣷⣼
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡼⣿⣥⡌⣿⣂⡀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣾⣿⢛⣒⣋⡀⠀⠈⠛⠋⣿⣷
⠀⠀⠀⠀⠀⣸⠏⣧⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⠾⢿⠋⠇⠙⡆⠀⢀⡾⠋⣷⠀⠀⠀⠀⠀⢠⡟⠻⣆⠀⠀⠀⠀⠀⠀⢀⣼⠛⣷
⠀⠀⠀⠀⠀⠻⠀⠿⠀⠀⠀⠀⠀⠘⠻⠿⠋⠈⠀⠈⠳⠤⠞⠁⠀⠀⠿⠀⠈⠳⠀⠀⠀⠰⠏⠀⠀⠟⠀⠀⠀⠀⠀⠰⠟⠁⠀⠟

⠀⠀⠀⠀⢀⣄⠀⢂⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⣤⡄
⠀⠀⠠⢶⣾⣿⣤⣼⠀⠀⠀⣤⣿⣷⣶⡶⠀⠀⠀⢀⣼⣯⣤⡤⠖⠒⠀⠀⢀⣀⡀⣶⡄⠀⠀⠀⠀⢿⠇⢀⠀⠀⢠⡴⣶⡷⠖⠋
⠀⠀⠀⠀⣿⣿⠀⠈⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⡟⣿⣿⠀⠀⠀⠀⠀⠸⢿⣼⡿⣌⠀⠀⠀⠀⣾⡟⠦⡜⠃⠀⠀⠀⢻⡷⣦
⠀⠀⠀⢠⡟⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠁⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⣦⠈⠁⠀⠀⠈⣿⣶⣄⠀⠀⠀⠀⢀⡾⠃⠟
⠀⠀⠀⠸⠇⠹⠆⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠴⠟⠀⠹⠆⠀⠀⠀⠰⠟⠀⠹⠆⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠠⠁⢴⡆⠀⠈⠀⠀⠀⠀⠰⣶⣠⡤⠶⣢⡾⠃⠀⠀⠀⠀⢀⣺⣟⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠘⠲⣿⠚⢁⡞⠀⠀⠀⠀⠀⣼⠿⣿⣾⣋⠀⠀⠀⠀⠘⠋⠉⣿⣿⠉⠉⠃⠀⠀⣴⠞⠛⣿⠁⠀⠀⠀⠀⠀⠀⠓⢾⡟⠋⣰⠇
⠀⠀⠀⢀⡼⠛⠋⠀⠀⠀⠀⠀⠀⣟⠀⠀⠉⢉⡛⠛⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠁⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠁
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠻⡟⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣤⣤⣬⣭⣿⣤⣤⣤⣄⠀⠀⠀⢰⡏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠑⠦⣴⡞⣳⠄
⠀⠀⢿⠽⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣦⣤⣾⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣀⣈⠙⠛⠻⢿⣿⣿⣿⢶⣶⣶⡲⣤⡢⣤⣀⠀⠈⠻⣿⢹⣿⡟⠻⢌⣛⣻⣿
⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣭⣗⣄⡀⠀⠻⠿⣿⣕⡮⠿⣿⣿⣿⣿⣿⣷⣦⣴⠿⣾⡟⠋⠀⠀⠛⣿⡇
⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣳⣦⣤⣀⠙⠿⣷⣗⡻⡛⠻⢿⣿⣏⠁⠀⠈⠉⠓⠦⡴⠞⠉
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣝⣷⣦⣄⣹⣿⣿⣷⣤⣉⠙⠻⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⡿⠿⠭⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⢿⣿⣻⣿⣿⠟⠁⠀⠈⠙⠻⢦⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⡀⠀⠀⠀⢀⣤⢖⣫⣙⠛⢿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣶⣿⣷⣾⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡉⠛⠷⣦⣤⡤⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣷⠙⣿⠧⠖⠛⠁⠀⠀⠀⠀⠈⠀⢨⠁⠂⠀⡴⠀⠀⠀⠰⠋⠀⠀⠀⠸⣧⠀⢠⠼⠉⠙⠃⠀⠸⠁⠀⠀⠀⣁
⣟⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠤⠻
⠋⠁⠁⠀⢀⣤⣀⢠⣤⣀⣀⡀⠀⠀⠀⢀⠀⠨⠔⠀⠀⠀⣟⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⢄⣸⣀
⢣⣈⡀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠃
⠸⢿⡏⠀⢹⣿⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣾⡟⠁
⠠⣶⠃⠀⠈⣯⣿⣿⣿⡿⡟⠀⠠⠎⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⠃⠀⠀⠀⢸⣽⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣶⣦⣄⠀⣼⣿⣿⣿⡆⠈⠀⠀⠀⣀⣀⠀⣾
⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⢸⣿
⢀⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣸⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣼⣿⣿
⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣄⢀⣤⣾⣿⣿⣯⡞
⠀⢸⣶⣗⠄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠀⠙
⠀⣈⡁⣉⠂⠀⠀⠀⠀⠀⠀⠛⠻⠛⠛⠻⣿⣿⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⡈⠻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⣀⣀⣀⣀⣀⣀⣔⣤⣤⣤⣤⣤⣤⡄
⣀⠀⠀⠷⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⣵⣶⠀⠀⢠⣿⣿⣿⣷⣤⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⣠⣮⣿⣻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣯⣿⢿⠿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣤⣀⣤⣤⣤⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠋⣛⢠⢸⣿⣿⣿⣿⠉⢻⡏⠙⣿⣿⠟⠙⠻⣿⣿⡟⠉⠉⠻⣿⡏⠉⠙⣿⣟⠉⣙⣿⣿
⣿⡉⠀⢸⣿⣿⣿⣿⡇⠈⡁⠀⣿⣿⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⣇⢠⣾⣿⣿⠆⢻⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿
⣿⣂⠀⠘⣿⣿⣿⣿⣇⣸⣿⣀⣿⣿⣦⣤⣴⣿⣿⣷⣤⣤⣾⣿⣿⣸⣿⣯⣤⣤⣼⣿⡿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣍⠉⢉⠉⡉⢉⣩⣭⢉⡍⢩⡍⠉⡉⣭⣍⣉⣋⠉⠉⠉⡉⠉⠉⠀⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠟⠋⢲⠂⠰⠆⢰⣼⣿⢿⡼⣿⡌⠀⠴⠶⠻⠟⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡺⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣿⠑⠰⢭⣽⣷⣴⣶⣤⣤⣄⢀⣀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣮⠳⠀⠚⠋⠉⠉⠉⠉⠉⠁⠀⡿⠿⣿⠂
⠀⠹⠟⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⣩⠀⣿⣿⡄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠑⠀⢤⣄⣀⠀⠀⠀⣠⣄⠀⠀⢸⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠿⠿⠂⠀⠈⠑⠯⠄⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⣀⡭⠵⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⣶⣶⡤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣼⣿⡿⣿⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠾⠃⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠤⢖⡂⠠⣿⣯⡿⠐⢯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣟⡳⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣤⣤⣄⣀⠐⠀⠤⠤⣀⣠⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣤⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠆⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠠⣀⠐⢄⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⠈⣷⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡄⠉⠀⠈⠢⠀⠀⠀⡠⠊⠉⠹⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣈⠀⠀⠀⠀⣰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⣿⣷⠄⠄⠒⠻⣿⣿⡿⠣⢄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣄⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠻⢏⠀⠀⠀⠀⢨⠁⠀⠀⠀⢸⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣠⣶⣶⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠠⣤⣤⣶⡀⠀⠀⢀⠌⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣀⠀⠉⢳⠢⠄⠄⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠙⢷⣌⣀⣀⣀⣤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡒⡪⢊⠍⢠⠋⠀⠀⠈⠛⠲⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡖⢳⡀⠘⢄⠀⠀⠀⠀⠀⡀⠑⢍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⢄⡋⠒⠄⣉⠉⠉⠁⠐⠢⡄⠈⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⠒⠤⢄⡉⢴⢤⠔⠒⠓⠒⡼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡄⠀⠀⠘⢣⣼⠞⠴⠐⢺⣝⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠁⣠⠞⠀⠀⡰⠡⠊⡌⠀⠀⡻⡏⠉⢹⡤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣙⠏⠁⠀⠀⠀⡰⠱⠁⢰⠀⡠⠊⠀⠈⠁⠊⠀⠀⠀⠉⠉⠉⠙⠛⠒⠈⠋⠓⠲⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠰⠁⡇⢀⠆⠀⠣⡀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡇⠀⠐⣎⠀⠀⠀⠈⠂⣄⠤⠊⣁⠔⠊⠀⠀⢀⡠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠃⠀⡘⠀⠑⢢⣤⠔⢉⡠⢓⣍⠀⠀⣀⠴⠊⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠀⠀⣃⡠⠒⣁⠤⠛⠓⣶⠋⣨⠖⢉⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⢠⢾⣶⠾⣁⠔⠊⠁⠀⢀⡮⢓⣽⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⡰⠟⡫⠕⠈⠀⠀⠀⡠⠒⣁⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡡⠔⠉⠀⠀⠀⡠⠒⣩⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⣁⠔⠉⠀⠀⢀⡠⠐⡩⠔⠉
⠀⠀⠀⠀⠀⢠⠒⠐⠂⣁⠔⠊⠀⠀⢀⡠⠒⢁⠔⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠨⣁⣈⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣿⣿⡟⢀⣿⣧⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⣠⣾⣿⣿⠿⠛⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠟⠁⠀⠀⠀⠀⠠⠶⠾⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠰⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠺⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⠃⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⣿⣿⣶⣆⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣭⣽⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣿⣿⡮⣻⣿⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢻⡇⢿⣿⡿⠟⠉⢭⡍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⢶⡁⠀⣿⡇⢰⣮⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣼⣿⣐⢿⠟⢀⣿⣇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠀⠀⠸⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣐⡤⠶⣔⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡴⠊⠃⠓⠚⠃⠉⠉⡷⢦⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⡃⠀⠀⠀⠀⠀⢸⠄⢊⡤⢞⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡛⢷⡤⡤⠤⢤⠔⣿⠋⢁⡠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⢹⡶⡧⢤⣴⢲⡗⣿⣥⠖⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⠷⣶⡚⠋⠀⣿⠁⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠇⠀⠈⠙⠦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢦⡴⠶⠶⣤⣄⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠘⠾⠳⢶⣒⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⣭⡭⣛⡶⠂⡀⠀⠀⠀⡆⠀⡐⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠊⠁⠀⠐⢄⠀⡠⠁⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣶⣶⣂⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡠⠋⠀⡐⠁⠀⠀⠀⠀⠀⣀⣀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠉⠻⢿⣦⡈⠢⠀⢀⠔⠁⡐⠀⠰⡐⠂⠐⠈⠀⡠⠚⢤⣴⠖⠶⠿⠛⠛⠉⠉⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣦⣰⠁⢀⠄⠀⠀⠠⠈⠀⠀⡠⠊⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠙⣿⠖⠁⠀⠀⠈⠛⢷⠄⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣕⡲⠲⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠸⠀⠀⠀⠀⠀⠈⠦⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⢂⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠄⠀⠀⠀⠀⠀⢰⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⣦⠀⠀⠀⠀⢸⣁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡼⠋⠀⠀⠀⠀⠀⠉⠁


⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣠⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⢿⣿⢟⣵⣿⡇⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⣰⡟⣴⡿⠋⢠⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠘⣡⠀⠀⠀⠀⠙⢻⡿⠛⠁⠀⠀⣌⢃⣿
⠀⠀⠀⠀⠀⢠⡿⢸⡟⠁⣠⣿⡟⢡⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡝⢛⡁⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⢈⠛⢫⢰⡆
⠀⠀⠀⠀⠀⠸⡇⣿⠁⣰⡿⢋⣴⣿⡿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⠞⠋⠀⠀⠀⠀⣤⣿⣿⣿⣯⣤⠀⠀⠀⠛⠳⡟⢁
⠀⠀⠀⠀⠀⠀⢷⢹⢰⠏⣴⣿⡿⠋⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣮⡴⠃⠀⠀⠀⠀⠹⢿⣿⡿⠿⠃⠀⠀⠀⠘⢦⣥⡾
⠀⠀⠀⠀⠀⠀⠈⠎⢊⢸⣿⠋⣀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⣠⠆⠀⠀⠀⠀⠀⢸⣿⡇⠃⠀⠀⠀⠀⠰⣄⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⢰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉⣰⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⡆⠉⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢷⡟⢰⡄⠀⠀⠀⠀⣿⡄⠀⠀⠀⢰⠂⢻⡾⠶⠋
⡤⣶⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⢦⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣼⣇⠀⢳⡄⠀⣿⡇⢠⡞⠀⣸⣷⣯⠤
⠳⠤⠃⢀⡴⠒⠲⣄⠀⡠⠖⠒⢤⡀⢠⠔⠒⠦⡄⠘⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⡿⠓⣦⣼⣷⠚⢿⣯⠅
⠀⠀⠀⢸⠀⠀⢀⢼⢾⣧⡀⠀⣤⡷⣧⣄⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠸⠉⠛⠆
⠀⠀⠀⠈⠳⠤⡿⠋⠀⠑⣷⢼⠚⠁⠘⢺⡦⠖⠃
⠀⠀⠀⠀⠀⠀⠙⠦⠤⠴⠃⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⢠⠒⢲⠀⡖⠲⠀⡔⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⢃
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣵⣿⡆
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣁⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢠⠀⡷⠠⡄⢤⠠⡄⢤⠄⠇⠀⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠘⠂⠙⠘⠃⠘⠊⠃⠘⠁⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡇⢠⠀⠠⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⢣⠀⡇⢸⡃⣻⠘⣇⣽⢘⣧⠘⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿
⠀⠀⡠⢄⠀⠀⢸⡀⢸⠀⡇⢀⠀⣇⠀⡇⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⣧⠬⠁⣠⡀⢀⡀⢼⠄⡄⢠⠀⡄⢀⡄⣷⢸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠙⠒⠶⠤⣄⣤⣤⣶⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣭⣭⣛⡻⠟⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⠋⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠨⣽⣛⣲⠦⣤⣄⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡒⠤⢭⣙⣛⠻⠟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⢿⣿⣛⡶⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣍⣀⠐⠤⢌⣙⡛⠿⠿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢭⡽⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣍⣒⣒⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣠⣼⣿⣿⣇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠟⠋⠉⠉⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣀⡀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⣷⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠓⠚⠉⠁⠀⠀⠀⡄⠂⣈⠽⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡉⡀⠠⠀⠤⣶⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢿⣿⣿⣿⣿⡿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⢠⠉⠾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣨⢉⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⣤⠈⢒
⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⢯⠄⢀⠀⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠼⠟⠄⠒⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠹⠄⠀⠀⠀⢀⣤⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠋⠀⠐⡢
⠙⠙⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⣾⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠻⠛⠛⠃⠀⠀⠐⠁⠀⠀⠀⠀⠘⠉⠠⡾⠊⠀⢀⣴⠡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⠟⢠⡄⠀⣰⣶⣆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢀⡀⠀⠀⠀⠀⠀⠴⠃⠈⠉⠀⠈⢁⢩⠛⢟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⢩⠿⣿⣿⣿⢿⣷⠀⠈⠝⠲⠆⣠⠀⠣⢄⣥⣤⣾⣴⣿⣟⡷⣮⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠈⢻⣿⡾⣿⣔⣂⠀⠰⠂⣀⣘⣧⣬⣽⣶⣿⣿⣿⣿⣿⣿⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠀⠀⠀⢀⣹⣷⣽⣿⣿⣷⣿⠿⠟⠛⠛⢻⢿⠋⣿⣿⣻⡿⣿⡛⣷⣿
⠀⠀⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⠀⢖⡞⣨⣤⣶⠶⠿⣛⣿⣿⢿⡛⠛⠉⠁⠨⢧⠀⠚⠺⠤⣧⠰⣿⣞⣽⣄⣽⣯⣿
⠀⠀⠀⠀⣀⡻⢿⣤⣤⠃⣀⣀⣀⡮⢸⣙⣯⠿⠿⠛⠋⠁⠀⠈⠃⠀⠀⠄⠀⠀⠀⠀⡀⠤⢤⣄⠙⣭⢿⣿⣿⠏⢁
⠀⠀⠀⠀⠉⠁⠂⣀⡠⠔⢛⡠⠁⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠐⠚⠂⠤⣬⡯⠟⣻⠿⣿

⠀⢀⡼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⢇⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡀⠀⠀⠀⠰⡷⠀⠀⣀⣀⡀⣴⡆⠀⠐⣿⡟⠳⠤
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢴⣶⣦⠾⠇⠀⠰⢻⣿⠛⠁⠀⠀⢾⣿⠓⠒⠚⠁⣠⡿⣇⡀⠀⠦⠽⢟⣛⡳⠆
⠀⠀⢸⣿⡇⠀⠀⢠⣾⣿⡟⠉⠀⠀⣠⡾⢿⡌⠉⠁⠀⢀⣼⠟⣶⠀⠀⢀⣼⠛⣦⠀⣤⣤⡿⣷⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠘⠛⠃⠀⠲⠚⠓⠋⠛⠀⠐⠛⠁⠀⠸⠃⠀⠀⠰⠛⠁⠀⠟⠀⠐⠛⠁⠈⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⣺⢗⣀⣿⠃⣀⡟⢃⣸⠟⣀⣸⠛⣀⡻⢇⣀⡟⢃⣸⡟⠀⠀⠀⠀⢀⣀⣀⠠⣶⠀⠀⠀⢾⠖⢾⣿⡋⡉⠛⠇
⠀⠀⠀⢿⡟⠛⢿⡿⠼⣿⡟⢻⣿⠋⢿⣿⠛⢿⡟⠻⣿⡛⠾⣿⠧⠀⠀⠀⠀⠀⢾⣿⡟⠀⠀⠀⠀⠀⠀⡏⠛⠀⡇
⠀⠀⢀⣸⠿⣦⡼⢿⡄⣼⣻⣦⡿⣦⢠⣟⣷⣸⢿⡄⣿⣻⣦⠿⣧⠀⠀⠀⢤⣤⡼⢻⡆⠀⠀⠀⣀⣀⣀⣇⣀⡀⡇
⠀⠀⠉⠉⠀⠛⠀⠘⠓⠋⠉⠉⠀⠙⠛⠉⠉⠁⠀⠛⠋⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠻⠛⠟⠛⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢀⡾⣇⡤⠖⠂⠀⠀⢠⣦⣤⣠⣦⠀⠀⠀⠀⣄⠀⠿⣦⠤⠤⠀⠀⠘⢛⣤⡤⠆
⠀⠀⠘⠛⢿⣧⠀⠀⠀⠀⠀⢠⣾⡟⡇⠀⠀⠀⠀⠈⠙⠛⣿⣆⠀⠀⠀⠀⢹⣧⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⢰⣶⠛⠁
⠀⠀⠀⢀⣼⠋⠻⣦⠀⠀⠀⣈⣿⣆⠃⠀⠀⠀⠀⠀⠀⢠⡿⠙⠳⣆⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠔⠉⠀⠀⠈⢻⣧
⠀⠀⠀⠛⠁⠀⠀⠈⠁⠀⠈⠉⠘⠃⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠉⠀⠀⠀⠀⠛⠀⠀⢀⠄⠀⠀⠀⠀⢰⡶⣾⠇
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢶⢀⣤⠀⡶⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠃⣿
⠀⢀⡄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣺⡇⠈⢫⣿⡞⠁⣠⡄⢰⠇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢯⣟⡁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢸⣿⡇⢠⣾⣷⠋⠀⠀⠀⠂⠀⠀⠀⣷⣶⣾⣿⣿⡇
⠀⠀⢸⣿⣿⠀⠀⠀⣠⣤⣟⡁⠀⠀⠀⠀⠀⠀⣿⣿⠉⢸⣿⡇⠿⣿⣿⠀⠀⠀⠰⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⢸⣿⡇⠀⠀⠸⠿⠶⣬⣥⡀⠀⠀⠀⠀⠀⣿⣿⢠⣼⣿⣧⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⣷
⠀⢀⣘⡛⢃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠋⠉⠁
⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣾⠀⡄⠀⠀⠀⠀⠀⢠⡆⣿⣆⡄⠀⠀⠀⠀⣬⡓⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢀⢰⣄
⠀⠀⠀⢰⣿⣿⣿⣆⠀⠀⠀⢰⣼⣿⢿⣿⣿⠀⠀⠀⠀⣎⣿⣰⠄⠀⠀⢠⣿⡿⣿⡆⠀⠀⠀⣴⣿⡿⢻⣧⡄⠀⠀⢀⣸⣿⢿⣠
⠀⠀⠀⠀⠉⡋⠊⢻⠆⠀⠀⠈⠿⠉⠘⠫⠟⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠈⢋⣀⣙⠃⠀⠀⠀⠋⠀⠀⠛⠟⠀⠀⠀⠘⠟⠀⠘⠟
⠀⠀⠀⠀⠀⣻⣦⠈⠀⠀⠀⠀⠐⣷⣶⡿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢶⣷⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣰⡆⢰⡆⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠀⠀⠀⣠⣸⣧⠀⠀⠀⠀⠀⢃⣿⠀⠀⠀⠀⢸⣷⡀⢰⣄⠀⠀⠀⠀⠀⣌⣿⣆⡀
⠀⠀⠀⠈⠻⣎⢷⢰⡄⠀⠀⠀⣼⣷⣿⡧⠀⠀⠀⢰⣿⡟⣿⠆⠀⠀⠀⠰⣫⡵⠂⠀⠀⠀⢮⣿⣿⣿⣿⣷⡀⠀⠀⠀⡿⠟⣿⣧
⠀⠀⠀⠀⢀⣨⠙⠼⠃⠀⠀⠀⠿⢉⡉⠀⠀⠀⠀⠈⣀⡘⠋⠀⠀⠀⠀⢀⣥⠀⠀⠀⠀⠀⠈⠙⠻⠀⠀⠹⠃⠀⠀⠀⢰⣤⡈⠏
⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣰⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⣿⣽⣠⣽⣿⡟⠃⠀⠀⢀⣠⠴⣴⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠚⠉⢩⣡⠀⢻⢻⣿⣿⣿⣁⣀⣤⣴⠋⣰⣦⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⡦⡀⠈⠉⠛⠻⣤⣿⠿⡇⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢁⡔⣾⠏⣿⠐⠦⣀⠀⠊⠁⠀⢡⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣯⣀⣤⠒⣳⣩⣅⡀⣏⡤⠀⠀⠁⠀⠈⡠⠃
⠀⢰⣿⣷⣄⡀⢀⣀⠀⠀⡜⢀⣀⡾⣿⣾⣦⣿⣽⣷⡿⠶⠶⠶⣤⣤⣀⣀⡄
⠀⠈⢯⣿⣿⣿⣿⣇⢩⣽⠛⠁⢿⣿⡟⠁⢹⣿⡿⠋⠀⠀⢠⢆⢙⢻⡷⠌⡇
⠀⠀⠈⠻⣿⣿⣿⣿⣽⣿⣾⣿⡿⠋⠀⠀⠀⠃⠀⠀⠀⡄⠇⠼⢠⢿⣦⣸⠃
⠀⠐⠠⠀⠀⠉⡙⠿⣿⣿⣿⣿⣷⣴⣶⣆⢀⡂⠀⡀⠔⠋⠡⡶⢠⠏⠉⠁
⠀⠀⠀⠀⠀⠂⣈⡓⠠⣉⠻⢿⣿⣿⣿⣿⣾⣽⣶⡴⠚⠃⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢥⣄⡈⠛⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣻⠿⠏⠐⠒⠀⠀⠄⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠈⠁⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠿⡄⢀⠀⠀⠀⠀⠀⣴⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⠟⣿⣇⠀⠈⠲⠈⠁⠀⠴⣾⣿⢻⣿⣦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠋⣿⣿⢸⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⠘⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠈⠃⠀⠀⢺⣷⠀⡏⢹⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣴⣿⠟⠇⠀⠀⠀⠀⠁⠀⠀⢾⢟⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡄⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠚⠏⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠣⢿⣿⣿⣿⠋⠛⠿⣿⣿⣿⣿⣷⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣟⠀⠀⠀⢿⣇⡄⠀⠀⠀⠈⠉⠉⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⣲⣶⣦⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣵⣿⣿⣟⣽⣯⣿⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠻⡿⣿⣆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠻⣿⣿⣿⣷⣽⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠈⠻⣿⣿⣻⣯⣟⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⣄⣀⠀⠀⣿⣿⣿⣷⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣶⣿⣿⣿⣿⣎⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣯⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢸⡌⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠁⠀⠀⢰⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠈⡝⠀⠀⢀⠝⠣⢀⣀⣀⠸⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠰⠲⠀⠄⡌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡀⠄⠂⠁⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⠔⠙⠉⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⡐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⠤⣔⠁⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⢀⠇⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠐⠊⠁
⠀⠀⠀⠀⠠⠊⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡀⠀⠀⠬⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠡⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠀⠄⢀⣀⣀⠀⠠⠀⠇⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢆⠀⠀⠀⠙⠿⠷⣶⣦⣤⡀⠀⠉⠂⢄⡀
⠀⠀⣰⡷⠶⢠⣀⡀⠤⠄⠒⢀⠂⢣⠀⠀⠀⠈⠀⠒⠒⠒⡶⠤⡈⠛⢿⣦⡀⠀⠀⠰
⠀⠀⣸⡠⢾⠁⠀⠀⠀⠀⠀⢸⠀⠈⢆⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⢆⠀⠙⣿⣄⠀⢀⡄
⠀⣴⢳⢃⣾⠟⠉⠉⠁⠒⠒⠛⠂⠠⠬⠷⠤⠠⠐⠉⠀⠀⠀⠀⠀⠈⠢⠄⠜⠛⠉⠁⢡⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠂⠀⠒⢠⣄
⣸⠃⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠠⣀⣲⡆⠀⠀⠀⠀⠀⢠⡾⠋⠙⢆⠀⠀⣠⠛⠣
⠛⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠁⠀⠸⡀⠀⠀⠀⠀⣾⡁⠀⠀⣼⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⡇⠀⠀⠀⠀⢿⡟⠉⠉⠙⢿⠛⠙⠢⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢰⣶⣶⣤⣤⣬⣷⣦⣀⣠⣏⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡙⠻⠤⣴⣚⣩⣤⣤⠟⠙⠛⠛⠃⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣞⣉⣁⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣯⣝⢿⣿⠿⣿⣟⣛⢾⢖⣛⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⢢⣶⣈⣧⢹⡄⠀⠀⠄⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠦⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⡅⢸⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣥⢿⡀⢩⣜⠔⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢿⡿⠛⣉⣌⠀⢀⡀⡇⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⣶⠾⠛⢹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⢍⣢⠠⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⠦⡀⡆⢸⣿⣿⣿⣇⣬⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣸⣹⣿⣿⡏⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣿⢼⣿⣿⣿⣿⣮⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠐⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠖⠂⠀⠤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⣿⣿⣿⣿⣿⣿⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣃⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣾⣿⣫⣿⣿⣟⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⠉⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡻⣿⣀
⣿⣷⣶⣦⣤⣀⣀⣉⣀⣀⣀⣈⣙⣷⣶⣦⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡿⢿⡾⣿
⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⠠⢤⣴⣶⣶⣶⡦⠀⠀⠀⠀⠀⢠
⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣿⣿⣿⣿⡟⠀⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣶⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⣿⣿⣿⣿⣿⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣀⣹⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⡿⠛⣿⣿⣤⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⡁⣀⣤⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠁⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠋⢁⣠⣤⠾⣾⣿⣿⣿⣷⢄⣠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠷⢦⠀⣠⣴⣶⣿⣿⣿⣷⣍⡻⠻⢟⡯⠚⡿⡿⢿⣿⣦⣄⡀⠀⣀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣴⠞⠉⠉⠀⠤⠻⢯⣁⣤⣴⣋⡁⠈⠘⠻⠦⠀⠐⠉⠀⠀⢛⣥⣾⣿⣿⣾⣿⣿⡿⢟⣷⣶⣤⣀⡀
⠀⠀⠀⠒⠒⠚⠛⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠀⠉⠓⠀⠀⠀⠀⠀⠀⠶⠾⠋⠹⠇⠉⠓⠨⠉⢡⠾⠛⠋⠻⠏⠉⠛⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⠈⠉⠙⠳⣦⡀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠉⠉⠛⢶⣄
⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣤⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣼⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢹⡏⠀⢹⣏⠙⠻⣦⡀⠀⠀⠀⠀⣠⡾⠛⢉⣿⠁⠈⣿⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣰⡟⠀⠀⠀⠻⣆⠀⠈⢻⡄⠀⠀⣼⠋⠀⢀⣾⠃⠀⠀⠘⣷⡀⠀⠙⣧⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠈⠛⢷⣤⣀⣀⣀⣿⣡⣴⠞⠋⠀⠀⠀⠀⠀⠙⠻⣦⣄⣿⣀⣸⣏⣤⡾⠛⠁⠀⠀⠀⠀⠈⠛⢶⣤⣽⣇⣀⣀⣤⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣄⣀⣠⣤⡶⠟⠉⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣀⣀⣤⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⣄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠄⣀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠆⠤⠊⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⣸⡟⠻⠻⠿⠿⠿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⠀⠒⠒⠒⠀⡠⢊⡁
⠀⠀⠀⠀⠀⢰⡿⠁⢀⠀⠀⠠⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⠁⠒⠀⠂⠠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠏⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠿⠿⢿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⡁⠀⠄⠀⠀⠀⠀⠁⣀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠈⠀⠀⠠⠈⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠉⠲⢄⠐⠀⣀⣀⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠴⢾⠃⠀⠀⠀⠀⠀⠀⠑⢦⡀⢀⠀⠀⠄⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣏⠀⡽⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣦⠀⣀⣈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣷⠄⢀⡀⠀⢀⣡⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣴⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠉⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡗⠒⢯⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣉⣉⣉⠉⢉⠠⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡀⠀⠀⠀⠀⣿⠃⣤⠎⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢒⡖⠒⠊⠳⠖⢲⡛⠁⠀⠀⠀⢸⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⢀⡞⠁⠀⠛⠂⠛⠃⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠤⣀⣀⡼⠀⠀⣀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠘⠉⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡴⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⡤⠤⠵⠶⠴⠦⠤⠤⠤⠤⠤⣤⣤⣬⣥⣤⣤⡽⠀⠀⠀⠀⠀⠀⣠⡼⡷⠖⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣃⣀⣀⣀⠀⠀⠀⠀⣛⣀⣀⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⠶⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⠛⠓⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣳⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠤⠤⠤⠤⠀⡀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠔⠙⣇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠃⠀⠀⠈⠙⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⢴⠚⠛⡏⠉⠉⠙⠒⠒⠦⢤⣀⠀⠀⠈⠑⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢴⡇⢈⣀⠀⢈⢦⣀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢿⡖⡢⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢾⡇⠃⡸⡿⠓⠢⢄⣀⡀⠀⠀⠀⠱⡀⡠⠃⢀⠤⠒⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠻⣾⠗⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠳⡀⠁⠀⠀⢀⠤⠹⠆⠀⠀⠈⠑⢦⠀⠙⠤⢄⣁⣀⡀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠚⠛⠛⠒⠚⠟⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢦⡀⠒⠴⠾⠇⠂⠀⠀⢀⣴⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠫⢁⠀⠀⠀⠀⠀⢀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣈⡑⡖⠢⠤⠤⠶⠞⠋⠀⠀⠀⢙⡦⡀⠀⠀⠀⣀⠀⠀⣀⣀⡀⠀⠉⠐⠂⠄⠊⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠑⠢⢤⡄⢀⣀⣀⠀⠈⠉⠒⠠⠤⢠⡤⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠘⠋⠉⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠳⣄⠤⢀⠑⠄⣀⡠⠄⠀⠀⣁⡠⡃⡧⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣳⡀⠀⠀⢹⡳⣖⠨⠥⠐⢖⡉⠈⠀⠉⠲⠖⠁⢈⢤⡁⢊⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠁⣡⠄⠀⡤⠓⠂⢀⠰⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⣨⠧⠬⢤⠤⠴⠦⢒⠒⠂⠀⠀⡠⠀⠙⡈⠀⢜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠈⠙⠊⡉⠷⠚⠒⢊⠇⠀⢣⢢⣄⣀⠀⣀⣀⡠⠴⠒⠋⠁⠀⠀⠈⢆⠀⠀⠀⠈⠀⠒⠈⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⡥⠤⡖⠒⠈⠁⠀⣀⣤⡁⠈⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠙⠒⠖⠒⢋⠁⢀⣡⠶⠚⠉⠉⠉⠉⠉⠙⠒⠒⠲⠦⡤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⡉⣡⣖⠐⠋⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⢀⠀⢷⢄⣀⡠⠎⠀⠀⠤⡔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠀⠀⠑⢠⡀⠀⠀⠀⢀⣈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⢄⣀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⣀⠀⠀⠑⠢⡉⠁⠀⠀⠈⠲⣀⢀⠠⠄⠀⠀⣁⠠⠄⠒⠀⠁⠀⢀⣀⠬⠗
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⢠⠀⠄⠄⠒⠁⠀⠀⠀⠈⢦⠄⠂⠉⠁⣀⣀⡤⠤⠒⠚⠉⠁
⠻⣍⠒⠢⠤⠤⠤⠄⠀⠀⠀⠠⠤⠄⠐⠒⢒⣘⡦⠤⣤⠊⠀⠐⠂⢈⣁⣠⡤⠤⣔⣢⡥⠴⠒⠚⠉⠉
⠀⣌⠛⠒⠶⠶⠶⠶⠶⠒⠒⠒⣚⣋⠩⠭⠔⠒⠒⠀⣉⣉⣉⠭⠭⠷⠒⠚⠋⠉⠁
⠀⠙⠦⣍⡉⠐⠒⠈⢉⣉⣉⣉⣠⡤⠤⠴⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉
⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⠛⠛⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠉⠁⠀⠀⢠⣤⣼⣿⠿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿
⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣄
⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣿⡿⠿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⡿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⣼⣿⣿⡟⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣶⣶⡆⣉⣐⢻⣿⣿⣷⠛⣡⡘⠒⠤⠴⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⢰⣆⣠⣤⣬⣭⣭⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣦⣬⣿⣭⣽⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⢉⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣴⣶⣤⣼⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⢀⠀⢀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⠏⠀⠉⠁⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⢠⣄⣚⣿⣿⠿⠿⠛⣿⠛⠿⠛⠛⠛⠛⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠃⠈⠉⠀⣠⣤⣴⡞⢛⣤⣄⣁⣀⣄⢀⣤⣴⣾⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣼⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⣆⣻⣿⣿⣾⢿⣿⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⣿⣿⣟⣷⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣹⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠐⠈⢰⠳⣄
⠀⠀⠀⠀⣶⠖⣶⠀⣠⣟⢴⠁⠢⠄⠀⢀⢿⡀
⠀⠀⣀⡀⠘⢈⡏⢰⡏⢑⠾⢀⠐⠀⠀⡂⠸⣷
⠀⢼⡁⢻⠀⡇⡇⢸⡔⡨⠄⢀⠀⠀⢀⡂⣸⡏
⠀⠀⠹⣄⢧⣥⣷⠘⣇⢋⠀⠀⠀⠀⠀⠀⡽⠁
⠀⠀⠀⠘⣌⢻⣼⠀⢻⡼⡚⠦⠀⡠⢀⣼⠃
⠀⠀⠀⠀⠈⣆⢿⡆⠀⢛⢦⣬⣰⣐⡿⠁
⠀⠀⠀⠀⠀⡟⢶⠘⣆⠀⠃⠀⡔⠁
⠀⠀⠀⠀⠀⡇⠘⣆⠈⢳⡄⢰
⠀⠀⠀⠀⠀⠃⠀⠸⡄⠀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⡤⠤⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠻⡀⠀⠀⠣⡀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠨⡆⣷⡀⠀⠀⠱⡀⠀⠀⡰⠁⠀⠀⡀⠂⠀⠀⠀⢀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⡀⠀⠀⠱⣄⡠⢅⠀⠐⠈⠀⠀⠀⢠⠊⠁⠀⢀⠴⡚⠉⠉⠀⠈⢇⠢⢄
⠀⠀⠀⠀⠀⢠⠀⠀⢹⡿⠛⢷⡄⠀⠀⠘⠅⠀⠱⡄⠀⠀⠀⠀⠀⠀⢄⡰⠁⠀⣴⣦⣤⠀⠀⠀⠁⢰⣧
⠀⠀⠀⠀⠀⢸⠀⢠⠜⠃⣦⡀⠹⢆⠀⠀⠈⢆⠀⠹⡀⠀⠠⠀⠀⠀⢠⣄⡀⠘⢿⣿⡿⠀⠀⠀⠀⠸⢿
⠀⠀⡔⠊⠃⢹⡀⠽⠄⠀⠀⠙⠢⠈⢂⠀⠀⠈⢢⡰⠓⡆⠉⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⠀⠀⠀⠈⣧⣀⠀⠈⠑⠂⠀⠀⠘⢧⡀⠀⠀⠡⣄⡙⠄⣀⣀⠀⠘⠣⢄⣀⣤⣆⠀⠀⠀⡄
⠀⠀⠢⠀⠀⣀⣇⠹⡀⢀⡀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠘⡜⡄⠀⡘⠒⠈⢡⡀⢻⣿⣿⡗⠀⠀⠀⠀⣶⡖
⠀⠀⠀⠀⠀⠀⠈⠒⠚⠋⠀⠀⠀⠀⠀⠀⢀⡚⠢⢴⠴⠓⠃⠀⠀⠀⠀⠀⠉⠢⢈⠉⢢⡀⠀⠀⠄⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠙⠻⣆⠀⠀⢀⣰⠟⠁⠀⠀⠈⠙⢿⣄⠀⠀⠀⣼⠋⠀⠀⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣴⢾⡿⠛⣿⣶⣦⠀⠀⠀⠀⢀⡴⢾⡟⠛⣿⣶⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢠⡞⢁⡾⠃⠀⠘⣦⡙⢷⡀⠀⣰⡞⢡⡾⠃⠀⠙⣆⡙⣷⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣿⠞⠋⠀⠀⠀⠀⠈⠻⢾⣷⣤⣿⠚⠋⠀⠀⠀⠀⠘⠻⢾⣧⣤⡶⠚⠋
⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⢻⡷⣷⡀⠀⠀⠀⣠⡶⣿⡟⠙⢻⡷⣦⡀⠀⠀⠀⢠⡶⣿⡏⠙⠛⢾⣦
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠻⣎⣻⡄⠀⣰⣏⡴⠏⠀⠀⠈⢳⣌⣷⡄⠀⣴⣯⡼⠋⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣡⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣸⠏⣹⡇⠀⢹⡍⠻⣆⠀⠀⢀⡺⠋⣼⠇⠀⢻⡝⢻⣄⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⣀⣼⣥⠾⠋⠀⠀⠀⠻⣦⣽⣇⣀⣿⣥⠾⠋⠀⠀⠀⠻⣦⣽⣆⣀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠿⣿⣤⣄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢠⡼⣣⡿⠃⠀⠘⣯⡻⣦⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠴⣿⠛⠉⠀⠀⠀⠀⠈⠻⢻⡷⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⢰⡒⢲⣿⠉⠛⠟⠛⢃⢻⡇⠀⠀⠀⡠⢠⡀⠀⢀
⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣿⠛⠿⢤⡃⠉⢰⡦⡀⠄⠊⠀⡀⡀⡰⡡⡑⡡⡡⢁
⠀⠀⠈⠃⠉⠢⣀⠀⠐⢤⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣀⠶⢉⠍⡸⠀⠀⠀⠀⠜⡐⠑⡔⢉⠜⠔⢠⢃
⠀⠀⠀⠲⢄⠀⠈⣈⠂⠈⠀⠉⠒⠤⣀⣸⣿⣿⣿⣿⣇⢨⣟⡳⠈⠀⠀⠀⢀⠎⠈⢈⢎⢀⢊⠎⡠⠁⠢
⠀⠀⠀⠰⢄⡈⠂⠁⠑⠀⠢⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣏⡇⠀⠀⠀⠀⠠⠂⠂⠀⣁⣀⡀⠔⡠⡑⡡
⠀⠀⠀⠀⠀⠈⠓⠆⢄⠐⠀⢄⢠⣠⣴⣾⣿⣿⠏⠙⢿⣿⣿⣷⡀⢀⣀⠤⠤⠤⠖⠋⠁⠀⠉⠒⠥⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠝⠛⠛⠋⡉⢀⣼⣿⣿⠿⢿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡉⠲⢤⣀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣛⠶⢥⣄⣀⡀⠀⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⡈⠉⠠⣄⣀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠲⢄⠉⠉⢍⡁⠀⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠈⠲⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⡿⣦⡀⣄⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡀⡀⠠⣄⡛⣴⢏⠀⠙⠛⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⡶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⡀⠰⢄⠐⢶⣄⢶⣄⠠⣀⢛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⠭⠀⣘⣇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⠻⠾⠿⠧⠴⢤⣰⣤⡀⠀⠀⠀⠀⠘⠷⠀⢰⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠹⢄⠀⠀⠀⠉⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠼⠃⠀⡀⠖⠀⡀⠀⣄⠀⠈⣷⣄⠀⠐⢀⠀⠐⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠆⠸⠢⣧⣱⠿⣤⣏⣀⠟⠀⠀⣾⡿⠛⣉⠉⠑⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠄⢐⣿⡿⠶⢤⡉⢠⡆⠀⡰⣽⡛⣹⣤⣴⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠄⢀⣬⠹⣦⠨⣿⠎⢀⠄⢀⡿⠃⢿⣩⠭⠭⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡼⢄⣲⠝⠁⣀⠄⠁⣴⡟⠃⡟⣤⠄⡀⠄⠂⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢀⣀⡤⠞⣁⣠⡴⠛⠙⠿⣦⢾⠆⢻⡀⠐⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠓⠚⠋⣿⣇⢈⠄⠀⠀⠙⠿⣧⣀⠠⠄⠓⠀⠈⠁

⠐⢆⣘⣟⠀⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠰⣦⡤⢖⣤⠶⢦⡲⢤⣴⡷
⠀⠀⢸⣿⡓⠚⠁⠈⠙⠒⢿⣿⠒⠦⠀⠀⠀⠀⢠⡿⠳⠿⠽⢿⣦⠀⠀⠀⠺⠛⢿⡏⠀⠀⠀⠹⣿⠟⠗
⠀⢰⡏⠉⢻⡄⠀⠀⠀⢰⡞⠛⢦⣄⡀⠀⠀⠀⣸⠻⡆⠀⠀⡾⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁⠀⠁⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣄⠀⢀⢰⣶⢀⠀⢀⡄⠀⠀⠀⠀⠀⡰⣶⡄⠀⠀⢤⣀⢠⣦⠀⢀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢼⡷⣤⣸
⠉⠀⠘⢻⣿⠛⠀⠈⠁⠀⠀⠀⢠⣿⡛⢮⡤⠀⠀⠀⠈⢻⣿⠚⠉⠀⠈⠙⣻⣿⢦⡼⠃⠀⠰⠤⢿⣾
⠀⠀⠀⣼⢻⡀⠀⠀⠀⠀⣶⣿⣧⣿⣿⡾⠋⠓⡄⠀⢠⠟⣿⠀⠀⠀⠀⢀⡿⢷⡀⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠻⠿⠟⠉⠀⠙⠤⠔⠁⠀⠸⠆⠘⠷⠀⠀⠀⠞⠁⠸⠇⠀⠀⠀⠀⠰⠋⠀⠟

⠀⠀⢰⡦⠀⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣻⣃⣸
⠀⠙⣿⣟⢛⠇⠀⠀⢿⣿⠛⠋⠀⠀⢀⣶⣿⠚⠋⠁⠀⢀⣴⣲⣼⠛⠀⠀⠀⣨⣏⣠⡆⠀⠚⢹⣿⣉
⠀⠀⡿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢸⣿⡀⠀⠀⠀⠀⢹⣿⡉⠒⠀⠀⠸⣿⣄⠉⠀⠀⠀⢀⡿⣽
⠀⠰⠇⢿⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⢠⠞⠁⠿⠀⠀⠀⢰⠏⠙⠧⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⣅⣽⡣⢄⡀⠀⠀⠀⠘⣷⣾⣉⡿⠋⠀⠀⢠⡤⢤⣿⡤⠤⡄⠀⠀⣀⣤⣤⠞⠁⠀⠀⠀⠠⣄⣽⡣⢀
⠀⠀⣻⡦⠞⠀⠀⠀⠀⠸⡏⠙⠛⠶⠶⠆⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⣻⡦⠞
⠀⣰⠃⠀⠀⠀⠀⠀⠻⡿⠛⠛⠛⠛⢿⠟⠀⠀⠀⢸⣿⡇⠀⠀⢀⣀⣀⣛⣿⣀⣀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠉⠉⠉⠩⠉⠉⠉

⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⢀⠔⠹⣄⠤⠔⠒⠚⠓⣄⠀⠀⠀⢀⣴⡇⢟⣫⣭⣵⣶⣶⡄⠀⠀⠀⣠⡎⠁⠀⠀⣀⣀⣉⢢⡀⠀⠀⡀⢾⣿⣿⡿⢟⣛⡻⢦⡀
⠀⠀⣏⠀⠀⠱⣄⠀⣀⣀⣀⠘⡆⠀⠀⢾⣿⣿⣜⢿⣿⠿⠿⠿⣿⡄⠀⢰⠃⠘⠒⠒⠋⠀⠀⠈⢣⢱⠀⢰⣿⣶⣭⣵⣾⣿⣿⣿⣎⢷
⠀⠀⡏⢦⡀⠀⢨⠏⠁⢸⡏⠉⡇⠀⠀⣮⡻⣿⣿⡗⣴⣿⣿⢸⣷⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸
⠀⠀⠱⡄⠉⢲⠞⠀⠀⣸⢀⡼⠁⠀⠀⠹⣿⣮⡟⣱⣿⣿⡏⣾⡟⠀⠀⠀⢣⡖⠉⠉⠒⠤⡤⠴⣣⠎⠀⠈⠟⣭⣭⣭⣝⡻⠿⢟⣵⠃
⠀⠀⠀⠈⠒⠥⣄⣀⠴⠗⠋⠀⠀⠀⠀⠀⠈⠋⠾⢿⠿⠟⠘⠉⠀⠀⠀⠀⠀⠙⠢⠤⣀⡠⠤⠚⠁⠀⠀⠀⠀⠙⠻⠿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀
⠀⠀⢀⠴⠿⠁⠀⠀⡹⠿⡷⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣦⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠂⠀⠀⠀⠀⣠⣾⣿⣷⣦⣉⠻⠄
⠀⠀⠀⠀⠀⣼⣾⣿⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⠁⠉⠈⣿⣿⣿⣿⡀⠀⠀⠀⠄⠀⠀⢀⣴⠗⠀⠙⠇⠀⠀⠀⣼⣿⣿⡟⠃⣸⣿⣷
⠀⠀⣷⡄⠈⠛⢿⠟⠁⠀⣴⣿⡇⠀⠀⠈⣻⣷⣤⣀⣤⣾⣿⠋⠀⠀⠀⠀⡀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⢀⣾⣿⣿⠏
⠀⠀⠹⠇⠀⠀⣀⠀⠀⠀⠟⠟⠁⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣦⠀⠀⠀⠀⠛⢦⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣷⣤⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠾⢿⣿⠯⠀⠈⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡉⠛⠋

⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣆⠀⠀⠀⡔⢹⠃⠀⠀⣠⠖⠛⠑⠄⠀⢀⣼⠁⣿⣿⡿⢋⣥⣶⣶⡄
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⢰⢠⠏⠀⠀⣼⠁⠀⠀⠀⢸⠀⢸⢏⣼⣿⣿⢃⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⠀⠀⠘⡏⠀⠀⠀⢿⠀⠀⠀⠀⡘⠀⠠⣾⣿⣿⣿⠸⣿⣿⣿⣿⡏
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⢄⡀⠀⠈⠧⣄⣀⠔⠀⠀⠀⠙⢿⣿⣿⣧⡙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣬⡉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢘⠤⠔⠒⠠⢄⡀⠀⠀⠀⣸⣿⢇⣠⣤⣤⣤⣀
⠀⠀⠀⠹⡗⢄⠀⠀⠀⡀⠄⠠⡀⠀⠈⢱⣄⠀⠀⠀⢀⣠⣤⣶⡄⠀⠀⠀⠐⠀⣰⠁⠀⠂⣀⠀⠀⠑⡄⠀⢸⣿⢣⣿⣯⣽⢿⣿⣿⣧
⠀⠀⠀⠀⠐⡀⠁⠂⠁⢠⣆⠀⠉⢀⢠⠀⢻⣿⣶⣾⣿⡏⡶⠿⢿⠿⢶⠀⠐⠀⠇⠀⠘⠀⠁⠀⠀⠀⡇⠀⠸⣿⣿⣿⣶⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠄⢀⣀⣈⡀⠤⠄⠂⠂⠀⠀⠻⠿⠿⣿⣷⡭⠟⠛⠛⠋⠀⠀⠣⣸⣄⠀⠀⠀⠀⠀⡰⠃⠀⠀⠻⠜⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⡿⢿⢿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⡿⠈⠻⣿⣿⣦⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡃⠀⠀⠈⠉⠛⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⣼⣿⣿⡿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⣾⣿⣿⡏⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⣿⣿⣿⡟⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⣿⠛⠛⠛⠛⠃⠠⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⠏⠙⠷⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣀⣤⠞⠉⠈⢯⠙⠳⣤⡀⠀⠀⠉⠻⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣻⣿⣿⣿⣶⣟⠉⠙⠶⢦⠀⠈⣳⠀⠘⠿⣷⣤⣀⠀⠀⠙⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⢹⠿⠿⠿⣦⡀⠳⢤⣀⠈⠁⠀⠀⠈⠛⢿⣷⣦⡀⠈⠁⠙⠿⠿⠿⣦⡀
⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠀⠈⢣⠀⠀⠀⠀⠀⠛⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠫⠀⢡⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠐⠀⢀⣴⣾⡿⣗⠂⠀⠀⡻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠢⡀⠀⠘⢿⣿⠦⣌⠀⠀⠀⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠑⠄⠀⠀⠀⠀⠀⠈⠿⣧⡈⠁⠀⢀⡏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⣠⣶⠊⠀⢀⠔⠀⠀⢈⣿⠀⠀⢺⣴⣒⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⠟⡏⠙⠀⠀⠞⡠⠀⠀⢸⠇⠀⠀⢈⠛⠿⣶⣒⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⢸⡆⠀⠀⠈⡷⠀⡈⠛⠻⠶⣤⣄⡀⢀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣀⣠⣼⡏⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠉⠉⠛⠤⠴⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⡿⠀⠀⠀⠀⠀⢸⣷⡆⠀⠀⣏⢀⣎⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠛⣿⠆⠀⠸⠟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⣆⠀⠀⣈⣀⠹⣦⠀⠀⠀⢻⡤⠒⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣬⣵⣾⣿⣿⣀⡇⠀⠀⠀⠀⣿⣇⡴⠁⠀⠰⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡏⠳⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠚⠒⣤⣿⣿⣿⢷⡄⠀⠀⠁⠀⠈⣼⣧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣾⣶⣷⣦⠀⠀⠀⣻⣿⣿⣿⣿⠋⠉⠒⠒⣶⣾⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢿⣾⣷⣾⣿⡿⣿⢿⡛⠠⠦⣀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠘⠻⢿⣼⣦⣠⣷⣤⣾⠟⢋⣁⠀⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⢿⣿⠻⢿⣟⠋⠀⠀⡀⠀⣶⣶⡾⠋⣉⠍⣉⣿⣟⣥⠶⠛⠛⠛⢅
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣠⣴⡶⢶⡶⠿⠁⠀⠀⠀⠀⠀⠁⠀⠀⡁⠀⠀⣿⣤⣿⣿⣧⣏⣀⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢳⣤⠀⠀⠀⠀⠀⠀⠀⣰⣰⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡟⠀⠀⢠⢀⣞⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⡁⠀⢀⡀⠀⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⡀⠈⢃⣤⡧⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠈⢻⣿⣿⣿⠿⠿⠛⢱⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⠦⣤⠛⠛⠻⢦⡀⡄⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⡖⠀⠀⠈⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠄⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡉⠙⠳⢶⣦⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠁⢸⣷⣦⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢰⠏⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⢸⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣾⣿⣿⠏⠀⠀⠀⠀⠙⠳⢶⠞⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠈⣿⠋⠉⠉⠉⠙⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⣠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡀
⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⡄⠘⢿⣿⠟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣤⣶⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⣿⣿⣿⡿⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣶⣶⣄⠈⢿⣷⣤⣹⣿⣟⣥⣴⣿⠇⠀⢀⣀⣀⡀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠸⣿⣿⣿⡿⠀⠀⠉⣿⣿⡿⣿⣿⠋⠁⠀⠀⣾⣿⣿⣿⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣬⣿⣯⣤⡀⠀⠀⣿⣿⣶⣿⣿⠀⠀⠀⠀⠙⣿⣿⠟⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⠀⠀⢀⣼⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⠀⣿⣿⡟⣿⣿⠀⠀⣾⣿⣿⣏⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡏⠋⠀⣿⣿⡇⣿⣿⠀⠸⠿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⣠⣿⣿⣧⣿⣿⣤⣄⠀⢸⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⣿⣿⣿⡆⣿⣿⣿⣿⠀⢸⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣧⣿⣿⣧⣤⣿⣿⣿⣷⣿⣿⣿⣿⣀⣸⣿⣟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢉⣡⡤⠉⠉⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠛⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠞⠋⠀⠀⠀⠈⠛⠢⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢻⣷⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⡘⠀⠁⠐⠇⡀⠀⢹⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡁⠈⠈⠀⠃⠀⠠⠇⡾⠃
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠞⡀⠀⢃⠀⠀⠠⠊⣼⠁⡇
⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠃⠀⠀⠀⠀⠀⢸⠂⠀⡞⠂⠤⠈⠘⠀⠀⠀⠀⡠⢞⢿⢠⠇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢈⣀⣠⠴⠏⠂⢠⡾⠋
⠀⠀⠀⠀⠸⣷⣆⣀⣀⣀⣀⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⣛⣿⣿⣿⣷⣤⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠈⠪⣒⣦⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠫⠟⢻⣉⣁⡿⢶⡲⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢛⣿⢹⠀⠈⡄⠀⠈⠛⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠈⠉⠀⠀⠀⠷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠃
⠀⠀⠀⠀⠀⠠⠠⠀⠤⠶⠚⠉⠀⠀⠀⠀⠀⣀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⢀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⠿⠟⠋⠉⠉⠉⠁⠈⢿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⣤⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠤⢀⠢⠈⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⡑⣠⣴⣿⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣿⠏⠁⣠⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢡⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢀⣠⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⡿⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⠟⠋⣠⣾⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠛⠉⠁⢠⣼⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⢀⣤⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠛⠁⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⠿⠛⠁⠀⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⢀⣠⣴⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠓⠛⠒⠛⠒⠛⠚⠓⠑⠛⠃⠙⠒⠛⠂⠚⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢶⣶⢶⡦⠤⢤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠙⠿⣽⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠠⣤⠈⢿⣾⣿⣿⣿⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⢸⣿⣿⣄⣤⠄⠀⠀⠀⠘⠻⣶⡿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡀
⠀⠀⠀⣀⣤⣴⣶⠴⢤⣄⣀⠀⠀⠀⠀⢻⣷⣤⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣀⣰⣿⡇⢿⣿⣇
⠀⢿⢟⣿⣿⣯⡀⠀⠀⠀⠉⠀⠀⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣠⣿⣻⣿⣿⣏⣤⣾⣿⣿⡀
⠀⠀⢻⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⡏⢻⣿⣇
⠀⠀⢰⣿⡏⢸⣿⣧⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠟⣁⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣇⣈⣿⣿⡄
⠀⠀⢾⣿⣷⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠋⣵⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⢿⣿⣿⣧
⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣥⣄⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠟⠛⠋⠀⠀⠈⠉⠉⠋⠉⠁⠀⠉⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣻⡟⠻⠿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠉⠻⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡛⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡭⣿⡄⠀⠀⠀⣦⣿⣢⣴⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠛⢻⣦⣶⣮⡹⠿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣢⠹⡟⢆⠻⠧⣴⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠙⠛⢲⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢢⡄⣇⣠⣴⣷⣴⣅⣏⠀⣛⣥⠴⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣟⣳⠦⡻⡿⠟⣦⣻⣿⣿⣿⡽⢟⣛⡋⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⣇⠘⢳⣰⣽⣿⣿⡟⠋⣕⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣷⢒⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣶⣄⡩⡟⠻⡿⢇⡸⣝⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠚⠿⣷⣷⣧⣰⣎⠳⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠠⢺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠃⠀⢀⣵⣿⣿⣆⠀⠐⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠐⡲⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣴⡇⠀⡀⠀⢣⠀⣀⣶⡿
⠀⠀⠀⠀⠀⠀⢀⠏⡠⢀⣀⣼⠀⣀⠀⠀⠀⠀⠁⠲⣄⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠘⠰⣿⣿⣿⣿⡴⢒⡇⠀⢸⣿⣿⡿⠳
⠀⠀⠀⠀⠀⢀⠂⠰⣤⢼⣯⣿⣞⣿⣏⡲⣦⣤⡶⡀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⢄⡁⠹⣿⣿⣿⣿⠛⣁⠀⢸⣿⠋⡀
⠀⠀⠀⠀⠀⠃⢠⠰⣯⡠⠬⢥⣽⢏⠿⣯⣿⠿⡇⢀⠀⣿⡄⠀⠀⠀⠀⢻⣿⣶⣤⣼⣷⣤⣀⣠⣄⣤⣽⣿⣿⣤⣧⣴⡿⠉⠙⠁
⠀⠀⠀⠀⠀⡀⠀⠈⠙⢷⠒⣚⣩⣅⡶⢿⣿⡇⢀⠇⢰⣿⡇⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠱⣀⠀⠙⠻⠁⠙⢧⠞⠛⠂⠉⠁⣀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠈⠉⠉⠉⠛⠿⠿⣿⢸⠁⢀⣦⣄⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠄⢀⣀⣤⣶⣾⠟⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⠀⠀⠈⠉⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⣶⣿⣿⣿⣿⣿⣿⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠰⠤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⣀⡀⢀
⠀⠀⠀⠀⠀⢀⠔⣡⣾⣿⣿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣿⣿⣦⡑⠄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⡡⠀⠀⠀⠄⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠄⠀⠀⠠⡙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠸⠿⠋⠁⠊⠀⠀⠀⠀⠀⢄⢀⡀⠢⢀⠀⠈⠂⠀⠀⠀⠀⠈⠀⢀⠠⠂⢀⢀⠀⠀⠀⠀⠀⠈⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢠⠀⠄⠀⠀⠄⢠⠀⠀⠠⠤⡄⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⢀⣀⠠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠋⠉⠑⢄⠀⠀⠀⢀⠜⠛⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣶⣶⣶⡎⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⣀⠠⣿⣿⣿⣿⣧⡀⠀⠀⢀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣼⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠙⡟⠉⠀⠈⠢⣾⡿⠀⠀⡀⠮⠒⠑⠀⠀⠂⠁⠁⠀⢠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⣄⠀⠀⢀⣐⠀⠀⠀⢀⡀⠋⢄⠢⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣿⣿⣿⣦⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠟⠻⠛⠙⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⣴⡶⣿⣿⠉⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣤⣤⣶⣿⠙⠿⠃⠉⠉⠀⠀⠄⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠝⠙⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⢿⣿⣿⣿⣿⣷⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢾⣭⢿⣷⡈⣿⢿⣿⣿⣿⣿⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣳⣿⣷⣿⣿⣿⣿⣿⣿⣏⢋⡙⠛⠻⢿⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡘⠤⠀⠀⠈⠈⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⠟⠛⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⢄⠀⠀⠀⠀⠀⠈⠓⠙⠇
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠫⠄⠀⠁⠰⢿⣿⣿⣿⣿⣛⣽⣿⣿⣿⣿⡧⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⣀⣠⡄
⠀⠀⠀⠀⣴⣾⣿⣿⠁⠀⠒⠀⠈⢡⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣗⣿⡆⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠉⣉⠁⠔
⠀⠀⠀⠀⠈⠟⠛⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠻⡝⠁⢸⣿⡿⠷⠀⠀⠀⢀⡄⢈⡀⠠⠀⣠⣄⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⢟⡇⢰⣇⢰⣧⡉⡀⠐⠰⠞⠊⠁⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢛⣿⣿⣦⠀⠀⠀⠀⠀⠀⠴⠖⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠛⢙⠋⣻⣿⣫⣽⣿⣿⣿⣿⣿⡿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣿⣆⠀⠀⣀⣠⣤⣴⣞⣝⣿⣿⡷⣿⡿⡿⠛⢏⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢼⢦⡿⢛⣻⣿⣿⣿⣿⠿⠏⠉⠁⠉⠀⠀⠀⠈⠀⠀⢀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⠿⠛⠉⠁⠙⠀⠀⠀⠀⠀⠀⠈⠃⠹⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠿⠛⠋⠉⠃⣿⠀⢫⡦⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠬⠁

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