Maritime Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠤⠤⠤⠤⠤⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢒⠒⣒⡒⠒⠒⠒⠒⠒⠒⠂⠀⠒⠒⠒⠒⠂⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⣄⣀⠈
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢉⡜⢨⣿⣿⣾⣷⣄⡰⢦⡀⠀⠀⠀⠠⢤⠒⠒⠂⠀⠀⠂⣀⡀⠀⢀⡀⠠⠀⣀⡀
⠒⢐⣉⣉⡀⠈⠤⠈⠒⠋⠀⠈⠒⠒⡿⡟⠠⠚⣫⢛⣽⣿⡿⣿⣟⠀⢩⣀⠀⠀⠀⠀⠈⠛⠾⣿⣷⣾⣿⣿⣿⣾⣷⣦⣤⠷⣲⢦⣬
⠚⠿⠟⠻⠟⠋⠐⠞⡿⢿⣿⣿⠶⠚⢚⡡⠔⣉⢶⣿⠿⣫⣾⣿⣿⣄⠀⠈⠛⢶⡀⠀⠠⡀⡀⠀⠉⠛⠻⠿⢿⣿⣿⣷⣯⣴⣵⣶⣤
⠀⠀⣀⣴⠾⠟⣲⡦⠤⠒⢀⡠⠔⣚⠥⢒⣩⠶⣻⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⢓⣻⠶⣷⢛⠛⠳⠒⠓⠤⠤⠂⠤⣤⠀⢀⡄⡀⠀⣠
⠩⠭⠔⣒⣨⡭⠖⣚⣛⣯⠵⢒⣉⡤⢞⡩⢶⡿⢿⣿⠟⣫⡿⣿⣿⣿⣜⢴⡤⡀⠀⠁⠬⢉⡳⠬⠷⠟⡭⢶⢤⡀⢤⡾⠘⠛⢯⡯⠶
⣒⣂⣭⡥⠶⡞⠛⣋⣭⠴⣚⣭⢶⣊⣥⣾⡷⠞⢩⡶⢟⢵⣾⢟⣿⣾⢿⣶⣵⡎⠠⠀⢤⡀⠁⠀⢀⠀⢀⡭⢀⢀⠀⠀⢀⠀⠐⢋⣉
⢥⣶⣶⠿⢷⣚⣭⠵⣒⣿⠿⣛⡭⠗⢫⡵⢞⡪⢵⣪⠟⢫⣾⠿⣛⣵⠿⣫⣿⠿⣿⣶⡴⣶⣦⣶⣶⣾⣶⣾⣧⣠⣥⣭⣭⣾⠟⣻⢁
⠹⠶⠞⢛⡩⠅⣒⣫⠭⣒⡯⣀⣈⠥⢈⣠⡴⠞⣋⣴⠶⢋⡵⠞⣋⣵⡿⢋⣡⣾⠟⣋⠔⣫⡿⢛⠕⢛⣿⣿⣋⣿⣿⣿⠟⣫⠟⢃⣞
⣶⣶⡶⢶⠶⣿⣷⣶⣿⣶⣾⣿⣶⣿⣿⣷⣶⣿⣿⣶⣶⣾⣶⣾⣿⣷⣶⣿⣿⣶⣾⣷⣿⣷⣶⣷⣶⣿⣿⣿⣷⣿⣿⣶⣾⣷⣶⣾⣿
⣿⣿⣦⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⡀⠀⠄⠀⠀⠘⠀⢇⠀⠀⠏⢛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣆⢨⣇⠀⠀⡇⠸⡄⠀⠀⠸⡆⠙⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⣇⠉⠉⠈⠉⠀⠐⢢⣀⡇⠀⠀⠀⠹⡄⠘⣆⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠸⣿⠅⠐⠶⠂⠀⠷⠀⠈⢧⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣆⠀⠘⢧⣀⣀⣘⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡿⠀⠀⠀⠀⠔⢰⣞⣿⠀⢄⠀⠀⠀⠀⢻⡤⢠⠴⠼⢿⣷⣄⣈⡹⠒
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠒⢹⡧⠖⢲⡏⣿⠈⠉⠙⠏⠘⣊⣥⢤⡴⠂⢸⡣⠂⣀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡟⠶⠶⡾⡷⡶⠮⣀⣷⣀⣤⣼⣤⣥⠿⠾⠿⣷⣶⣿⣿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣧⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡉⣠⣴⣶⣶⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⡎⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣄⣸⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡄⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⡏⠉⣁⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠶⢾⣟⣿⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡃⠀⠀⠀⠀⠈⠣⣀⣠⣇⣀⡜⠃⠀⠀⠀⠀⠀⠀⠸⣭⣽⡿⠿⠃⠀⠀⠀⠀⠀⠀⠉⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⣀⣠⡤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠐⠆⠂⡗⠂⢔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡠⠊⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣽⣿⡿⠃⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⠴⠤⢞⢉⡦⠤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠢⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⠦⣀⣏⡀⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⣦⣠⣀⠀⣠⣤⡰⠂⠀⠀⠀⠀⠀⠀⢰⣶⠀⠂⠀⠀⠀⠀⠀⢠⣼⡉⠽⠍⣹⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿
⠀⠀⠀⠀⠀⠀⢸⠁⠙⡊⣛⣛⠁⢸⡆⠀⠀⠀⠀⣴⣿⢋⣭⢻⣷⣴⡆⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⣎⠻⣿⠁
⠀⠀⠀⠀⠀⠀⢈⡵⠻⠞⠉⠻⠷⢎⡀⠀⠀⠀⠀⠹⠿⢮⣯⡾⠿⠙⠃⠀⠀⠀⠹⢿⠂⢤⠄⢺⡿⠁⠀⠀⠀⠒⠲⢮⣟⣓⣨⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣮⠗⠀⠀⠀⠀⢀⠀⠀⣼⣿⡀⢀⡀⠀⠀⠀⠀⢠⠚⠈⢀⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠐⣄⣿⣿⣿⣿⣣⠔⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣄⣀⣈⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠛⠟⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⣆⠀⠀⠀⠀⠀⢫⡛⠿⠿⠿⣛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣤⣾⠟⢻⡟⠙⠋⠀⠀⢀⣠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⡄⠀⠀⠀⠈⡄⠘⠛⠛⠋⢇⣀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠀⠀⠀⠀⠀⢁⠐⣀⡴⠖⠛⢇⣤⣾⣿⣿⣿⢿⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⡴⣿⠍⠀⠋⣼⣿⣿⣧⣹⣿⣵⣿⠿⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠙⠿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⢠⣔⣤⣤⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⠀⠀⢰⡋⣴⣿⣯⣥⡌⠻⣄⡀
⠀⠀⠀⠀⡀⠀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⢁⣤⠀⣴⣷⣿⣿⣿⣿⣿⣿⠿⣀⡈⠙⠒⠒
⠀⠀⠀⠀⢉⡽⠟⠉⠋⠙⣛⡉⢉⡉⢉⡉⣁⠀⠀⠀⠴⠞⠋⠵⠟⢻⣿⣿⣿⣿⣿⣿⢿⢿⢟⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣵⣷⣌⡉⠥⠾⠿⣶⣤⣄⣄⣀⡀⠀⣈⣙⣛⡯⠅⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠈⠉⠉⠉⠓⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠁⢈⠟
⠀⠀⠀⠀⠀⠀⢀⢀⠄⡀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡠⠒⠠⠖⡠⠒⠈⠉⢀⣴⠕⠉
⠀⣘⠣⡀⠀⠀⠀⡎⠀⠈⢀⣈⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠔⠈⠀⠀⠀⢀⠁⠀⠀⠀⠈⠀⠐
⠀⠁⡀⠈⠢⣤⣀⢀⡴⣲⠿⠛⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡦⠀⠀⡄⠀⠀⠀⢀⠄⠂⠀⠀⠈⠀⠀⠀⢠⠎
⠀⠐⢌⠂⠀⢀⣀⢻⡿⠁⠀⠈⣠⠁⠉⠒⡉⠂⠀⠀⠀⠀⠙⠆⠀⠛⠀⢂⠙⠀⠀⠀⡠⠠⠀⢀⠤⠂⢐⠀⣰⠋
⠀⠀⠀⠈⠐⠂⠉⢻⡗⠀⢀⡄⠧⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⣀⠀⠄⠁⠀⠀⢀⠜⢡⠀⠀⣨⡾⠋⠁⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡄⢰⠀⠀⠀⠀⠀⠀⢬⠒⠖⠒⣉⠝⠚⠁⠠⠔⠁⠀⠀⢰⠃⠀⠀⡟⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⠥⠔⠂⠀⠀⠀⠀⠀⠀⡌⠀⡆⢸⠖⠩⠄⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠐⢸⡔⠀⢄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠆⠀⡰
⠀⠀⠀⢠⡾⠤⣌⢀⠀⠀⠸⣤⠸⠦⣐⣤⣀⠀⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢊⣥⡆⠀⠀⠀⣰⠁
⠀⠀⠐⠜⠀⠀⠹⠢⣉⡀⠀⠀⠁⠒⠠⢌⡩⠬⠅⠀⣤⣀⣀⣀⣤⣤⠤⡴⠖⠹⢀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢥⣀⠀⢈⠁⠠⣐⢚⡥⠾⠃⠀⡆⠠⠤⠖⠪⠀⠀⡆⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⣉⠈⠉⠐⠁⠀⠀⠀⠀⠀⢠⠀⢀⣤⣤⠤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠁

⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⡆⣤⠄⢀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠃⠬⠡⠌⠂⠘
⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠠⠤⢄⣀⠀⣼⡄⣁⣠⠤⠐⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⡤⡤⡤⠤⠴⢀
⠀⠀⠀⢸⣿⣿⣿⣿⢟⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢐⠷⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣯⣴⣿⡿⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⢀⣤⠀⣀⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢡⣄⣉⡤⠍⠤⡄⢛⡊⠅⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⣴⣷⠂⡏⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣷⣶⣶⡶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣸⣿⣿⠇⡷⠦⠀⠀⠀⢀⣤⡾⣻⣿⣿⣿⣿⢿⡳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣾⣶⠄⠀⠀⠀⠀⠀⢀⠄⠀⣰⣿⡿⢫⠞⠑⠀⠀⠀⠀⠻⣫⡾⣻⡟⣿⢹⡟⣎⢿⣮⡟⠁
⠀⠀⠀⠈⠛⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⠊⠙⠄⠀⠀⠀⠀⠀⠺⣿⣱⡟⣸⣿⢸⣇⢻⣮⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢿⠿⢾⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⠛⢶⣄
⠀⠀⠀⠀⠀⡠⢠⢠⠀⣷⠒⡔⢮⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢉⣤⣼⣿⣗⡀⠈⢣
⠀⠀⠀⠀⠈⡴⣣⠯⡦⢸⠰⡘⢦⡉⠁⠀⠀⠀⠀⠀⠀⠀⢤⡲⣿⣮⢿⡹⣷⣶⣤⣀⠀⠀⠀⠀⢻⣿⠿⠶⣦⠈⢻⡄⠀⠁
⠀⠀⠀⠀⢀⠼⢣⢧⡇⡇⣇⠹⣄⠉⠀⠀⠀⠐⠛⠻⢿⣿⣿⣿⣿⣿⣧⢷⡻⣿⣿⡿⠛⠁⠀⠀⠘⢿⣛⣋⣛⢷⣰⡷⢀
⠀⠀⠀⠀⠈⠢⠁⡾⠁⠇⠘⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣭⣷⠟⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⡀⠠⠄⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⢀⠀⠀⢰⡅⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠐⠂⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⡇⠀⠀⠀⣀⣀⠀⢀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡉⣥⠀⠒⠁⠀⠀⠉⠀⠸⠉⠉⠁⢠⠅
⠀⠀⠀⠀⠀⠀⢀⣤⠄⠋⠓⢀⠎⠠⠈⢁⡀⠀⠀⠀⢀⣌⠓⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣧⠀⠀⠀⠀⢀⠆⠀⠀⠀⠰⠄⠠⠤⣀
⠀⠀⠀⠀⣠⣤⠸⠀⠀⠀⠈⠉⠀⢀⠈⠉⢁⡀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡈⠈⠀⠀⠉⠀⠀⠀⠀⠑⢀⠀⠀⠀⠠⡄⠨
⠀⠀⣠⠂⠀⠀⡀⢀⣠⡀⠀⠠⠆⠤⠲⠛⢿⠓⠚⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣐⡂⢀⠈⠀⠴⠀⠀⠀⠀⠀⠀⠊⡀⠀⠉⠀⠃
⠀⠀⠉⠀⠀⢒⠀⠻⠤⠇⠀⣀⡤⣤⡐⠆⠀⠀⠀⠀⠠⠤⠚⠉⠀⠉⠉⠀⠝⠁⠉⠀⠈⣡⠶⠀⠤⠀⠀⠀⠀⠀⠀⢘⡂
⠀⠀⠠⠀⠀⠈⠁⠁⠀⠀⠘⠁⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⡀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠙⠀⠀⠀⢀⠀⠀⠒⢉⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡼⡆⣀⠀⡙⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢒⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡟⠁
⠀⠀⠗⠀⠈⠁⠙⠉⠛⠃⠀⣀⣶⠁⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⣂⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⢠⡀⠀⠀⠠⠛⠗⢠⠟⠁
⠀⠀⠀⢠⣀⠱⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⢠⡾⠆⢀⠴⠲⠛⠛⠁⠀⠚⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠛⣄⠀⢱⠀⠛⠉⠀⡴
⠀⠀⣶⠆⠉⠀⠀⠀⠀⡜⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠐⠂⣾⠒⠐⠀⣠⣤⣬⡤⠀⠀⠀⠤⠀⠀⠆⠀⠀⣀⠀⠀⠀⠀⢸⡧
⠀⠀⢤⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢁⠀⢀⣠⠄⢀⡇⠈⠀⣀⣀⠷⠴⢾⣃⡀⠿⠤⠾⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠁⠂⠀⠀⠀⠠⠄
⠀⠀⡄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣠⠆⠀⠀⠉⠀⠀⠁⠠⠄⠀⢀⣀⣀⢠⠀⠀⣀⠀⠀⠠⠖⠈⠓⠛⠁⠀⢀⡤⠆⢠⣤⡀⠀⠀⠃⠀⢀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣴⢄⣵⠔⠾⠃⠉⣯⠉⢷⡔⠉⢹⠛⢳⡆⠀⠀⠀⠀⠀⣤⡼⠦⡤⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠼⠁⢀⡴⠛⠇⠀⣽⠀⠀⠤⠐⠛⠛⢛⡶⠤⠚⠃⠻⠿⢲⣿⡦⠶⣆⠀⠀⣴⠃⠀⠀⠀⠀⠀⠉⠀⠀⡁⠸⡿⠳
⠀⠀⠒⠀⠀⠀⡆⠀⡀⠀⠀⠀⠄⠀⣠⠊⠀⠀⣧⡄⣿⠀⠧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⠸⠗⠿⠲⢿⣂⣸⣏⣆⢰⣄⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⡁⠀⢃⠀⠀⠀⢠⠞⠁⢀⠔⠀⠀⡄⣻⠁⠴⠂⢸⡟⠙⣄⠤⠐⡾⡗⢰⠟⠀⠀⠀⠀⠈⠉⠁⠈⡀⢸⠃⡄⠉⠀⡀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠁⠀⠈⢡⠀⢠⠆⢀⠜⠁⠀⠀⠀⠀⣹⡀⣴⠛⠛⠋⠈⠉⠀⠀⠈⠃⠸⡛⣖⢲⣷⣷⡶⢂⢶⣎⡅⢸⠀⢁⠀⠀⠡⠀⣹⡇
⠀⠀⠀⠀⠀⠀⢰⣀⡀⠀⣀⠋⠀⠋⠀⠀⣀⠄⠊⠉⢿⠉⠁⠀⠀⠀⢀⠆⠀⠀⠀⠀⢰⡗⠉⠉⠉⢋⡵⠾⠺⠗⠋⢸⠀⣞⠀⠀⢠⡦⢸⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠛⠦⢤⣄⠁⠀⠀⠀⣺⣷⠒⢸⡗⣶⠻⣶⠄⣶⢶⢄⡘⣧⣀⠀⠀⢈⣀⣀⣀⣤⣴⣾⣇⣭⣥⣄⠁⢀⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠀⠛⠲⡄⠀⣈⠛⠗⠲⢞⡛⡛⢿⣻⡗⠟⠠⠧⢴⡷⣼⣼⣧⣹⣧⣼⣲⣿⣇⠇⠾⠛⠟⡿⠿⢏⡹⠋⠀⡼⠃⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⣠⣀⠂⠀⠀⠀⠘⢠⠀⠀⢀⣷⡄⢀⣀⢨⠿⠶⠚⠱⠟⠒⠚⠚⠛⠛⠛⠚⠛⠽⠁⠈⣉⠅⣰⠚⠛⠃⠘⠁⠀⡴⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠎⡀⢉⡆⠀⠀⠀⠀⠀⢃⠀⠊⠉⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠒⠂⠀⠀⠙⢿⣤⡀⠘⠄⠀⠀⢠⡾⢁⠠⠁⠀⡀⠀⡀⡅
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠓⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⡀⠀⡀⠀⠀⠦⠀⠀⠱⡀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢀⣈⡀⠀⠀⠀⡰⠋⠀⣀⠀⠀⠟
⠀⠀⠀⠀⠀⢀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠒⢄⡀⠀⠈⠇⠀⠀⠠⡀⠀⠿⠀⠀⠀⠁⠀⠀⣠⠀⠀⠀⠁⢠⣿⠀⠘⠀⠠⠊⠙⠀⠀⢰
⠀⠀⠀⠀⠠⠈⠐⠀⠀⠀⠀⠀⢀⡀⠀⠁⠠⠄⠀⠀⠁⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠀⠀⡠⠾⠏⠀⠀⠀⢠⠀⡀⠘⠛⠋
⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠒⠂⠀⠀⠀⠇⠈⠷⠇
⠀⠀⠁⠀⣀⠀⠀⠀⠀⢀⢀⡑⠠⣈⠈⡃⠀⣀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠛⠒⠒⠒⠂⠿⡉⠀⠁⠀⠀⠀⢠⠶⢶⢀⢠⡤⣤
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠋⠉⢁⠴⠀⠸⣅⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⡶⠀⠀⣠⡄⠐⠀⠀⠀⣈⠀⡄⠸⠟⠀⠘⠁
⠀⠀⠈⠠⠞⠁⠀⠀⢠⠆⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢌⠉⢳⡄⠀⠀⠀⠀⠤⠞⠋⢾⡲⠲⣄⠀⢁⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠒⠂⠀⠀⢀⠀⠠⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⢀⡿⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠈⠶⠾⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠤⠄⠀⠂
⠀⠀⠀⠄⠀⠀⢀⠀⢠⢚⣶⠄⠃⠀⠀⠀⠀⠴⠁⠆⠀⠀⠤⠀⠄⠛⠁⠀⠀⠀⠀⠀⠠⠟⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⠦⠀⢀
⠀⠀⠀⠀⣤⢀⡁⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠈⡀⠀⠐⠀⠀⠈⠦⠔⣂⠀⠈⠔⢀⠀⢀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠩⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠁⠊⠁⠀⠐⠃⠈⢋⡐⠃⠀⠠⠄⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠖⠛⠀⠀⠠⠀⠈⠀⠃⠀⠆⢀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠒⠐⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠐⠛⠂

⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠟⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠿⠿⠟⠛⠛⠛⠻⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⡀⠀⣸⠿⠿⠿⠄⠀⠀⣠⣶⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣿⣿⣿⣿⣧⣴⣧⠙⠜⢡⣷⣶⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠿⠛⠋⠁⠹⣿⡏⢱⣿⣺⣷⣉⣻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠉⠉⠉⣉⣩⣽⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣶⣦⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣠⣤⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠀⠀⠀⣀⣀⣤⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠉⠉⠉⢀⣠⠄⢀⣴⣶⣶⣶⣶⡶⠋⠘⠛⠛⠛⠛⠉
⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣿⣿⠋⠀⢀⣴⣶⣿⣿⣿⣿⣿⡿⠃⠀⠈⠉⠉⡉⠉⠉⢀⣠⣤⣤⣤⣴⠃
⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠁⣠⣴⣶⣶⣶⠿⠁⠀⠐⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⣛⣻⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠀⠄⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠄⡄⠁
⠀⠀⠀⠀⠀⠀⣀⡀⡠⠤⠤⠁⠀⠀⢤⣀⠶⠂⠀⠀⠀⠀⠀⡜⠩⠧⠀⡁⠀⠀⡠⠫⠍⠤⠃
⠀⠀⠀⣀⡔⠉⠀⠒⠉⡀⠲⡂⠀⡀⠷⠀⠀⠀⠀⠀⠀⠒⠂⠃⠀⠀⠂⠁⠄⠀⠈⢀⠀⠄
⠀⢰⠀⢀⠄⡠⡀⠠⠀⠐⠚⠟⠒⠀⠀⢀⡀⠀⠤⣄⠤⠂⣀⠀⠂⠀⠀⠀⠀⢅⠈⠈⠁
⠀⢈⠀⠀⠄⠀⠁⠔⠊⠚⠁⣀⠔⠊⠉⠀⠤⠀⠀⠁⠀⠀⠰⠉⠀⠀⠀⢠⠤⠜⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⡰⣆⡀⡓⠂⢀⠔⠈⠀⠀⠀⠰⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢀⠀⠈⡤⡀⠀⠀⠀⣀⠀⣘⠁
⠀⠈⡀⢀⠀⠈⠁⡐⠋⠀⠀⠀⠀⢠⡇⢀⠠⠧⡄⠠⠴⠲⠂⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢸⠩⠼⢁
⠀⢰⠋⠉⠀⢀⡜⠀⠀⡀⠀⠀⠀⠀⢠⠆⠂⡖⢀⡀⣤⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈
⠀⢸⠀⠀⢀⠎⠀⠀⠀⣰⠀⠰⠂⢸⠀⠠⠒⠉⠙⠒⠈⠈⠁⠀⠅⡤⠀⠀⣀⠀⡁⠀⠀⠀⠀⠀⠐
⠀⠘⠀⠀⠰⠀⠀⠀⠀⣷⠀⢀⡤⢼⠄⠖⠐⣇⣳⠄⢸⠚⣆⠀⠀⠀⠠⠴⠤⠀⠈⠀⢀⡀⠀⣠⣀⢸
⠀⢈⠀⠀⡇⠀⠀⠀⠐⠁⡠⠋⡅⢸⠂⡔⠀⠉⠉⠃⠈⠉⠉⢩⣿⣙⡄⢠⠇⢀⡀⠀⠀⠀⢠⠘⠛⠁
⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⠈⢙⡏⠕⢐⣖⣂⠀⢴⣆⠾⠀⠀⠀⠈⠉⡉⠈⠃⠀⠀⠀⣸⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠐⠠⠀⠄⠀⠀⠀⢤⡇⠏⠈⠉⠁⠂⠈⠉⣁⠄⠿⠿⣀⣿⠡⢰⠀⠀⠀⡀⢿⡀⠠
⠀⠀⠀⠀⠉⠏⠀⠒⠲⠤⣄⠁⠀⢸⣧⡂⡶⡔⣷⢀⠤⢄⣣⡀⠀⢈⣀⣀⣤⣼⣚⣦⡐⠁⡟⠀⢀⣀
⠀⠀⠀⠀⠀⠑⠀⠈⠑⠄⠀⠙⢙⠹⠽⢿⣟⡫⣼⣾⣾⣾⣿⢷⣿⠿⠦⠞⣉⣿⡯⡏⠀⠌
⠀⠀⠀⢀⡄⠁⡆⠀⠀⠈⠂⠐⠛⠷⠂⠉⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠺⣅⠁⠄⠀⢀⠎⠉⠀⠀⢀⢺
⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⢄⠀⠂⠀⠱⡀⠀⠀⠀⠀⠀⠀⠤⢀⠀⡰⠃⠠⠀⠘⠁
⠀⠀⠐⡐⠀⠁⠀⠀⠐⠀⠂⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠈⠀⠀⠀⠿⠁⠀⢤⠀⠠⠒⠁
⠀⠠⠂⢀⠀⠀⠀⠀⠁⠀⡀⠀⠈⠐⠀⠄⡀⠀⢀⠀⠀⢀⡀⠀⠀⠒⠉⢊⠀⠀⠈⠈⠋
⠀⢀⠄⠀⠀⠀⠄⠒⡨⠠⣆⠐⠀⠀⠀⠀⠂⠠⠀⠀⢠⠀⢀⡀⠀⠀⠘⠉⡁⡾⠙⠃
⠀⠈⠐⠃⠀⠠⠆⢠⠃⠀⠀⠀⠀⠀⠀⠐⣍⢣⠀⠀⠀⠒⠋⠳⠖⢦⢀⡔⠄⠀⠀⠀⠀⠀⢀⠄
⠀⢀⡠⠀⠀⠀⢠⡀⠢⠀⠀⢀⠖⠀⠀⡀⠀⠞⠀⠀⠀⠀⠴⠀⠐⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠠⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠐⢐⡦⡀⠂⠄⠀⠁⠀⠀⠀⠀⠀⠐⠋⠈⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⢢⠤⠀⠀⡀⠐⠀⠨⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢴⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢠⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠐⣠⣧⣤⢦⣤⣧⣤⣶⣶⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⣿⣿⣴⠾⢛⣿⣷⣿⣯⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠛⣿⣿⣿⡻⢁⣠⠾⢛⣿⣿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⣿⡟⣠⡴⢛⣥⣨⠖⣉⣠⣴⣿⣿⣷⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⣿⣟⣿⣷⣦⣿⣛⣀⣾⠿⢛⣛⣡⣾⣿⣿⣇⡀⠠
⠀⠀⠘⠷⣴⣦⣤⣤⣶⣶⣌⣻⣿⣷⣿⡉⠹⠿⠛⠋⠁⠀⠀⠉⠉⠉⠛⠙⢿⣯⣤⡀
⠀⠀⠀⠀⠀⠈⣻⡭⠋⠛⠛⠋⠉⠛⠛⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢀⡀


⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⣀⣀⣉⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⡴⢻⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠜⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠏⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⣀⣤⠞⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠛⣹⡟⠁⠀⠀⣼⠛⠋⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢠⡟⠀⠀⠀⢸⡇⠀⠀⠀⣤⣼⣿⣷⠦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⠀⢠⡟⠀⠀⠀⠀⢻⡿⠿⠟⢻⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡄⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⠞⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣤⢶⣶⠋
⠀⠀⠀⠀⠀⢀⡞⠁⢠⡟⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡾⠞⠛⠉⠙⠀⠀⣿⠀⢀⡀
⠀⠀⠀⠀⢀⡾⠁⢀⡾⠀⣠⠀⢠⡇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⣼⠃⠀⢸⠃⣼⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣾⣿⣶⣇⠤⠄
⠀⠀⠀⢠⡟⠀⠀⣿⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠿⠷⠾⠖⢲⡟⠋⠉⠉⠉⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⣠⣤⣤⢴⣶⡗⠊⠉⠉⠁
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⡿⠛⣻⠛⠹⠋⠉⠉⠰⠁⡞⠁
⣄⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣰⠋⠀⠐⠁⠀⠀⠀⠀⠀⠀⢸⣧⢠⡤⠤
⠈⢳⣄⠸⣿⣷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠙⢷⣌⠻⣿⣷⣄⡀⠀⠀⠘⣷⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣆
⠀⠀⠀⠀⠻⣷⣌⠙⢿⣿⣶⣿⣿⣿⡆⠀⠀⠀⠙⠿⣶⣶⣿⠶⣿⡟⢿⡟⠛⠛⠛⠟⠛⠛⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠈⠛⠿⣿⣿⣿⡉⢳⠐⠲⡤⢬⣀⣈⡂⢿⡅⠈⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠠⠂⠠⡉⠛⠷⣼⣀⣸⡧⢸⣇⣹⠇⢾⣇⡀⠰⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⣀⠀⠀⠈⠉⠛⠻⠧⣼⣄⣻⠀⠉⠒⣯⣥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡂⢻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⣎⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⢲⣦⣤
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠀⠒⢿⣿⣧⠻⣄⣠⣄⡀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠂⠀⠉⣛⠛⠻⣻⡿
⠀⠀⠒⠀⠒⠈⣨⡅⣠⡦⣶⠊⠀⠈⠙⢷⡿⠁⠠⠙⠒⠛⠶⠦⠄⡀⠀⠐⢂⠀⢒⣈⢉⠓⢂⣿⠇⠀⠂
⠀⠀⠀⠀⠐⠶⠋⠀⠙⢥⠀⣴⣦⣤⣄⣀⣁⣤⣠⡀⠀⢠⡄⢀⣤⡤⡀⠀⠈⣉⣉⣾⣷⣾⠟⢋⣀⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⡩⠟⠋⢩⣭⣀⣻⣿⡯⣿⠈⠳⠌⠛⠿⣯⠭⠭⠀⡁⠙⠛⠓⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⢄⠠⠊⢔⡠⠄⠺⣏⠑⠀⠀⠉⠛⠂⠀⠀⠒⠀⢠⠦⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶⠞⢯⠀⠀⢸⠀⠀⠀⠀⢀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣀⡀⠸⠔⠖⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠊⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⣀⣀⣠⣤⠤⢤⠄⠰⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣾⠟⢛⠉⣄⣿⣤⣼⡶⠚⠉⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣭⡥⠼⢶⡿⠿⠟⠉⡇⣽⠹⠄⠀⢀⣠⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠀⠁⠀⠀⢀⣀⠀⠻⠀⠈⠿⠛⡛⣻⡿⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣤⣶⣶⠟⠛⢉⣹⠀⠀⠀⠀⠈⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣭⠀⠀⠀⠀⠀⠈⢦⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠉⠹⠉⠙⠛⢿⠀⠀⡀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡀⠈⠀⠀⠀⠠⠀⣸⠀⠀⢠⣰⢃⣀⣄⣀⣀⡐⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⡄⠈⠉⢰⣶⡖⠊⠽⠿⢿⣦⣈⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⣧⠀⠀⠀⢸⣿⣿⠘⠧⣤⣄⣘⣿⣿⣦⡀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⡆⠀⠀⢰⣿⣿⠀⠀⠸⣿⣿⡄⠉⠉⠙⣷⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠣⠄⢸⣆⢸⣷⡐⠀⢸⣿⣿⣾⣤⠤⠽⣿⣿⡦⣢⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠰⠖⠻⠷⣦⠼⢿⣛⡛⠛⠒⣿⣿⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⣷⣞⣿⣿⡿⠟⠛⢉⠠⠀⠈⢡⡀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣿⡁⠀⠀⢀⡈⠁⠚⠻⠃⣰⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⢤⣽⣯⡭⣠⣶⡉⠛⠛⣛⣀⣃⣀⣠⣤⣦⣤⣤⣥⣴⠶⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠹⠿⠋⠓⠰⠿⠯⠭⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡷⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠘⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣰⠲⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢦⠠⠋⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢶⣤⣄⣀⠀⢰⣟⣻⣟⣿⣶⣶⣶⣤⣼⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⣼⣩⡙⠲⢾⣽⡏⢉⣛⣿⣿⣏⡙⢻⠿⡇⣤⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⣿⣿⣎⠁⠀⠀⠾⠯⠉⢻⠿⠿⢿⣯⣿⣿⣿⣷⣷⠀⠀⠈⢇⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣶⣿⣖⠤⢤⣀⠀⠀⠀⠁⠒⠦⠤⢭⣭⣽⣿⡿⠶⠶⠛⣾⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢻⣿⣿⣿⣿⣯⡀⠀⠀⠈⠀⠐⢢⠤⢤⣒⣤⡿⠶⣿⣿⣶⣾⣏⣩⣝⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠁⣀⡘⣿⣿⣿⣿⣷⠦⠤⢀⡒⢠⠄⢤⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⡠⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⡟⣶⣶⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⣤⡀⠻⣿⣿⣿⠃⢀⠴⠳⠾⠛⠿⢿⣿⡿⣽⣿⣿⣿⣿⡿⢟⣿⢲⣿⣿⣿⠃⣠⣶⣶⣾⣷⣤⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢙⣆⠘⣯⠁⣀⠀⣠⣶⡷⢎⣀⡀⢰⣶⣷⣤⣈⠻⠿⠿⢿⣯⣤⣶⡇⠈⣠⣽⣿⣿⣿⣷⣭⣭⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⣴⣶⣿⣷⢶⣶⣤⣄⠀⠀⠀⣀⣠⣴⣾⣿⣤⡏⠰⣿⣿⣿⣿⠀⢦⣭⣥⣾⣯⡙⠿⣿⣯⣝⣿⣿⣿⣿⠿⠠⣽⣿⣿⣟⣯⣥⣶⣶⣭⣭⣥
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡿⠶⣶⣿⣿⣿⣿⣿⣿⠇⣴⣿⣿⣿⣿⡷⠘⠿⠟⠛⠿⣟⣶⣮⣭⣭⣽⣿⣭⣤⣶⢤⣬⣽⣿⣿⣿⡿⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⠏⣠⣔⣶⣿⣟⣿⣿⣿⠋⠀⢿⣿⣿⣿⣿⣐⣒⡓⠀⠀⠀⠀⠈⠉⠙⠛⠛⠟⠛⠋⠁⠀⠀⠁⠀⠉
⠀⠀⢀⣤⣤⣤⣼⣯⣭⣭⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠕⠀⢻⣿⣿⣿⣿⣿⣛⡂
⠀⠀⠀⠉⠉⠉⠉⠛⠛⢿⣿⣿⡿⠟⣋⣝⠻⠋⠉⠀⠀⠀⠀⠀⠙⠻⢿⡿⢿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡵⠀⠀⠀⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠠⠊⠀⠀⠀⠀⠀⢸⠄⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⡜⠀⠀⠀⠀⠀⠀⣀⡾⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⠞⠀⠀⠀⢀⡴⠞⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⠋⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⠷⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣤⣀⣠⠤⠔⠶⢾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠿⠶⠊⠛⠉⡋
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠂
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⢿⠷⠒⠛
⠀⠀⠀⠀⢀⠏⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢤⠶⠄⠀⢺⡟⠋⠀⠀⠀⠀⠀⠸⠆⠀⢀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡀⣀⣠⣤⣤⠴⣶⡶⠖⠛⠛⠉⠉
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠒⠶⠂⠉⠙⠉⠉⠉⠀⠀⢀⡜⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⢶⡀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠾⠛⠚⠛⠁
⠀⠻⣶⣄⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠈⢿⡷⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠙⣜⢳⡈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣹⣦⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠈⠳⡙⢦⡀⠉⠲⣄⣀⣀⣀⣀⣀⣷⣀⣀⡀⠀⠀⠛⠓⠶⠶⠾⠻⡛⢻⡟⠉⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡻⣄⠠⠈⠙⠫⢌⡉⡉⠙⣏⠉⢻⠛⠓⡶⢶⣶⣠⣀⣀⠉⢸⡇⠀⠸⣷⠄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣜⢧⡀⠀⠀⠈⠙⠛⠷⢉⣄⣸⠀⢀⣿⣀⣸⠋⠙⡿⠳⢾⣇⠀⠀⠰⠾⠄⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡳⣄⠀⠀⠀⠀⠀⠀⠉⠛⠱⠶⢿⣠⣽⣟⠁⡇⠀⢸⡏⢻⢦⣄⠲⠖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⢧⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠹⠷⢤⣼⡇⢨⢸⠁⠛⣷⣟⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⢳⡌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢾⣅⡀⠺⢻⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡹⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣾⣿⣿⣿⣿⣿⢶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⢘⣷⡙⣆⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠀⠀⠈⠠⠄⠉⠿⠿⠿⢿⣾⣿⣿⡇
⠀⠀⢀⡀⠀⢀⡠⢄⠲⠏⠁⣀⠀⣀⠀⠤⠙⠿⣿⣆⢹⣷⠟⠛⠳⣄⣀⣀⡀⠀⠀⠀⢀⠈⠀⠂⣠⠀⠀⠀⣀⠀⢀⡀⡀⣠⣿⠁⢀
⠀⠀⠀⠉⠀⠀⠁⢀⣼⡆⢠⣾⠿⠿⠇⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠉⠀⠙⠛⠋⠰⢄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠆⠈⣃⣿⡟⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠺⠿⠉⠒⠊⠙⣄⣠⡴⣶⣤⣤⣄⣀⣀⢀⣦⣀⣀⡀⠀⢀⡀⠀⢀⣤⡴⣇⠀⠀⠀⠀⣀⣀⣠⣵⣶⣾⠿⢏⣩⡉⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢿⡟⢀⢝⣻⡿⠛⠚⣉⡉⠁⢶⣿⣟⢻⣏⠉⢷⣌⠛⢿⣿⣟⣛⡙⠉⠑⠀⠍⠲⣾⡿⠬⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠤⠤⠤⠾⠿⠛⠁⢡⠀⢀⡐⠩⣛⣷⠻⢟⠣⠩⠙⡮⢦⣄⢉⠀⠂⢉⡉⠍⠁⡤⣀⢈⡀⠤⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣄⡀⠢⠈⠂⠈⠳⠔⠃⠈⠙⢷⣄⣀⣀⠀⠀⠁⠀⠉⠓⠂⠀⠀⢦⠘⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢄⡀
⠀⠀⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡿⣙⢒⡹⠁⠈
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠰⠦⣤⣤⣀⣈⣀⣀⣀⣀
⠀⣿⣿⣿⣿⣿⢯⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠁⢸⣿⡇⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠍⠉⠁⠈
⠀⢻⣿⣿⣿⢷⣫⣾⣿⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠙⠿⠿⠿⠿⠟⠃⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠐⣀⠒⢀⡙⣊⣉⠷⠟⡛⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣶⣶⡷⠬⠅⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡅⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⢀⣴⡆⢰⠋⠀⠀⠀⠀⠀⠀⣴⣦⣤⣶⣤⣶⣦
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⣿⣿⢘⠦⠀⠀⠀⠀⣠⣴⢿⣿⣿⣿⣿⣿⡷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⣶⠄⠀⠀⠀⠀⠀⢀⠀⠀⣼⣿⠿⣡⠚⠂⠀⠀⠀⠘⢟⣵⢿⡿⣿⢿⡟⣏⢿⣬⡟
⠀⠛⠷⠾⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠊⠳⠀⠀⠀⠀⠀⠰⣿⣫⣿⢡⣿⢸⣧⢻⣎⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡿⢾⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢠⠖⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠄⠊⠀⢀⢀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡙⢷⣄
⠀⠀⣠⢀⢤⡇⢸⠊⢎⠢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣴⡶⠿⣯⡀⠀⠳⡀
⠀⠀⠠⢫⢾⢷⢸⡎⣎⠳⣌⠀⠀⠀⠀⠀⠀⣀⣀⣮⣻⣿⣎⢷⢻⣷⣶⣤⣄⡀⠀⠀⠸⣿⢿⣿⣷⡄⠻⣆⠀⠁
⠀⠀⡔⢫⢏⡏⣿⢣⠘⢧⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣮⣯⣿⣿⠿⠛⠉⠀⠀⠀⠻⣯⣩⣭⡻⣼⡿⢀
⠀⠀⠀⠀⠸⠃⠻⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠲⠖⠓⠲⠲⠤⣀⠀⠀⠈⠁⠀⠄⠀⠐⠀⠄⢀⣀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠀⠀⠀⣰⠏⠁⢀⠀⠘⠂⠐⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢄⢀⡖
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠠⠀⠈⠄⠈⠱⡆⠀⠀⢀⠀⢠⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⢀⣿⠀⢀⠀⡀⠀⠀⠀⠋⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣢⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣤⣄⠛⡀⢀⣶⠞⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣠⠀⠀⠁⠇⠀⠀⢠⣰⠞⠁⠀⠀⠰⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⠀⣀⠀⠀⠈⠂⠀⠉⢓⠁⢶⢃⡤⠋⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠉⠐⠶⠶⠚⠚⠃⠀⠀⠀⢠⡀⠀⠠⠤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠠⠊⠷⢀⡀⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠀⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠒⠀⢈⢥⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢐⣀⡴⢀⣀⠀⠠⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⡀⠐⠚⡀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠄⠈⣉⡀⢏⢠⠄⠄⠀⢐⢨⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢵⠀⠘⠃⠀⠸⠇⠀⠀⠀⣀⠀⠀⠄⠀⠈⠁⠂⡠⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⣆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⠒⠂
⠀⠀⠀⢀⡀⠀⣬⣔⡶⠶⠒⠇⠐⠾⠉⠇⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠈⢣⠀⣹⠈⠁⠀⠀⠙⠀⠀⠀⠈⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡐⢤⠄⠤⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠦⡰⠐⠆⠀⣀⢰⣄⣤⠴⢦⡼⠁⠀⠀⠀⠁⠀⠀⠀⠠⠠⠀⠒⠀⠢⠀⠁
⠀⠀⠀⠉⠀⠁⠀⠚⢸⣿⠟⠃⣿⠃⣈⣭⠁⠀⠀⠀⠀⠄⠀⠀⡀⠀⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠴⠆⠀⠈⠀⠲⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⠀⠀⣀⠀⠀⠀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⡰⠂⣀⠚⠈⠃⠠⡄⢀⡄⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣈⣀⣤⡦⠐⡣⣄⣆⠠⡦⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠘⠳⡌⢷⡐⠻⠟⠤⠉⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣉⣀⡅⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠀⡀⢀⢀⣤⣤⣶⣧⣌⣿⢠⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠰⠆⠉⠀⠉⠀⠀⠀⠰⠆⠀⠀⢀⠀⠘⠠⡄⠀⠀⠉⠛⠉⠉⢯⡽⠛⠉⠊⠰⣸⠠⣦⣠⣷⣤⢀⣶⣦⠀⡀⠀⠐⠀⣊
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠃⠀⡘⠆⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⡍⠀⠋⠈⠈⠠⠉⠀⠁⠀⠀⠁⠀⠋⠓⠴⠭⢸⣥⣾⣷⣾⡄⠡⠀⠀⠀⢿
⠀⠀⠀⠀⢀⡀⢀⡐⠁⡞⠀⠉⠁⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⡄⠉⠀⠁⠀⠀⠀⠂⠀⠀⠉⠐⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠋⢹⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠉⠀⢴⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⢸⣄⠀⡀⡀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠲⠀⠀⡎⠀⠀⠈⠀⠐⠟⢻⡓⣤⡀⠰⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡤⠀⠀⠀⢀⣀⠐⠐⠀⡀⠧⠀⠡⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠖⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧⠀⠀⠦⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡟⡜⠅⠀⠀⠀⠀⠀⠀⠘⣦⡤⠄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠐⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⡆⠀⢠⠀⠀⠈⠹⠃⠀⠀⡤⠀⠀⢀⠘⠙⠻⠁⠀⠀⠢⡄⠠⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡘⢀
⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⢀⡀⠴⠒⠒⠀⠀⠀⠀⠀⠀⡉⠐⢶⡀⠒⠀⠀⣀⠀⠀⠈⠠⠄⠀⠁⠦⠀⠀⠀⠐⠠⠁
⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠔⠂⠠⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠋⢻⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣤⣀⣤⣴⣿⣿⣿⣿⣿⣿⣦⣤⣀⣠⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣿⣿⠿⠛⠛⠋⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣢⠤⢀⣀⠀⢀⡤⠀⠈⠀⠀⠀⠀⠺⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⡍⠁⠀⠈⠑⠾⢃⣄⠁⠀⣀⢀⡀⠀⠀⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣍⡁⠐⣄⢀⠕⠀⠀⠀⠀⠀⠀⠉⠓⣌⣡⣻⣿⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠾⠟⣀⣴⣶⢶⣦⣄⠀⠀⠀⢹⠁⠿⠿⢆⣄⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⠛⠸⢼⡿⢹⣄⠀⠃⠀⠀⠀⠀⠙⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⠔⢈⣀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠮⢷⠖⠈⠙⣦⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠓⢦⡄⣀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢲⣄⠋⠿⠀⢀⠂⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠙⢧⡈⠒⠻⣉⣾⠷⣲⡶⠂⣤⣤⣄⡀⠀⠀⠀⠀⠀⢸⠀⠈⠉⡅⠣⣨⣆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠙⠶⠊⠉⠙⠾⠟⠀⢦⣞⣽⡟⢹⡟⣦⣄⠀⠀⠨⡄⠁⠀⠈⠀⠈⠻⠓⠬⣱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⣸⠿⣿⡷⣾⠙⣾⡏⢑⣄⠀⠇⠀⠀⠀⢠⠀⠀⠀⠐⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⢰⠄⢤⣄⠀⢀⠞⠋⠀⠀⠉⠻⢤⣹⣧⣼⠀⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠨⠗⠊⠁⠠⠊⣰⣤⣀⠀⠀⠀⠀⠀⠉⠛⠎⠁⠁⢸⠛⠲⡀⡀⢰⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣤⡀⠀⠀⠀⣀⠤⠈⠛⠉⣠⠄⠀⢀⡄⠀⠀⠋⣧⡈⢉⡂⠠⣀⠀⠀⠀⠀⠈⠟⠾⣄⡀⢻⠃⢸⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢞⣀⠀⠤⢄⠀⠀⠀⢠⠎⠀⠀⠌⠀⠀⠁⠠⠬⠁⠪⡑⠦⡀⠀⠀⡀⠤⠀⠉⠐⣎⣛⠈⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠓⠚⠉⠲⢦⣀⡵⣄⡀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢦⡙⣆⠘⠑⠦⣀⠀⠀⣇⠈⠙⠒⠤⣄⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠔⠁⠀⠉⠀⠀⠀⢠⠃⠀⠈⠋⠐⠒⣤⣀⠀⠀⠀⠀⢻⣤⡀⠀⠈⢢⡀⡧⢀⣠⡤⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⡄⣠⣾⡅⠀⠀⣠⡄⡀⠀⢀⡏⠀⠀⠠⣀⠀⠀⡇⠁⠑⢄⠀⠀⠀⠀⠈⠒⢠⠀⡶⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⣟⢻⡄⠀⠀⠀⡴⠁⡆⠀⠀⠊⠀⠀⠀⠀⠈⠉⠒⠣⠄⡀⠈⠁⣠⢅⡀⠀⠀⠀⠀⠘⠒⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⡛⢧⠆⢀⡞⠀⠀⣇⡀⢀⣪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⠹⣀⠀⠁⠢⡀⠀⠘⠂⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠰⠁⠘⣶⠿⠶⡄⣤⣯⣀⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠂⠠⠌⢢⡁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣰⡔⢻⡴⣴⢾⣿⣯⠎⣽⣿⣄⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠠⢽⣟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣛⣶⣤⣿⣵⣾⣃⣯⣼⣿⣿⣿⣿⣿⠹⢹⣦⡀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⡴⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣄⣠⡔⣹⣶⣄⠀⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡄⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠢⠤⠐⠧⠸⠧⠤⠤⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿⡻⣃⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡵⢂⣠⣤⣾⣀⡀⠀⠀⢀⣤⣄⣈⠙⠛⠛⡻⠿⠛⠙⣛⣋⣀⣈⣫⣆⣄⡰⢶⣶⡶⠈⠛⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠿⠿⠛⠀⠈⠾⠿⠿⠛⠓⠂⠡⠄⠈⠁⠚⠛⠛⠛⠛⠋⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⡤⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⠁
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣤⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢤⡤⢚⣁⡈⠉⠛⠿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠈⢻⡿⠀⠉⠙⠉⠂⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠅⠃⢩⣦⣤⡤⠀⠀⢁⣤⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠤⠤⠤⠀⠠⣭⣭⠥⠖⠓⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣈⠉⠀⡀
⠀⠀⠀⣠⡴⠶⠒⣒⣦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡁⠀⠇⠀⠀⠀⠠⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠓⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡄⢀⡠⣦⡞⢁⢀⡀⢻⡿⣾⣸⣾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠏⠀⢡⣤⡶⣆⡄⠠⠷⠞⢡⡏⢙⣻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⢶⡄⠀⢸⠀⠈⠒⠐⠒⠑⠾⠶⠦⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⣄⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣴⣾⣿⣿⣦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠆⠀⠀⣶⣿⣿⣿⣭⣛⠻⡷⣿⡇⣆⢀
⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣮⣡⡄
⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣶⣿⠀⣄⠈⠛⠻⠿⣿⣿⡟⠋⠉⠙⠉⡿⣶⡇⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡲⡄⣽⡏⠀⠀⠀⠀⠀⠀⠉⠛⠹⣳⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⣿⣷⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢻⡾⢿⣷⡀⠀⠀⠀⠀⠀⢠⡠⠆⠀⠀⠀⠀⠀⣢⣄
⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠃⠋⠙⠛⠻⠉⠰⣌⠛⢶⣤⣤⣤⣶⣿⣅⣤⠀⠀⠀⠀⣾⣇⣄⣧⣄
⣿⣿⣿⣿⣿⣟⣋⣩⣥⣄⣀⣀⣠⣶⣾⣧⣴⣶⣾⣦⣀⠓⣄⡉⡛⠿⢿⣿⣿⣿⣿⣤⣠⣀⢨⣙⣛⣿⣿⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⣶⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠿⠿⠻⢿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣰⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⠄⢨⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣶⣶⣶⡄⡿⢿⣙⣇⢻⣿⣆⣿⣿⣿⣿⠇⠀⢀
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⠄⠀⠀⠀⠈⣿⣿⣿⢿⡟⣟⠁⡧⢻⠘⣿⠈⣿⣯⣹⡟⢻⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠐⠉⠛⠀⢀⠠⠀⠀⠠⠟⠃⢸⡁⣾⠁⠁⢰⢈⡿⢀⠸⣿⣿⢄⡠⢿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠉⠙⠈⢠⡆⠄⠀⠀⠀⠈⠀⠀⠀⢸⡗⠃⠀⡀⢼⣸⢃⣿⡆⢹⣿⣾⡅⣾⠅⡀⠈⢷⠤⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠄⠀⠀⡀⠀⠀⠀⠉⡇⠀⠆⡃⣸⠇⢸⣿⡇⢸⣿⣏⣠⣿⣢⡇⠀⢰⣀⠀⡄⢉⡹⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⢄⠂⣼⡿⡇⠀⣤⣤⢉⢙⠛⠲⠧⠈⡡⠃⠀⠠⠘⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠁⠈⣰⣿⣷⢧⠸⣯⣿⣡⣾⣿⡟⢀⣟⡁⠀⠀⠐⠘⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⣸⢠⣿⠏⣿⡺⢞⡇⣌⠛⢣⠀⠀⠂⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠰⠀⣿⡿⣿⠠⡃⠸⢁⡴⠟⠀⣿⠇⠀⠘⠀⣧⡴⠂
⠀⠀⢳⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⣏⡿⠃⠁⠘⡡⢖⠀⠄⢩⠆⠀⠀⠀⠇
⠀⠀⠘⠀⡖⠉⠀⠀⣠⢦⣤⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡠⡠⢀⡰⠛⡯⢅⠀⠀⠀⠑⠀⠀⠀⠈
⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⣿⣿⣿⣿⡯⣷⣿⡿⢓⠲⣖⠢⠀⠀⠀⡩⠈⣤⠟⠀⠀⠡⠀⣠⠂⣄⣂⢀⠀⣦⠀⠐⠀⠀⠀⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣏⢿⣿⢩⢸⣿⣻⢠⠘⠀⣠⠀⣀⡤⠊⡠⠊⠀⠀⠀⠀⠒⠸⠀⣴⣌⠙⢾⡧⢹⣤⡄⠤⠀⠔⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⠵⢾⠿⣁⢸⣿⢿⢀⠔⠋⠹⠃⡀⠀⢘⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣹⣿⣆⠙⣯⡇⣡⣷⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠜⣿⠴⠫⢸⠀⠀⠁⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡹⢟⠹⣧⠈⡟⢿⡇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢸⠀⠀⠰⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⣿⣧⠘⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢪⠹⡄⢻⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢱⢨⡀⠀⠀⠀⠉⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⡀⠁⠀⠀⣠⠔⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠰⡄⢀⡴⠞⠙⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣐⣾⣄⣷⣾⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣾⣾⣷⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⡈⢻⣿⣿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠋⣿⣟⠿⠟⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣸⣦⠀⠘⣇⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⡈⠻⠃⣰⠟⣯⣴⣶⣄⣶⣦⡶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣿⣇⢠⣿⣿⣿⣇⡀⣀⡀⠀⠀⢠⠚⠿⢟⠋⡙⠻⢿⣟⠷⢿⣷⣾⣿⠋⢿⢟⣹⢿⢄⡸
⠀⠀⠀⠀⠀⣠⣀⠀⠀⣠⣤⡂⢀⠀⠻⣟⣾⢀⣎⣉⠙⠻⣿⣿⣆⠀⠀⠴⢆⣭⣤⣱⣦⣄⡹⣿⣒⣶⣶⡗⠲⠆⠙⣀⣴⣤⣄⢀
⠀⠀⠀⠀⠰⠛⠿⠂⠀⠁⢟⣽⠿⣷⡄⣿⢿⣾⣿⣿⣷⣤⡋⢩⠟⠀⠀⠒⠘⠛⠀⠉⠛⠙⠛⠿⢮⣥⣵⣶⣦⡶⠾⠋⠛⢹⡿⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈

⠀⠀⠀⠀⠀⠀⢀⣠⣦⣴⣶⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣦⣤⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠁⠀⠉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⢟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣷⣾⣶⣿⡿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⡇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢻⡏⠻⠟⣷⣦⣤⣦⣧⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣠⡀⢀⠐⠄⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡬⣼⣣⣄⣼⡟⣿⣿⣿⣤⣤⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠻⡿⣷⣬⡆⠂⢹⣿⡏
⠀⠀⠀⠀⠀⠀⡀⢸⠁⠀⣀⠀⡈⠻⢿⣤⣇⣿⢹⡟⠹⣟⠋⡙⢿⣦⠀⠀⡀⠀⠀⠀⠀⠀⢀⡠⣤⠼⠟⠁⠘⠀⢋⡟⠠⠃⢯⣿⠁
⠀⠀⠀⠀⠀⣠⣤⠿⠿⠛⠛⠹⢛⡷⠞⣽⣛⣿⣿⣷⠾⢃⣿⠓⠀⠀⠀⠀⠀⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠐⠀⢀⡼⠃
⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠙⠻⣷⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠐⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣿⡿⠿⣷⣦⣄
⠀⠀⠹⣿⡌⠙⠻⢿⣶⣤⡀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠹⣿⣄⠀⠀⠈⠙⢿⣷⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢀⣀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡷⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⢀⣾⡟⠁⢀⣠⣴⣾⠟⠉⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⣾⣿⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣠⣴⣾⠿⠋⠁
⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣃⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣆⣼⣏⣝⣻⣶⣄⣶⣤⣴⣦⠀⠀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣿⣯⣿⣿⣿⣿⣿⣽⣿⣧⠿⠃
⠀⠀⠀⢀⠊⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⠃
⣢⡬⠞⠉⠀⠀⠀⠀⠸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣥⡴⠃⠀⠀⠀⠀⠀⢠⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠁⠀⠀⠀⠀⠀⠀⠂⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠹⣷⣀⡀⡒⣴⣿⣻⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠃⠀⢀⢄
⠀⠔⢸⣖⣷⣻⣍⢻⣿⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⣀⠃⠀⠀⠀⠈⠱⠀⠄⠂⠀⠂⠨⠒
⣤⣤⣼⣿⣽⣧⣿⣟⣿⣾⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⠞⠁⠀⠀⠀⠀⡄⣀⣸⣿⣿⢻⣿⣿⣶⣴⣶⣶⣤⡆⢶⡆
⣿⣾⣿⣽⣿⠽⠗⠛⠛⠛⠉⠛⠛⠛⠿⣿⣿⣿⠟⠁⠀⠀⠀⠄⠀⢀⢀⡹⣿⣿⣿⣿⣿⣻⣿⣿⣟⡿⠟⠉⠁⠈⠀⠁⠀⠀⠈
⣿⡯⠝⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢉⠀⠀⠀⢤⡀⢦⣀⢀⣄⢳⣻⣿⣿⣿⣿⢿⣷⣷⡟⠁
⠋⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⡘⢤⣰⣀⡀⠴⠄⢀⡀⠰⣦⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡷⣿⡇
⠀⠀⠀⡀⠀⠒⢑⣆⣾⡟⣿⣿⣿⣿⣿⣿⣷⣀⣨⣽⣷⣿⣿⣿⣿⢿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡦⡀
⢀⣀⢀⡁⠀⠀⠒⣌⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣷⠦⠄⠀⠀⠀⣠⣤
⢷⣿⣿⣿⣷⠎⣯⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠙⠙⠛⠛⠛⠛⠻⠟⣋⡉⢉⣀⣀⣉⣩⣅⣀⡀⢶⣦⣐⠷⣶⣮⣽
⣿⣿⣻⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⣀⣤⣴⣦⣄⡀⣷⣶⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣦⣭⣻⣿⣿⣿⣿⣿⣿⣮⣝⡛⢶⣭⡙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣟⣻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣌⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣀
⠀⠀⠀⠚⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡶⠂
⠀⠀⠀⠀⠀⢀⣴⣾⣶⣿⣿⣿⡿⢿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣶⣏⠝⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣻⣿⣇⠀⠀⢀⡀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠂⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠁⠒⠈⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠻⣿⡟⠟⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣷⣶⣤⡀⠀⠀⠀⠋⣀⠀⢠⣼⣦⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣭⣥⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⡆⠈⠁⠀⠀⠀⠘⢿⡏⠃⠀⠉⠀⠈⠉
⠀⠀⠀⠰⠾⠿⠿⠻⠿⠛⠟⠛⠛⠉⠛⠞⠻⠶⠆⠀⠀⠴⠻⠿⠷⠿⠿⠿⠟⠿⠞⠖⠒⠒⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡶⢶⣶⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⣟⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢤⣤⣴⣶⠿⣿⣿⣿⣧⣼⣿⣿⣿⣧⡀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⡍⠛⠀⠀⠀⢀⣤⣤⡄⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠸⣿⣿⡅⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⣀⣴⣿⣿⣿⡌⠧⡀⠀⠀⣄⠀⠀⠀⠀⢀⡴⠟⢛⣿⣧⣀⣀⣠⣴⣂⣀⣀⡀
⠀⠀⠀⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣴⣾⣶⠀⠀⠀⠀⠀⠀⢁⣾⣿⠇⠀⢀⣼⡟⡋⠁
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣛⠓⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠁⣀⣴⣿⡿⣿⡀⠀⠀⢀⡀⠀⠀⠈⢿⠁⣤⣾⣾⣿⣻⣷⠖
⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣍⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⣵⣿⣿⣿⣿⣷⣿⣷⣤⣾⣿⣏⠀⠀⠀⠀⠐⠿⣿⣿⠾⠛⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⣟⣿⣿⣻⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣻⣿⣻⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠈⠐⠀⠀⠀⠊⠀⠀⠀⠀⠒⠖⠂⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣀⠠⢤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⠔⠋⠀⠀⠸⠈⠉⠀⢀⡴⠋⠐⡷⠋⠉⠀⢀⠴⡃⢸⠖⠉⠃⠀⢀⣤⠊⡹⠋⠉⠀⠀⠀⢀⠴⣾⡿⠋⠙
⠀⠀⠀⡏⠀⠀⠀⠀⠘⠤⣠⠴⣻⠓⠰⡈⣇⣀⣠⠔⣏⢦⡇⢻⣀⣀⡠⣴⠉⡀⠀⠳⣄⣀⣀⡠⠖⠇⢨⢳⠷⣀⣀⣀⡄
⠀⠀⠀⡇⠠⠀⢘⠰⠆⡀⠐⠈⢿⣧⠗⠢⠧⣠⠃⡰⣿⣿⠇⠘⣿⢠⡧⠛⠲⠇⡄⠠⠒⢼⡄⠠⠀⢀⢾⢀⡶⠊⣩⡷⡀
⠀⠀⠀⠇⠀⠀⠠⠀⠈⠀⠀⢀⠸⠃⠀⠍⠷⠁⠀⠤⠟⠁⠠⣅⠿⠊⠀⢸⣸⠄⠃⠀⢀⠘⠔⠀⠀⢿⡠⠊⠀⠰⣼⡧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠂⠀⠄⠂⣴⣂⠀⠂⣠⠄⠐⡠⢀⢀⠀⠀⠀⡀⡀⢀⡄⠀⢠⠀⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠂⠚⠘⠀⠀⠁⠈⠀⠃⠀⠀⢓⠟⠀⠀⠀⠀⠂⠃⠈⠐⠃⠈⠘⠐⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠶⠦⠄⠉⢙⣳⣤⣀⣀⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⢛⣣⡤⠶⣚⡛⠉⢩⡄⠀⣀⠀⠶⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠞⠋⠁⠤⢀⣈⠀⠠⠄⠀⠀⠉⠀⠀⠈⠀⡴⠁⠀⡴⣢⡒⡄⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢫⠀⠘⠃⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⠾⠚⢁⣠⡴⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠾⣿⣿⠤⠄⣲
⠀⠀⠀⠀⠀⠀⢀⣤⣾⢃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣛⣋⠙⠓⠒⠄⠀⠀⠙⠄⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⢰⢏⢠⡇⠈⢀⣀⡤⠀⠀⠀⠀⢀⣀⣀⣼⣯⡡⠄⠀⠀⢀⣄⣀⣀⣀⣀⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⢸⣧⢸⠇⢠⡯⢒⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠙⠺⠶⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢻⡄⠀⠛⠗⠋⣰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠇⢸⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⠏⠀⠐⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡶⠖⠊⠁⢀⠈⣦⠞⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠐⡄⠘⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢸⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡼⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡗⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠲⣤⣤⣀⢀⠀⠀⠀⣀⣠⣦⢠⣶⣀⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡗⡏⣯⣼⡆⢿⣿⡼⣿⣿⣿⠃⠀⠄
⠀⠀⠀⠠⠀⢸⠆⠀⠀⠀⠀⠀⣠⢔⡤⣶⠀⠀⠀⠠⠚⢻⡟⢹⠏⢿⠅⡇⠏⢸⡇⠘⣿⣿⢟⠙⣻⣇
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠠⠄⠀⢸⢠⢈⢁⠀⠀⠁⠀⠀⠉⠁⢨⣶⡿⠀⠆⣧⣾⢣⣆⠹⣧⣯⠔⣹⡇⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢼⠀⠀⠀⠀⠀⠀⠈⠸⡇⠀⡀⡆⣵⡇⣾⣿⠀⣿⡿⠃⣼⡅⡤⡀⢙⠉⢀⠠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠁⠂⢄⠀⠀⠃⠀⠀⠇⣋⠆⣿⣿⠂⢉⡙⠒⠿⠿⢥⡆⠸⢺⢼⠀⠋⣿⢿⣶
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⣰⣿⣽⠀⣸⣯⣟⢰⣶⡦⠈⣶⡃⠠⢀⢀⢨⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⣴⣯⡿⢹⠃⣹⡿⣷⣿⣿⣇⡸⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠸⣿⢿⡟⢨⠁⠋⣰⠿⠃⣿⡇⠈⠃⢸⣆⡤
⠀⠈⢦⣀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡻⢋⡟⠈⠀⣀⠥⠀⠠⢙⠄⠀⠀⠸⠋
⠀⠀⠘⢸⡹⠁⠈⢀⢰⣤⣀⡀⣀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠤⢀⡴⢹⡉⠂⠀⠀⠏⠀⠀⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠁⠈⣿⣿⣿⣿⡾⣿⣿⠟⡂⢶⡶⠂⠀⠀⣨⠈⡤⠋⠀⠈⠀⢀⡖⠠⣆⠁⡀⣔⠀⠐
⠀⠀⠀⢨⡀⠀⠀⠂⠀⣿⣿⣻⡩⢺⣿⡏⡆⠑⣈⡄⢀⠰⠊⠠⠊⠀⠀⠀⠈⠂⠍⣠⣦⡉⠻⣧⣸⡤⣶⠇⠠⠊
⠀⠀⠀⠘⡇⠀⠀⠀⠈⠰⠻⠟⣥⢸⡿⣧⠰⠚⡅⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣧⣿⣷⡈⢯⣇⣽⡿⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⡿⡚⠁⢉⠀⠀⠄⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢟⡇⠻⠉⣷⡄⢽⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣰⠀⠀⠃⠀⠀⣀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠁⠃⢀⠀⡝⣿⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡊⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠘⡇⠸⠃⠁⠁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⢸⠐⠂⠀⠀⠀⣠⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠰⣊⡠⠬⠓⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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


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

⣀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⠀⠀⡆⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⢀
⢹⠄⡄⠀⠛⠀⠀⢻⡄⠀⢸⡇⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⠸⡇⠀⠙⠁⠀⠀⢸⠀⠀⠀⠀⠰⠇⠀⢀⠀⢿⠀⠀⠀⣯
⠀⠰⡽⠀⠀⢰⠀⠈⠀⠀⠀⠁⠀⠀⠀⡷⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢘⡄⠈⠀⠀⢰⡄⠀⠀⠀⢮⡆⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠆⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢲⠇⠀⠀⠀⠀⠀⢀⡀⠀⠸⡂⠀⠀⣵
⢸⡀⠀⢰⡄⠀⠀⠀⠀⠘⠁⠀⠀⠸⡄⠀⢸⡆⠀⠀⠀⠀⢴⠀⠨⡇⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⡔⡄⠘⠆⠀⠘⠃⠀⠀⣀
⠈⠀⠀⠘⠃⠀⢠⣆⠀⠀⢰⡀⠀⠘⠃⠀⠀⠀⠀⣀⣀⣀⣿⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠐⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⣠⠀⢀⡄⠀⠋⠀⠀⢈⣀⠤⣠⢶⡶⢒⠩⣀⠦⠅⠒⠈⢋⢩⠅⡒⠒⠶⠦⣤⣄⣀⣀⡋⠀⠀⠀⠀⠀⠀⢸⢣⠀⠸⠆
⠀⠀⢷⡇⠘⠃⠀⣀⡤⠾⢁⣰⡪⢃⠡⡲⢐⡩⠔⠊⠀⠀⠀⠋⣂⢥⡐⠠⠄⠀⠀⠈⠑⠾⣬⡉⠒⠢⠤⣀⠀⠈⠛⠀⠘⠃
⢰⡀⠀⠀⢀⠔⣪⠅⠁⢠⢊⠕⡪⢂⠥⣊⠅⣊⠁⠀⠀⠀⠀⠀⢨⡈⡡⠐⠂⠭⡄⠀⠀⠈⠈⠙⣔⠡⠂⠤⠉⠲⢄⡀⠀⢸⡄
⠘⠃⢠⠜⠥⢂⠂⠀⢠⢃⢔⡡⣊⠕⡪⢐⠩⠀⠀⠀⠀⠀⠀⠀⠀⡆⠅⢂⠡⠄⠐⣀⠀⠀⠀⠀⠈⢫⢔⡨⢐⡀⠀⠙⢆⠈⠁
⠀⡴⡳⣄⠬⠀⠀⠀⣎⢔⠡⠪⡰⢝⠨⢐⠬⢐⠀⠀⠀⠀⠀⠀⠀⠳⠞⡀⠌⢑⣀⡀⠀⠀⠀⠀⠈⠄⢳⡪⠐⡠⠄⣀⠢⡳⡀
⣼⠚⠃⠉⠚⠒⠤⣀⠇⡡⣠⠮⠴⠆⠚⠓⠚⠐⠖⠴⢤⣄⡀⠀⠀⢈⢵⡀⢡⣐⣒⣮⣍⣙⣀⣂⣀⠤⠎⢗⣡⡴⠷⠒⠓⠓⠷
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠳⢤⣸⡵⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⢠⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⣉⣁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠠⠄⢂⠔⡠⢂⠄⣊⠔⡠⠐⡠⢒⠄⡠⢂⠄⡠⢂⠔⡠⢂⢔⠠⡠⢀⠄⡀⢄⠄⡠⢀⠄⡀⡀⢀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠁⠈⠐⠁⠋⠐⠁⠊⠖⠑⠋⠊⠐⠁⠊⠐⠁⠊⠒⠕⠡⠪⠂⠕⠡⠊⠔⠡⠊⠞⡕⠰⢢⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⢛⡿⠛⣻⠶⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⡰⢋⡴⢊⡠⠖⣚⡿⢦⣄⠀⠀⣠⠞⡩⠊⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣵⣵⣧⣎⣴⢋⡴⢊⡡⠴⣶⠟⠀⣼⠏⡴⠁⢠⠃⢘⡿⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠉⢥⣀⡀⠀⠐⠒⠯⢭⣛⠷⣯⣔⣺⡇⠀⣰⠏⡼⠁⣰⠏⢠⠏⢔⣽⠋
⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠈⠑⢮⡻⣿⡀⢠⣿⡼⢃⡴⠋⣰⠏⢠⡾⠁
⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⣤⡀⠀⡀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠙⠌⠛⢾⣿⣴⣯⣵⠞⣡⢠⡟⠁
⠀⠀⢰⡿⣧⣀⠀⠀⠀⢿⣶⠏⠀⢳⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣑⣒⣉⠀⣾⠁
⠀⠀⠀⠻⣬⡉⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢗⡒⠂⠀⣿
⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠴⠁⠀⣠⣿⣿⢦⣄⠀⠀⠀⣠⣾⡋⠉⢻⣷⣦⣍⠲⢄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣭⣄⣀⣤⡾⠋⠹⢮⣓⣽⣷⣤⡾⢏⡻⢝⢷⣄⢻⣷⣌⠳⣦⡑⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡿⣿⣟⠻⡛⢿⡻⣟⢦⣽⡶⠿⠋⠀⢻⡜⢷⡈⢳⡄⠀⢭⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣯⠳⣌⠢⡝⢿⣿⠁⠀⠀⠀⠀⠈⢿⡄⢻⡄⠹⣄⠳⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢞⣷⣌⠳⣌⢺⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⡄⠘⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⣿⣗⢌⣣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣜⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠳⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡤⠶⠒⠚⠛⠛⠒⠒⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠈⢉⣀⡤⠤⠴⠤⣤⣄⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡾⠉⠀⣀⡤⠤⠤⢄⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⠈⠉⠉⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⢠⠔⠋⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣄⣤⣄⣀⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⣀⡤⠖⠛⠋⠛⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠉⠳⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠇⠀⡠⠊⠁⠀⠀⠀⠀⠀⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡈⠉⠛⢦⡄
⠀⠀⠀⣠⠖⠋⢀⣼⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠉⠳⣄⠀⠹⣆
⠀⢀⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢰⡄⠀⠀⠀⠀⠈⣧⠀⢹⡄
⠀⡾⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠰⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠾⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡇
⠀⠻⣄⠀⠳⣄⡀⠀⠀⣀⡴⠞⠉⣭⠗⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⡞
⠀⠀⠈⠻⠦⣬⠉⠉⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⠉⣹⠶⠤⠤⠴⠚⠉⣠⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⣀⣠⠴⠋⠀⠀⠐⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠤⠶⠞⠛⠦⣤⣀⣀⣀⣀⣤⣴⠖⠋⠁⢀⡬⣭⣭⣅⣤⣤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣀⡀⢀⡟⠀⠀⠀⠀⡾⠒⣒⣚⣭⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠈⠉⡿⠁⠀⠀⠀⠘⠛⠋⠉⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠴⠖⠛⠋⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⠃⠀⠔⠂⠀⠠⡀⠀⢀
⠀⠀⠀⠀⠐⡀⠀⢰⣩⡟⠀⠱⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠈⠓⠛⠋⠀⣐⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠅⡀⠀⠀⠑⠀⠒⠤⣤⠊⠀⢀⠔⠀⠠⠤⢰⠀⠀⠀⡠⠄
⠦⡹⢦⠄⠀⠀⠘⠀⠇⢀⡴⠋⢔⡂⡄⠀⠈⢀⣲⣿⣻⡀
⠀⠀⣸⠀⠀⠀⠀⢸⠁⡌⠀⠀⢛⣽⠟⢊⣴⣿⠿⠿⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄
⡄⢀⡁⡄⠀⠠⢠⡯⠆⣀⡉⣩⣟⣡⡶⠟⠃⢀⣀⠀⠆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠡
⠄⢈⠎⠀⢠⢆⣾⢷⢏⠋⣰⢟⣿⠋⠀⠀⡴⠋⠁⠙⡤⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⡴
⢠⡏⠀⠠⠃⡼⠁⠀⠁⣼⢃⡼⠁⠀⠀⢠⠁⠀⣾⡶⠇⠀⠰⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡴⠁⢠⡃
⣸⠃⠀⠀⡼⡁⢀⠀⣼⡣⢸⡇⠀⠀⠀⠀⢃⠀⠉⠒⠂⠉⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣀⠴⠾⢤⣀⠨⠁
⡇⠀⠀⠾⠚⡐⠁⠀⠟⢣⠸⣇⠀⠀⠀⠀⠀⠃⠀⠀⡴⢺⠿⡦⠀⣀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡠⠎⠻⠇⠊⠀⢀⣀⠀⠙⠇
⡏⢠⠀⠄⣬⠀⠀⠐⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠀⠉⠈⠡⢾⡷⠾⠤⠀⢀⠀⠀⠘⢻⡿⠃⡀⠒⠈
⣧⠠⡊⢀⠃⠀⠀⢒⠀⠀⠀⠀⢧⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⢁⡳⠀⠀⢀⠊⠱⣄⠴⠁⠀⠀⠀⠀⠈⠓⢆⠀⠀⠝⠊⠀⣄⡀
⡟⠃⠅⡜⠁⠀⠀⠸⣄⠀⠀⠀⠌⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⠡⠾⠋⠉⠑⢶⠊⢁⡀⢿⠀⠀⠀⠀⠀⠀⣤⠀⣠⠇⠐⠀⠀⠀⠀⠈
⡧⠀⠘⠀⠀⠀⠀⠀⢯⠰⢄⠀⠈⠒⠂⠙⠂⣀⣀⣀⣀⠀⠀⠀⢀⣀⣤⡴⠒⠉⠀⠀⠀⠀⡀⠀⠈⢩⣀⠼⢿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠐
⣷⠀⠀⠀⠀⠀⠀⠀⡾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠭⠁⠀⠀⠀⠀⠀⠀⢊⡠⠤⠤⢀⠀⠻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢻⡔⠂⠀⠀⠀⠀⠀⠆⠘⢧⡤⠶⠤⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠤⢤⡀⠛⣄⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠃⢡⠀⠘⠠⠀⠀⠀⠘⠀⠈⡝⢧⣁⠉⠐⢝⠀⣄⠀⡀⠀⣉⠖⠀⠀⠀⠀⡠⢊⡁⠄⠀⠀⢀⠀⣻⡆⣽⡀⠀⠀⠙⢖⡀⠂⠀⠠⠤⠠⠒⠋⠉⠀⠑⢤⡀
⠀⠈⢀⢀⠇⠀⢠⠈⢀⠀⠐⢅⠀⢉⠛⢷⣤⣀⣀⣠⠀⠊⠁⣀⣀⡤⠔⢉⠐⠁⠀⠀⠀⠀⠀⠀⠀⢽⣽⡇⠀⠀⠆⠀⠙⠲⠐⠀⠀⠀⢉⠁⠂⠀⠀⠀⠈
⠂⠐⢜⣌⠀⠀⠈⠀⠘⠄⠐⡀⠑⠄⠀⠀⠈⠉⠋⠙⢋⠟⢉⡼⢁⡄⠈⠥⠄⠀⠀⠀⠀⠐⠈⠀⠀⢀⣬⣣⠀⠀⠀⠂⠀⠀⠀⢄⡀⠉⢉⣫⣅⣍⡀
⠀⠀⠀⢹⣇⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠄⢈⣀⠤⠊⠀⡜⠀⠎⠀⠀⠀⠀⠈⠄⠀⠀⢀⠠⠒⠈⠁⠈⠉⠓⠐⠂⠀⠄⠑⠖⠒⠂⠈⠀⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠙⢷⡄⠀⠀⠐⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠒⠀⠀⠠⡀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀
⢀⡐⠂⠀⠀⠀⠻⣄⡀⠀⠘⣄⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⣌⠑⠒⠛⠛⠁⠀⠀⠀⠉⣦
⠉⠁⠀⠐⢄⠀⠀⠈⠙⢦⣀⠈⢮⡑⠀⠀⢀⡀⠂⠤⢔⡀⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠉⡤⠀⠀⢀⠐⢒⡄⠀⠀⣾
⠀⠀⠀⠀⠈⢢⠑⢤⣀⠀⣈⠽⠷⠿⠶⠤⠤⠬⠵⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠘⠿⠇⠠⠐⠁
⠀⣀⣀⡀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣈⣐⣀⠀⢀⠀⠀⡀⢀⣀⣀⡀⠀⣀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠓⢦⡤⠶⢤⣀⡀⠀⠀⠀⣠⠴⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠀⡴⠋⠉⠉⠛⣦⣰⣏⠐⠀⠀⠀⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⣠⠤⢼⠁⠀⠀⠀⠘⠁⠀⠈⠉⠀⠂⠀⠀⠀⣼⠶⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⡞⠁⠀⠀⣀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣠⠞⠉⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡴⠋⠁⠹⣆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂⣁⠀⠀⠀⠀⠠⠀⠀⢀⡼⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠐⠲⠤⠴⠒⠀⠀⠀⠲⠀⠀⠀⣠⠋⠉⠓⠤⠄⠀⠀⠀⠐⠒⣉⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠠⠀⠈⠑⠶⠖⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠖⠋⠡⡀⠀⠀⠀⠀⢠⠀⠀⢀⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠲⣦⡀⠀⣀⡼⠻⣤⣙⠢⣌⣀⡥⠆⠀⠤⣁⢀⣀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠉⣉⢠⢂⠀⠉⠉⠉⠙⠦⣤⡤⠞⠉⠉⠹⣄⠀⠀⠀⠀⠀⣀⣀⣠⠏⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⡟⢰⠇⡎⡞⠀⠀⣶⡇⠀⢀⡄⢠⠄⣾⡇⠀⠉⠳⠦⠤⠴⠚⠁⠈⢁⠀⠂⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⢁⡟⡘⠸⢀⠃⢰⡟⠀⠀⣼⢁⡟⣸⡾⠀⠁⣠⡄⠀⢀⡤⢰⡇⡜⡞⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠇⣼⢡⢁⠃⠌⠀⣽⠁⠀⢰⠇⣼⢡⢷⠃⠄⠠⣿⠁⠀⣼⠃⡞⢰⢱⠁⠄⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡟⢰⠃⡎⡎⡘⠀⣰⡟⠀⢀⡟⢰⡇⣾⡟⠀⠀⢰⡇⠀⢰⡏⣸⠃⡇⡆⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡾⢁⡏⡰⢸⢡⠃⢠⣿⠁⠀⣼⠁⡞⣸⣽⢁⠃⢰⣿⠁⠀⡾⢀⡏⡸⡸⢀⠁⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠇⡼⢠⢃⠇⠆⠀⣾⡇⠀⢰⡇⣸⢁⣿⡏⠄⠀⣼⠇⠀⣸⠇⣼⢠⢇⠇⠌⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⡏⢰⠃⡄⡘⠀⠀⡸⡟⠀⢀⡿⢀⠏⣸⡿⠰⠀⢰⡿⠀⢀⡟⢠⠇⡎⡼⠠⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡼⢁⡏⠸⢠⠁⠀⢠⣻⠁⠀⣼⠃⡜⢠⣻⠃⠀⢀⢿⠃⠀⣼⠁⡜⠐⢱⠃⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠇⡼⢠⢃⠇⡀⠀⢾⠇⠀⢰⡏⢰⠃⣞⡟⠀⠀⡜⡎⠀⢰⡏⢠⠁⠄⡎⠀⠀⡘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡎⢐⠃⠎⡌⠀⠀⢈⡾⠀⠀⣾⢀⡇⡸⣼⠁⠀⠰⡽⠀⠀⡾⠀⠇⠰⢰⠁⠀⢀⠱⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠇⡼⠀⡎⠸⠠⠀⠀⠀⠾⠃⠀⢸⠃⡼⢠⢣⠇⠀⠀⢣⠇⠀⢸⠇⠸⠀⠀⠇⠀⠀⠀⠃



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣳⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣩⣾⣿⣫⣽⣭⡽⠁⢻⣿⣿⢷⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⣿⣿⠓⠋⢥⣁⢠⣲⣮⣭⣟⡋⠙⠾⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⣿⡿⢡⣬⣻⣿⣿⣿⣿⠋⠁⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣠⣾⣟⡵⢈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣯⣿⠏⠀⠾⢽⣿⣿⣿⣿⠀⣠⣴⣶⡲⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣠⣿⣿⣗⠀⠉⣤⣿⣿⣿⣿⡻⡿⠛⠋⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠔⠉⠤⣲⡞⠿⠛⠉⠀⣄⢀⢶⣿⣿⡿⢇⣁⠤⠤⠴⣿⣿⣿⣷⡀⠠⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢀⠀⢶⣎⣷⣿⣿⣿⣿⡏⡓⠀⠀⠀⠀⠀⢨⣭⣻⣿⣿⣷⣶⣶⠟⢛⡿⢟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⡧⢋⠠⠀⠀⢼⣿⣧⣶⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠈⠝⢋⣤
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⣭⣽⣿⣷⣶⣟⣃⡉⠀⠀⣀⣤⣼⣿⣿⣿⣿⣿⡿⠷⠉⠉⠀⠀⠀⠤⠶⣺⣿⠟
⠀⠀⠀⠀⠀⠈⢻⡿⠿⢿⣛⣿⡿⣿⠿⣥⡐⠀⠀⠀⠈⢻⣿⣿⣾⢿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⣴⣶⣦⣀⠀⠀⠨⠈⠃⠀⠀⢀⢠⡄⢀⢰⣶⣿⣿⣿⣟⠷⠉⠻⠿⣖⠀⠀⠀⠀⠀⢀⣤⣴⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⣯⠄⡤⣸⣖⣶⣦⣼⣿⣿⣼⣷⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠠⠖⣛⡛⠉
⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠻⠍⠛⠛⠛⠓⠒⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠿⠿⠿⠿⠿⠟⠋⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⢀⡄⠀⠀⠀⠀⠀⠲⠁⢸⠀⠀⠀⢸⠈⢃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠀⡀⠀⠀⠀⢁⡀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠓⠤⠈⠉⠈⠁⠂⣀⠠⠄⠀⡀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⢰⠀⠀⠀⠀⠠⠅⠉⠈⠉⠉⢈⠝⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠀⠀⢈⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⡐⢩⠀⠀⠀⠀⠈⡁⠀⠄⠀⠀⠐⠈⠁⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠺⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⢠⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠁⠐⠒⠈⠀⠄⠀⠉⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠠⠀⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠈⠀⠩⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠁⡖⢁⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠒⠆⠚⠉⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠚⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠀⡄⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠘⠀⠤⠀⠒⢀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⡎⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂
⣡⣤⣀⣀⣀⣠⣀⣀⣀⣀⣀⣠⣤⣄⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠒⢶⠶⠒⠢⠤⣄⡀⠀⣀⡤⠤⠒⠒⠛⠉⠉⠉⠉⠉⣩⠗⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠲⣶⡒⠋⠀⠀⠀⣀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⣤⣤⡀⠉⠓⠲⠤⠤⠤⣤⣄
⠀⠀⠠⣤⡔⠃⠀⠀⢀⡴⠊⠁⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⣿⢸⣿⣷⠀⠀⠀⠀⣠⠖⠉⡟
⠀⠀⠀⠈⡇⠀⢀⡴⠋⠀⠀⠀⠈⢀⡢⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⢠⠞⠁⠀⡜
⠀⠀⢰⡛⠀⣠⠎⠀⠀⠀⠀⠀⠁⠀⠁⠘⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣞⠀⠀⠀⠃
⠀⠀⠀⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⣸⣀⠀⠀⠳⢦⣀
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠤⠭⢽⣛⣒⣛
⠀⠀⢠⡇⠀⠀⠀⠀⣤⠞⠉⠉⠀⠀⠀⠀⠀⠀⠑⢄⡈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡤⠔⠊⠁
⠀⠀⣸⠀⠀⠀⠀⢸⡗⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠻⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣲⠶⠒⠚⠉⠁
⠀⠀⣯⠀⠀⠠⠈⡄⢄⠀⠉⠛⠒⠒⠒⠊⠁⠀⠀⠀⠀⢀⣤⢾⢫⠃
⠀⢰⢿⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠘⠀⠘⡆
⠀⡼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠸⡄⠀⠀⠀⢳
⢀⡇⠀⢳⠀⠀⠀⠀⠀⠀⠠⠡⠐⠀⠀⢀⡴⠋⠀⠀⠀⠙⠦⣀⡀⠈⣇
⣸⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⡏⠀⠀⠀⠸⣆⠀⠀⠀⠀⡀⠀⢠⠞
⠻⣄⠀⠀⠀⠸⣆⠀⠀⢰⠁⠀⣿⠀⠀⠀⢶⠤⢤⣀
⠀⠈⠙⠒⠶⠴⠋⢧⡀⣾⠀⠀⢹⠀⠀⠀⢸⡄⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⡀⠀⠣⣀⣀⣸⢃⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠈⠉⠙⢮⢖⡡⢠⣼⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⢸⣿⠦⠤⠀⠀⠀⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⢍⣑⠒⠀⠀⠀⠀⠀⠀⠈⢣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠒⠶⢤⣀⣀⣀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢸⠁⠀⢀⡴⠋⠉⠀⠉⠙⢾⡁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢇⣀⠤⠐⡲⠊⠁⠀⢠⠖⠛⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢇⠀⠀⠀⠀⡠⠤⡀⠀⠀⢀⠀⠈⢢⢀⠖⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠘⢦⣀⡔⠁⠀⠀⠀⢸⠁⢄⠀⠀⠀⠀⠹⣄⠀⠀⠀⣴⠁⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⣐⠠⣉⡇⢠⢋⡁⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⢤⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢹⡍⠁⠀⠫⡈⠀⠐⢤⠜⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⡀⠈⠓⢧⡀⠀⠀⠀⠀⠀⣠⠞⠀⡴⣻⢟⡳⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠓⢤⣀⠈⠑⡏⠁⠀⠀⢇⠉⠆⠙⠲⣉
⠀⠀⠀⠀⠀⢀⢆⠀⢀⣠⠇⠀⠀⠀⠀⠀⠈⢉⠉⠙⢦⡀⠀⠀⢹⠈⡀⠀⠠⡉⠢⢄⡀⠀⠀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠁⣿⠀⠀⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⡇⠠⠀⠀⠈⠲⣄⢈⣑⠊⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢄⣀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⠚⠀⠀⡀⠀⠀⢈⠀⢳⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠠⠜⢀⡼⠆⠀⣀⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠈⢆
⠀⠚⠁⠀⠘⠀⠀⠠⠁⡸⠀⠀⠀⠀⠀⢠⢪⠀⠀⠤⠆⠀⠀⢀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⠀⠀⠒⢄⠀⡜
⠀⠀⠀⠈⢠⠂⠀⢤⣠⠇⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⠀⠜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠢⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⠊⠀⠀⠀⠀⢸⠀⠀⠻⢦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢀⡴⠓⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠭⢉⡁⠀⠀⠀⡀
⠀⠠⣂⠁⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣢⣄⣰⣤⢻⣿⣴⣶⡴⡀⠴⠀⠀⠠⠐
⠀⣲⠿⣗⠁⠀⠀⠄⠀⡠⠀⠀⡀⢯⡻⣽⡿⢏⣵⣬⠏⠙⠁⠐⠈⠄⠀⠀⠈
⣾⠁⣶⡹⣦⣀⠂⠀⠈⠀⠀⠸⡏⣮⡇⡷⠃⢈⢱⡇⠌⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⣐⢀⠠⠈⠠
⠙⠞⢿⣿⡿⠟⠋⣠⠤⢤⣿⡁⢿⡟⢧⣇⢰⣾⡀⠸⡦⠀⠠⠀⠂⣤⠐⣄⠀⡀⠀⢠⣶⠆⢛⠣⠀⠀⠐
⠀⠀⠀⠈⠰⣖⠊⠁⠀⢩⠧⠍⠰⠂⠉⣅⢨⡀⢄⣲⣽⢷⡴⣷⣤⣿⣻⠿⠋⠀⠀⡟⢁⠘⢿⡷⣇⣮⣁⣀⢀⠀⠠⠑⢄⠈⠀⠠⡀
⠀⡀⢀⣻⡦⢌⣙⣒⠒⠛⠒⠁⠄⠠⣦⢽⣞⠓⠈⣮⣿⡶⠀⠈⡫⠅⢠⣖⣤⣤⠜⠀⠹⢿⣶⣯⣦⣭⣭⡅⢠⡗⣠⣦⣅⣩⣷⣴⡿⠿⠄
⠀⠈⠀⠉⢉⠉⣉⠉⠆⡀⣮⠹⣀⣶⣿⢺⣍⣙⣒⣪⢞⠁⣶⣄⠻⢷⡬⣭⣥⡄⠠⡤⠴⠶⢶⠓⠷⠶⠶⠦⠫⠶⣜⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⢠⠸⡦⣹⠳⣔⣦⢮⠙⠛⠛⣋⣉⣉⠫⢅⡐⣯⣳⣬⡿⡏⣲⡉⡓⠈⡓⣤⡤⠶⣞⢿⠻⣿⠗⠊⢰⢶⣻⡛⣻⡛⢏⠛⠛⠁
⠀⠀⠀⠀⠀⠉⠓⠦⣄⠸⣷⣦⣌⣛⠓⠲⠮⣽⡲⠿⠲⡿⣮⣻⣤⠽⡿⣿⣒⠛⠾⠿⠖⠚⠛⠉⠠⡖⢄⣈⠲⢶⡐⢄⠝⠾⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠖⠉⠉⠁⠀⠀⠀⠉⠉⠁⠉⠈⠀⠀⠀⠀⠀⠁⡉⠙⠷⠖⠒⠒⠲⢶⢻⣯⣹⠯⠭⠓⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⢿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣩⣱⠏
⣀⣤⡶⣶⣿⣿⠿⠿⠯⣻⡟⠃⣰⡇⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⣷⣶⣦⣤⣴⣶⣿⣿⣿⡶⠋
⠹⣝⡗⠈⠯⣉⣀⡐⠒⠛⢣⡿⣿⣇⠀⠤⠦⠢⠠⠩⠉⠁⠸⠈⠠⠑⠤⠨⠅⠁⠉⠩⠭⠌⠈⠄⠁⣈⡚⢁⡒⠤⠄⡨⠭⠥⣭⣤⣿⣿⡏
⠀⠈⠻⢤⣀⢈⣡⠥⢤⣶⣤⡉⡸⢌⡄⢀⡀⠀⠀⠀⠀⠄⠢⠐⠒⠐⠐⠄⠀⠀⠀⠀⠀⠢⠔⠄⠐⠂⠂⠒⠒⣂⣠⣴⠶⠶⠶⣬⣿⣿⣿⣄
⠀⠀⠀⠀⠈⠑⠒⠦⢤⣭⣿⣶⠀⣀⢠⡤⠀⠉⠁⠀⠂⠀⠒⠂⠐⠒⠒⡈⠉⠁⠉⠁⠐⠂⠀⣀⣀⣠⡤⠖⢋⣭⠞⠁⠀⠀⠀⠈⠳⣟⠻⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⣿⠷⠿⠿⠾⠶⠒⠚⢿⣿⣿⡟⠛⠛⠛⠉⠉⠉⠛⠿⣯⡦⣲⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⢛⣉⣥⣤⣤⠶⠶⠦⠤⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣾⣟⣥⡶⢚⣋⣩⣥⣴⠶⠶⠦⠤⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠉⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠛⠛⠻⠿⣶⣶⣶⡒⠒⢒⣠⣾⠏⠀⠀⠀⠀⠀⠀⣠⡾⠛⣡⡴⠚⠁⣰⡿⠁
⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣭⡀⠀⠀⠀⠀⠀⣠⡾⢋⡴⠟⣁⣤⠀⣼⡟⠁
⠀⠀⠀⣠⡿⠋⣠⠤⠤⣄⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⠾⠋⠀⣩⠶⠛⣉⣀⣼⡟
⠀⠀⣼⠟⠀⣼⣥⣈⣷⡌⡆⠀⠀⠘⣆⠀⠀⠀⣀⣴⣶⣶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⠁⣿⡇
⣠⣾⠏⠀⠀⠹⣌⣛⣛⡴⠃⠀⠀⠀⣿⠀⣠⠾⡋⠁⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠛⠛⠓⠻⣿⡄
⢻⣷⣤⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣿⠘⢑⣪⠀⣾⠏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⡛⠶⢤⣰⡿⠃
⠹⢷⣬⡛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠰⠶⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣄⠀⠀⢮⣙⠦⣄⣿⡇
⠀⠀⠈⠙⠷⣦⣀⡀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⡾⠋⠁⠀⠀⠀⠙⢷⣌⠢⣍⠳⣄⠻⣧
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣟⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣾⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⠳⣌⡙⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⢿⣿⠿⣿⠷⠶⠶⠶⠿⠿⣿⡻⢟⡻⢟⡦⠭⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⡙⠢⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡻⣕⡘⣷⠀⠀⠀⠀⠀⠘⢿⣦⣙⠲⢭⣙⡒⠂⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⣸⡧⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣬⣉⣛⣓⣨⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠉⠙⢳⣶⠞⠃⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢐⡢⢐⣲⣤⣄⠀⣤⣼⣿⠃⠠⣠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⣀⡤⢾⠀⠀⠀⠈⠻⠿⠏⠉⠉⠀⠐⢀⢋⠅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣠⠾⢋⠆⠊⠠⠄⠀⠀⡠⠴⣀⠠⠒⠒⠊⡨⢀⡕⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⡠⠞⢁⡎⠁⠀⠀⠀⡠⠒⠆⠀⢚⡡⠒⠉⠥⠌⢱⡂
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠐⠉⠁⢠⠤⣀⠤⠚⠁⠀⠈⣾⣥⡢⢀⠚⢁⡱
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⡘⠤⠃
⠀⠀⠀⣴⣯⡤⠤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣤⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣐⣾⣄⣷⣾⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣾⣾⣷⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⡉⢻⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠋⣿⣟⠿⠟⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣸⣦⠀⠘⣇⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⡈⠻⠃⣠⠟⣯⣴⣶⣄⣶⣦⡶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣿⣧⢠⣿⣿⣿⣇⡀⣀⡀⠀⠀⠠⠾⠿⢟⠋⡙⠻⢿⣿⠷⢾⣷⣾⣿⠋⢿⢟⡹⢿⢄⡸
⠀⠀⠀⠀⠀⣠⣀⠀⠀⣠⣤⡂⢀⠀⠻⣟⣾⢈⣏⣉⠙⠻⣿⣿⣆⠀⠀⠴⢂⣭⣤⣹⣦⣄⣻⣿⣒⣲⣶⡗⠲⠆⠙⣀⣴⣤⣄⢀
⠀⠀⠀⠀⠐⠛⠟⠂⠀⠉⢟⣽⠿⣷⡄⣿⢿⣾⣿⣿⣷⣤⡋⢹⠟⠀⠀⠚⠈⠛⠂⠉⠛⠙⠛⠿⢮⣥⣵⣶⣦⡶⠾⠋⠛⢹⡿⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋

⠀⠀⠀⠀⠀⠀⢀⣠⣦⣴⣶⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣦⣤⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠁⠀⠉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⢟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣷⣾⣶⣿⡿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢻⡇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢻⡏⠻⠟⣷⣦⣤⣦⣧⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣠⡀⢀⠐⠆⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡮⣼⣧⣄⣼⡟⣿⣿⣿⣤⣤⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠻⡿⢷⣬⠆⠃⣹⣿⡏
⠀⠀⠀⠀⠀⠀⡀⢺⠁⠀⣄⠀⡈⠻⢿⣴⣆⣿⢹⡟⠹⣟⠋⡙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠼⠟⠁⠚⠁⢋⡟⠠⠃⠿⣿⠁
⠀⠀⠀⠀⠀⣤⡤⠿⠿⠛⠛⠹⢻⡷⠞⣽⣛⣿⣛⡷⠞⢃⣟⠓⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠡⢂⡠⠤⣉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠷⣍⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢰⠏⠀⢀⡜⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣦⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⢻⣄⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠙⠛⣶⣶⠶⠶⠶⠶⠶⠶⢾⣷⡀⠀⠈⢷⡀⠀⠀⢀⡤⠒⢀⡁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⠟⠋⠁⢀⠀⠀⠀⠀⠐⠂⠀⠈⠛⠛⠶⣎⣳⡄⢠⠏⠀⣾⠋⢉⡷⠈⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⡩⠋⠁⢀⠄⠈⣀⠤⠤⠐⠒⠒⠒⠀⠠⠄⣀⠀⠈⠻⣿⣿⡄⢰⣇⣤⠼⠶⠾⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⠞⠀⢀⠄⢁⣔⣭⣴⠶⠒⠛⠛⠛⠓⠶⢦⣄⡀⠈⠐⠠⣈⢻⡷⠋⣡⣤⣤⣄⠀⠀⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⠊⠀⢐⣡⠾⠛⣩⠏⢀⣠⣤⣤⣤⣄⠀⠀⠀⠈⠙⢷⡤⠀⠀⡿⠀⣾⡇⠀⠀⠈⠻⣦⠈⠢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡄⠀⣼⠟⠁⠀⢰⠃⣰⠋⠁⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠹⣄⠀⢿⣀⠙⠛⢹⡆⠀⠀⢹⣧⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡼⠁⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢄⠀⠘⣆⠀⢉⠙⠛⣿⠀⠀⠀⣸⢿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢼⡇⠀⠀⠀⠀⠀⢧⡙⠦⠴⠚⢹⠀⠀⠀⠀⠀⢻⡆⠀⢣⠀⢻⡄⠀⢂⠀⣻⣇⡀⢠⡟⢸⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⣄⠀⣠⡤⠲⢦⡈⠒⠒⠒⠃⠀⠀⠀⠀⠀⠘⡇⠂⠀⠀⢸⡷⠖⠛⠉⠉⠉⠙⠛⠷⣿⣇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠈⠙⠛⠋⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠏⠀⢀⣤⡤⢤⣤⣤⠤⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⠀⢰⡏⠀⡆⠀⠈⣿⣷⣄⠀⠀⠒⠤⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⡀⠸⣇⢀⣧⠔⠒⣾⣿⡏⠑⠄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⠂⢹⡶⢤⣍⣡⣤⣴⣿⠏⣿⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡎⠀⣼⠁⠀⢠⠀⢠⣿⠇⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡜⠀⣼⣇⣠⡤⢶⣤⣿⣿⡀⠀⣸⠀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢠⣎⣤⠾⠛⡉⠤⠀⠀⠀⠌⢉⠛⢳⡇⠀⠀⠀⠀⠈
⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠒⠶⠶⠾⠿⣯⣤⣶⣶⠾⠛⢉⡠⠐⠁⢀⡠⠔⠒⠒⠂⠠⢈⣾⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠛⠻⢶⣷⣤⣤⣴⣊⣥⠴⠶⠖⠒⠶⠶⣶⠇⠀⠀⠀⠀⠀⠀⠀⣸⠃
⣀⡀⠒⠀⠀⠀⠀⠀⢀⣉⣭⣭⡿⠿⠛⠉⠀⡠⠂⡠⠔⠈⠀⠀⠀⠀⠀⠢⡀⠀⠀⠄⠀⢐⣤⣤⠿⢛⠩⠝⠂⠈⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠂⢠⠏
⡛⠿⠿⠷⠶⠶⠶⠶⠿⠛⠋⠁⠀⠀⠠⠐⠁⠄⠊⢀⣠⡴⠶⠶⠾⠟⠛⠒⠾⢶⣶⣾⡟⠛⢉⠄⠊⣁⠤⠒⠈⠁⠤⠤⠠⣼⠏⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠻⣶⡄⠀⠀⠄⠀⠀⠀⠀⣠⠶⠒⠛⢦⣄⠠⢊⡴⢋⣁⣠⣀⣈⠀⠀⠀⠀⠀⠐⠨⡉⠛⠿⣿⣦⣮⣀⡀⠀⡀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⢀⡀
⠀⠈⠻⢷⣆⡒⠂⠀⠀⠈⠷⠞⠋⣷⠀⢸⣦⡏⣴⠟⠉⠉⠉⠉⠛⠶⣤⣀⠈⠐⠠⠀⠁⠢⠄⡉⠙⠛⠿⠿⠿⠿⣿⠟⠁⠀⠀⠁⠀⠀⢀⣴⡟⢡⠂⣠⠞
⣤⡀⠀⠀⠉⠛⠳⢦⣤⣤⣤⣤⠾⠋⠀⣸⢿⡅⢧⡠⢀⣴⢶⡆⠀⠀⠀⠉⠛⠶⣦⣤⣀⡀⠀⢀⠀⠀⠀⠀⣴⠞⠃⠀⢀⡴⠀⠀⢀⣴⡿⢋⠔⢁⡴⠋

⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣭⣭⣭⣭⣍⣉⡉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣫⠥⣶⣶⣶⣾⡯⢭⣙⠓⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣶⣟⡛⠉⠉⢙⣛⣻⢯⣗⠦⣙⠷⢄⡈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢢⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣟⡉⠉⢹⡄⢴⣟⣿⣟⢷⣮⠙⠪⣻⢦⣙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⢰⢲⠇⠈⠀⠁⠈⠀⠀⢀⣴⣶⣶⣦⣄⡀⠀⠀⠈⠛⠋⠀⢠⡇⡸⣿⣾⢡⡆⠹⣵⣄⠈⠳⣮⠑⢦⣉
⠀⠀⠀⠀⠀⠄⠀⡱⠛⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣭⣄⡀⠉⠛⠻⣷⣦⡀⠀⠀⢀⡼⠀⢉⡩⢪⡞⣀⣉⣉⣿⣷⣤⣌⣳⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠿⡤⠈⠹⡄⠀⠀⠀⠈⠛⠶⣦⠞⣡⣔⣩⣾⠿⣻⣿⣿⣥⠤⣄⠉⠛⡝⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡒⠝⢧⡀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⣾⣷⠾⣛⠿⣫⡵⣻⠟⣫⠽⣿⣿⡌⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣞⠀⢀⡀⠙⢷⣤⣤⠏⠀⣀⣤⡾⠿⠛⣉⠤⢊⠵⢋⠔⡽⠁⢸⠁⣻⣿⣾⠷⢁⡴⢁⠜⡠
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⡻⣝⠛⠛⠿⢿⣶⣿⡿⠷⠿⠛⢛⣛⣩⠥⠔⢒⣩⠴⢚⢁⡴⢋⡜⠁⣰⡞⢦⣙⢿⣍⣛⣛⣊⣡⣖⣥
⠀⠀⣠⣴⡿⡻⠝⣚⣉⡙⠊⣿⠢⢭⣑⣲⣶⣬⣿⣿⣿⣿⣯⣭⣴⣶⣒⣩⣩⣤⣴⣴⣯⠶⠿⢶⣿⣿⣧⣤⣭⣳⣿⣷⣶⣾⣟⡛⠫
⠰⠟⣱⢋⠊⠔⠉⣁⣍⣉⣴⣿⠴⠚⠉⠉⠥⢖⣛⣩⣙⣛⣿⣿⣯⣭⣭⣭⣄⣤⣤⠤⠶⠶⠖⠲⠤⠤⢄⣀⠀⠉⠉⠙⠛⠛⠿⠿⠷
⢀⢰⠃⠔⠈⠀⠀⣯⣿⠿⠟⡁⠄⣢⠴⠚⢋⣉⣡⣤⣤⣶⣶⠖⠛⠛⠟⠿⠟⣛⣛⣛⡛⠿⠟⠻⠿⢶⣶⣤⡭⢽⣶⣶⣶⣤⣤⣤⣤
⠈⢸⠀⠐⡐⢄⡁⠒⣂⣈⣤⠶⣚⣁⣤⡴⢞⣿⣭⣽⣿⣿⣿⢿⣷⣶⣦⣤⣄⣀⣈⣉⣉⣉⣉⣉⣭⣭⣤⣤⣭⣐⣒⠦⢄⣀⣀⣀⣀
⢈⠸⣇⠀⠉⠦⢍⡉⣉⣥⡴⠿⢛⣩⡵⠾⠛⠉⡉⠁⢀⣀⣠⣴⣶⣶⣿⣿⣿⣷⣤⣤⣤⣴⣞⠫⣅⣶⣶⠦⣌⠙⢧⡈⠍⠐⠛⠟⠟
⠐⠀⢙⣷⣤⣤⠶⠛⠉⢀⣠⡶⠛⠉⠀⢐⣨⣴⣒⠋⠩⠁⢈⠔⠑⠡⣟⡿⡯⠛⢦⣙⠻⢯⣈⠥⠚⢹⡀⣾⣼⡁⠀⢳⡀⠁
⠘⠛⠛⠉⠁⠀⢀⣠⠶⢟⠉⢐⣠⣴⣊⣉⣀⣬⣍⣁⣒⣮⣀⣤⣀⡬⠜⠋⠁⠀⠀⠙⠻⣔⡛⠶⣄⠀⠙⢮⣳⢶⣮⡤⠛⢦⡀
⠈⢀⣀⣀⣤⢖⣋⣁⡤⠴⠚⠋⡉⠩⠄⠀⠒⠐⠒⠒⠒⠬⣭⣍⣉⡛⠛⠶⠦⣤⣀⡀⠀⠀⠉⠙⠛⠻⢦⡀⢌⡙⠛⠻⠷⢶⢿⣷⣶
⠸⠿⠟⠛⠛⠛⠋⢉⣭⣭⣉⣙⠛⠒⠲⠶⠤⠤⠤⠴⠖⠒⠒⠒⠛⠛⢛⣛⠋⠭⠽⠿⠷⠶⣶⣤⣀⡀⠀⠉⠳⣽⣧⣅⡠⠄
⠠⡤⡤⠴⠶⠚⣋⣉⣉⣉⣉⣉⣉⣛⣒⡒⠶⢤⣤⣤⠴⣖⣒⣂⣉⣡⣤⣤⡤⢤⠤⠄⣒⣒⡈⢛⠿⣯⣿⡛⢲⣶⣿⣿⣿⣮⡄⢀⣀

⠀⠀⠀⠀⣠⣶⣿⢶⡦⣄⠀⠀⠀⠀⠀⠴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠷⠾⠁
⠀⣀⠀⡀⢻⠸⣿⢸⠃⠘⠓⠒⠄⢀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠀⠈⠁⢸⠀⡇⠘⢸⡇⠀⠀⠀⠘⠉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣈⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⣶⠀⣆⢸⡧⠀⠊⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⡄⠰⠘⣿⣁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠙⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⢩⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣉⣀⡤
⠀⠀⠀⠀⠀⠀⢠⣶⠤⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⣠⣤⠴⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠿⠛⠻⠿⢶⡆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠆⠀⠀⠠⣶⣛⠉⠠⠄⠐⠒⠒⠛⢛⣫⣇⠘⠀⠀⠀⠀⠀⢰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣛⡛⠿⠿⠿⢛⡛⠉⠁⠀⠀⠀⠀⣠⣦⠈⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢠⠘⣿⡷⠂⢤⡆⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠄⠁⠀⢹⣇
⠀⡤⠤⢤⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢰⠀⠞⠆⢰⠀⡆⢸⠋⢠⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢃⢴⣿⣶⠀⠹
⠸⠁⢠⣶⣿⣯⠈⠉⠀⠀⡀⠈⠁⠀⠀⠀⠒⠤⢄⡀⠀⠀⢸⡀⢸⠀⢆⠀⢸⠀⡇⢸⠄⠘⣄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣾⣿⠑⣠⣦⡀
⠈⠰⣿⣷⣿⣿⠀⠀⠀⠀⠡⠈⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⢸⠀⢸⠀⠀⢣⢸⡤⡇⢸⠀⠀⠈⠆⠀⠀⠈⠁⣀⠄⠀⠀⠀⠀⠀⢀⣉⣼⣿⣿⠏
⠀⠀⠈⠛⠿⠃⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⠀⢸⠀⠀⠈⢸⠀⡇⢸⠀⠀⠀⢸⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠀⠸⠀⠦⠜⠸⠀⠇⠸⠄⠘⠤⠎
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣄
⠀⠀⠀⠀⠀⡀⠘⢿⣿⡌⠙⢿⣿⡄⠐⣦⣤⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣤⠀⠀⣠⠄⢸⡿⠁⣼⠋⢈⡷⠂
⠀⠀⠀⠀⠈⠋⠀⢸⣿⡿⠀⢸⣿⡇⠀⣼⣿⡧⠙⠿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡷⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠐
⠀⠀⠀⠀⠀⠀⠰⠿⣿⣁⣴⣿⡟⠁⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⡾⠟⠀⠀⠻⠀⢸⣧⠀⢿⡄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠁⠀⠀⠀⠀⠀⢀⡀⠆⠠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢀⣴⣇⠀⠀⢻⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⠀⠀⠸⡟⢂⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⠻⢉⣾⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⠀⠰⣷⢼⣷⡀
⠀⠀⠀⠀⠚⢿⣷⡄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⢿⡶⠖⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⣧⢼⡿⠁
⠀⠀⠀⠀⢤⠄⢹⡇⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⣴⣦⠉⠝⠃⣄⡀⠀⣠⣶⠇⠀⠀⠀⠀⠛⠃⠤⠐⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙⢁⠆⠿⢟⣻⠡⣾⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡿⠶⡗⠘⣿⠇⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣠⣤⣬⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠍⠁⠀⢀⣀⣬⣿⣿⢦⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠃⢤⣀⣤⣾⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⢟⠟⣴⣾⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠟⠉⠻⡻⣙⣸⡿⣿⢃⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⠀⠀⠀⠀⠈⠉⠉⡰⢁⣾⠿⠿⠿⣿⡦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠭⠒⣒⠢⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢸⣻⡄⠀⠀⠀⠀⠢⠄⠀⠀⠨⠔⠁⠀⠚⢁⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣷⡙⢦⣀⣄⠓⠦⡤⠄⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⡟⠁⠀⠀⣤⣶⡿⢟⡛⠋⢉⣿⡿⠛⠲⠤⣭⣍⣃⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⣿⣿⣿⣿⠁⠀⠀⠀⠉⠻⢿⣿⣷⣶⣿⠾⠁⠀⠀⣀⣤⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣤⠶⠚⠙⣿⠿⠟⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠀⠀⠀⠀⠘⣿⡄⡀⣘⣧
⠀⠀⠀⡀⠀⠀⢠⣾⠖⠶⠦⡤⠘⣿⣷⣄⠀⠀⠀⠀⢠⡟⠁⢀⣠⣤⣤⣶⣶⣶⡎⠻⣿⣿⣿⠠⠶⢄⡀
⠀⠀⣾⠀⣀⢠⣮⠓⢶⣂⠀⢀⣉⣉⠛⢿⣶⣀⠀⠀⠘⢷⣶⣿⣿⣿⡭⣭⣽⣏⣀⣠⣶⣿⠶⠒⠉⢁⢣⢀⠇⢠⠷
⠀⠀⠘⢇⢽⡾⣟⡳⣶⣿⣟⠲⠾⢭⣉⡉⢙⣿⣿⣷⣤⣾⣿⣿⣿⣶⣾⣿⠟⣿⣥⡶⠾⠋⢅⠤⢒⣫⢏⡞⠜⠁
⠀⠀⠀⠀⠙⠟⣾⣿⣿⢿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣟⣭⣵⣶⣶⣶⣶⣾⠻⣵⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠛⠻⠟⠛⠻⠿⢿⣿⣟⣛⣛⣛⣻⣿⣿⡿⣿⣿⣟⣿⡿⠛⠿⠛⠙⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢦⣤⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣽⣿⠿⠓⠛⠃⠀⠸⢝⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⢀⣤⣀⠀⠀⠀⠐⠲⢦⣤⡉⠁⠐⢮⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠁⣠⣞⢻⣤⣿⣷⣶⣤⡈⠳⢄⡈⠻⣷⣤⢄⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡇⠀⣿⣿⠟⠋⠉⠉⠙⢿⣿⣿⣦⡀⠻⣆⢪⣻⣎⢧⡀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣾⠷⢾⡿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣾⣆⢳⣿⡎⣷⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠘⣿⡇⣿⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣿⣿⣿⣿⡿⢠⣿⢿⣿⠃⣸⢇⡄
⠀⠀⣠⠴⠞⠛⠛⢛⣛⣻⣿⣿⣷⣦⣄⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⢛⣥⣾⣿⣿⣿⣿⣿⡿⢁⣼⢯⣿⠋⣰⢏⠞⣼⣷⣤⠀⡰⠃
⠀⠉⠀⣀⠤⠖⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣦⣄⠀⠉⠉⠙⠛⣛⣛⣛⣋⣩⣥⣶⣾⣿⡿⢟⣿⣿⣿⠿⢋⣤⡾⠫⠟⣡⠜⠵⣫⣾⣿⡿⣣⠞⡡⢊⡀
⠀⠀⠀⠀⠠⠴⠒⣚⣛⣛⣛⣛⣳⠶⢤⣍⡙⠻⢿⣦⣄⡀⠤⣬⣭⣭⣭⣭⣍⡉⠥⢶⣞⣛⣋⣭⣴⠾⠛⠉⠀⠀⠀⣀⣤⣾⡿⢛⣡⣾⢕⣪⠞⠋
⠀⠀⠀⠀⠀⠈⠉⢉⣩⣭⣭⣭⣭⣝⣛⣿⠿⣷⣤⣈⠙⠻⢷⣦⣌⡙⠻⢿⣿⣿⣦⣌⠉⠉⠁⠀⠀⠤⠤⠴⠖⠚⣛⣋⣩⡴⠾⣛⣩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠶⣶⣾⣿⣿⣷⣾⣭⣝⣂⠀⠉⠛⠻⢿⣶⣬⣭⣛⠻⠿⢶⣦⣤⣄⣀⣀⣀⣀⣀⣨⣥⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⣭⣭⣛⣒⠂⠀⠀⠉⠉⠛⠛⠿⠶⠶⠶⠬⠭⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⠷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⠿⢷⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣄⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠛⠿⣶⡄⢀⣤⣶⣶⠶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠶⠶⣶⣤⣄⠸⣿
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠻⣿⣿
⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠙⠻⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢀⣠⣾⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟
⠀⢠⣾⡿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⣿⡏⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷
⠀⢿⣧⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⠀⠀⠀⠀⣀⣴⣾⠟⠉⣿⡆
⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⣿⡇⠀⠀⢀⣾⡿⠋⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆⠀⠈⢿⣦⣄⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢹⣿⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⢸⣿⡁⠀⠀⠀⣸⣿⠁⠀⠀⠈⠻⣷⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⣿⡇⠀⠀⠻⣷⣦⣤⣾⠿⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣾⣷⠀⣠⠆⢄⣀⣠⣾⠻⣷⠀⠀⠀⠀⠀⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢹⣶⣼⣿⡛⣾⣿⣄⠈⢿⣿⣿⡇⣿⣶⣦⣀⠀⠀⢀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⡄⣀⣀⣾⣿⡛⠉⠳⡌⠿⠃⢀⣻⣿⣿⣿⣼⣿⣇⣿⣧⡔⠫⠀⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠻⠆⠀⠀⠠⡘⡄⠀⠈⡏⢹⡗⠛⡿⢿⣿⣿⣿⣧⣤⣰⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠘⠛⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠃⠈⠁⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡘⣸⠀⣿⢠⣿⣿⣦⣄⠄⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⢀⣴⣒⣠⣶⡿⠟⠻⠛⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⣸⣧⣿⣿⣿⣿⣿⡀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠛⠋⠋⠛⠛⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡟⠠⢱⣿⣿⣿⣿⣿⣿⣿⣦⡞⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠈⢀⠀⠀⣴⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠈⠀⠀⢿⡟⢏⣿⣿⣿⣍⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠠⠿⠖⠚⠛⠛⠛⠛⠻⣿⡶⠶⢿⣥⣤⠀⠀⣀⣠⣤⠀⠀⠀⢚⣿⡿⠃⠀⠀⠀⠀⢀⡀⠀⣠⠔⠀⣰⡿⠻⣿⣿⠟
⠀⠀⠀⠠⠔⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⣀⣀⣒⣀⡀⠀⠐⢶⣬⠁⠀⠀⠉⢹⣤⣶⠉⠉⠉⢴⣦⠖⠁⢀⠔⠛⠻⠟⢛⠶⡟⠋⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⢤⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣉⡿⠛⠀⠀⠉⠥⢀⣀⣠⡶⠤⢀⣾⠗⣀⡤⢀⠀⠠⣾⣿⣴⣶⡟⠁
⠀⠀⠀⠀⠀⢠⣤⡾⠋⠁⠀⠠⣄⠀⠀⠀⠀⠈⠓⠲⠦⠤⠤⠀⠀⠈⠁⢀⣬⣭⠤⣤⣤⣤⣭⣤⣤⣶⣯⣵⣿⣿⣿⣿⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⣿⣅⡀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠉⠂⠀⠀⠀⠀⠈⠔⠀⠒⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡐⠂⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣴⣄⣠⣶⣦⣤⠀⠀⠀⠀⣃⡀⠀⠴⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠳⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⡿⣿⣟⠁⠀⠀⠀⠀⠙⠿⠆⢠⣮⣳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣠⡾⢡⣿⣿⣿⠟⠉⠀⠀⣿⠟⠀⣰⡿⠀⠀⣠⡀⢰⢿⣿⣿⣿⡧⠀⠀⠀⠀⠐⠀⡀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⣿⣿⣿⡟⠀⠀⠀⠲⠁⠀⠀⠟⣁⣶⣾⣿⡋⠾⢿⣿⠏⠀⠀⠀⠠⡀⠄⠀⠀⡧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⡀⠀⣰⢰⣴⣿⣿⣿⣿⣷⣿⢹⣿⣧⣰⡷⠂⣄⠈⠃⢾⣿⡿⢿⣿⡛⠜⢠⣿⡏⠀⠀⠀⠀⠀⣏⠠⡗⠀⠄⠁
⠀⠀⠀⢰⣶⣿⣿⡟⣴⣿⣯⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣇⠠⣴⣦⣳⠟⣿⡁⡸⢿⣷⣶⣾⣿⣧⠀⠀⠀⠠⠀⠁⠂⠘⠑⠆
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⡻⣧⣽⡻⣷⡌⠻⣿⣷⣜⢷⡝⣄⣻⣿⣿⣿⣿⣧⠀⠀⠂
⠀⠀⠀⠀⠀⢿⡿⣿⣿⣷⡟⢯⢿⣿⠿⢿⣿⡿⢿⣷⢾⣛⢿⡆⢴⣷⣌⠙⡻⢷⣝⠙⢿⣯⡛⢧⣝⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⣿⣾⡲⠿⣧⠻⣿⡻⠶⣭⡊⢛⣷⡈⣬⡻⢟⠻⣦⣰⣮⣛⠦⠨⣽⣄⡛⣢⠛⢿⣿⣿⣗⡶⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠁⠈⠁⠙⠻⠐⠢⣍⠑⢬⡻⡎⣻⠮⢿⣎⣺⠻⣿⣝⡳⢦⡾⣿⣿⣿⡛⠻⣭⣭⣛⣉⣿⣿⣿⡿⢶⡶⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠉⠈⠁⠈⠙⠛⠂⠓⠀⠙⠛⠻⠷⠾⠋⠉⠛⠙⠀⠈⠉






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

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣖⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣄⣀⣤⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠉⠘⠹⢿⠆⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⣸⣭⣷⣿⣾⣯⣧⣽⡷⢆⢻⠅⠀⠀⠀⠀⠀⠀⠀⡀⠶⠶⠶⠿⠾⠦⠥⣦⢠⡾⡇
⠀⠀⠀⢀⠊⠐⠀⠀⠀⠀⣀⡀⠶⠂⡀⢀⣿⡇⠀⠀⠀⠀⠀⢴⣿⣿⣿⡿⢿⣛⣫⡷⠚⠀⠘⠿⠀⠀⠀⠀⠀⢀⡔⣨⠀⠂⣠⣴⡰⠖⠛⣒⡢⠉⢽⡇
⠀⠀⠀⠘⠇⠀⠀⠀⠌⠛⠋⠠⠶⠟⠃⠀⠙⠻⠀⠀⠀⠀⠦⠀⠉⠛⢁⡒⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠈⠀⢈⡉⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣳⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠠⠒⠌⣉⠈⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣿⣾⣿⣿⣿⢿⣿⣿⡠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠐⠀⠀⠁⠂⣀⣀⠍⠄⡏⠠⣾⡇⠀⠀⠀⠀⠀⠀⠠⣊⠈⢁⡸⢿⣟⣿⡿⣻⢯⠈⠐⠻⠏⠀⠀⠀⠀⠀⠀⠐⠡⠄⠀⠀⠿⣿⣿⣿⣿⣤⣲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠚⠀⣠⠆⠰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⣺⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⠸⢿⣼⣿⡏⠉⠚⠛
⠀⠀⠀⠀⠀⠀⠑⠤⠄⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣴⣤⣄⣀⢠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣤⣾⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢿⡏⢻⡝⣏⢠⡇⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣴⡯⢉⣭⡍⢹⡿⢅⣠⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠶⢤⣀
⠀⠀⠀⠀⢺⡆⢸⣇⡸⡇⣿⢸⣗⣒⠱⡟⠀⠀⠀⠀⠀⠀⠀⣘⣻⣇⢸⡟⣻⣼⣇⡞⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠠⣨⢿⡟⠙⣿⠿⠿⠧⢴⣶⡆
⠀⠀⠀⢤⢐⡇⠘⣷⣾⠟⢻⣼⡏⠀⠑⠿⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⠿⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⣼⠁⠐⠶⠷⠒⠀⠈⠛⠁
⠀⠀⠀⠀⠉⠉⠱⣿⣧⠀⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡠⢤⠤⣄⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⣄⣠⣤⡀
⠀⠀⠀⠀⠀⡀⢈⣿⣿⣿⣿⣿⣿⣦⡤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠐⠒⠚⡻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⡿⣿⣿⣿⣿⣿⣀⣴⣶
⠀⠀⠀⠠⢀⡄⠴⠿⣟⣿⣍⣙⡄⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠠⠊⠈⢁⣠⠒⠄⠉⠀⡈⠀⢰⣅⠄⠀⠀⠀⠀⠀⢠⡞⠻⠃⠈⠓⣹⣾⡽⣼⡏⠿⢿⣿
⠀⠀⠀⠦⠀⠀⠀⣨⣍⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⣤⢀⣻⠗⠚⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣃⠱⢤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠈⠃⠀⠀⢀⣻⡄⡡⣽⠫⡤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢩⣯⠽⡽⣿⡷⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠿⣤⣴⡆⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡞⠀⠐⠻⠛⠋⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡆⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠀⠀⠀⠠⠄⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠁⠙⠀⠀⠁⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠰⠆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣶⣦⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣽⣿⣿⣿⡇⠀⠀⠀⠀⣀⣠⣤⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣤⣀⠀⣀⣄⣀⣀⣰⣶⣶⣶⣾⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠉⠙⣷⡀⠀⣀⣀⣀⣀⡀⢘⣭⣿⣿⣯
⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣷⣤⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⡠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢋⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⣽⠃⢴⠛⣃⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⢉⡀
⠀⠀⠀⠌⠉⠄⠠⠄⠄⠀⢲⣇⠀⠀⠠⢀⠀⠐⠉⠡⠘⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠄⡍
⠀⠐⡋⠀⠀⠠⢀⡀⡄⠀⠘⡉⢀⠀⠀⠈⠃⠀⡀⠀⠀⠀⠉⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢠⡈⠂⠀⠀⠀⣠⠚
⠀⠀⠁⠀⠀⠁⠤⠀⢀⡀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡩⠀⠀⡄⠀⠀⠀⣐⡖⠡⡄
⠀⠀⣿⡀⠘⠂⠁⠀⣼⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡇⠁⠀⢀⠃⠀⠀⢀⠞⢉⠀⠀⠁
⠀⠀⢸⠀⠀⠀⠂⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⡞
⠀⠀⢠⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⡅⠀⣸⠀⠀⠀⣀⡀
⠀⠀⠀⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠧⠶⠛⠛⠛⠛⠛⠋
⠉⣩⡍⣹⣾⠻⠓⣾⡗⠙⠒⠉⠙⠛⠛⠿⠿⠻⠋⠉⠉⠛⠛⡉⣛⠛⠉⠉⠛⠛⠋⠙⢋⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂
⣴⢱⣿⣿⣷⣇⣶⣾⡗⢠⡀⡀⠐⠒⠀⡀⢀⠀⢀⣀⣀⡀⠈⠉⣤⣾⣯⣠⡀⢀⣠⣄⣠⣴⡤⠤⠐
⣷⣿⣿⣿⣿⣿⣿⣿⡇⣸⣟⣴⣦⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠺⠟⠻⠻⠿⠿⢦⠝⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⣿⣿⣿⣿⡿⠿⢿⣿⠂⠿⠿⠿⠋
⠉⠀⠉⠉⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢅⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠊
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣶⣴⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣤⣤⣠⣀⣀⠀⢣⠀⠀⣀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⢿⠿⡿⠳⡿⣿⠛⠛⣿⣿⡳⣾⣷⣶⣶⣿⣿⠂⠀⠀⠀⢀⡠⠔⣲⡇
⠀⠀⠀⠰⢾⠟⠿⠛⠟⣽⡿⣆⣻⣤⣴⠿⣿⠀⢩⢿⡏⠀⢻⣿⡿⣿⡿⠝⢿⠟⠿⣿⣶⣦⡋
⠀⠀⠀⠙⠪⢍⠐⠚⢥⡍⠞⠠⠬⠛⠿⠒⠛⠀⠈⠿⠄⠀⠑⠋⠀⣈⡀⠀⠀⠀⠀⠛⢺⣄⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠄⢀⣀⣄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠂⠨⠄⠈⠄⠀⠀⠀⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠛⠒⠪⢥⣂⢄⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⠀⣿⠿⠶⢮⣑⠮⣒⢄⡀⠀⠀⠀⠀⠀⢀⣤⠶⠒⣒⣒⡒⠤⢄⣀
⢉⣉⣉⣉⡓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⢸⡆⢿⡀⣴⣶⣬⣙⠮⣓⢭⡲⠤⣤⣤⡴⠟⢁⣴⡿⠋⢉⣉⣉⣓⠲⢭⣓⡲⠤⣤⣀⣀⣀⣀⣀
⠛⠋⠉⠉⠙⠛⢦⣑⢄⠀⠀⠀⠀⠀⣼⠃⣠⣾⠇⢸⡇⢹⣿⠟⠛⠷⣮⡓⢬⣓⠶⢤⣤⣶⠿⠟⠀⣰⡿⠋⣩⣭⣙⡓⠦⣍⡓⠶⢤⣤⣤⣤⣤
⠿⠿⠿⠿⠿⠷⢦⣌⠳⣕⢤⡀⠀⣰⠇⣸⠏⢁⣠⣾⡇⢸⣿⠀⢠⣶⣤⣙⠳⣬⣛⠶⣤⣤⣤⣤⣾⡿⠃⣸⣿⠛⠛⠛⠷⣦⣍⡓⠶⣤⣤⣤⣤
⣶⣶⣶⣶⣴⣶⣦⣬⡓⢮⡳⢬⣙⣉⣰⡟⢠⡿⠋⣁⣤⣾⡿⠀⣼⣿⡟⠛⠿⣦⣝⠻⢶⣤⣬⣥⣤⣤⣾⣿⠃⣠⣾⣶⣶⣤⣉⠻⢷⣦⣬⣭⣭
⣤⣤⣀⣉⣉⣉⣉⣉⡙⠳⣭⡳⢬⣉⡋⢀⣾⠃⣾⡿⠋⢉⣀⣴⣿⡟⢀⣴⣶⣦⣙⠻⢶⣤⣍⣉⣉⣉⣉⣀⣴⣿⠟⠋⠉⠛⠿⢷⣦⣤⣉⣉⣉
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣭⡓⢮⣝⣛⣃⣼⡿⠀⣾⣿⠿⠛⠋⣠⣾⣿⡿⠿⠿⢿⣦⣍⣙⣛⣛⣛⠛⠛⠛⢉⣠⣶⣿⣷⣶⣤⣌⣙⠛⠛⠛
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⣦⣭⡓⢮⣝⣛⠛⢁⣼⡿⢁⣴⣾⣿⠿⠟⠋⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠛⠛⠻⠿⠿⠿⠿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣙⠳⢮⣭⣛⡛⠛⣁⣾⡟⠉⣀⣤⣤⣴⣶⣾⣿⣿⣿⣷⣦⣤⣴⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤⣤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠶⣦⣍⣙⠛⠛⢋⣠⣾⠟⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⢤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⡀⠐⠈⠉⠛⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⡆⠀⠀⠀⢂⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢨⠀⣾⣷⠀⠀⠀⠂⠀⠀⠀⠀⠈⠳⡄
⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⡄
⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⡘⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠁⠢⣀⠀⠀⠈⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣴⣶⡿⠳⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⠁⠚⠲⠦⣄⠀⠀⠀⠀⠘⠋⠉⠉⠙⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠠⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⢿⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠆⠀⠀⠈⠙⠦⠆⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⡰⠊⠐⣆⠀⠀⠀⠀⡆⠀⠀⡎
⠀⠀⠀⠀⠀⢀⠉⠀⠀⡎⠀⠀⠀⠸⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠈⡃⠀⡀⢦⣀⡀⣠⡃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠁⠀⠀⡜
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣤⣤⣄⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⠂
⠀⠀⠀⠀⡀⢀⣭⠃⠀⠀⠠
⠀⠀⠀⢀⠀⠁⠁⠠⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢄⡦⡄⠀⠀⠈⡇⢀⠔⢡⠄⠉⠀⢀⣴⣶⠁
⠀⢠⠀⠀⠀⢠⠃⠀⢀⣻⡷⢃⣵⠿⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠆⢄⠄⠀⢀⣆⡁⠤⣰⣯⡶⠋⢁⡠⠤⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⢠⠏⠀⠌⡼⠙⠈⣴⢡⠏⠀⠀⡏⠀⣤⡝⠀⠈⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⢠
⡼⠀⠀⣜⠁⠀⣸⠡⡟⠀⠀⠀⠣⠀⠓⠒⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⠦⣄⠈
⡇⠀⠘⠡⠀⠀⠋⠂⢻⠀⠀⠀⠀⠁⠐⠒⠹⠻⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⢀⡴⠃⠙⠀⢀⣤⡄⠈⡇
⡁⡀⠀⠇⠀⢠⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡚⠉⠁⠀⣄⢀⠙⠉⠉⠀⠢⢄⠀⠘⢥
⡟⠈⡰⠀⠀⠸⡄⠀⠀⠈⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡠⠴⠚⠑⠂⣔⢀⢸⡃⠀⠀⠀⠀⣤⢀⡇⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⢳⠑⠀⠀⠁⠀⠑⠠⠤⠄⠀⣀⡠⠤⠖⠊⠁⠀⢀⠀⠀⠀⠈⣄⠚⠇
⢧⡠⠀⠀⠀⠀⠋⢧⡀⢀⡀⢀⠀⠀⠀⢀⠀⠈⠈⠀⠀⠀⠀⠀⠠⠂⣁⡀⠂⠈⠀⠀⠄
⠈⡄⠀⠀⠀⠀⠀⠀⠝⢶⡀⠀⢔⠀⣀⠀⠉⡲⠀⠀⠀⢀⢔⠀⠀⠀⠈⠙⣆⣢⠀⠀⠐⠄⠀⠀⠀⠀⠔⠚⠃
⡀⠀⡀⠀⠀⠂⠀⠀⠀⡀⠍⠛⢦⣤⣤⣄⣈⡀⣤⠔⠈⠀⠁⠀⠀⠀⢀⠀⠸⢿⠄⠀⠀⠀⠑⠂⠀⢀⣀⣀⡀
⠀⠀⢲⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡡⠁⡘⠁⠂⠀⠈⠀⠀⠀⠀⢀⠀⠀⠳⠧⠀⠀⠀⠀⣀⡲⠀⠐⠀⠀⠢
⠀⠀⠀⠻⣎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⢢⣀
⣀⠀⠀⠀⠘⠦⣄⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠱⡉⠉⠙⠉⠀⠀⠈⢳
⠀⠀⠈⠢⠠⡀⠈⠓⢤⣝⣄⣀⣐⠪⢉⣂⠤⠄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣤⡆⠀⠜
⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣛⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣤⣴⡲⢄⠀⢀⣀⣠⢤⢤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠲⠭⣝⢦⡙⣆⡽⠖⢋⠙⡀⠀⠈⡀⡇⠀⠀⢀⣴⣶⣦⡁⠁⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡒⠉⠛⠳⢄⡨⠛⢋⡔⠅⠄⣼⣆⠁⠀⠀⠀⡇⠀⠀⢸⡃⣇⣀⣿⠀⠀⠀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠭⠬⣭⡉⢢⣔⠁⣂⠵⠫⢊⣤⢞⠁⢹⡄⠀⠀⢀⡁⠀⠀⠈⠓⠶⠟⠁⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠒⠒⠒⠦⣼⠏⠀⠉⣁⣤⠶⠋⢕⠅⡡⢸⡇⠀⢠⠞⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠗⣛⣉⣉⣩⡏⠁⠈⢘⡩⠿⣷⣄⠀⠅⠊⠔⣼⣷⣄⠡⠀⠀⠀⠙⢦⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⢸⣋⠁⠤⣴⣏⠀⠀⠈⠈⠀⡐⠀⠻⣦⡀⢐⡴⢻⠏⡍⢳⠀⠀⠀⠀⢸⡆⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠭⠛⠶⢦⣄⡀⠀⠛⣧⠞⢔⢁⠄⠀⠀⣇⡴⠚⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⢻⡏⠣⡄⠘⡖⡠⢂⢄⣠⠴⡿⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⣤⡶⣴⠦⠄⣀⢿⠀⡘⡆⡇⢠⠖⠋⠀⠀⡇⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠘⠁⠈⠙⢦⠀⠀⢀⠙⡆⠡⢻⡷⠁⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢇⠀⠀⠀⢸⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⠶⠾⠓⣻⠶⣤⠀⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢁⣾⣿⠿⣤⡛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⢻⣿⢿⡶⣽⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡎⣧⠹⡀⠹⣟⢿⡫⢍⡙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠃⢹⠀⡇⠀⠘⣧⠙⢄⠉⠢⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞⠀⡄⢀⡤⢼⣧⠀⠑⠤⣈⣑⢨⣑⣦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⡼⠁⠀⠃⡸⠀⠀⠈⠑⠦⠤⠤⠤⠌⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣟⡜⠁⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢐⠞⡠⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢞⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣒⣀⠍⣍⠏⢫⠉⣉⠙⢢⡾⠛⢒⡀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡏⠡⠠⠈⠃⢈⠒⠉⢄⡰⡘⠏⠀⠢⠈⢒⡨⠏⠀⢀⠯⢲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠮⠃⠁⣃⠀⠁⠀⢠⠊⠑⣀⠀⢐⠀⠥⠀⠤⠀⢰⡧⡀⠀⢹⠂⠀⡠⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡀⢻⢀⠆⠀⡀⢇⠋⠈⢠⠌⠀⠁⢢⠔⣬⠙⣨⢥⠘⣮⡀⠎⠐⠖⠐⠀⠄⢉⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⡤⠀⠀⠀⡰⠆⣀⠒⡠⡯⣔⡡⠖⠤⠐⢆⠃⢠⠀⡎⣙⡃⢰⠋⡇⠂⠀⢛⠈⡀⠈⢬⡡⢜⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠠⠀⣄⠁⠂⣺⡑⠺⠁⠓⢩⡄⣒⢒⠲⢍⠂⠰⢡⠧⠉⠉⠁⢽⡘⣀⠉⡤⡈⠠⠈⠠⠴⠀⣡
⠀⠀⠀⠀⠀⠀⡰⠛⢛⢣⡉⢊⡡⠜⠓⠛⠋⠁⣟⠛⠀⠆⠀⠍⢈⠀⠀⢙⡤⠂⠈⠀⠲⠔⢨⡓⠌⠠⠀⠜⢤⠉⠠⠀⠏⠀⠃
⠀⠀⠀⠀⠀⢠⠝⠃⠈⠀⠀⠀⠉⠑⠶⡇⢁⠀⠇⠀⠀⢠⣀⢀⢚⣃⣴⣾⣄⡚⢀⠅⠢⢠⡔⡟⠐⡀⢻⠮⠔⠛⠋⠐⠀⠲⢌⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣈⠗⡤⠗⠊⠈⠁⠀⠀⠰⠀⠀⠈⠉⠘⠀⢴⡑⠳⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡄⠀⠀⠀⢸⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢇⠀⠀⠀⡔⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⡦⣤⣞⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⡰⣶⡄⢀
⠀⠀⠀⠀⠀⢀⣤⣦⣰⣿⣿⣤⡿⢿⡿⣾⣾⣿⡜⣷⡼⣷⣄
⠀⠀⠀⠀⠀⠶⠖⠿⣿⣿⡿⠿⠇⣿⣾⣿⣿⣿⣷⣿⣟⣿⣿⣷⣀⣄
⠀⠀⠀⠀⠀⣰⣆⣿⣦⢀⡀⣦⣤⡝⢿⣿⣿⣿⣿⠟⠟⣛⣿⣿⣿⣿⣷
⠀⠀⢀⣠⣾⣿⣿⣿⡿⣼⣇⣿⣿⣿⣼⣿⣿⣿⠃⠀⠀⠈⠉⠈⣭⣹⠯⣤⡀
⠶⢿⣛⣭⣿⣿⢿⣻⣿⣿⠞⠻⢿⣿⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣷⢀⢈⠃
⠿⢛⣛⣫⣽⠶⠟⠋⡁⢠⣤⣿⣦⢠⣭⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⣈⣉⣁⣀⣤⡤⠖⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⣋⣥⡶⠟
⣙⣉⣩⣭⣤⣶⣾⣿⢿⣿⣿⡿⣿⣿⡿⠛⠉⣨⡙⣿⣿⣶⣄⡀⠀⠈⠉⠉⠉⠉⠉⠁⠤⠖⢛⡉⢥⣔⣦
⣉⣉⣉⠉⠉⠉⠒⠛⠋⠽⠓⠛⣋⣁⠴⢠⣿⣿⣿⣯⣻⣿⣿⣿⣿⣶⣖⣒⣀⣀⣘⣈⣉⣁⣀⠉⠩⠁⠂
⣤⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣛⣫⣥⣾⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠯⠭⠭⠭⠭⣭⣤⣭⣭⣭
⠀⠤⢤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣽⣶⣶⣶⣶⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⣭⣥⣤⣤⣤⣬⣤⠄
⠀⠀⠀⠀⠐⠒⠿⠿⢿⣿⣭⣭⣿⣿⣿⣶⣶⣖⣒⣚⣛⣛⣯⣭⣭⣭⣽⣿⣿⣿⣿⠿⠿⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠉⠳⣶⠟⠉⠛⢶⡶⠛⠉⠉⠉⠉⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡟⠀⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⣴⠿⠛⠀⠀⠀⠀⣀⢀⡝⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⣀⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠬⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢦⢠⣀⣔⣬⡺⠛⠁⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈⢻⡄
⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠢⡄⠮⠓⠋⠁⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⠀⠀⠀⢈⣀⢀⣠⡴⠎⠀⠀⠀⠀⣀⠠⣴⡍⠀⠀⠀⠀⠀⢀⣿⣀
⠀⢸⡇⠀⠀⠀⠠⡀⢀⡠⢀⠎⠀⠀⠀⠀⠀⠀⡠⠀⠈⠒⠙⠞⠓⢉⠀⠀⢠⣠⡀⣄⣦⣴⣿⣿⡿⠋⠀⠀⠀⠘⠎⠳⠿⠿⠊⠀⠀⠀⠀⠀⢸⠀⠉⠉⠛⢷⡄
⠀⠀⠻⢦⣤⣤⡄⠉⠀⠉⠀⢄⠀⠀⡀⢄⠠⡔⢊⡄⡀⢀⡦⣄⡴⠄⢠⡐⢤⣙⢻⣿⠿⣫⣾⠛⡱⣄⢦⡠⣰⣵⣦⢂⡄⠀⠀⠀⠀⠀⠠⡰⡅⠀⠀⠀⣠⠼⣿
⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠁⢈⣤⣤⢾⢯⡳⣝⡲⣝⡲⠭⢃⣼⢯⣷⡿⢋⣡⣾⡿⣷⣷⠙⠮⠻⠟⠛⢉⣥⣿⢭⡇⡄⢐⢦⡪⣕⠿⠃⢀⡠⠖⢉⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣦⣍⡂⠓⠌⣓⣩⡀⣀⡀⣀⠓⠝⠳⠝⢒⣉⣴⣿⣿⠟⠉⠀⠋⠉⣠⡾⣟⢮⣿⡿⢲⣖⠿⡛⣝⣾⡟⣰⣏⣪⣕⠎⢚⢁⣬⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⣷⣀⣑⣈⡙⡿⢿⢿⢿⡻⣽⣿⣥⣶⠖⠀⣠⡾⣏⣽⣮⡿⢏⣴⣟⢮⣓⣽⡾⢟⣴⡿⣤⣥⣶⡾⢋⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⢷⣌⠈⢯⣷⣻⣾⡿⠋⠀⣠⣾⢻⣶⣤⣤⣴⣾⣿⣏⣈⣓⣉⣥⡶⣿⢯⡷⠾⠛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡽⠋⠀⣠⣾⣿⣿⣿⡿⠿⠟⢛⣿⣿⠿⠟⣽⠿⠷⠶⠶⠶⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⢦⣄⢀⡰⠋⠀⠀⠔⠛⠛⠉⠁⠀⠀⢀⣴⣟⣋⣤⠴⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠋⠀⠀⣹⠟⠀⢀⣀⣤⣴⣶⣾⠏⠀⢀⣴⠿⠋⠥⠅⠂⠀⢀⣤⣾⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠿⠃⠀⠀⠀⠀⣠⣼⣧⣶⡿⢟⣿⠁⣲⠞⠁⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⡐⢶⡄⢄⡐⠻⡿⠯⢠⡿⠋⢁⡼⠋⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢶⣤⣥⣴⠞⠛⠛⠋⠀⣠⠟⢁⣴⡿⡫⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⠟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⢤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡾⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡟⠁
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠉⠁⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠙⠛⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣤⠶⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣰⠟⠁⠀⠀⠀⠀⠙⣦⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠳⣤⠶⠛⠛⠛⠶⣄⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⣷⠶⠶⠶⣤⣸⡇
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠉⢻⡟⠛⠛⠛⠛⠛⠛⠛⢻⡛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⣠⡴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠈⣻⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠋
⠀⠀⠀⠀⠀⣾⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⣠⣴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢁⣼⡄
⠀⠀⠀⠀⠀⠻⣦⣤⠾⠁⠀⠀⠀⠀⢠⡟⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡆⠀⠀⣠⡶⠟⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠸⣧⣀⣠⡟⠀⠀⠀⠀⡀⠀⢀⣴⠛⠁⢀⡇⠀⢰⡏⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⢹⡇⠀⠈⠉⠁⠀⠀⢀⣠⡼⢿⡀⠸⣧⣀⣀⣼⠃⠀⠀⠛⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⠟⠉⠀⣸⠁⠀⠈⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⠀⠀⠹⣆⣀⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⡉⠝⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⢀⠔⠈⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡠⠊
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡿⠟⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⢹⣿⣿⠟⠉⠀⠀⠀⢳⡄
⠀⠀⠀⠂⠀⠀⠀⠀⡀⠔⠀⠙⠀⠀⢀⠀⠨⠟⠁⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠁⢀⣴⣿⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⢰⠔⠛⠛⢿⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠋⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠒⠚⠓⢦⣀⣼⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢀⣠⠤⠤⠦⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⠶⠶⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⠲⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⡽
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠠⢾⡁
⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡋⠀⠀⠀⠀⠀⡀⠹⡆
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠐⠐⢀⣀⣀⡴⠃
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠳⡤⠖⠁
⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡱⢦⣄⠈⠦⠂⠂⠀⣀⡴⠋⠳⣄⣈⠘⠁⠀⠀⠈⣀⣤⠞⠙⠶⠶⠞⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠈⠙⡛⠛⠛⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠃⠀⠀⢰⡄⠀⠀⢸⠀⠀⠀⠀⠀⢲⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡆⠀⠀⠀⠀⠘⠃⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡆⠀⠀⠀⢠⡄⠀⠸⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣦⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣦⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘