Exploding head Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠿⠟⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠦⣤⡀
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠻⣯⣁⡀⠀⠀⠀⠀⠀⠀⢀⣨⡿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢧⣄⣀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣤⣼⣟⢻⡟⣿⠛⣿⠟⣿⠟⣿⣦⣤⣤⡴⠂⠀⠀⠀⠀⠀⢀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣍⠉⠀⠀⠙⠲⠶⠶⠿⣿⣿⣆⢻⡈⣷⢸⡇⣽⢀⡟⢰⠇⣼⣿⣿⠷⠶⠶⠚⠁⠀⠀⢉⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⢶⣖⠀⠀⠀⠀⢉⢻⡘⡇⢹⠸⡇⢻⢸⡇⣿⢰⡏⠁⠀⠀⠀⢀⣦⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠘⡇⠻⠘⠀⠃⠈⠘⠀⠃⡼⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⢶⠶⠛⠛⠛⠛⢶⡶⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡴⠛⠙⠁⠀⠀⢠⣄⣀⣀⣀⣀⣤⠀⠀⠀⡙⠉⠳⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠠⠤⠼⠟⢻⡏⣿⢹⡏⢹⢉⡏⣻⠛⠻⠦⠤⠄⠀⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠙⠓⠒⢾⣤⣤⡄⣘⡇⢹⢸⡇⢸⢈⡇⣿⡀⣠⣤⣼⠷⠒⠛⠉⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⠓⠦⠀⠀⠀⠀⠈⠙⠉⠁⠈⠀⠁⠘⠀⠀⠉⠉⠉⠁⠀⠀⠀⠠⠖⠛⠉⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣠⣷⣀⣠⣶⣄⣠⣶⠀⠀⠀⠀⠀⠀⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠶⠶⠶⠶⠶⠶⢿⠉⠉⠀⠙⠉⠀⠉⠉⡿⠶⠶⠶⠶⠶⠶⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠒⠤⣌⣀⡀⠀⠀⠀⠀⢀⣀⡡⠤⠒⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠓⠀⠀⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⡀⠀⠀⣠⣴⠶⠞⠛⢷⡄⠀⠀⠀⠀⣰⡟⠛⠶⠶⣄⣄⠀⠀⢠⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⡾⠛⠁⠀⠀⠀⣦⠘⣷⣤⣶⣶⣶⡿⢀⡤⠀⡀⠈⠙⠛⢦⣾⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣰⠟⢀⠀⢠⡀⠸⡄⢸⣦⣼⠇⠀⠀⠹⣧⣼⠁⣰⠃⣠⠀⠀⡈⢻⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⣀⣤⣽⡾⠟⢧⣀⣿⣴⠟⠛⠉⠀⢀⠀⠀⠀⠈⠙⠻⢷⣴⣏⣠⡾⠿⣶⣧⣄⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⣡⣶⠿⠛⠉⠁⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⠈⠙⠻⣷⣦⣜⠇
⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣽⣥⣬⣭⣿⣥⣿⣯⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢠⠟⣆⠀⢠⠞⢦⡀⠀⢀⡴⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡏⠈⠓⠤⠏⠀⠈⠳⠏⠀⠀⠹⣤⠞⠀⠙⢆⠀⢀⡴⠋⢳⡀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⢷⡴⠚⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠬⠷
⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⡴⠦⡄⠀⠀⠉⠉⣹⠇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡆
⡠⠒⠒⠒⢄⠈⠢⡀⡄⠀⠀⠀⠿⡀⠀⣃⠜⠀⠀⠀⠀⠀⠿⠶⠚⠋⢹⡏⠀⠀⢠⣄⠀⠀⣠⣤⡀⠀⠀⠀⣀⠀⠀⣠⢦⣀⠀⢀⡏⠙⠳⠦⠿
⢇⡀⠀⠀⢈⣧⠀⢘⠃⠀⠀⡸⠀⢇⠀⠁⠀⠀⠀⡰⢄⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠈⢳⡴⠋⠀⠙⢦⣠⠞⠉⠳⣤⠏⠀⠈⠓⠾⠃
⡌⠀⠀⢠⠁⠈⢇⢸⠳⡀⢠⠃⠀⢸⢀⡠⠒⡏⣴⠥⠚⠀⠀⢀⡾⠦⣸⠇⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠁
⢑⠶⠀⢠⠀⡄⢸⡌⠀⠱⡎⠀⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⢀⡾⢁⣤⣌⠀⣼⠁
⢘⠀⠀⠘⡀⣡⢸⡇⢠⠀⠠⠒⢤⠐⣀⠀⠘⢂⠽⠋⠀⠀⡾⣡⠟⠀⠉⠛⠃
⠈⢢⠄⠀⠐⠾⠤⠀⠔⠀⠀⠀⠀⠀⠘⠀⢾⠁⠀⠀⠀⢸⡷⠋
⠀⠸⣄⣀⣀⠀⠀⠔⠁⢂⣤⠐⠄⠀⣠⡕⡈⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠙⣄⠀⢄⠀⢀⣁⡀⠀⠀⠈⠐⠀⡇
⠀⠀⠀⠀⠐⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢀⡗⢀⡐⠀⠀⠀⠀⠐⠈⠁⠒⠐⢮⣠⡄
⠀⠀⠀⠀⠀⠳⢤⠄⠀⠊⠒⠀⠀⠀⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⢽⡁⢀
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠀⢲⡁
⠀⠀⠀⠀⠀⠀⣀⠔⠃⠠⠀⠀⣀⠀⠀⢀⠊⠉⡢
⠀⠀⠀⠀⠀⠀⠙⠚⠓⠒⠦⠤⢀⣀⡩⠛⠦⠤⠃
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣰⠳⡄⢠⠞⢦⠀⢀⠶⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣰⠃⠈⠑⠃⠀⠈⠋⠀⠀⠳⠋⠀⠙⣄⡠⠊⠈⢧⢀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹
⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⡴⢤⠀⠀⠙⢛⡗⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣇
⠐⠉⠉⠰⡀⠀⣴⠀⠀⠘⢃⠀⡗⠁⠀⠀⠀⠘⠛⠉⠁⣿⠀⠀⡶⢄⢀⡴⠦⡀⠀⣠⣄⠀⡴⠲⢄⣀⡇⠉⠓⠛
⡱⠂⠀⠐⠳⡀⡦⡀⢀⠇⠈⠀⢀⠠⢀⣌⡥⠀⠀⣠⡀⣿⠀⣸⠁⠈⠋⠀⠀⠘⠚⠁⠀⠙⠃⠀⠀⠉
⠡⠤⠀⠀⠀⢡⠃⠱⡌⠀⠀⠛⠁⠘⠈⠁⠀⠀⣰⢃⣌⡃⣰⠃
⢈⠀⠀⠀⠤⠼⠁⠄⠀⠢⠔⢤⠀⢋⠜⠋⠀⢰⣣⠎⠀⠙⠃
⠀⠙⠀⠀⠙⠂⡐⠁⠀⡀⠀⠈⠈⢃⠀⠀⠀⠿⠃
⠀⠈⠐⠲⡀⠈⠀⠀⠛⠈⠀⠘⠣⠨
⠀⠀⠀⠀⡏⠀⠄⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡇⠠⠁⢀⡀⠀⠀⠀⠁⠘⠥⠆
⠀⠀⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠂⡤⠊
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠂⠈⠉⢲⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠀⣔⣉⣀⠀⠀⠀⠀⣀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠛⠉⠉⠁⠀⠀⠉⠉⠙⠷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠓⢦
⠀⠀⠀⠀⢀⡤⠶⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢦⣀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠂⠀⠀⠀⠀⠀⠐⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣹⡶⣦⣤⣤⣤⢤⡶⣾⣁⣀⣀⠄⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠹⣏⠃⠀⠘⠲⠤⠶⢿⣿⡍⣧⢻⢸⡄⡇⣼⢰⢏⣿⣿⠷⠦⠴⠂⠀⠈⢉⡟
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢶⣀⣀⣀⣈⢻⢸⠸⡞⡇⡇⡏⡜⣸⢁⣀⣀⣀⣰⠖⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢘⣀⣀⣤⣤⣤⣁⣁⣃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠞⠉⠉⣉⠀⠀⠀⢈⠉⠉⠳⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠁⠁⣤⣴⠶⣶⢻⢹⡟⣿⣿⣳⠶⢦⣤⠄⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠙⠛⠚⠷⢶⣂⣼⢸⣘⡇⡇⠏⣿⣀⣶⠾⠟⠛⠋⣀⣤⠄
⠀⠀⠀⠀⠀⠠⠤⣤⣀⡈⠁⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠊⣉⣠⡤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡟⠾⠛⠶⠋⠓⢚⣧⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠠⢄⣁⡀⠀⠀⠀⠀⣈⣁⠤⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠓⠀⠐⠋⣁⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣆⣀⣴⠶⠛⠋⠘⣧⠀⣀⡀⣰⠏⠉⠛⠳⢦⣀⢠⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⠋⠀⡀⢰⡀⢧⣸⡟⠛⠻⣟⣰⠃⡴⠀⡀⠈⢻⡃⠀⠀⢸
⠀⠀⠀⠀⠀⣰⠏⠀⣀⣬⣷⠾⣄⣹⣤⠷⠞⠉⠀⠀⠀⠈⠛⠶⣧⣞⣀⠾⢶⣧⣀⠀⠘⣧
⠀⠀⠀⠀⢀⣫⡴⠟⠋⠉⢀⠀⠈⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠈⠙⠻⢶⣼⡃
⠀⠀⠠⠖⠋⠁⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⢠⠈⠂⠀⠀⠀⢲⡦⠄⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣦⠀⢀⣤⣄⡥⢠⡄⠀⠀⠀⠀⠀⡸⠇⠀⣼⠶⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠀⢀⣠⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠠⠻⠿⠆⠐⠋⠉⠐⠈⠿⠱⠀⠀⡆⠀⣦⡀⠛⡀⢀⡦⢄⡒⢲⣆⠀⡠⠀⠀⠀⠀⠀⢀⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠊⣠⠊⣠⣿⢦⡌⠡⠀⠀⠀⠀⠀⠴⢀⣾⠇⠔⠛⠃⠜⡀⣳⠐⠀⣻⣿⡷⢿⡇⢀⡤⢈⡴⠞⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠊⡁⣨⣿⣧⡀⠀⢀⡤⠧⣀⣤⡀⠂⡿⠃⠀⠺⣏⡐⠖⠀⠄⠀⠔⣻⣏⠓⠘⢿⣿⢟⡫⠤⡤⠀⠉⠘⠁⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠛⠁⣿⠁⣿⣿⡷⠀⠀⠛⠁⠀⣸⣿⣇⠐⠉⠀⡟⢡⣽⠟⣛⣿⠀⠠⠛⢠⣿⣌⠹⣿⣻⡇⢀⣄⠀⠀⡀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡿⣠⢀⣠⠁⡸⣿⡟⠃⠀⣄⠀⠀⠀⢹⣿⣿⣿⣁⡈⣠⣼⣿⣾⠀⠙⠁⠀⠀⠀⢍⣻⡜⢿⣿⡩⠃⢭⠄⠈⠀⠀⢀⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⢧⣧⣿⣇⣶⡷⠛⢃⢀⡀⠊⣤⣄⡀⠹⡿⠿⣉⡇⣸⠀⢀⠟⣉⠷⠶⣦⡴⢋⡄⠒⠛⣄⢈⣿⡎⠑⠀⠀⠀⠀⣒⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠲⢾⣀⢺⣍⣿⣾⡿⠋⠻⢦⣿⡹⠓⣶⣿⣦⣼⣧⣠⣤⣿⠛⢁⡤⢴⠅⠸⢸⣿⣦⠀⣿⡾⣇⠹⢠⣤⡞⡇⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣄⢸⣿⣿⣿⢊⣿⣅⣠⣴⡿⠟⢁⡶⡾⠁⠙⢻⣿⡟⣙⢓⡐⣈⣠⡼⠁⠀⡿⢻⣿⠶⣿⢾⡟⣴⠏⠋⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣺⣽⣧⣿⣟⡏⣉⣤⢽⣥⠗⠋⢷⣴⣶⣦⠘⢿⣿⡍⢉⠙⠙⣿⠿⢿⣟⣵⣛⣹⢿⣫⣾⢁⣩⡤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠶⠉⣧⣽⣿⠿⠿⣿⣿⣿⢟⣴⣏⣨⠟⠿⡧⢆⣸⡟⠁⠈⠳⣚⣡⣤⠞⣿⣿⡾⠁⣿⠟⢹⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⣿⣿⣿⣧⢠⣄⢻⣟⣮⣅⣨⣿⣦⣶⣾⡌⢻⣿⣎⡤⣶⢫⡌⠘⣿⡋⠉⠀⠈⡁⢠⠐⢯⡀⢐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣉⣾⣿⣿⣿⣍⣻⣦⡙⢿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⢀⣹⡇⠹⣆⣩⠟⡀⣠⡾⣡⢀⣿⣷⣇⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢿⡏⠘⢋⣿⣿⣟⣿⡶⣿⠧⢛⡁⠙⠻⠿⠟⠿⠿⠿⠋⢉⣀⠾⢙⡅⠈⢧⣈⣻⡷⠟⢿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠈⠀⡟⠙⢻⣿⣶⣷⣄⡰⣦⣭⠀⠐⠳⣶⣦⣤⠄⠰⣿⣧⣼⣿⡧⢀⣠⣞⣽⡻⢦⣤⣾⣿⣟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠙⢿⣿⣿⣿⢿⣿⠦⢠⣤⣭⣵⣤⣤⣨⣭⣭⣶⣶⣿⣟⠋⢩⢇⡘⢿⣷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⡆⠀⠀⠀⠙⢿⡿⣿⣿⣷⣾⣿⣭⣭⣿⣿⣿⣷⣿⣿⣿⣿⡉⠀⠀⠈⠁⠈⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣦⠀⠀⠀⠀⠻⢮⠛⠿⣿⣿⡤⢿⣷⠿⠋⣩⣽⣿⠋⠀⠉⠓⢤⡀⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⣿⣃⣀⠀⠀⠀⠙⢤⣀⣸⣋⡉⠁⣤⡾⠋⢠⠇⠀⠀⠀⠀⠀⠉⠳⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣸⡈⢿⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚⠳⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠨⡇⢸⡅⠀⠀⠠⠄⠀⠀⠀⠀⠀⠖⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠃⠀⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⠃⠀⢠⡷⠶⠦⠤⡤⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⠀⣿⠚⠦⠘⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠱⣾⣿⣤⣑⡂⠯⢀⠲⢤⣀⣤⢞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⣴⣦⢘⣿⣿⣿⣿⣷⣶⣤⣤⣍⣁⣈⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⡞⠉⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠧⠈⠀⠀⠘⡁⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣖⢀⡦⣀⣹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠈⠁⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⠋⠈⡇⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠼⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡆⠀⡈⠉⢱⠀⠀⠀⠀⠈⠣⡄⠁⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠓⠊⠀⣠⣀⣠⡀⠀⠑⠒⠙⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠈⠀⠘⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢚⠳⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⢏⡀
⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡩⣽⠗⠶⢤⣄⡀⠀⠀⠏
⡤⢤⢠⡇⡾⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠳⣄⡀⠈⠙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡀
⠙⣜⢾⣇⡿⣡⠞⠀⠀⠀⠀⠀⠰⣶⠦⣤⣸⣓⢦⡀⠉⠓⠦⣄⠀⠀⠈⣿⢦⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡙⡆⡼⡰⣣⢤
⠀⠈⡏⠋⢱⠧⠤⠤⠴⠒⢒⣒⣛⣻⡄⠀⠀⠀⠀⠈⠢⡀⣰⠋⠳⣄⠀⡿⠉⠋⠉⠉⠀⢸⡧⠤⠤⠀⣀⣀⣀⣷⠛⠱⢟⡥⠞
⠀⠀⠑⠲⡾⣟⠋⠉⠉⠉⠉⠉⠹⣸⣃⣠⡤⠖⠂⠀⠀⠘⠃⠀⠀⢈⣿⣤⠀⠱⣤⣀⡀⢸⠟⠉⠉⠙⠛⠓⠒⣦⢠⠊
⠀⠀⠀⠀⠙⢬⠇⠀⠀⠀⠀⠀⠀⠉⠉⠀⢧⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⡟⠀⠀⠹⣄⠉⠉⠀⠀⠀⠀⠀⠀⢰⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠒⠶⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⢸⠃⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣏⣠⣇⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠛⡛⡛⢦⣄⡀⠀⠀⠀⠒⠒⠒⠛⠛⡿⠛⠛⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠛⠉⠀⠉⠙⠓⠒⠒⠒⠶⠖⠒⠳⠶⠶⠶⣖⣋⣽⡟⠉⢛⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣀⠤⠒⠊⡉⣩⣍⣑⠢⠤⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⣄⣤⣁⡄⠠⣔⣭⠾⠿⢿⣷⣤⢰⣦⢆⣀⣏⢈⠙⣦
⠀⠀⠀⠀⠀⠀⣠⠞⢉⡥⣻⡁⡇⣧⡖⠿⠉⣇⣽⣿⠋⢀⣴⣿⣿⣁⣿⠉⣻⢧⠄⢺⣽⣿⣬⡇⡉⠉⠒⢦
⠀⠀⠀⣠⠔⠐⠃⢐⠁⣨⣙⣷⣧⣻⣼⡚⠛⡕⢺⡏⠀⢸⣿⡻⠗⣹⣿⣷⠥⢲⣿⣿⡟⣸⣿⢦⣟⣠⣶⡄⠷⠤⣀
⠀⢀⡞⠀⡀⠌⣻⣶⡾⢿⢿⣇⢻⣿⢷⣷⠟⢁⠺⣧⠀⠘⣿⣪⢞⢭⣠⣿⣽⣿⣷⠀⠀⣉⣻⣿⢟⡯⠉⣻⡰⣱⢈⢳⡀
⢀⡸⠀⠻⣷⣶⣬⣚⣭⣭⡿⠟⠀⠀⣾⣿⡟⢛⢝⡻⢦⣄⣀⠉⢉⣩⣽⣿⣷⣶⣟⣴⢀⣿⣿⣮⣶⠿⠻⠗⢻⡵⡿⡡⣱
⡜⢀⢐⣝⣳⡙⠟⡽⠋⣩⣿⣿⡅⣤⠘⣧⣸⣿⣽⣺⣽⣛⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣸⣯⣷⡛⢲⣾⣉⣠⣿⣄⡞⣟⣸
⣧⣼⣳⣽⣿⣄⢀⡹⣶⣽⣻⢯⣧⣿⣷⡘⣿⣿⡿⠛⠛⠛⠛⠉⢉⠛⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣶⢀⡯⠁⣙⣿⣾⣿⠃
⠈⠻⣭⣷⣾⣿⡆⣿⣬⣻⣿⣿⠿⣏⠡⡈⢣⢈⡳⣸⣀⡟⢹⣶⡠⠓⣄⢂⠀⠠⣾⣯⡈⠙⠛⢿⣿⣿⣱⡆⣿⡿⠉
⠀⠀⠀⠈⠛⠻⠷⣿⠟⡋⠴⢀⠀⡹⠃⢶⠘⢻⡿⣿⣼⣶⣝⠛⠁⣀⣘⣊⣡⡏⣿⣿⠞⠒⠛⡠⣄⠻⠛⣇⠁
⠀⠀⠀⠀⠀⠀⣠⢛⣴⣴⣢⢘⡶⠓⢶⣶⣿⣽⡇⢹⣿⠿⣷⣿⣿⣿⣿⣯⡉⠀⠛⠿⣾⡦⠈⣻⢿⡆⡤⠇
⠀⠀⠀⠀⠀⠀⢇⡛⠄⢭⣁⣨⣝⢀⣸⣼⡿⠟⠃⠀⠀⣴⣿⣿⠿⢿⠿⡿⢁⡄⣾⣿⣿⣿⢿⣯⠟⣸⠁
⠀⠀⠀⠀⠀⠀⠈⢧⣞⣱⣜⣹⣿⣿⣿⡿⣿⣿⣷⢰⡄⢻⣯⣝⡂⣬⣾⣣⣾⡇⣿⣿⣿⠧⠴⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠹⡿⢿⣿⢻⠇⣯⣿⢿⣌⢻⣿⡿⣿⠿⣟⢻⣡⡏⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡈⣿⠀⠀⢸⣏⣸⢹⣿⡆⠃⡘⠇⣉⣬⣧⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡀⠃⠸⠳⣾⢉⢃⠘⡡⢦⠎⢛⡶⠛⠀⡟⢀⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣵⠈⢍⢦⣧⡀⠘⠀⠀⣿⠀⠀⠀⣬⠀⠁⢘⡟⡩⠁⡧⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠁⠻⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠀⠠⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠋⢲⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠄⡁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⡄⢿⢂⣀⣀⣀⣁⠀⠀⠀⠈⢀⡠⣀⣀⣤⡂⢀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⢠⢟⣵⡶⢶⣦⣕⣰⣐⢊⣵⣾⡿⣶⣿⢣⣽⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⣿⢻⣽⠩⣹⡯⣻⡿⠙⢋⠏⣟⣐⣿⡿⣸⡟⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢫⣿⣮⣛⣿⣛⣵⣧⣷⣶⣜⣺⠿⢿⣭⡔⠛⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠿⣿⣷⣈⣍⣼⣿⣿⣿⣿⡟⠳⠈⠀⣉⣛⣠⢆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣤⣿⢿⣽⠛⡟⠪⠯⠉⡱⠈⣼⠛⣿⢿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⠀⣿⡈⢀⢀⢀⠠⠤⠨⢿⠀⡿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢧⣷⢔⡶⠶⠤⠔⣒⣓⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣌⠻⣿⣟⡻⣟⠉⢩⠈⣵⠏⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡖⠉⠙⠻⠦⠕⠁⣠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⢢⠀⡄⠀⡠⠄⠒⠒⠒⠂⢄⡀⠀⠀⣠⡀⠀⣠⠀⠀⡐⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹⠶⠚⠰⠷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠈⠙⠢⡰⠑⠔⠃⢀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡒⠤⣀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠤⢊⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠢⠀⠱⡀⠈⠀⠀⠀⠀⠀⠀⣠⠴⠲⠆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⢀⠎⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣢⣠⡔⠀⠀⠘⣄⡧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠄⠀⠀⠀⢠⢁⠊⢀⣀⣀⣀⣀⣀⡛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠉⠉⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠑⠢⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡴⠁⠈⠉⠑⡄⠉⠒⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠐⠁⠀⠉⠐⢲⠊⠁⠉⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢺⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⣷⡖⢲⡖⣶⢉⣄⣷⠂⠤⣤⢴⡎⠁⠀⠀⠀⠀⠀⠀⣀⠰⡆⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⣉⠵⠚⠁⠀⠀⠀⢀⠄⢒⣘⣛⠣⢼⡇⠟⢸⣿⠻⢸⡆⢿⠜⢓⣀⣐⠂⢄⠀⠀⠑⢄⠀⠈⠁⠉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣀⠀⢀⣴⠞⠗⠈⠀⠀⠀⠈⠢⡑⣄⢸⣿⠀⢸⡗⡁⠊⠀⠀⠀⠈⠲⡕⣄⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠻⣿⡿⠋⡎⠀⢀⣾⣿⣷⡄⠀⠀⡝⠻⣿⡠⠞⡅⠀⠀⣴⣿⣷⣆⠀⠘⣼⣿⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⡇⠀⠘⢿⣿⣿⠇⠀⢀⡇⠀⠀⠀⠀⣇⠀⠀⢿⣿⣿⠏⠀⢀⡧⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⣠⠞⠠⣶⣤⣶⠀⠘⢦⡀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠋⠁⠀⣀⣙⣛⣉⡀⠀⠀⠉⠓⠒⠚⠛⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣉⣁⣤⡤⢤⣤⣄⣉⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠈⠉⠉⠉⠉⠁⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠷⢶⠶⠛⣿⣶⠶⠊⢡⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⣙⣶⣦⡾⢠⠀⠼⠏⠀⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣇⣀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣤⣴⣾⣿⣿⣾⣶⣴⣴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠙⠉⠃⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⣿⢁⣴⣶⣤⣤⡴⣦⣴⣶⣦⣄⣿⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡿⢸⣿⣿⡿⢿⣧⡘⠿⢿⣿⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣿⣷⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡞⢳⣦⠈⠛⠉⢠⣾⠙⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠙⠛⠻⠛⠘⠉⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡆⠀⠀⠀⠀⠀⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠂⠤⠀⠀⠐⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠢⣀⣀⣀⡠⠊⠀⠀⠀⠀⠲



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡴⠄⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⠢⠘⢠⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡜⠗⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠚⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡿⠛⠛⠁⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⢀⣤⣄⡀⠈⠁⣀⣀⡀⠀⠀⠀⠀⠘⣷⠀⠀⠀⡄⠀⡀⠀⢢⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⠶⠖⠁⠀⠀⠀⢀⡀⠀⢉⣽⣦⠀⠀⠀⣠⣄⠉⠻⢿⡉⠀⠈⠳⣤⣄⣴⠾⠃⠀⠀⠀⠠⠖⠁
⠀⠀⠀⠠⡶⠒⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣾⣿⡏⠀⠀⣸⣿⣿⠀⠀⠀⠙⢦⠀⠀⢛⡉⠀⠀⠀⠀⠄⠉⠢⡀
⠰⠆⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠈⢣⡀⠙⠁⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢷⡀⠘⠃⠀⠀⠀⠘⠀⠀⠐⠁
⠀⠈⠲⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⢉⡀⠀⠀⠀⠀⠀⠘⣧⠎⠀⠀⠀⠈⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠖⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠙⢦⣀⠀⠈⠑⠒⠒⠒⠒⠒⠈⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠉⠳⢦⣤⣀⠀⠀⠀⠀⣀⡤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⠀⠀⢠⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢠⡌⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠳⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠙⢦⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠈⢷⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣧⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣩⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠆⠉⢄⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⢆⠈⠱⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠐⢲⡄
⠀⠀⠀⠈⢆⡀⢣⡄⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡀⠀⠁⠀⠀⠄⠊
⠀⠀⠀⠀⠈⠱⣄⢡⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠊⢠⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢀
⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣠⣤⡄⣠⣄⣀⠀⠀⠀⠤⠠⢤⣤⣤⣤⣄⢀⣀⣀⡀⠀⠀⣤⡤⢠⣤⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣤⣤⢤⠀⠀⠀⢀⣤⣤⣤⣤⡖⣠⣴⣦⣴⣤
⠀⣝⢿⣿⣿⣾⣿⣼⠁⠐⡀⠠⠀⠀⠈⠁⠟⣼⠋⠀⠉⢹⣷⠀⠘⣦⡙⣡⡄⠀⠁⠀⢀⠀⠀⠴⣶⣿⣿⠉⠠⠁⣀⣴⢿⠛⠁⢛⡉⣾⣿⣿⣿⣿⣷
⠀⠛⢩⠻⣟⠟⠁⠉⠁⠠⠄⠀⢀⣦⠀⠀⠀⠈⣻⣦⣤⠈⠯⠠⠀⢹⡏⣥⣴⠞⢂⡴⠧⠛⢶⡤⡾⠋⢁⣴⣾⣷⠿⠋⠈⠀⠒⠈⠈⠹⢻⡿⠵⠋⠾
⠀⣄⠀⠁⠈⢤⡀⣄⠂⠀⠀⣴⣿⣿⡀⠀⠀⠈⠁⠋⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡈⠻⠄⠀⠈⠁⠈⠀⣠⣾⡄⠼⠂⢀⡠⠒⠉⢀⠀⠀⢀
⠀⣿⡿⠒⣤⣀⠀⠀⠈⠛⢷⣴⣾⡈⠁⠀⡠⢹⠀⡆⠀⢠⠄⠀⠀⠀⠀⠠⣴⣄⠀⠞⢃⠀⠈⠑⢢⣤⡀⠀⠀⠀⢈⠹⣥⠆⣀⠀⠑⠂⣀⠀⠉⠣
⠀⠉⢠⡞⠷⣟⢷⡄⠀⠀⠈⠙⡇⠁⠀⠀⠛⠆⠘⠣⣶⡆⢠⣄⠀⠈⢁⣼⠋⠸⡧⠀⡀⠀⠀⠀⢫⡼⣿⠀⠀⢠⠈⡦⢀⠀⠉⠀⠀⠀⠉⠀⢠⣀⣀
⠀⣾⢷⣄⠐⠚⠗⠁⠢⠈⢣⣄⠀⠀⠀⠀⠀⠀⠀⠔⡆⠁⠈⠛⠤⠄⠈⠉⢿⡆⠀⠀⠀⣠⣔⣤⡄⠀⠀⠐⠚⠠⡈⢹⣶⣶⣆⠦⣄⡀⠀⠴⠊⠉⢻
⠀⠾⣿⣭⠇⠀⠀⠘⣶⡄⢐⣠⡌⠀⣶⣦⣤⣀⣀⣾⡧⠤⠀⠀⠆⠀⠀⠀⠈⠀⠀⣠⡜⡿⠋⠈⠟⠀⠀⠰⡄⠀⠀⠘⢏⠁⠈⢳⡬⠗⠃⠀⠀⠐⣃
⠀⠛⢮⡉⢠⠛⢤⣄⠈⢳⠚⠻⠂⠂⠀⠈⠀⠀⠀⠙⡄⠀⠀⠀⠀⢤⡀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⣴⠴⠃⠀⠀⣀⠁⠀⠷⢶⡟⠈⢀⠀⠀⠀⠄⢋
⠀⣁⢀⠁⢤⢇⣠⡽⣀⣈⠆⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⢰⣦⡀⠀⠰⡦⠄⠺⠋⠀⡄⠄⠈⠉⠀⣀⠀⠉⣸⠀⠁⠄⣠⣆⣤⣿
⠀⠛⠛⠳⣬⠈⢩⠀⠉⠁⠀⠀⠀⡦⠰⠃⢺⡄⠀⠀⠀⢀⢬⡄⠀⣙⠀⠁⠙⠀⠀⠀⠀⣀⣠⣤⡔⠀⠀⠃⠀⠀⣀⠠⠁⠀⣠⠀⠀⠀⠀⣥⠛⠉⠻
⠀⠠⢆⠀⠈⢷⣆⠀⠀⠀⠀⠀⢠⠀⠸⠀⠺⠟⠀⢀⠀⠀⢲⣄⠀⡉⠀⠀⠀⠀⠰⠀⢸⣿⠽⢟⡱⠚⡂⠀⣀⣀⠀⡰⠳⢤⡜⠋⢲⠂⠀⣤⠈⠃⠈
⠀⠑⢉⣠⣄⠀⢫⣤⣶⠟⠁⠀⢀⠴⠒⠀⠘⢿⣆⠘⢀⠀⢀⣈⢿⡏⠀⠠⢀⠀⢀⠀⠘⡀⢦⡤⠀⠐⡀⢀⣍⣿⣟⡑⢷⠀⠀⠀⠀⣵⠞⠁⠀⡠⠄
⠀⣴⠟⠻⠻⣶⣄⠿⢟⣀⡄⠘⠃⠶⣦⠀⠀⠀⠀⠀⡈⠡⠘⠷⡾⠛⠀⠀⠀⠀⢢⣤⣾⠅⣮⠁⣀⢘⠿⠿⠋⠋⠉⠀⠀⠱⠀⢀⠀⠛⠰⢠⣼⡁⢀
⠀⡉⢀⣸⣰⠆⣿⣿⣇⠘⢿⡀⠀⠀⠹⣧⠀⠀⠀⢀⠁⢴⡖⠐⠀⠀⠀⣡⣤⡀⠀⡀⠒⢶⣿⠇⠈⠉⠀⠀⠀⠀⢀⡐⠀⣀⡴⢋⠀⠞⠁⢈⡽⢶⣿
⠀⢻⣯⣿⣧⠺⢿⡏⠛⠑⢢⠆⠀⠀⠀⠘⠠⠆⡀⠈⠀⠀⠓⢦⡀⠘⠋⠙⢄⡉⢠⡄⠀⢸⡇⢀⠀⠰⠄⠈⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⢀⣩⠶⠛⡱
⠀⢀⣍⠼⠃⠀⢸⠅⠀⠀⠀⠀⠀⠐⠓⠄⠀⠈⠁⠉⠀⠀⢁⡄⠓⢄⠀⢃⡞⠀⡌⠀⠀⠊⡆⠀⢃⠀⠀⠀⠀⠀⠀⢠⣫⠀⠀⢀⣤⡼⠛⡔⢿⠏⢰
⠀⠛⠏⠛⠳⠄⠀⠒⠂⠶⠀⠀⠀⠀⠀⢰⡇⠀⢲⣆⠀⠐⢿⡉⠀⠀⠀⡿⠁⡄⢠⠐⠛⢦⡙⢲⣴⣆⠀⠀⠀⠀⠀⣸⠁⡞⠙⠿⠟⠧⠘⠃⠀⢰⣶
⠀⠰⠀⠀⢈⠂⠈⠒⠈⠀⠠⣤⠆⠀⠀⠀⠠⣦⡈⠛⢣⣽⣆⠀⠂⠊⠀⠁⠀⠘⠃⣀⣈⡴⠁⢸⠑⠈⠓⠰⠶⠚⠉⠀⠂⠀⠒⠠⠀⠀⣤⠶⡻⣯⡙
⠀⠛⠀⣀⠈⠀⠤⠶⢶⣿⣀⣹⠀⢙⠆⠐⠂⠘⢻⡛⠟⡱⠾⣶⠀⠀⠰⠀⠡⠀⠀⠘⢟⠀⢠⠄⢲⠀⠀⣀⠀⣠⠀⠀⣀⣦⠀⠀⠠⠄⠩⠐⠩⢨⠁
⠀⢻⡗⠙⠧⢄⠀⠀⠀⣀⣁⣠⣤⠹⠆⣿⣄⡄⠘⠃⠊⠛⣦⠈⣠⡐⡀⠀⠀⠀⣠⠆⠘⡄⢸⠂⠈⠃⠈⡁⢨⠍⠀⢤⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⢀
⠀⣿⣏⣠⣀⣀⠀⡖⠻⢟⠻⣌⡱⢿⣶⠞⠻⢿⠄⠑⡧⣾⣿⣷⡏⣁⣿⠠⢀⡒⢰⣆⣼⠃⠀⠶⠀⠀⠾⠷⢾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿
⠠⠏⠀⡴⢆⠨⡶⠶⠆⢀⠀⠒⠓⠂⠐⠂⠶⠇⠀⡀⠲⣄⡘⣿⣿⣇⣸⡗⣌⠙⠷⠏⣁⣵⠀⠐⠦⣀⣤⣤⡀⣁⣴⠾⠖⠒⠀⠀⠀⢀⣠⡞⠀⠛⠿
⠠⠾⠓⠶⠂⣶⠾⣷⣀⠈⣀⢌⢡⣁⠀⠀⠀⠀⠈⠀⠀⠈⡏⠻⠿⠿⠇⠈⠋⠀⠂⠀⠙⠛⠉⠀⠀⠛⠛⢭⠥⣹⠂⠰⠀⠄⠀⠀⠀⠈⠛⠀⢀⡘⠻
⠀⡶⠂⠖⠚⠁⡉⢠⠟⠉⠉⠀⠈⢿⣴⠃⣦⠀⢀⡀⠀⠀⠀⠻⣄⣀⢀⡀⠸⠷⡈⠀⠀⠀⠀⠀⠀⠉⢩⣄⠀⡀⠘⠃⠀⠀⠀⢈⡀⠘⠿⢿⣧⡝⢿
⢀⣴⠟⣠⠁⠈⣳⠋⣀⣀⢸⣇⢀⡘⠋⢠⣿⠛⢿⣿⣁⣄⣀⠀⠈⠁⠈⠀⠁⠀⠀⣀⡤⢤⣄⠐⣶⠋⠀⠈⠀⢁⡀⠀⢠⣿⣷⣮⡛⣦⡐⣦⣙⠿⣶
⠈⣴⢿⣵⣿⣾⣵⣦⠈⢁⣼⡇⢸⡇⠀⣾⡏⣠⣿⢿⡏⢉⣹⣷⣦⣄⠀⠀⠀⠀⠉⣀⡐⢦⡀⠁⠃⠀⠀⠀⠀⠈⣿⣷⣮⣿⣯⣿⣿⣿⣿⣙⢿⣷⣌
⠈⣵⣿⣿⣿⣿⣿⣇⣼⣿⡟⠀⡾⠀⣸⠏⠀⢹⣿⡏⢸⣿⣿⢈⣿⡟⠀⠀⠀⠠⠴⣿⣿⠦⠛⠀⠀⠀⠠⠀⠀⠀⠘⣿⣿⣿⢿⣿⣿⣿⣯⣿⣮⣿⣿
⠀⣿⣿⣿⣿⣿⣟⣿⣻⣿⠀⡸⠁⣠⠏⠀⢰⠃⠙⠓⠉⠁⠀⠈⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⣿⡟⣿⣷⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⡴⠃⠀⣰⠟⠀⠀⠀⠀⠀⢀⣴⢋⣡⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⢠⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⣿⣿⣟⢻⣿⣿⣿⡿
⠀⣿⣟⣠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠈⣀⠴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠈⠉⠛⢿⣿⣿⣿⣷
⠀⣿⡿⠿⠟⠛⠋⠀⠉⠙⣇⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢱⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠉⠉⠙⠛⠋⢻⣿⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡀⠀⠀⠀⠀⢦⣀⡀⠀⢻⡍⠀⠀⠀⠀⠀⠀⣈⡿⠏⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠙⣧⠀⠈⢳⣤⡀⠀⠴⠶⠞⠋⠀⠤⠀⠀⠈⠀⢿⡀⠁⠀⠀⠀⣀⣴⣒⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠳⣤⡀⠈⠉⠁⠐⠈⠀⠀⠀⠀⠀⠁⠀⠀⠘⣧⡄⢠⣴⣾⣿⠋⠉⠙⠾⣿⣷⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣆⣶⣶⣤⣶⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⡀⣸⡧⠀⣀⠀⠻⢿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣤⣘⠿⠧⣤⣦⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⠛⠓⠛⠙⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠟⠛⠟⠛⠻⠛⠛⠿⠐⠛⠛⠙⠓⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠶⢦
⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠺⣏⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢠⡇⠀⣰⠏
⢠⠞⠉⠁⠀⠀⠈⠙⢦⡀⠀⠈⠳⣄⠀⣷⠀⠀⠀⠀⠀⠀⢰⠏⢷⠀⠀⠀⢸⣁⡞⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⢳⣿⠀⠀⠀⠀⠀⠀⡿⠀⠸⡇⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠳⣄⣀⡀⠀⠀⠀⠀⣠⢾⣷⡀⠀⠀⢠⡉⠀⠀⠀⠀⠀⣸⠇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢳
⢰⠟⠉⠀⠀⠀⠀⣰⠃⡎⠈⢷⡀⠀⣾⠻⡄⠀⠀⠀⢀⡟⠀⠀⠀⢸⡆⠀⠀⠀⢀⡤⠾⡗⠀⣰⠏⢀⣠
⣿⠀⠀⠀⠀⠀⢠⡏⢠⠀⠀⠘⣧⠀⣿⠀⠹⣄⠀⠀⡼⠁⠀⠀⠀⠀⣇⢀⣤⠞⠉⠀⣸⠃⢸⡷⠞⠋⠁
⠸⣦⣀⣀⠀⠀⢸⡇⠈⠀⠀⠀⢹⡄⣿⠀⠀⠹⣆⣸⠃⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢀⡏
⠀⣴⠋⠁⠀⠀⢸⡇⠀⡇⡇⠀⠸⣧⡇⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣀⣀⡀
⢸⠇⠀⠀⠀⠀⠀⣧⠀⢣⠁⠰⢠⣿⠇⠀⠀⠀⠀⠀⢠⠶⣄⠀⢀⡴⡆⠀⠀⠀⠀⠿⠖⠚⠋⣩⠟⠃
⢸⡄⠀⠀⠀⠀⠀⠸⡄⠸⣾⠛⢾⡿⠀⠀⡟⠢⣄⡰⠃⠀⠈⠓⠋⠀⠹⢶⡆⠀⠀⠀⣠⡴⠛⠁
⠀⠻⢦⣤⣤⠀⠀⠀⠙⣄⣯⡇⠀⢀⣀⣀⣷⣀⣬⠥⡤⠤⠤⠤⠤⠤⠤⠾⠶⠶⠲⣟⠁
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠈⠙⠉⠉⠉⠀⢨⠁⠀⠀⢀⣀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⢻⡄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⢠⠞⠉⠉⠉⠙⢦⡀⢰⠃⢀⣀⡀⠉⢆⠈⣇
⠀⠀⠀⠈⠛⠒⠶⠶⣾⠁⠀⠀⠀⠘⡆⠀⢾⠀⠀⣾⣷⠄⠈⡇⢸⠀⠸⠿⠇⠀⣸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢠⡇⠀⠘⣆⠀⠈⠁⢀⡴⠁⠀⠳⢄⣠⡤⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠙⡆⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠠⢶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠤⠤⣤⣀⡹⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠦⠤⣄⣀⣀⣀⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠀⠀⠀⢹⠀⠀⠙⣦⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠓⠀⠀⠀⠀⠻⢀⠀⣠⡼⠁⠀⠉⠀⠀⠀⠀⠀⠘⠂⠈⠉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣏⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠘⠃⠀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣤⠴⠚⠁⠙⠛⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀⡴⠶⢿⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢄⡀⠈⠛⠛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠐⠆⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠈⠉⠉⠉⠒⠒⠒⠶⠤⠤⢤⣀⣀⣀⣰⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠶⠶⣤⣄⣀⡀⠀⠀⠀⠀⢀⡿⠳⢦⣄⣀⣰⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠖⠚⠉⠙⠒⠋⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠤⡄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄⣀
⠀⣠⠤⠤⢤⡀⠀⠻⣅⠀⡇⠀⠀⠀⠀⢠⣆⠀⠀⢸⢀⡜⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⡎⠀⠀⠀⠀⠈⢦⠀⠈⠣⣿⠀⠀⠀⠀⡏⠸⡄⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢳⣤⠄⠀⠀⢠⠾⢦⠀⢠⣌⠀⠀⠀⢸⠁⠀⣧⠀⠀⠀⠀⣀⠀⢠⠟⢧⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡞⠀⠀⠀⢠⠃⠆⠘⡆⢸⠙⢆⠀⢀⠏⠀⠀⢸⠀⣠⠔⠋⡟⢰⡧⠔⠊⠀⣇
⠱⣤⡄⠀⢸⠀⡁⠀⢸⣸⠀⠈⢦⡌⠀⠀⠀⠈⠋⠁⠀⢰⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⢰⠃⠀⠀⠸⠀⢁⠀⢸⣿⠀⠀⠈⠁⣀⠀⠀⣀⠀⠀⠀⣮⡤⠴⣶⠆⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠊
⢹⡀⠀⠀⠀⢧⠀⠞⠺⠇⠀⡗⢄⠜⠀⠙⠚⠈⢶⠀⠀⣀⡴⠊⠁⡔⣢⠀⢰⡒⢒⡄⠑⠦⢤⣤⣤⠤⠴⠚⠳⠶⠤⠤⠖⠚⠁
⠀⠙⡶⠂⠀⠀⠳⠶⠤⠐⢲⠓⠒⠒⠒⠒⠈⠁⠈⠉⠉⢇⠀⠀⠀⠉⠀⠀⠈⠛⠉
⠀⠀⢇⠀⠀⠀⠀⠀⠀⡤⠊⡰⢂⣭⡕⢄⢠⠃⣴⣮⠢⠸⡄
⠀⠀⠈⠉⠉⠻⡄⠀⠀⢰⠀⠇⠀⠛⠃⠸⠘⡄⠈⢉⡴⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠆⠀⠙⡀⠈⠒⠒⠚⠀⠀⠀⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢻⠄⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣸⠃⢈⠉⠁⠀⠀⠀⠀⢀⠊⠀⠈⠑⠏⠉⢯⣀⣤
⠀⠀⠀⠀⠀⠀⠹⣄⣈⡁⠀⠀⡟⠂⠀⠂⠀⠀⠀⠀⠀⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠐⠄⣀⠀⢀⣀⠄⠙⠢⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠻⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢊⠀⠠⠈⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠑⠀⠐⢲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣈⣉⠐⠒⠒⠒⠂⠤⠤⣠⡎⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⠦⣤⣀⡴⠋⠉⠓⠶⠃
⣿⣿⠟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠘⢿⡦⠀⠀⠀⣠⡆⠀⠀⠰⠖⠀⠀⣀⠀⠸⠃⠀⢰⡿⠃⠀⠀⠀⠘⠃⠀⢀⡄⠀⠀⠀⠼⠋⠀⠀⢀⠄
⠀⠀⠀⠀⠀⣀⠀⠘⢧⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⡾⠀⠀⢀⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠘⠦⠀⠀⠙⢦⡀⠈⢷⠀⢨⣿⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⡇⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠈⠃⠀⡀⠀⠀⢿⠀⠀⠀⠿⣿⠂⠀⠀⡄⠀⠃⠀⠠⠏⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠞⠋
⠀⠈⠁⠀⠀⠛⠷⣄⠀⠁⠀⢀⠘⠄⠀⠀⠀⠏⠀⠀⣾⠃⠀⠀⠀⠀⣰⠗⠀⠊⠁⠀⠔⠀⠀⠀⠀⠀⠀⠴⠤
⠀⠀⠀⠰⠒⠂⠀⠈⠀⠀⠀⣿⡄⠀⠀⠻⡇⠀⠀⠰⠏⠀⢸⣦⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠳⠀⠀⠀⠀⢀⣴⡄⣀⡀⠿⠃⠐⠃⣠⡾⠂⠀⠀⣴⣶⠄⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠙⠶⠄⠀⠀⢸⣷⣿⣿⣿⣿⣧⣤⠀⠀⠘⠛⠋⠀⠀⢀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⠋⠉⠻⢿⣿⣿⣶⡶⠀⠀⣀⣤⠄⠈⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡿⠛⠿⣦⡀⠀⠀⢹⣿⣿⣷⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⢻⣷⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠑⠲⠾⠿⠟⢛⣥⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠿⣿⣿⣿⣧⣀⠀⠀⠀⢠⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⣠⡈⠙⠻⣿⣿⣷⢀⣴⠟⢁⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣬⣘⣁⣀⡄⣽⣿⣿⢋⣴⡿⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠙⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋
⣿⣿⣟⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⠀⠒⠶⡦⠠⣄⢰⡴⡀⠀⠀⢀⣶⡶⠀⣠⡶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⢋⣀⠐⣴⡖⠀⡼⠶⠘⠀⠛⠛⠱⢛⠷⠋⠴⠛⣡⠀⠁⠽⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡥⣦⡄⡘⢿⠁⠸⠷⣤⣦⠀⣴⡻⣎⡡⠉⣶⡄⢄⣟⡰⠈⠐⢠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠉⢨⣸⡆⢦⣄⢰⡇⡈⠋⠈⠛⣠⠿⠳⠂⣨⠁⠻⣛⣷⡖⠃⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣢⣯⢇⠛⠁⢀⡄⠀⠁⠀⢈⠀⡠⠅⠤⠀⠫⡄⠒⠡⠙⠍⢠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢦⢀⡀⠀⡃⢻⠀⠀⠥⡈⠀⠀⢀⠾⠻⠄⢀⣀⡥⠄⠀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢮⡏⡄⠘⢄⠀⠁⢀⠲⠋⠀⠀⠰⣺⠠⠐⠋⠉⠀⠀⢈⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡗⢛⠁⠀⠀⠀⠀⠀⠀⠠⠀⠩⠀⠄⠉⠀⠀⠀⠀⡀⠀⢀⠀⡯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⡀⠀⢂⠲⢠⡃⠀⠈⠠⠀⡓⠲⠂⣀⠄⠑⠚⠋⣈⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⡐⠚⣒⣠⢷⣀⠹⢫⡗⣀⠀⠀⡈⡀⠔⠀⢢⠔⠐⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⢠⡏⡙⢫⠛⠹⠋⠙⣻⣿⣷⡾⠟⣃⠡⣨⣙⢣⡴⠀⠀⣠⠬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣩⢻⡳⠒⡚⣦⢤⣀⡀⠻⣍⣁⠵⠀⠉⠀⣘⣟⡿⠖⣀⢀⠶⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣺⣧⠆⣾⣽⣿⣶⣦⡀⠀⣴⣯⡋⠃⠀⠰⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠙⡸⢡⠟⠛⠉⢻⠄⠀⠉⠉⠀⠀⡀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣟⠈⠀⠋⠀⠀⠐⣋⡃⠈⠀⠀⠀⠀⠀⠀⢸⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠉⠉⡀⠀⠀⠀⠀⣶⠶⠶⠶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⢩⠀⠀⠠⠟⠁⠀⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡀⠀⠀⠆⠉⠀⠀⠀⠀⠀⠘⣴⣾⡉⡌⠛⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠛⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠶⠕⠶⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠠⠅⠰⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⠃⠠⠄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⡇⠀⠀⠀⠰⠂⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⢂⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡸⠁⠤⠄⠙⠋⠀⠀⠀⢢⡈⠁⡸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡈⠁⡸⠀⠀⡿⢏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠉⠁⠀⠀⠘⣧⡼⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡄⢤⡀⣭⠱⣄⠀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠊⣭⠀⡤
⠑⢂⡈⠻⠆⠈⢶⠉⣨⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⡅⠈⡶⠃⡰⠟⢁⡀
⠷⢄⣉⠒⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠬⠽⠿⠿⠿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠁⠒⣋⡠
⠀⠀⠉⠙⠢⠄⣀⣀⡸⠛⠦⢀⡀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠴⠚⢇⣀⣀⠠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⡀⠀⠈⠃⠦⣀⠀⢠⡤⣤⣈⣙⣛⠲⠶⣦⡤⠤⠤⢤⣤⠶⠖⣛⣉⣁⣤⢤⡄⠀⣀⠴⠚⠉⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠉⠋⠛⠀⣀⣀⠀⠛⠉⠉⠀⠀⠀⠀⠀⣟⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡴⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢳⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⢿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⢸⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣾⢸⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣹⠋⠀⠀⣍⣹⣼⡇⡅⠀⠈⢙⣏⠻⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣾⣭⣷⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢿⡤⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠄⠀⡌⠂⢠⠀⡀⠀⠁⠐⠶⠖⠀⠀⠀⡀⠸⡆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⢀⠀⠒⠀⠄⣀⣸⣛⠁⣹⡽⡛⢳⡀⠀⣤⡆⠠⡴⠊⢀⣽⣎⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠴⠃⢄⣉⠁⠀⠰⢖⣫⢞⠛⢻⡞⢻⣿⠷⠾⣶⣷⡉⢀⣲⠀⠐⣠⠟⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⠀⠀⠐⠋⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠇⢻⣿⠀⠒⠛⠛⠁⠐⠀⠑⠛⠀⠀⢒⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣫⣶⣰⣤⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⢯⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢿⡿⠋⡼⣿⡟⢸⠇⠙⠀⠀⠁⠀⠉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⡠⣄⡀⠀⠀⠀⣀⣴⣿⣿⣪⣴⠋⢀⠞⣴⠋⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⣿⣷⡿⠝⠛⢻⣶⣾⢟⡿⣻⣿⠟⠁⡴⠋⣼⠋⠀⢀⣾⠀⠀⠀⠀⠀⠀⣀⣄⠀⢀⠺⣝⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⣹⣏⣿⣷⣷⣤⣶⣿⣿⡿⣫⣾⣿⠁⢁⡜⢀⣾⣇⡆⢸⣿⣿⠀⢄⠀⠀⡆⠀⣿⣏⡆⠈⢦⡈⢻⣿⣦⡀
⠀⠀⠀⠀⠼⠿⣿⣿⣿⣾⣿⡿⢿⡟⠿⢿⣿⣿⣯⣾⠛⠹⠿⣶⡞⢀⠈⢿⣿⠁⣿⣿⣿⠀⢸⣄⣀⢱⢸⣿⠋⢹⢂⣤⣻⣮⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠻⠏⠉⠉⠀⠙⠲⣬⣟⠾⡟⠃⠀⠀⢠⣽⡿⠞⠀⠘⠉⠶⢽⣿⣿⠂⣿⣿⣿⡿⠟⠉⢦⠘⠋⢀⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠛⠙⠛⠻⠛⠛⠛⠿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠤⠀⠀⡤⠔⠠⠤⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⣀⣀⡠⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠶⠶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⠠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠤⠠⢄⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⡶⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⣤⡔⠋⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠹⣿⣿⣿
⣿⣿⣿⣿⡇⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢹⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠟⠛⠛⠉⠉⠛⠛⠿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠐⠒⠒⠀⠀⠤⠴⠶⠄⠀⠐⠒⠛⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠒⠒⠂⠀⠀⣀⣀⣀⣀⣀⣠⣀⠀⠀⠐⠒⠚⣏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠣⣄⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⢀⡨⢿⣿⣿⣿⣿
⣿⣿⡿⠋⠁⠀⠀⠈⠑⢦⣀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⠶⠋⠀⠀⠙⠻⣿⣿
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣿⣶⡾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⣿⣿⣄⣼⠟⠀⣠⡆⠀⣠⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣀⡀⠀⢠⡀⠈⢷⡄⠈⣿
⣿⣿⣿⣿⣄⣰⣿⣀⣼⣿⣿⣿⣶⣶⡦⠤⠤⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠤⠤⠀⢸⣏⣁⣴⣿⣿⣄⠈⣷⡀⢈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠲⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⣸⣀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠙⢿⡴⣤⣤⠴⠦⠤⠤⣬⡉⠀⣿⠃⠈⢫⣤⣤⣤⣤⣤⣤⣬⠙⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠐⠛⠁⠉⠁⠐⠒⠛⠛⠛⠋⢀⡹⢆⠘⠛⠉⠁⠀⠀⠀⠀⠀⠃⠒⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⣀⠀⠀⠀⠀⠀⣀⡀⠒⢉⣿⣄⠑⠂⣀⠀⠀⠀⠀⠀⠀⣀⠐⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⡇⢸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⣠⣶⣼⣧⣤⣄⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠻⠿⢿⣆⡀⠀⠀⠀⠀⠀⠀⣼⡟⡁⠁⠀⠈⠉⠉⣙⠻⣷⣯⡀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢸⣿⠙⠋⠀⠀⠀⠀⠀⠙⠓⠘⢻⣇⠀⠀⠀⠀⠀⠀⠻⣿⣷⡏⠀⠀⠀⠀⠠⣷⣿⣾⠃
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠰⠆⢠⣾⠀⠀⠀⣴⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠷⣦⣤⣠⣿⠀⠀⠀⣶⣾⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣤⣤⣬⣿⣿⣆⠀⠀⠻⣿⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢐⡿⣷⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄⠀⢹⡧⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢘⣣⣿⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠃⠀⠈⣿⠀⠀⣤⣶⣶⣤⣄⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣾⣷⣆⣶⣄⠀⠀⠀⠀⠀⠀⠙⢧⣀⠻⠟⠛⠿⠿⠃⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣦⣿⡾⠟⠛⠛⠛⠙⠛⢿⣷⡄⠀⠀⠀⠈⠙⠛⢶⣶⡶⢶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⣰⣿⡅
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⠀⢀⣀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡁⠀⣀⣀⠀⠀⣠⣴⣶⠿⢿⡟⢋⣤⠚⠻⣧⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡟⢁⡈⢡⡄⠀⠀⠉⠃⠀⢸⣿⣄⠀⠀⠀⣾⣷⣴⡟⠀⠘⣷⣾⣿⣿⣧⣾⠿⠟⣁⣰⠶⠀⠿⢿⣤⣀⠀⣬⣽⣿⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠈⠁⣸⠇⠀⠛⠁⠀⠀⠸⣿⢿⣧⠀⣸⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⡶⠛⠋⠁⠀⠀⠀⠀⠀⣉⠓⠦⢭⣧⢀⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⢿⣶⡦⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣟⢻⠄⣷⠀⠀⢰⢞⣉⠃⠀⠀⠹⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⢻⢟⣿⣿⣿⣿⣿⣿⢹⣿⢻⣏⠀⠀⠉⠀⢠⣿⠃⠀⠀⠈⠉⠀⠀⠀⠀⣹⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⠃⠀⢀⣴⣿⣿⣿⣿⠇⢎⡎⣽⣿⣿⣿⢿⣿⡇⣿⡄⢿⣆⠀⠀⠀⣸⣷⣾⡟⠀⠀⠀⠀⠀⠀⣠⣿⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣦⣤⣀⣤⣿⠟⠛⠻⢿⣿⣆⠘⣰⣿⣿⣿⣿⣿⢿⠇⠈⠀⣿⡏⡟⢹⢃⡟⠉⢸⠟⢻⣟⣛⣿⣷⠞⢻⡟⠁⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠉⢿⡏⠁⢨⡿⠀⠀⠀⠀⠻⣿⣤⠟⣡⣿⣿⣿⣿⡄⠀⠀⢠⣿⣇⣀⡉⢸⡇⠀⠉⠀⠀⢨⣿⠋⠀⢠⡟⠁⠀⠀⠀⠟⠹⣆
⠀⠀⠀⠀⠀⢰⡟⠀⡄⠀⠀⠘⢿⣶⠟⠁⠀⣤⢀⠀⠀⠙⣧⣰⣿⣿⣿⣿⣿⣿⣤⣤⣴⣿⣿⣟⠿⣿⣿⣿⣿⠖⢷⣾⡟⠀⣰⡿⠇⠀⠀⠀⢀⡄⠀⢻
⠀⠀⠀⠀⠀⣿⣧⣠⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣷⡶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⡏⠻⣧⠈⢿⣿⢀⣿⣇⣀⡀⠀⠀⢿⡇⠀⠈
⠀⠀⠀⠀⣴⠏⢠⣿⠀⠀⠀⠀⠀⠀⠠⢀⠰⠆⠸⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⠿⣇⠀⣹⣧⠈⢻⣿⣿⣿⡿⢿⡆⣴⣾⠃
⠀⠀⠀⣰⡟⢀⡾⢻⠀⠀⢠⠀⠀⠀⡄⠘⡄⢀⠀⣿⣿⣇⠀⢿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⠁⠀⠚⠁⣠⣿⣶⣿⣿⣧⠀⢻⣿⠏⠀⢸⣧⣾⣿⣾⣷⣶
⠀⠀⢰⡏⢀⣾⠁⣼⠀⠀⠚⠀⠰⠚⠇⠀⠛⠘⢂⣸⣿⣿⠀⠘⣿⡟⣽⣿⣿⣿⣽⣿⣿⡇⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠁⣾⠃⢠⣿⣿⠿⣿⡿⠿⣿⣿⣷⣶⣾⣿⠻⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣟⣛⣿⣿⣿⣷⡀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢰⣿⢀⡇⠀⢸⣿⣫⣾⣿⣿⢶⣿⣿⣿⡝⣿⣿⠀⣿⣿⣿⣆⠈⣿⣿⣿⡿⣣⡿⣫⣿⣿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣿⠷⣿⣿⣿⣿⣿
⠀⢸⡇⢸⡁⠀⣾⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⡾⠻⣟⣿⣿⣿⣿⠿⠛⠻⣦⢹⣿⢟⠉⢻⣷⣦⡝⢿⣿⣿⣿⣿⣿⡿
⢠⡟⠀⢈⣻⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⠋⡁⠀⠘⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⠀⠀⣰⣿⡆⣿⣿⣇⠀⠸⣿⣿⡤⣿⣿⣿⣿⠏
⢠⣴⣦⣾⠏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣷⣿⡇⣀⢴⣼⣿⣿⣷⢟⣿⣿⣿⣿⡏⠀⣴⣿⣿⠿⣿⣿⣿⣆⠀⢻⣿⣷⣿⠿⠛⠁
⠋⡿⠻⡿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⣷⢸⣿⣷⢿⣷⣿⣼⣿⣿⣿⢲⣿⣿⣿⣿⣿⡇⢠⡏⠀⠀⠀⠀⠉⠙⢿⡆⠸⣿⡋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⢃⣩⡾⣿⣿⣿⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⢿⡀⢿⡇
⠀⠀⠀⠀⠀⢼⣿⣏⣭⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⣿⣿⣿⣿⣿⡿⣋⣴⣿⣿⣿⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⢻⡏⠀⠀⠀⢹⡧⠀⢻⡏⢸⣿⣿⣿⢿⣿⣿⣿⣋⡾⢋⣿⣿⣿⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⠀⠀⢸⣧⠀⣻⡇⢸⣿⢟⠅⣸⣿⣿⣿⣿⡴⢛⣵⣿⣿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠀⠀⠀⢸⣿⠀⣿⣇⣸⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣛⡙⣧⣼⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣧⣼⠟⠳⣦
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣷⣾⣷⣤⣴⣶⣿⣿⣶⣿⡟⠛⠋⠁⠀⢹⣿⣿⡿⠿⢛⣿⡎⢿⣷⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣄⣩⣤⣾⣿
⠀⠀⠀⣠⡾⠛⠛⣻⡟⢹⣿⡿⠋⠀⢉⣙⠃⢹⣿⣇⢀⣀⣀⢀⣾⣿⣿⣄⠛⠻⠿⠷⠾⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣒⠚⠛⠟⠛⠛⣛⣉⣿
⣀⣀⣀⣙⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣉⣉⣀⣉⣀⣀⣈⣉⣉⣁⣀⣀⣀
⣿⣿⣿⢿⠻⢿⡿⠻⡿⢿⡿⢿⠻⢿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣷⣴⣧⣦⣾⣤⣼⣿⣤⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠠⠶⠍⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⢴⣤⣄⠀⢙⣇⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠀⠀⠀⠀⠀⣨⣟⡍⢀⠇⣠⢆⣴⣶⣖⣠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⠉⠉⠁⠀⠀⢰⡏⢿⠀⠀⠀⢠⠀⠀⢉⡀⣼⣴⣱⣷⣾⣿⣿⣿⣿⣿⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠽⣦⣤⣴⡯⣤⠴⠋⣸⣿⠟⢉⣥⡀⠀⠉⠉⣿⣶⣾⣿⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠋⠀⠀⠀⣼⠏⠁⣠⣤⣤⠀⠀⠀⠀⠀⣽⣿⣿⣟⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⠳⣄⡀⠀⠀⠀⠀⡋⢀⢿⣿⠇⠀⡆⠀⠀⠠⡄⠙⠛⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠐⣠⡿⠁⠀⢠⡟⠙⢢⣠⠂⠸⠂⣿⡀⠀⢀⡰⠀⠀⣿⣷⢰⡀⠀⡷⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡶⠞⠛⠁⠀⢀⣴⠟⠁⠀⠀⢹⡀⡀⠀⠈⠙⠛⠁⠀⠀⠘⣍⠁⢨⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⢻⣷⣦⣤⡴⠞⠛⠁⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⢷⣄⠀⠀⡙⠻⠃⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⡇⠀⠀⠀⢻⣷⡈⠛⠒⠀⠀⠀⢀⣤⣾⣿⠁⣠⣶⡶⠠⡄⢈⠛⠿⠛⠁⠀⠀⠀⠀⢄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⣿⡀⠀⠀⠀⠹⣮⡁⠀⠀⠀⠀⠻⢿⣿⣷⠐⢿⣽⣳⣶⣀⡜⠑⠦⠀⠀⠀⢀⣒⡐⠂⠀⠸⣦⣌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⡆⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⣹⣷⣤⠄⠀⠀⠉⠻⣟⣷⡇⢠⡴⠋⢁⣿⠶⠠⠠⠧⣹⡧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣹⣄⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠘⠻⣿⣿⣄⡀⠀⠀⠀⣉⣉⡠⠊⠀⢀⣾⠟⢶⣶⣶⣦⡀⠀⣰⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⠿⣳⣼⣦⣄⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠉⠉⠁⠙⢿⠟⠉⠁⣀⣠⣴⡿⠃⣎⣀⣀⣀⠀⠀⣰⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠣⠜⠂⠉⠙⢳⣬⣟⣲⢤⣀⣀⠉⠛⠳⢶⣤⣀⣀⠀⠀⠈⠉⠛⠛⢶⠶⠖⠚⠛⠛⠛⠛⢥⣄⠀⢈⡟⠛⠉⠁⣰⣯⣽⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠔⢋⡠⢀⠠⠄⡒⠬⠳⠙⡫⢝⡳⣿⢽⣏⣿⣒⣶⣤⢭⣙⣛⣻⣶⣶⣶⣮⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢏⣹⡿⣿⡿⠞⠋⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢀⣴⡻⠝⡂⢥⡐⠬⢓⠨⢐⢈⡁⣄⢢⣥⣬⣦⣍⣭⣿⣿⣿⣿⣿⣿⠿⣍⠷⢉⣣⣤⡾⠿⢓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠤⠒⠋⠁
⠀⠀⠀⣰⡿⢅⡲⠍⢂⠡⣐⣨⡴⢞⠯⢒⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⡿⠶⣳⣶⣾⣉⠫⠕⢢⡑⢮⡳⠬⢛⣄
⠀⠀⡴⠏⠐⢡⠀⢌⡠⠕⠋⠡⢚⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣟⡋⢀⣡⣴⠶⣛⢃⣍⠦⣔⠶⠟⠛⣛⣦⡀
⠀⠀⣷⠐⢈⡀⠈⣠⠴⣲⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣦⡷⠛⠋⠈⡑⠬⡑⢊⣉⣤⡽⢿⡀
⠀⢀⣿⢂⣡⡲⣝⣶⣿⠻⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠺⣷⣼⣷⠿⢟⡻⢍⡳⢸⡇
⠀⠸⣯⢿⣿⠿⢛⢩⠲⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⣧⣙⣦⣽⣶⢿⣿⠃
⠀⢸⡿⣻⡷⠞⣺⣿⢾⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡹⢌⣣⣴⣟⣿⠋
⠀⣷⠞⣿⣤⠞⢽⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⣷⢾⣻⣿⣿⣼⡇
⠘⣷⢺⣯⢔⡪⢼⡟⣧⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⠿⣿⢟⣿⠁
⠀⢹⣯⢲⠟⡩⣴⠟⠺⢑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡻⢝⢢⣽⣟⣧⡿
⠀⢨⣿⡾⢓⠭⢂⣵⠞⢪⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢜⢣⣝⣾⣿⡿⢛⣸⡇
⢀⣼⣷⣻⣷⣾⠿⢶⣶⣿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡿⣟⣣⣼⣿⣿
⠸⡀⢉⣿⣿⣧⣤⣤⣸⣿⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠻⣧⣿⡾⣿⣵⣾⣿⡄
⠀⠈⣿⣾⣟⣛⡛⠿⣟⠻⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⢿⡟⣉⣀⣼⣁
⠀⣼⣿⣿⣭⣭⣿⣷⡚⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣦⣤⣤⡶⣻⡶⣋⣉⣽⡇⢀
⢰⡿⠟⠉⠀⠀⠀⠀⠈⠙⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣯⣄⣡⣿⣷⣏⣠⡴⠶⣧⣄
⡏⠑⠶⢤⣤⡀⢀⠀⠀⣀⡸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠛⠉⠑⢦⡀
⠘⠢⠤⠤⢀⣀⣀⣀⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣶⣶⣤⣤⣬⣍⣙⠛⠻⠿⣡⣀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣤⢭⡭⠽⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢻⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⡀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡞⠁⠀⠀⠀⠀⠀⠀⢠⣿⠻⣦⣀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⣠⣤⠉⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⢶⣿⣟⡋⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠉⢀⣤⣤⣄⠀⠀⠀⠀⣸⣶⣦⣤⡈⢿⣍⠛⢷⡄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⣠⡾⠋⠉⢛⣛⠳⣄⠀⣼⠋⠀⠀⠈⠙⢾⣿⣦⡀⠳⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢠⣤⠶⠟⣫⣾⡟⠀⠀⣼⡏⠀⠀⠀⣿⣿⡷⢸⣦⣿⡀⣾⣿⡆⠀⠈⣿⣿⢳⣄⠀⠀⠀⠀⠈⠻⣦
⠠⠤⠤⠤⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⠀⢀⠹⣇⠀⠀⠀⠀⢉⣠⡟⠁⠻⣷⣌⡉⠀⠀⣰⡟⢹⡆⠙⣦⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⠀⠀⠈⠻⠾⣿⣿⡿⠿⠛⡁⠀⠀⠀⠈⠻⣿⢿⣿⣿⡾⠛⣷⣤⡜⣧⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣠⣿⠂⠀⠀⠀⠀⠹⢩⡿⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣦⡀⠀⠀⠀⠀⢸⣇⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢶⣾⠃⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣈⠻⣦⣄⠀⠀⠀⠻⣤⠙⠻⣿⣦⡶⠀⠀⣠⣶⠾⣿⠟⠻⣿⠟⠛⠻⣿⠛⠛⣿⠿⢷⣦⡀⠙⣧⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠶⠶⠶⠶⠿⣿⠋⢀⣤⡾⣿⠁⠀⢿⡀⠀⣹⡆⠀⢰⣿⠀⢀⣿⣀⣀⣾⣿⠀⣿⡇⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢤⡀⢰⡏⠀⣿⣿⣧⣼⣷⣴⡿⣿⡟⢿⠟⢷⣾⣿⣾⠛⠿⣇⠉⠻⣿⢷⣿⣧⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠻⣾⣧⣀⣈⣽⠟⢁⡾⠃⠀⣿⠁⣾⠆⢸⣿⠀⠙⣷⡀⠙⣷⠀⢻⡆⢻⡏⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠀⣾⠃⠀⢰⡇⠀⠛⠀⠈⣿⢧⡀⠈⠓⠀⠸⠇⠘⠿⠘⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡤⣄⢻⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣸⡇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣻⣶⠆⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣯⠀⠀⠀⠀⠀⠸⣿⣤⠶⠶⠶⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣤⣤⣼⣿⣡⡾⠛⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣤⣤⣄⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠞⠋⠉⠀⠀⠀⣹⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠈⠻⣭⡙⢷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⡶⠾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⠿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣾⠏⠁⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⢿⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣤⠶⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠠⣏⠁⠀⠀⠀⠀⠀⠀⠂⢄⡀⠈⠀⢄⠀⠉⠠⢀⢀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⢀⣴⣿⠟⠁⠀⠙⣄⠀⠀⠄⡀⠀⠀⠀⠀⠉⠓⠄⡀⠀⠀⠀⡀⠉⠻⣷⡄⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⠏⠀⠀⠈⠀⠈⢢⣀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⣂⡬⢷⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠔⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡴⠖⢀⣠⠄⣛⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢀⡀⠉⠀⠔⠈⠀⠄⠋⠀⠀⠀⠠⠀⠀⠀⠐⠤⠀⠀⠀⠀⠄⠤⠀⠀⠛⠁⠛⣛⠄⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠈⣿⡶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢿⣿⣿⣿
⣿⠿⠿⠋⠀⢀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⠀⠈⠻⢿⣿
⣧⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠁⠀⠀⠀⠀⠀⠀⠒⠀⠙⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡖⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣧⣾⣷⣤⠄⠀⠠⠀⠀⢀⣀⣀⣀⡀⠀⠠⠄⠀⠀⣦⣴⣿⣄⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⠀⠀⢀⡀⣀⢀⣀⣀⣄⣀⣀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠍⠛⠛⠛⠻⠇⡿⠠⠟⠒⠒⠒⠩⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣤⣿⣤⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠠⠄⣠⢀⡆⠀⠀⠀⣍⡈⠙⠀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠼⠅⣠⡿⠿⠷⠷⠴⠶⠀⠠⠶⠟⠡⢬⢱⠾⢇⢀⣀⣿⣿⣿⣶⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠺⢇⣶⣾⢬⣼⣿⣿⡿⠟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣶⣬⣿⡿⠀⠁⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡞⣿⣦⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣿⡿⢩⣿⣿⣦⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡃⣼⣿⣿⣿⣦⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⢹⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡃⠿⠿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠦
⠀⠀⠀⠀⠶⠛⣻⠋⠓⢠⣀⠐⢻⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⡆⠀⠀⠛⠘⢿⣿⡛⣿⡟⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠚⠃⠴⢂⣈⣥⣤⡬⠷⣿⡿⠟⠻⠉⠀⠀⢀⡟⠇⠀⠀⠀⠃⠀⠽⣷⡈⢷⡄⠁⠀⢿⠛⠻⠟⠋⠉⠛⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢵⠀⠴⡓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣆⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣿⡇⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⡤⠞⠁⠀⠀⢀⣴⡿⠟⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⠁⢀⠜⠀⢀⣤⣤⣾⡟⠉⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣶⡾⠟⠋⠁⠀⢸⡟⠀⠁⠀⠚⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣇⣀⣀⡀
⠀⠀⢰⠟⠀⠀⠀⣠⡟⠁⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⢠⣾⣿⡟⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠈⠁⠈⢳
⠀⠀⡻⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡤⡼⣷⣴⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⢻⣷⣴⣶⣶⠶⠶⠶⣦⣀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠃
⡠⠞⠁⠀⠀⠀⢻⠀⠀⠀⢀⠞⠁⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠘⠀⠀⠎⠀⠀⢀⣀⣠⣤⡶⠶⠟⣿⡟⠋⠉⠙⣿⣿⣆⠀⠀⠀⠀⠙⠷⠶⣤⡀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⠤⠤⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣇⠈⠉⠛⠿⢿⣧⣄⠀⠀⠓⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠿⢶⣶⣤⣤⣤⣄⣀⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢠⡞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⢻⣆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⣫⣿⣿⠟⠀⠀⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⠁⣠⣾⣿⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣾⣿⠟⠁⢻⣇⠀⠀⠀⠈⠻⣆⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣸⣿⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣆⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠞⢻⣿⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠷⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢴⡁⠀⠀⠀⡀⠀⠃⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠋⠀⠀⠑⠄⢄⠀⠀⠑⢄⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⡀⠀⣄⡸⠀⠀⠀⢠⠞⠑⢤⣤⣶⣿⣶⣶⣷⡤⠤⠟⠀⢀⠀⠒⠒⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠈⠀⠀⠀⠀⡰⠃⠀⠀⠀⠛⠿⠟⢿⣿⣿⢿⣆⠀⢐⣉⡠⢀⣀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠉⠉⠁⠉⠝⢢⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡀⠻⣧⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠒⠀⠰⠀⠐⠊⠉⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣬⢷⡀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢈⠥⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠽⣧⢳⡄⠀⠀⠀⣆⠁⠀⢀⣠⣠⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠻⣄⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠠⠀⠀⠈⢀⠃⡀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⠀⠀⠀⠀⣠⢃⠁⠀⠀⢸⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠘⣦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⣀⡴⠁⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠻⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⣠⡟⠀⠀⠀⠀⢦⠀⠀⠀⠘⢆⠀⡀⠀⠀⠈⣝⡻⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣠⠾⠋⠐⠀⠀⠀⣠⣾⠟⠁⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡀⠈⠀⠱⠈⣿⣯⡓⣬⡳⣄
⠀⠀⠀⠀⠀⠀⡴⠟⢁⣴⠟⠁⢀⡜⠀⠀⣰⣾⠟⠁⠀⠀⠀⣀⣾⡟⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⢹⣦⣀⡀⠀⣸⣿⣿⣾⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠑⢾⣏⣁⣀⣠⡞⠀⣠⡾⠟⠁⠀⠀⠀⣠⣴⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠲⢦⣴⣴⣾⣿⣿⣿⣤⣤⣀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⡶
⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⡿⠃⠀⠀⠀⠀⠘⠛⣿⣿⠟⠀⠀⠀⠀⠀⠸⢿⣿⠋
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀⡀⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⢀⣾⣿⣇⣀⣀
⠀⠀⠀⠀⠛⢿⣿⣷⣦⠀⠀⠀⠀⠘⣻⣿⣿⣟⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠉⠁⣸⣿⡟⠁
⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡶⠒⠛⠛⠙⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠏
⠀⠀⠀⢠⡇⡰⣊⠩⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡟
⠀⠀⠀⢸⡇⠁⠁⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠒⠒⠲⢦⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠛⠛⠉⠉⣹⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣹⠇⠀⠀⠀⡜⠀⣀⣀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⢉⡟⠀⠀⠀⠀⣼⠃⠀⠀⢀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣯⣤⣄⠀⠀⠀⢲⡏⠀⠀⠀⠀⠉⠉⢉⡽⠃⢀⢀⡼⠁
⠀⠀⠀⣿⣀⣠⣤⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣏⣴⡶⠟⢋⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣢⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⣁⣀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠟⢛⡿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢠⠏⠀⠀⠀⠀⠿⠛⠒⡖⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟
⠀⠀⠀⠀⠀⢸⡇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠠⡇⢷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡴⠶⠖⠒⡒⣲⠆
⠀⠀⠀⠀⠀⢰⣶⠛⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢀⡇⠈⢣⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡶⠛⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⢠⡶⠞⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡞⠉⠀⠀⢀⢂⡴⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠀⠀⢠⣿⣃⣀⣀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣠⣭⠇⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⠉⠉⠉⣩⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⣤⡄⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠠⣞⠃⠀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠓⢲⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⢠⠞⠀⠀⠀⢰⡯⠶⠖⠚⣿⠀⠀⠁⣴⠋
⠀⠀⠀⠀⠹⣆⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⡀⢸⠀⠀⠀⠀⠀⠙⠋⢹⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡄⠀⠀⠀⠀⠀⢀⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠹⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠒⠀⠉⠀⠀⠀⠀⠐⢲⡤⠀⠀⠀⠀⠀⠀⠀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢢⠀⠀⠀⠀⠒⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠲⣦⠀⣷
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡀⣀⡀⠠⠀⠘⠀⢀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢇⡄⡀⠈⣠⠀⠈⢩⡔⠃⣀
⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠆⠀⢀⠀⠀⠀⠈⠉⠀⠃⢉⡠⠏⠘⠉⠠⠴⠚⠁⠞⠃
⠀⠀⠀⠀⠀⢸⢻⣿⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⢠⣶⠀⠀⢸⠀⣿⠀⢸⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠤⡤⣤
⠀⠘⣿⠀⠀⢸⢠⣿⠀⠌⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣞⢷⡀
⠀⠀⣿⡄⠀⠘⢸⡿⢊⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⢄⢸⢿⣧⣷⡀⠀⠀⣀⠀⣀
⠀⠀⢿⣧⠀⡞⢸⣿⠟⠁⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⡶⣋⡄⠀⣀⣼⠇⣾⣿⣛⢯⣿⣷⣾⣯⢁⠈⠢⣄
⠀⠀⠘⣿⣦⡗⢸⣿⠀⠀⢸⣼⡇⠀⣀⠀⠀⢀⠠⠀⢀⠎⠀⣼⡿⠋⠀⠀⡿⠋⢸⣧⡈⠛⢿⣿⡿⣿⣷⣿⣷⣀⣉⢻⡶⣦⣀
⠀⠀⠀⠘⢿⣇⢾⡏⢀⣄⢸⣿⠃⠀⠸⣷⡀⢉⡤⠚⠉⢀⡾⠋⠀⣰⣧⠞⠀⢼⠙⢿⣿⣦⡺⣿⣿⣮⣿⣿⣿⡿⠋⠻⣧⡿⣿⣿⣆
⠀⠀⢀⣶⠀⣿⢸⡇⢸⣿⢸⣿⡆⢀⣤⢿⡗⠉⠐⣠⡾⠋⠁⠀⣰⣿⢏⠄⡆⢠⠸⣄⠈⢿⣷⣿⣿⡿⠛⠉⠁⠀⠀⠀⣽⠿⣿⣄⡈
⠀⠀⠘⣿⣆⣿⣿⠁⣸⣿⣼⣿⡇⣿⣿⣿⣿⣶⠟⣫⣤⣶⠶⠋⠉⣠⠎⢀⡁⢸⡆⠨⣦⡈⠿⠟⠉⢰⢸⡇⠈⠁⠀⣰⣿⡀⠀⠉⠃⠑⠂
⠀⠰⣾⣿⣿⢿⣿⡇⠈⡻⣾⣿⠃⣤⠙⠿⠿⠗⠛⠛⢻⣃⣤⠈⣰⣟⣀⣾⡇⠘⠛⠀⠉⠀⠀⠀⣶⠸⢿⠁⣾⢠⣿⣿⡛⢻⣶⢤⡀
⠀⠀⠀⠀⣈⣿⣿⡇⠀⠙⢻⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣥⣾⣿⣷⠿⠋⠀⠀⠀⠀⢠⢠⣄⠀⢹⣄⣿⣰⠇⠈⠙⢻⣿⣿⣿⣿⣧
⠀⠀⢀⣈⣠⣿⣿⣿⡓⠶⣿⣿⣿⡿⠀⠀⠀⢀⣀⣀⡀⠀⠸⠿⠏⢠⡀⠀⠀⠀⠀⠀⠸⣇⣿⠀⠀⠙⡿⠋⠀⠀⠀⠀⠈⠙⠛⠛⠻⠟
⢰⠂⠉⠉⠻⣛⡙⣉⣙⠓⠿⢿⣅⠀⠀⠀⠀⠨⠽⠟⠋⠀⠀⠀⠀⠈⣿⣦⣤⣀⣀⣤⣄⣸⣿⣤⡤⠼⢿⣶⣦⠙⢟⠗⣶⢷⣤⣤⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⠿⠒⣘⠛⠿⠿⢷⣶⣴⣦⣤⡀⠀⢀⣀⠀⠀⠙⠿⠟⠛⠀⡈⢙⣿⠋⠀⠀⠸⣿⡿⣷⡆⠀⢀⣀⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠁⠀⠉⣙⣯⠁⠀⠈⠛⢯⣤⣤⡤⠀⠀⠀⠀⠹⠿⡿⢦⣄⣤⢀⣿⠷⠛⠋⠙⠛⠿⠿⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠶⢀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⣡⡞⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⣀⠐⠆⠀⣤⢀⣂⠀⠀⠀⠀⢈⠁⠀⠘⠓⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠲⠀⣠⣬⡿⠿⠿⠷⠤⠤⠶⠄⠀⠠⣤⣦⠉⣤⢀⡉⣤⣼⡁⠀⠀⠀⣤⣤⣶⣦⣀⣴⣤⡟⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢠⡮⢐⠉⣈⠠⠆⢶⣶⣾⣿⣿⣿⠏⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘⢡⡼⠇⣅⠾⣶⣽⣟⣿⡻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠡⢂⣀⣵⣿⡟⠀⠀⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠟⢿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡈⠃⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠂⣿⣶⣀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢴⣶⡶⠃⢶⣿⣳⡀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢱⣿⡇⠃⠀⣿⣿⣿⣿⡀⠸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢛⡟⣱⣥⡟⣿⠀⠘⠻⡿⣿⣟⣿⡆⢹⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⣼⠟⢚⣻⠞⣐⣿⣿⣷⣿⠀⢀⢰⣏⢹⣿⣿⣧⡌⢿⣿⣿⣝⣿⣶⣄⡀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⠁⠁⠀⢠⣚⠧⠼⠩⢈⢋⣵⣶⣿⣿⣿⣿⡇⠀⠈⣾⣿⡆⣿⣿⣝⡄⠸⣿⣯⣿⣧⣍⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡴⣺⢶⡛⠃⡀⢈⡅⣵⣦⣶⣼⣯⣿⡟⠁⣶⠏⣸⠟⢳⣷⡘⢿⣿⣿⣦⠈⢿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⣀⣤⣴⡶⠀⢃⡀⠉⠊⢉⣀⠀⠌⠁⢸⣧⣿⣿⣿⣡⡿⣿⣥⣾⢃⠀⠋⠀⢸⡎⠹⢶⣝⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⡛⠛⠿⠶
⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠲⠤⠀⠠⠺⡿⢛⢡⣖⡥⣪⣿⣏⠐⠙⢫⢋⡆⠀⠈⠀⠀⠁⠀⠺⠻⣧⠹⣿⡝⠳⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⡻⣍
⠀⠀⠀⠀⠈⠙⠀⠐⠀⢀⣀⢠⣤⣤⢄⠠⣽⣫⡽⠿⠛⠍⠀⠀⠀⠀⢸⢠⠇⠀⠠⠀⠈⠇⠀⠐⣿⣆⠘⢷⡄⠈⠃⠈⡻⠿⠿⣿⣿⡟⠻⠿⠮⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠐⠫⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⠀⣰⣿⣿⡶⠛⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠼⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢈⡽⠟⠁⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⠀⠀⠀⠀⠐⠁⢠⢌⠁⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⢃⠔⠃
⠀⠀⠀⠀⢀⢀⡘⠂⡰⠀⠀⠀⠀⠈⠀⡪⠐⠁⠀⠀⠀⠀⠈⣱⠃⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢴⡩⠇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠠⣸⣁⣀⡀⠀⠀⠀⡠⠋⢠⣎⠁⠀⠀⠀⣠⠊⢘⡂⠖⠀⠀⠀⠀⢠⠎⣠⠄
⠀⠀⠀⠀⠰⢓⢖⡴⠋⠀⠀⠀⠈⣩⡿⠒⠁⠀⠀⠀⠀⣩⠏⠋⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠰⠎
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⠁
⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⡶⠾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣀⣀⡄⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠉⠉⢀⡇⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿
⠀⡀⠀⣠⡶⢻⡄⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⠀⠀⠀⠀⠀⠘⣿⠙⠛⠳⠿
⠀⠱⠞⠁⠀⠈⢷⣀⣴⠟⠉⣷⠀⠀⣠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⣤⣄⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⣧⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠄⡀⣄⠐⡌⠉⠉⠑⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠐⠆⠀⠄⠈⠑⠈⠀⠈⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠤⠤⠔⠒⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠿⣦⡀⢠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠎⠈⠁⠀⠀⠂⡀⣀⠀⠀⢀⢻⠀⠀⠱⠈⢏⡳⠤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢴⠀⡰⠃⠀⢁⠊⠕⠁⠀⢠⠋⠀⠆⠀⠀⠐⠀⠀⠷⢀⠄⢡⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⣁⢀⠔⠇⠐⠂⠀⠀⡸⠀⠀⠃⠀⠀⣴⣄⠢⠠⠤⠐⠤⣠⡔⠢
⠀⠀⠀⢀⠐⠁⠀⠀⠀⣠⠋⠀⡁⠀⡘⣀⠀⠀⢠⠄⢸⠀⠀⠐⠂⡰⠓⣌⢢⠐⠶⠀⠀⠢⠀⣀⡤⣈⠶⢤⡀
⠀⠀⠀⠀⠃⠀⢀⠐⡈⢀⡰⣊⠀⠈⠉⠉⢀⡴⠁⠂⠘⠀⠀⠀⠀⠤⠀⠊⢸⠀⠰⠄⠀⠀⠀⠈⠁⠀⠀⠈⣙⢦⡁
⠀⠀⢠⠀⠀⠀⠀⢀⠡⠂⠀⠁⠀⠀⠀⣠⠎⠐⠆⠀⢑⠀⠀⠀⠀⠀⠀⠀⢬⠄⠀⠀⠀⣠⠀⠀⠀⠈⠓⠀⠀⡀⠍
⠀⠀⠂⡠⣤⣀⠈⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⣸⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠼⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⣆⡠⣄⢠⠀⠀⠅⢠⡄⠀⡀⠐⢀⠎⠁⠓⠤⣀⠂⢀⡰⠚⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠄⢀⡠⠤⠤⠥⠀⠂⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠉⠲⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠖⠈⠑⣤⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡀⠀⠀⢀⡘⠈⠙⢦⣀⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣩⡷⣦⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠂⠀⠀⠀⣠⡟⠀⠀⣠⡶⠿⠄⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠰⠿⢶⡄⠀⢀⣠⣾⠋⠀⠀⣿
⡄⠒⢤⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⠷⠶⠋⠁⢸⡇⠀⠀⣿⠀⢀⠐⠒⡀
⠁⠰⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠘⠃⠐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠶⠶⣦⣸⡟⠀⠀⠀⢰⠏⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠿⠄⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⠁⣤⠀⠀⠛⠁⠀⠀⠀⠘⣄⠈⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⢀⡶⠀⠀⢸⣿⡄⠀⠀⡟
⡏⢀⠜⠀⠀⠀⢸⡄⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⢿⣀⡿⠀⢸⡇⠀⠘⢃⠈⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠇⢠⡝⠃
⠀⠀⠀⢀⣀⣰⠳⢆⠀⣠⢄⠙⢶⡠⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠁⠘⣾⠁⠀⠀⡠⣄⠀⣠⠞⣦⣀
⠀⠀⠀⠻⣍⡙⠳⠈⠛⠁⢸⠁⠈⠻⣆⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋⠀⢂⡿⠁⠀⠀⠈⠣⠈⠟⠁⠾⢛⣉
⠀⠀⠀⠰⣯⣁⡀⠀⠀⠀⠘⠧⣤⣄⠉⢳⣄⠀⠀⠀⠈⠉⠙⠓⠒⠶⠒⢶⡤⠀⠠⡤⠶⠶⠶⠞⠋⠉⠀⠀⠰⣹⠞⠁⠀⣀⣤⡸⠇⠀⠀⠀⠀⣉⣽
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠚⠲⠦⣄⡀⠙⠋⠻⠷⠌⣡⣄⣀⣠⣶⡶⠶⠶⠾⣄⠀⢀⡴⢶⠶⠶⢶⣆⣀⣀⣀⣴⠿⠔⠛⠋⢀⣀⠴⠖⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⡄⣀⡀⠀⠁⣹⡇⠀⠀⠀⠂⠀⠤⠌⠀⠀⠀⠀⢸⡏⠈⠀⢀⣀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠻⢷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⡿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀⠀⡀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠻⠟⠻⠟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣾⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⣿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⢸⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⣟⠛⣿⠀⣸⡛⣿⠿⠶⣦⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣷⣶⣶⣦⣤⣟⣀⣻⣤⣤⣶⣶⣾⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠒⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠖⣀⣤⣴⣶⣖⣀⣤⢦⡀⠀⠀⠀⠀⠀⢰⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠚⠁⣼⠷⠋⠁⠈⠉⠉⠉⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⢾⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣿⡈⠿⢁⡤⢤⡀⠀⠀⠀⠀⢀⡴⠒⠲⣄⠀⠀⠀⠀⣿⠞⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡇⢠⡏⠀⠀⢳⠀⠀⠀⠀⢸⣁⣀⠀⠈⢧⠀⠀⠀⠀⠀⣼⡤⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠸⣯⣷⠀⢸⠀⠀⠀⠀⢸⣷⣼⡆⠀⢸⠀⠀⠀⠀⠀⢩⣤⡀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣠⠇⣼⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⡼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢤⣖⠒⠒⠓⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⣼⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸⣤⠤⠤⣤⣤⣈⣷⠀⠀⠀⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⡾⠀⠀⠀⠉⠉⠉⠉⠀⠀⣰⠃⠀⠈⡿⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢉⡟⢹⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⡞⠁⢀⣠⣤⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡞⠀⢸⡄⠀⠸⣄⠀⠀⠀⠀⣀⡴⠋⠀⢀⡰⢋⡴⠚⠉⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠁⠀⠀⠹⣄⡀⠈⠙⠓⠚⠋⠁⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣼⣀⣀⣀⠀⠀⠉⠓⠲⠶⠶⠖⠚⠋⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠒⠒⠲⠶⠶⠤⠤⢤⣽⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠙⣷⡎⠀⠈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⠀⠀⠀⡠⠤⠽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣶⣾⣄⡀⢀⣸⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠶⠚⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⢀⣴⣦⢀⡞⠀⠠⠤⡀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠄⢀⡴⢋⡀⣿⣼⠃⣴⣄⠀⢈⡀⠄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣆⠀⢘⣚⡭⠚⠻⠄⣙⠋⡘⠉⢻⣷⣿⠿⠶⠤⢀⡀⠹⠷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠀⢠⡿⠛⣿⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⡇⠀⠀⠀⠀⠈⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⢀⠜⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷
⠀⠀⠀⠓⠒⠛⠉⠳⣤⣴⣿⣦⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣀⡀⢀⣠⣤⣴⣶⣿⣷⣦⣴⠿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⢰⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⣼⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣼⣏⣀⡀⠀⠀⠀⢰⣿⣯⣤⡄
⠀⠀⠀⠀⠰⢫⣿⣿⢏⣠⠄⠀⠀⠀⠀⢰⡿⢀⣠⡄⠀⠀⠀⢰⣿⣿⣶⠀⠀⠀⠐⠛⣻⡟⠀⠀⠀⠀⠞⢻⣿⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢛⣿⣿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠚⢁⣿⣿⠃⠀⠀⠀⠀⠀⠟⠋⣽⡟⠀⠀⠀⠀⠀⣸⣿⣿⡶⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠟⣿⡿⠁⠀⠀⠀⠼⠿⣿⠁⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣹⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣶⣶⣶⡶
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣼⣿⣿⣿⣶⡦
⠀⠀⠀⠀⣰⣿⡟⠀⠀⣀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣀⠀⠀⠘⠛⢻⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣤⡄⠀⠀⢀⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠛⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠾⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⡶⠀⠀⠀⠀⠉⢹⣿⣿⠃
⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⢅⠢⡀⠀⢆⠑⢄⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡑⠢⣄⠳⡀⠑⢄⢣⠈⢢⡆⠱⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡢⢄⠀⠈⢢⠀⠑⠬⣆⠀⠳⣇⠀⢻⠀⢣⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠢⡉⠒⢄⡱⡄⠀⠈⠳⠀⠈⠂⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⢄⡲⣬⣢⣄⠉⠺⢆⠀⠀⠀⢀⣀⠀⢤⠂⢀⣴⠆⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠊⠻⢍⡓⠄⠀⠀⣴⠉⠀⠀⠀⠠⠃⣈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠆⣐⠒⠒⠷⢄⠀⠁⠀⠀⠀⠁⠀⠀⠀⢀⠗⠀⠀⠈⠀⣀⠔⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⣁⡀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠠⠀⠈⠉⠂⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠀⠀⠀⠘⢣⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠄⣀⠃⠀⣠⠔⠒⠒⠢⢄⡇⠀⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠁⠠⠀⢠⡄⠀⠁⠀⠙⠢⠤⠤⠒⠊⠙⠢⠤⢤⠔⠊⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠁⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⣀⡀⠁⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣟⣓⣒⣂⠒⠋⠑⠢⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡎⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡜⠉⣠⣤⡈⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣄⡈⢟⡄⢇⣀⠀⢀⣀⠀⢠⡀⠀⢀⡀⠀⢀⠔⠉⣀⠀⠀⠀⡜⢀⡊⠁⢀⣈⠢⠄⠠⠜⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠑⠻⠆⠀⠀⠈⣉⠁⢤⡈⢑⠀⠉⠁⠸⡀⠁⡰⠎⠀⢂⠀⠀⣏⠉⣉⠝⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠂⠜⠁⠀⢀⣯⠋⠀⠀⠀⠀⠈⠉⠁⠘⠢⣈⣦⡀⠈⣉⡤⠖⡡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠆⡀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢙⡛⢄⠀⡰⢁⠞⠐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠜⠀⠀⠈⣂⠉⠁⢓⡀⠤⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠑⠒⠢⢤⢤⠄⣆⠀⠀⠀⠀⣀⡧⢠⠀⠀⠉⢉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⠄⠑⠒⢶⠀⠀⠀⠀⠠⠦⣼⣘⠀⠈⠈⢑⠀⠘⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣖⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣢⣔⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠀⠉⠉⡠⡶⠖⢶⠘⢻⠉⠀⣈⢽⡈⠸⢒⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠐⠊⢥⣍⣬⢷⣒⣒⡒⠈⠅⠰⠖⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣳⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⢀⣀⠀⣀⣀⠀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠄⠈⠁⠐⠲⠌⠑⠒⠊⠉⠀⠁⠉⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠟⠛⢿⡟⠋⠁⠀⠀⠀⠀⠈⠸⣟⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⣀⣀⣀⣌⣥⣤⣿⠆⣠⣾⣿⠴⠤⠿⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣶⣴⣤⣤⣶⣶⣶⣤⡄⠈⠒⣦⣬⣷⣶⠶⠶⣤⠤⣤⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠈⢦⣾⣿⠁⠈⠉⢧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢸⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⣟⡄⠀⠀⠈⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣾⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡀⠀⠀⠀⠀⡟⡔⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⣡⠀⠀⢠⡀⡷⢄⣠⠄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡅⠠⣤⠀⣿⠁⡾⠴⠀⠰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠓⠀⠛⠛⠓⠊⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


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

⠱⠄⠈⢁⣀⣵⡙⠳⡿⠄⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⢀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢕⣤⠀⠀⣠⠊
⢀⣴⣕⠄⠛⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠉⠀⠛⠀⢀⡀⠀⠀⠀⡔⢠⠀⠀⠀⠘⠁⠀⠜⠁⠸⣆
⣝⠂⠙⢀⣴⠄⠓⠀⠀⠀⣠⠰⠲⠲⠆⠁⠀⠀⠴⠀⠒⠠⠀⠀⠈⠀⠀⠀⠀⠀⠁⠤⡄⠀⠀⠀⠀⠠⠀⠀⠈
⡙⠂⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡚⠶⠄⠀⠀⢀⠥⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠘⠉⠁⢢⠀⠀⠀⠈⠀⠀⢸⠒⠒⢀⡀⠀⠀⠀⠁⠀⠔⠀⠀⠀⡠⠚⠁
⠀⠀⢠⠀⠤⠰⠇⠀⠀⠀⢰⠃⠚⠁⠠⡄⠀⠀⢁⠇⠰⠀⠀⠀⢰⠤⠐⣀⠸⠂⠒⢀⠰⡄⠀⢀⡀⠀⠀⠀⠠
⡰⠈⠙⠂⠀⠈⡱⠔⠶⠤⠌⢠⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠸⡀⠀⠉⣠
⠳⠦⠀⠀⠀⠀⣁⠀⡤⠬⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠄⠀⠠⠅⠀⠠⠒
⣦⠀⠀⠀⢰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⠂⠀⠀⠉⠉
⠀⠈⡁⠀⠘⠠⠂⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⡤⠤⠄
⣤⡦⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡶⠖⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣼⡷⠖⣋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⢴⡶⠟⣋⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠢⣌⠙⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣆⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⣠⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠻⣿⣿⡄⢰⡣⠔⠃
⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢠⣄⠘⢿⣧⠀⢀⡤⠤⣄
⠀⠀⠀⠀⣾⣧⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⡄⠈⠓⠲⠊
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣆⠘⠫⢳⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠐⣾⠁⠀⠀⠀⡤⠦⣄⣀
⢎⣉⡭⠖⠈⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹⣦⡧⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠸⣦⡴⠶⣼⠁⢠⠏⠉⣿⡀
⠀⠀⢀⠠⡄⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⣿⢧⠀⠙⡖⢧⣠⠜⠉⣹
⠀⠰⣅⡼⠁⠀⠀⢹⡿⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣄⠀⠀⠀⣿⠘⣧⠀⠀⠀⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⢠⣷⠞⠁⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣩⠕⠋⠉⠀⣠⣟⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⡞⢁⠈⢳⡴⠋
⠀⠀⠀⠀⠀⠀⣞⠉⠛⠻⡄⠈⢷⡈⠙⢲⣦⣤⣤⣤⣤⣤⣴⣖⣿⠽⠛⠀⠀⠙⢯⡀
⠀⠀⠀⠀⠀⣰⠛⠑⠒⢲⠓⠀⠈⠉⠉⠉⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠘⣶⠒⠒⠛⠀⣀⣤⠤⠤⠤⠤⠾⠷⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠙⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⡟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣀⣤⠴⠚⠛⠉⠁⠀⠀⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣅⠀⠀⠀⠀⣀⣠⡴⠖⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢤⣄⣀⣈⣹⣟⣛⣉⣿⠀⠀⠀⠀⣧⣤⣤⡿⡠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⠀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣇⢀⠀⠉⠉⠭⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠇⣦⣻⠶⠤⢤⣤⠴⠞⠀⠀⠀⠀⠀⠳⠖⠒⠒⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠙⢷⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣼⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠐⠈⠉⠉⠑⠢⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠙⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠂⠀⠀⠤⠄⠀⠀⡄⠀⠀⠀⠑⠖⠖⢀⣀⠄⠀⠀⢀⣄⣀⠄⠀⠀⠈⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠒⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⢀⡉⢣⠀⠀⠀⠀⠐⠦⠤⠄⢀⣀⡀⠀⠢⠤⠔⠊⠒⠃⠀⣠⢀⠄⠀⠀⢠⠀⠀⠀⠈⢢
⠀⠀⠀⠰⠊⠁⠀⢀⠀⠀⣈⣧⣀⠉⡉⠀⠙⠓⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡜⣠⠀⠀⠀⠀⠁⢳
⠀⢀⡤⠂⠀⠐⠊⠹⣄⡀⢀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⡀⠀⢀⡜⠀⠀⡀⠈⠹⠄⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠀⡴⠀⠀⠈⠀⢤
⠀⠊⠀⠀⠒⣆⠀⠀⣀⡍⠀⠀⠀⠑⢪⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠠⣄⡀⠐⠆⠀⠒⠁⠒⠒⠋⠀⢀⠦⠔⠄⠈⠇
⡀⠀⠀⠀⢤⡈⠁⠀⠰⡁⠀⠸⣄⠀⡀⠑⠂⢄⠀⠰⣄⡀⠄⠀⠀⣀⠀⠀⠣⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⣰⠤⠂⠈⠋⠁⢀⠄⠀⠀⡀
⢇⠱⠄⠀⠀⣉⠁⣠⠀⠉⠀⠀⢀⠉⡀⠀⠀⠀⠉⠁⠀⢀⠀⣉⡉⢀⣀⡠⢄⣈⠉⣉⣁⡀⠠⣄⣠⡄⠀⠀⢤⡀⠀⠀⠈⠉⠁⠠⠖⢊⡁
⠈⠢⠀⡄⣄⠈⠀⠻⠀⠀⠙⠤⠈⠉⣁⡜⠐⠋⠈⡐⠊⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⣀⣜⠃⠀⠀⠀⠠⠴⠒⠊⠀⠄⠈⡄⠊
⠀⠀⠀⠈⠢⡁⠀⠤⡀⠄⠐⠂⠀⠀⠀⠈⠋⠉⠉⠀⣿⠀⠀⠀⢰⠀⡇⠰⠀⠀⠀⠀⡼⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⣀⠀⠚
⠀⠀⠀⠀⠀⠉⠒⠂⠐⠤⡍⠉⢀⣐⡊⡐⠒⠀⠀⠖⡿⠀⠀⠀⢸⠀⢇⡇⠀⡈⠠⢸⠃⠒⠓⠒⠂⠀⠀⠉⡈⣁⣀⡠⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠑⠄⠀⢺⡀⢈⠸⠈⡇⠸⠀⢃⠇⡆⡿⠖⠤⠒⠐⠊⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡆⡀⡇⡄⠠⣸⢰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⡇⠁⠀⡇⡏⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡼⢇⡠⠴⢧⣇⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠌⠉⠁⠀⠀⠀⠀⠀⠁⠘⠋⠳⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⢠⣴⠦⣴⠠⢤⣤⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠲⠄⠠⣿⡇⣿⠀⢸⠀⢸⢣⠔⠢⠤⠆⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⡿⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⡇⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢳⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠴⢦⠀⠀⢈⠤⢤⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠣⠀⠀⠀⠀⠋⠑⡀⠀⠀⠉⠀⠀⠁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠀⠀⠔⢢⡤⡈⠂⢀⡀⢩⠤⠒⢤⢤⡀⠙⠀⠐⠳⡄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠠⠄⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠐⠆⠀⠀⠑⠲⠀⠈⠓⠀⠈⠈⠋⠂
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣶⣶⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴⢒⠒⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠙⠳⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠹⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠐⠀⡀⠀⠑⠠⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠑⣄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⢀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⢒⠹⣿⣿⣿⣿
⣿⣿⣿⠋⠀⠀⡠⠀⠀⢀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⡀⠭⠚⠁⣉⠃⠘⠿⠿⠿
⣿⠟⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⡄⠈⠀⠀⠀⢀
⣿⣿⡗⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣠⣴⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠛⠛⠻⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠀⠀⢁⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⣀⠀⢀⢸⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⣁⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣚⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣹⢉⡧⡁⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠐⠁⣸⡇⠀⢄⠀⢈⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⡠⠒⠀⣰⡀⡇⠀⠀⠑⠀⡽⣗⣬⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠚⡁⢈⣤⢂⡠⠂⠀⡀⠉⣇⠘⢂⡀⡈⠛⢷⣤⡀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⢀⣤⣒⣚⣯⣄⡉⣒⣁⣸⣿⣿⣃⣢⣖⣜⣀⡩⠵⠟⠋⠁⢀⣀⢍⡙⣝⣢⣤⣀⣀⠈⣙⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠎⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠲⠾⡀
⢰⠤⠤⣀⣀⠤⠒⠊⠁⡠⠤⢄⡈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢃
⠀⠳⡄⠀⡠⠀⠀⠂⡄⠀⣤⠀⠱⠀⠀⠳⣀⡀⡄⠀⠀⠀⠀⡀⠀⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠸⢤
⠀⡰⠁⢸⠀⢠⣦⠀⢀⠀⠉⠀⠐⠀⠀⠀⠀⠐⢳⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠘
⢀⠃⠀⠀⢄⡀⠀⡠⡎⠀⠀⣁⣀⡀⠀⠀⠀⠀⢉⡇⠀⠀⠀⠀⠀⣠⠴⡄⣠⠔⡆⠀⠀⠀⠀⠀⠀⠀⢡
⣸⠀⠀⠀⠀⠀⠀⣠⠄⠀⢊⠉⠀⠘⡄⠀⠀⠀⡠⢋⢤⣠⠊⠱⠜⠁⠀⠛⠁⠀⢡⠔⠒⠀⠀⠀⠀⠀⢸
⠉⡲⠀⠀⠀⠀⢀⠉⢐⠡⠁⠘⠀⠊⠀⡀⠰⣋⡠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⡘⠀⠰
⠀⠉⣟⢠⡄⠀⠀⠉⠁⠀⠀⠀⣀⠔⣎⠷⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠃⢀⡔
⠀⠀⠀⠈⠓⠒⠚⠉⠉⠉⢒⠯⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠖⠁⣀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠆⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⡦⣀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⢰⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠘⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠎⡇⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣸⠀⠀⠱⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡴⠋⠀⡇⡰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⢶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢟⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢍⠉⠃⠀⠀⠀⢀⠄⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣺⣯⣶⠄⢀⠤⠄⠂⠀⣀⠀⠀⠔⠃⠀⠐⠤⡀⠐⠄⠀⢀⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡝⠟⠉⡀⠎⠉⣶⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⢀⡈⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣫⡞⠀⢼⠀⢀⡀⣷⡀⠀⢄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡇⠊⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣾⣦⣼⣶⣤⣽⣿⣿⣶⣾⣷⣤⣿⣿⣦⣄⣀⣀⣤⣄⡀⢀⣀⣁⡀⠀⠀⠁⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣝⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⡿⠿⠶⣶⣦⣴⢀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠳⣭⣿⣥⣝⣿⣷⣶⣿⣿⣿⣿⣁⡀⠀⣀⡀⢠⣿⣿⣿⣾⣿⡿⠿⣿⣫⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⢸⠿⢿⣿⣷⣿⣏⠉⢻⡟⣿⣇⠉⠋⠀⠀⡟⠃⣼⠟⠉⣹⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣬⡉⢿⡙⢿⣆⠀⠁⣿⣿⠀⠀⠀⢠⡇⢰⠏⢀⣾⣿⠟⣡⣾⣿⠁⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠲⣝⠆⠙⢧⠀⣿⣿⠀⠀⠀⠈⡇⠀⣠⡿⢋⡴⡚⠱⠂⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣭⣙⡛⠂⠁⠀⠀⢰⣿⣿⠀⠀⠀⠀⢱⡐⠋⠀⠑⠊⠁⠔⢛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣭⣽⣛⣛⠟⠃⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⢧⠀⠀⠀⠴⠒⢛⣛⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣾⣿⣿⣿⡄⠔⠒⢂⣀⠜⠆⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠁⡀⠀⠀⠈⠺⣿⠟⠛⠻⣿⣦⣴⣈⡀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⣁⣀⣀⣀⣈⣙⢿⠶⣶⣀⣀⣠⡤⠈⠛⠉⠙⠛⢿⣿⣁⣙⣒⣲⣷⣶⣶⣀⣾⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣟⣹⢛⣟⣻⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⠾⣥⣵⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣿⣿⣞⣿⣷⣿⡿⠿⡿⣻⠲⣖⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣩⣿⡗⣿⣿⣷⣶⣿⡟⠁⣶⣾⡧⣻⣄⢵⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢿⡿⣿⠿⣾⣿⣷⣜⣿⣷⢽⣶⣾⣿⣾⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣸⣿⣿⣷⢮⡉⠿⣿⠍⠿⠆⠾⠟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢸⣿⠉⢗⣿⣕⢽⣦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠀⡀⠀⠀⠀⣸⣻⢻⠀⠘⣧⢿⣤⡈⢫⣮⣍⣿⣦⣀⡀
⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⣀⡾⠀⢰⠇⣴⡿⢱⠀⠀⠀⡟⣾⣿⣕⢶⣮⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠛⠚⠛⠓⠋⠀⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⠻⠷⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣹⣿⡆⠀⠀⠀⠈⣷⠀⢰⣇⢈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢉⣿⣿⠇⠀⠀⠀⢰⡏⠀⠘⢧⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⡇⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢀
⠀⠀⢣⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠈⠻⢷⣤⣀⡀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⢀⡜
⠀⠀⠀⠳⣄⠀⠀⠉⠻⢶⣤⣀⣀⠀⠀⢀⣀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠉⠁⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⢿⠶⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣁⡀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⣗⠀⣀⣀⣠⠼⠟⢻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠈⠉⠉⠛⠛⠓⠒⠒⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⣿⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠘⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⢈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠳⣦⠶⠛⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⡤⣤⡴⠶⠶⠶⠶⢦⠤⢤⣽⣇⣀⡀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⠀⢸⣿⠀⠀⠀⠀⢀⡴⠞⣿⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⣿⠛⠶⣄⠀⠀⠀⠀⠘⢿⣄
⠀⠸⠀⢸⣟⠀⠀⣀⣀⣸⣇⣀⣿⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣸⣄⣀⣀⣠⣤⣿⣤⣤⣼⣦⣤⣤⣤⣤⣴⣿⣷⣤
⠀⢠⠀⠘⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠓⠖⠛⠓⢦⣠⠤⠤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠑⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠦⢤⠀⠀⠀⠀⠀⢀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠀⠁⠀⠀⠈⠓⠒⠦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⣀⡠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠶⠶⠶⠴⠒⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣀⠀⣄⣤⢀⣄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡟⢸⣿⠸⣿⠈⠙⢄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢰⣷⢻⣿⠀⢻⡳⣄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣿⠃⠈⠁⠀⠀⠹⡜⡆⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⠓⠃⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⣼⠁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣙⣦
⠈⠉⠉⠉⠉⠁⠻⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⣀⣀⣤⠤⣤⣀⣀⡀⣸⠋⠉⠉⠉
⣦⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣦⣬⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣽⣥⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣴⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿