Manure Emojis & Symbols

⠀⠀⠀⠀⣀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⡗⠀⠀⠐⠀⢶⣶⣆⠀⠐
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⢠⠀⠀⣤⣶⣶⣽⣿⡶⠐⠛⠛⠺⢦⣧⣶⣿⣽⡍⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣷⣜⢗⣿⣿⣿⣿⣿⣿⣧⡄⠀⠐⢀⠀⣿⣧⣾⣿⣿⣿⣾
⠀⢹⣷⣿⣿⣿⣿⣿⣿⣴⡙⠫⣿⣾⣥⣿⣻⣿⣿⣿⣿⡻⣾⣿⣾⣵⠘⣿⣿⣿⣿⣿⣷⡇
⠀⠸⠛⠛⠟⠻⠛⠛⠛⠛⠗⡿⣿⠟⢿⣿⣯⠛⠿⠟⠻⠛⢻⠟⠋⠉⠀⠉⠛⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣼⣶⣬⣾⣿⡃⢠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣻⣿⣶⣿⣿⣇⣀⠀⠀⢸⣦⢀⣿⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⣿⣿⣯⣿⣿⣿⣿⠷⢠⣏⣿⣿⣿⣿⡇⢠
⠀⠀⠀⠄⢄⡀⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣿⠋⢛⣡⣶⣾⢿⣿⣿⣿⢹⡟⠟⡄
⠀⠠⢄⡞⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣆⣿⡄⡆⠀⠀⠀⢀⠂
⠀⠐⠆⠐⠀⠉⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣾⣏⣿⣿⡆⣿⣇⣾⠀⡂⠀⠐
⠀⠈⠁⠀⠉⠘⠀⠀⠀⠀⠰⣾⢾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⠛⡇⣿⣽⣿⠀⠈⡀⠲⣦
⠀⠀⠠⠄⠤⠀⠀⠀⢀⡀⠠⢺⡟⣻⣿⣿⣿⣿⣿⡿⣿⡇⠸⣟⠫⡟⢻⣿⠻⡷⠴⠂⠁⠐⠛⢬
⠀⢘⠐⠲⠒⠤⣤⣄⣈⣀⣰⣾⣷⣾⣿⣿⣿⣿⣿⣏⣿⡇⣀⠉⠀⠳⢻⣿⣟⣼⠂⡀⢁⠀⠈⠉⠀⠂
⠀⠀⠂⠒⠀⠢⠤⣄⣁⡘⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣇⢰⠌⠝⡄⢿⣿⣽⠀⠀⠉⡀
⠀⠀⠈⠁⠀⠞⠲⠀⠈⠁⠙⠛⡛⠻⠿⢿⣿⣿⣿⣿⣿⣧⣷⡟⠾⢄⣠⣼⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣇⠫⠼⠛⡿⣿⣿⣿⣦⣦⣀⠩⢿⣿⠉⠈⠀⠀⡆⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠛⠻⠿⠿⠿⠿⣿⠿⠚⢈⣥⣿⣿⣽⣿⣿⣿⡗⠻⠇⠻⣞⠆⢹⠀⣤⣿⣿⣿⣟⡽⠆
⠀⢸⡧⣢⠀⡒⠢⣀⠈⠀⢰⢤⡄⠀⠀⠠⠀⣁⠉⠻⠿⠿⢿⣿⣿⣿⣿⣷⠾⠒⠈⣂⡘⣿⣿⣿⣿⣿⣿⠲⡲⠂
⠀⠀⠀⠁⠈⠐⠒⠥⠭⠶⠆⠤⠤⠈⠀⠀⠚⠐⠄⠀⠐⠀⠐⢂⡀⣎⡉⡾⠆⡄⢿⡿⣿⣿⣿⣿⣿⡗⡄⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠁⠩⡄⣸⡍⠙⠋⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠒⠀⡀⠀⠀⠀⠀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠒⠀⠒⠚⢶⠒⠤⠤⠤⠤⠔⢲⠖⠒⠒⠒⠒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠈⠣⠀⠀⠀⠀⠠⠋⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡀⠀⢀⡠⢄⣀⠀⠀⠀⠀⠀⠀⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⣨⡋⠓⡇⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢳⠀⠀⠏⠀⠸⡀⠀⠀⡠⠤⢤⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠑⢄⢸⢰⣸⢠⠔⠉⢩⠃⠩⢍⡉⠀⠀⣰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠷⢧⢹⢡⠷⠯⠔⢁⠐⠤⠤⠜⣠⠖⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡃⠀⠀⠀⠀⠉⠓⣤⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠩⣀⡬⠑⠢⡄⠀⠀⠐⠛⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡰⠀⠔⠄⠰⡀⠘⢦⠀⠀⠀⠀⠀⠀⠐⠊⠉⢑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠐⠐⠀⠒⠐⠒⠈⠀⠒⠒⠒⠒⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠰⣶⡄⠀⠀⠀⢀⣀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⡞⠁⠀⠙⢆⠈⠀⢀⡴⠋⠉⠉⠙⡆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠘⡆⢀⡞⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢠⠝⢺⢠⡠⠀⣠⠴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠚⠁⣀⠙⠒⠒⠉⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠨⣐⡡⠀⢠⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⢀⣀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠼⠁⠈⡳⢲⠒⡎⢻⡾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣃⣠⠈⠉⠉⢉⡽⠃⠀⠀⠀⠐⠶⠾⠽⠂
⠀⠀⠀⠀⢰⡶⠀⣀⣀⣓⣦⠀⠀⠠⢴⠚⠀⠀⠀⢰⣞⣓⠒⠒⣶⢀⣀⣀⣀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⡄⠀⠀⡴⠊⠀⠀⠀⠀⠘⠤⠤⢄⣀⠀⠀⠀⢠⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⠀⠀⠑⠤⡤⠤⠒⠢⡄⠀⠀⢧⣄⠀⠀⠀⠀⠉⠉⠀⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠐⠯⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠲⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠭⠛⠃
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠖⠛⠓⠲⢤⡀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⡠⠞⠃⣀⣠⡞⠁⠀⠀⠀⠀⠀⡹⠞⠉⠁⠀⠀⠉⠉⠛⢦⣀⣤⣤⣀⠀⠀⠰⠤⢤⡀
⠀⠀⠀⠀⠞⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠱⡀
⠀⠀⠀⢀⡄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠐⠓⠦⡀
⠀⠀⢠⠃⢠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠓⠛⠲⠦⣄⠀⣹
⠀⠀⠈⠢⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡨⠂
⠀⠀⣠⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣇
⢠⠞⠁⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠉⢧
⠻⠦⠤⠤⠴⠾⠦⣔⠊⢳⠤⢤⡎⠀⠀⠀⠰⣀⡤⠒⠒⠒⠢⣄⠀⠀⢀⡴⠒⢲⣀⠔⠒⠲⡄⡴⠋⠀⢈⡷⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠚⠓⠒⠒⠒⠚⠿⣄⡀⠀⠀⠀⠀⠉⢙⣮⣤⡤⢴⣧⣤⣤⠤⠿⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⡄⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣤⣤⣄⡘⣀⣴⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣩⠿⣿⣧⡘⠁⠀⠉⠀⠀⠀⡄⠀⠀⠀⠀⣗⠀⠀⠀⣿⣿⣶⣿⣾⣷⣿
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣇⣰⡾⠿⢿⢿⡵⡐⣦⡀⠀⠋⢀⣀⡤⠰⣭⣄⠀⠀⢸⣟⠉⠑⠀⠀⠅
⠀⠒⣐⢲⣔⣠⣳⣆⣺⡷⠶⢰⠅⢀⠄⣄⢀⠘⠀⢀⡼⠀⠄⠀⠠⣾⡿⠇⠦⢍⣁⣀⣉⠁⠶⣅⠸⣷⡀⠚⠃⠀⠀⠂⡉⠈⠁⣀⠀⠋
⣠⡼⠿⠃⠈⠙⢹⠩⠟⠙⠁⠉⠈⠡⠀⠀⠁⠐⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣾⡄⠙⢷⣄⠀⠰⡀⡄⡁⠠⢀⠀⡒⣀⠙⠃⠀⠃
⠉⠁⠐⢿⠶⣛⠀⠄⠀⠀⠀⠠⠀⠀⢀⡄⠀⠀⠀⠆⡉⠂⠀⠀⠀⠈⠉⠀⠀⢨⣿⣿⡿⣿⣿⠋⠀⠀⠘⢿⡇⠀⢙⠈⠁⡤⠀⠈⠉⠙⠟⠂⠄⠀⠄
⠀⠀⠁⠀⠈⠉⠀⠀⠈⠤⠀⠀⢡⠾⠃⠁⠆⠀⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⠈⣿⠃⠀⠀⠀⠀⠘⣷⠀⠈⠀⠀⠑⣈⠁⠀⠀⠀⠀⠀⠀⢠⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⢁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠨⠀⠀⠀⠀⣸⣤⣿⡀⠀⠀⠀⠀⠈⠱⠶⠚⠛⠳⢢⡄⢀⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠟⠀⡆⠀⠀⠀⢠⣿⣿⣿⠗⠀⠀⢤⣄⠠⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠓⡋⠀⠆⠀⠀⣧⣿⣿⡇⠈⢠⡥⣤⣤⠌⢛⠾⢃⠒⢀⠀⠀⠉⠍⠐⠂
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠰⣟⣿⠗⠉⡄⠀⠀⣾⣿⣿⠓⢿⣀⢸⣤⡄⣱⡀⠓⠈⠛⠂⠀⠐⠀⠀⠤⠆
⠀⠁⠀⠀⠀⠀⠀⠀⡴⢶⣶⣆⣀⣀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣄⠀⢀⡰⣉⣀⣽⣄⣀⡰⠆⠀⠂⣾⣿⣏⡖⣂⢰⣶⣶⣠⣙⣀⡀⠄⠘⠀⢂⣤⣌⣁⡀
⠸⣿⣀⣾⣶⣾⣧⣤⣾⡿⢿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⣰⠗⠞⠋⠿⣿⣿⡗⠊⠠⠂⣿⡿⡋⢹⣿⣿⣿⣿⣾⣴⣶⣔⣾⠶⠷⠼⠿⠿⣽⠿⠷
⣶⣿⣿⣿⣿⣿⣿⣿⣻⣷⣶⣽⣻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣥⣶⣿⣿⢿⡃⡀⠀⢠⣿⠇⠁⢸⣿⡿⣿⢿⣿⣿⣥⡈⠉⢀⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣷⣧⣯⣽⢿⠻⣷⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢿⣿⠇⠀⠀⠉⣿⠄⠀⢀⣬⢇⣷⣶⣲⢾⣿⣿⡦⠀⠀⠤⠴⣶⣆
⢟⢿⢻⡿⣋⣿⡿⣩⢩⣧⣶⣿⡷⢆⡈⣯⡶⠽⣿⣿⡿⢿⠿⢻⣷⡻⢿⣿⢿⠷⠀⣞⠣⠀⠀⢸⣷⣀⣀⡄⣿⡷⠬⢹⠋⠀⠩⠉⣁⠀⠀⠀⡀⠀⠀⠠
⣿⣿⣿⣿⣼⣿⣿⣟⣷⠿⢿⣿⡿⠶⠚⣞⢳⡔⢿⠿⣿⢷⣿⣿⣿⣶⡄⣷⣷⠛⣶⣤⣉⡛⣓⡺⢧⡿⢿⣿⠫⣀⢢⣀⢠⣴⢢⠴⠧⣙⣷⡀⠙⣇⣀⡀
⠀⢁⣩⣉⣿⡟⠿⣟⡀⠳⠎⣸⡃⢀⣻⠋⢻⡙⡿⢯⣅⣻⢿⣟⣛⣋⣼⣧⡴⠾⣿⣿⣿⡿⡿⠻⠀⠀⠀⠀⢛⢻⣛⠳⠷⠒⠖⠞⠒⠀⠀⠀⠐⠾⠏⢁
⣾⠿⢻⠟⣛⣛⠚⠻⠿⢿⣦⣈⠀⠈⠁⠂⠉⠩⠁⠘⠛⣿⣍⣟⣻⠳⠉⠬⣟⢻⣟⠛⠿⠃⢤⣤⣤⣤⣤⠸⠗⢠⣌⠃⠋⠉⠀⠀⠀⠀⠆⣀⠘
⠀⠤⠐⠒⠉⡀⣀⣀⡀⠀⠂⠉⢽⣻⠀⠀⢠⣤⣤⣴⡄⠀⣘⠃⠈⠀⣹⣍⠩⠏⠉⠀⠀⠀⠀⠉⠙⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠤
⠀⠀⡀⡀⠘⠿⠿⠟⠃⠀⠤⠌⠈⠁⠀⠀⣠⣄⡈⠉⠀⠈⠉⠃⠉⠀⠀⠈⠀⠀⠀⠀⣈⠀⠁⠀⠀⠀⡀⣲⣾⣾⣷⣶⣾⣿⣶⣶⣤⠀⠀⠀⠀⢀⠄
⢁⣘⣽⣧⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣀⠀⠈⣀⡄⠐⢐⠀⠀⠀⠀⠀⠀⢤⠄⢢⠼⣿⠿⠿⠿⠿⠻⠿⠿⠟⣿⣿⡟⢲⣒
⠛⠛⠛⠛⠿⠛⠂⠠⠀⠀⠀⠠⣀⠀⠀⢀⣀⠄⠉⠉⠀⠀⠀⠡⣿⡎⠠⠄⠀⠈⠀⠀⠀⠀⠀⠤⠀⢀⡠⢀⡀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉⠗
⠀⠀⠐⠒⠂⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠘⠁⠈⠁⠲⡮⠷⠲⠿⠿⢿⠿⡷⣤⠤⣄⠋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡠⠶⠒⠤⠤⠤⠀⠀⠀⠀⠀⠐⠀⠈⠛⠒⠂⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⡰⠦⠤
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠠⠶⠂⠀⠠⡶⠃⠀⠀⠀⣌⠙⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠿⠷⠀⠀⠀⠀⠀⣇⠀⠀⡜⠁⠀⢀⠀⠀⠚⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⡼⠀⠀⢹⠄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡄⠀⠀⠀⠀⠀⠀⠏⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠐⠀⠀⠈⠐⠘

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣛⣏⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠉⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠉⠁⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠘⠋
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡴⠛⠿⢿⣷⣿⣦⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣧⣴⣶⣼⣿⡆
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⢐⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⡆⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣭⣭⣭⣭⣭⡌⢩⣍⣛⣛⣛⣛⣛⣛⣛⣋⣭⢁⣭⣭⣭⣭⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣮⠻⣿⣿⣿⣿⣿⣿⣿⡿⣣⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⣿⣿⣿⠿⠿⣛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⣻⣿⡧⢛⣛⠻⢰⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⣿⣿⣋⣐⡛⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡜⣿⣿⣿⢣⣿⣿⣷⢸⣿⣿⣿⣿⠟⣛⣋⣛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠐⣿⢻⡇⣿⡿⢟⣫⣭⡭⣹⣷⣭⣭⣭⣶⣿⣿⣿⡿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⢷⡄⠀⠻⠸⣱⢏⡴⢿⣿⡟⣱⣿⣿⣶⣮⣍⡻⣿⣿⡿⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠉⣄⠀⢰⡟⢈⣒⣛⣥⣞⠿⣷⣭⣽⣫⣭⣴⠟⣩⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢈⣾⣿⣿⣿⣿⣿⣷⣮⣍⡛⣿⢋⣥⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣏⠻⠿⣹⣷⣶⣭⡻⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢹⣿⠿⠿⣿⡟⣽⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣫⣥⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠴⠾⠿⠾⠿⠾⠿⠦⠶⠿⠿⠎⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄⠀⠀⣀⣤⣤⣤⣤⣄⡀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠉⠻⣿⣷⣾⡿⠟⠛⠛⠛⠿⣿⣿⡿⠛⠉⠉⠉⠙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⣠⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣦⣀
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⢿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⠈⠙⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠟⠋
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠟⠛⠋⠉⠉⠉⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⢀⠀⠀⣠⣾⣷⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣿⣷⣾⣿⠟⠁⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⣴⣦⠀⠀⣠⣿⣿⣿⣷⡆⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⣿⣤⣾⣿⠟⠉⠉⠁⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⣿⣿⣥⣤⡄⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⣠⣾⣿⠟⠛⠛⠛⠛⠃⣀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⣀⣀⣤⣴⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⢸⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡄
⠀⠀⠀⠸⣿⣧⣀⣀⣴⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣷⣀⠀⣀⣾⣿⠃
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠙⠻⠿⣿⠿⠟⠁
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠷⠶⢶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣀⣀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠉⢹⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣀⠀⣠⣴⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⠙⠻⣯⠀⢀⣿⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠁⣼⠇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢠⡿⢀⣿⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣀⣀⣼⣇⣼⣧⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣿⣦⣬⣭⣍⣩⣭⣥⣴⢿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣀⣿⠁
⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣾⠃⠉⢹⣿
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢀⣾⠋⠉⠙⣿⠿⠿⠷⣦⣤⣼⣯⣀
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢸⡏⠀⢀⣾⠋⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⣦⡀
⠀⠀⣾⠇⠀⠀⠀⣾⡿⢶⣄⠀⠀⢿⣤⣾⠏⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣿⠁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⢀⣿⠀⠀⠀⠀⢻⣇⠀⠻⣧⢀⣿⠉⣠⡶⠶⣶⡄⠀⠀⠀⢸⡇⠀⠀⢰⣟⣴⠟⠀⠀⠀⡀⠀⣰⡆⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢸⡏⠀⠀⠀⠀⠀⠻⢷⣤⣿⣸⣷⡾⠛⠷⠶⠛⠁⠀⠀⠀⠈⣿⠀⠀⣼⡿⠋⠀⠀⠀⢸⡇⢀⣿⠀⣾⠆⠀⠀⠀⠀⢸⡇
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⠿⠿⠶⠶⣦⣄⣿⠁⢸⡏⢠⣿⣀⣀⣀⡀⠀⣿⠁
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠙⢿⣤⣿⣷⠾⠋⠉⠉⠉⠛⠛⠿⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⢹⣇⣠⣤⡶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡴⠾⠛⠛⠛⠛⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠈⣿⡉⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣈⣛⣷⣾⣟⣋⣉⣁⣀⣀⣉⣉⣉⣉⣁⣀⣀⣉⣙⣻⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈
⣿⣿⡟⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠂⠉⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣀⠄⡀⢶⡄⠀⢀⣀⣀⢠⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠀⠈⠄⠀⡔⠉⠀⠈⡇⠁
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⢦⠀⠀⡸⠾⠀⠀⡠⣔⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⢀⣽⠈⡠⠈⠂⠀⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠶⠒⠚⢦⣀⠈⠢⣀⠀⠀⣨⠏⠀⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠱⠔⠉⢱⢲⢒⣲⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⡄⠀⠀⢀⠂⠀⠀⠠⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡄⡈⠆⠀⠐⡎⠁⠀⢀⠾⠅⠈⠦⠤⠄⢀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠣⣀⡀⠠⡀⠉⠠⡓⠀⠀⠘⠓⠀⠒⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⡭⠀⠀⠀⠀⠀⠀⠈⠆⠲⠉⠀⠀⠀⠀⠀⠀⢨⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣀⡞⠁⠀⠀⠈⡳⠒⠉⠉⠉⠓⢦⣀⣠⣀⠀⠘⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃⠀⠀⠀⠀⢀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠧⠤⢤⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠴⠤⣔⢢⠤⡤⠁⠀⠑⣀⠔⠂⠂⢄⠀⠀⠤⠒⣀⠔⠂⢀⣔⣉⣱⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠛⠋⠓⠤⣤⣤⠄⠠⠿⠒⠒⠳⠖⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⢹⠋⠙⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣦⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⠀⢀⣠⠖⢋⣉⣄⣠⠟⠃⣿⣙⠳⢤⡀⠀⢠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣰⠋⣠⠞⠉⠀⣉⣀⣀⣀⠀⠉⠓⢦⡙⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⣇⠘⢁⡞⠁⠀⣠⠞⠉⠀⠀⠈⠙⢦⡀⠀⠹⡄⢳⡀⠀⠀⡇
⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⢠⡇⠀⠀⢸⡄⡾⠀⠀⣀⣁⣤⣄⠀⠀⠀⠀⢈⡧⠀⠀⢹⡈⣇⠀⠀⡧
⠀⠀⠀⠸⡄⠀⠈⠙⠲⣄⠈⣇⢰⡆⣸⠁⣡⠖⠋⠁⠀⢰⠇⠈⠉⣿⠉⠉⠀⠀⠀⢸⡇⣿⠀⠀⡇
⠀⠀⠀⠀⠙⢦⡀⠰⢤⡘⣆⠸⣼⣷⠏⣰⢃⡴⠆⢀⡤⠃⡄⠀⠀⢸⡄⠀⠀⠀⠀⣸⠁⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⠲⠦⠽⠿⣄⢸⡇⣰⠿⠯⠴⠚⠋⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⣠⠏⣼⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡷⠃⠀⠲⣄⠳⣄⡀⠀⠈⠓⠒⢋⠀⠀⣠⠞⢁⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⢧⣄⡀⠀⠈⠳⢤⣉⣳⠆⣠⠖⠛⠛⣋⣥⠖⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠋⢀⠀⠀⣄⠙⠶⠤⣄⠀⠈⠉⠀⠉⠙⠲⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡞⠁⠀⣀⠀⠈⠓⠚⠁⠀⣀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡷⢤⣀
⠀⠀⠀⠀⣰⠚⠁⢀⣀⡼⠀⢠⠖⠲⡄⠀⢧⣀⡄⠈⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⢳⡄
⠀⠀⠀⠐⢧⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠒⠚⠋⠉⠁
⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⣆⠀⡆⠢⢰⠀⡖⠀⢶⠀⣆⡦⠠⠀⣔⡀⠀⢔⠀⠰⠂⢰⢰⢠⠂⢢⢰⡐⢠⣂
⣀⣀⣀⣀⣀⣀⣀⣥⣄⣧⣜⣸⣄⣇⣼⣉⣧⣧⣝⣤⣤⣥⣤⣀⣤⣇⣠⣄⣼⣸⣜⣤⣜⣸⣙⣬⣤⣄⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠒⠒⠲⠶⠶⠒⠒⠚⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠤⠤⠤⠤⠤⠤⢿⡤⠤⣤⣤⣀⣀⣀⣠⣤⡤⠤⢤⡧⠤⠤⠤⠤⠤⠤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢦⡀⠀⢀⣠⠴⢚⡉⣑⠶⢰⡖⠦⣀⠀⠀⢠⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠁⡴⢋⡴⠚⠉⠛⣁⡀⠈⠉⠳⣌⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⣆⠘⣱⠋⠀⠀⠀⢠⠿⡟⢦⠀⠀⠀⠳⡌⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣄⡀⠀⠀⡟⠀⠀⢸⠠⡇⢀⣠⠤⠤⡄⠀⡇⠘⣇⠀⠀⠀⢳⢸⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠳⣄⢹⡘⡇⡾⢀⡞⠉⠀⠀⡸⠁⠀⠁⠀⡿⠀⠀⠀⢸⢨⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣈⡳⣼⡆⢻⡟⢁⣯⢔⣋⡤⠞⠁⢸⡀⠀⣸⠃⠀⠀⢀⡏⣸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣬⣧⠞⠉⣉⠡⣄⠀⠀⠀⠷⠞⠁⠀⠀⣠⠞⡰⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⡀⠀⠈⠳⣌⡓⢦⡀⣀⠴⠶⠶⢋⡥⠚⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⢉⠀⠀⡉⠳⠤⣄⡀⠉⠛⠀⠙⠲⠶⠋⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡀⠈⠓⠚⠁⢀⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠠⠤⠇⠠⠞⠳⠄⠘⠧⠤⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠄⡀⡀⡤⠀⡄⠀⠄⠀⠀⠠⠀⠀⠠⠀⡤⢀⠄⠀⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠰⠀⠃⠅⠀⠘⠘⠈⠈⠀⠀⠈⠀⠀⠁⠀⠀⠸⠐⠀⠀⠄⠙⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⣄⣀⢊⠥⠀⠐⠤⠤⢄⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡶⠤⣈⠑⠢⣀⡀⠀⠀⠠⠴⠥⠤⠤⠤⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠉⠒⠤⣌⡑⠒⠧⢄⡀⠀⠀⠀⠠⡁⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡲⣇⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢄⣉⠒⠠⢄⡈⠲⢤⠀⣠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡜⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢢⡄⠉⢙⣯⠤⠀⢸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣰⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⡎⠉⠀⠀⡇⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡠⠃⠀⢀⢂⡀⠘⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠞⠀⡇⠀⠀⠀⡇⠀⡇⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⠀⠀⠂⠀⠤⢀⠆⠀⠃⠠⠀⠀⠀⠀⠀⡜⢠⠏⠀⠀⠇⠀⠀⠀⡇⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠥⣮⡀⠀⠀⠀⠀⠄⠂⢀⡴⠖⠛⠡⠄⠀⠀⠤⠀⡜⢠⢏⠟⠈⠀⠀⠀⠀⠀⡇⢸⠀⠀⢺⢤
⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠁⠀⠀⠈⠀⠉⠐⠀⠬⠀⢀⠄⠀⠀⢰⠁⡞⠁⡄⠁⠈⠀⠀⠀⠀⡇⢸⠀⠀⠊⠸⡀
⠀⠀⠀⠀⠀⢀⠜⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠋⠂⠠⠀⡐⠀⠀⠀⠀⢃⢸⠠⠀⠀⠀⢡
⠀⠀⠀⡠⠊⣡⠖⠁⠐⠒⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠃⠈⠐⠠⠀⠀⠀⠀⠀⠀⢸⠀⡆⠂⠀⠀⠀⠱⡀
⠀⠀⠀⠘⠢⢄⣉⠒⠤⠤⢀⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⠀⣌⡀⢎
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠦⠤⠌⣉⡐⠒⠢⢄⡀⠀⠀⠀⢸⢁⠎⠀⠀⢀⠄⠀⠀⣀⡠⠔⢀⣉⠈⠉⠀⢸⠤⠆⠼⠙⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢋⢒⣢⣌⠑⠢⢄⡆⢸⣃⣒⠀⡚⣓⡉⠭⠄⠒⠊⠁⠀⠈⠉⠁⠉⡀⠀⣸⢙⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⡠⠔⠛⠑⠢⢄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠴⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⢰⣦⣤⣤⣤⣤⣤⣤⣤⣴⡄⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⡈⣽⣟⣛⣛⣛⣛⣛⣻⣯⢁⣭⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣜⢿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡻⣿⣿⡿⢟⣫⣬⠝⣿⢛⡻⢿⣿⣿⢏⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡾⣫⢞⣭⣶⠴⠿⢿⣶⣭⡳⣝⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡜⣿⢣⣿⣟⣵⣿⣿⣿⣮⢻⣿⡜⣎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⠀⠀⢿⢻⡇⣧⢟⣫⣭⡯⣽⣯⣩⣭⣵⣿⣿⢹⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡳⠄⠘⠘⣼⠳⢟⣿⢟⣵⢻⣿⡇⣿⣿⣿⡟⡼⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢦⠸⣣⣤⢵⠶⣿⣿⣘⠿⣣⣿⣿⢟⡵⣣⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠿⣿⣮⣙⠮⡝⡿⢟⣍⣫⠵⣫⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡻⣿⢻⣮⣭⡻⣿⣾⣿⣶⣭⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡯⣻⡿⡾⢿⡟⣿⣿⣜⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣭⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠶⠿⠿⠿⠾⠿⠶⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢢⠠⡄⠀⠀⠀⢾⣿⣶⣴⡃⢀⣎⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠤⢤⡬⣄⡹⠦⢀⡚⣿⣿⠟⠝⣲⣿⣝⢢⠀⠀⠀⠀⣠⠦⢀⡀⢴⣦⣹⣇⠼⠇
⠀⠀⠀⠀⠀⣰⡾⢉⣽⣿⣷⣶⡾⣽⣴⡶⣿⣿⣧⣿⠿⠋⠀⠀⠀⣿⣶⣾⣿⡀⡛⠛⠿⣀
⠀⠀⠀⠀⠇⠙⠃⡐⠻⣿⠛⢻⣿⣏⡉⣥⢁⡐⣴⣅⠸⣧⣤⡤⠀⠙⣻⠿⠿⠷⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣕⣌⠙⠓⡠⣤⡀⠀⢠⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢯⣭⣥⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡿⣻⣿⣷⣦⢇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢠⡼⡯⣝⡿⠛⠉⣯⣿⣿⣿⣾⣽⡶⢳⣾⣿⢝⣿⣿⣿⣻⣻⠿⣷⡀⣀⣈⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡤⢶⢠⣀⣤⣮⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⡂⣿⣿⣟⣟⣈⡁⠙⢿⣿⡽⠡⡀
⠀⠀⠀⠒⣫⡷⣾⢸⣋⣴⠾⢿⣿⠿⠛⢉⠛⣿⣿⣿⣿⣿⣿⡷⢦⣛⣯⠜⣿⠛⣽⣷⣀⣉⠁
⠀⠀⠀⠀⠴⠞⢳⠞⢛⡷⠗⠈⡟⠀⢁⣤⠉⠟⢿⣿⣿⣿⣟⠁⡈⠉⠀⠀⠋⠈⡿⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡤⠀⠀⠀⠘⠀⠄⠄⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⠖⠻⠶⠶⠶⠤⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⢘⣇⣀⣸⣇⣸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⢻⣄⣀⠀⠀⠀⠀⠀⣀⡿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⡟⠉⢉⣩⣍⡉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠙⠛⠛⢛⣋⣉⣀⡀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⡄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠛⠛⢉⣉⡉⠙⠛⠿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢳⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣤⡶⠟⠋⠉⠉⠉⠉⠛⠶⣤⡈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠀⠀⠀⢣⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⢀⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢠⡟⠁⠀⠀⣠⡴⠿⠛⠛⠻⠷⣦⡀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⣇⣴⠞⠛⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⢸⡿⠀⢠⣿⠁⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣼⡿⠁⠀⣠⠶⠦⣄⠀⠀⠙⣿⡶⠶⠶⣾⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⣾⠁⠀⠀⠈⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⣇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠹⣦⣀⣀⡼⠁⠀⠀⣼⠇⠀⠀⠻⠷⠦⠼⢿⡄⠀⠀⠘⢷⣄⣀⠀⠀⢀⣠⣴⠟⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⣠⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠚⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡲⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⡜⢀⠀⠀⠶⠃⠁⢠⠀⠀⠻⠆⠀⢀⠘⡴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠆⢀⠛⠄⠷⣠⠐⠀⢸⢀⠐⢠⡰⠇⡜⢃⠀⠾⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⡷⠀⠛⣢⡘⡂⠀⣌⠃⠀⡀⡀⣰⢼⡌⠔⡀⠀⠀⢋⡄⠀⡚⣠⡚⠃⠐⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠛⣠⠄⢀⠡⣀⠀⡀⢙⣆⠀⠘⡃⠁⣸⠃⡀⡛⠀⢀⣞⠁⡀⢀⡠⢁⠀⢤⡘⠋
⠀⠀⠀⠀⠀⠀⣠⡄⢠⡤⠘⠛⠉⢇⠈⢲⡁⠹⢶⡇⠀⣿⠀⢠⣷⠾⠁⣱⠊⢀⠏⠙⠛⠠⣤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠠⠀⢶⡖⠾⢧⣘⠋⠛⠢⢽⣆⣿⣠⣾⠥⠚⠋⢛⡠⠿⠖⣶⠆⠠⠀⠀⠍
⠀⠀⠀⠀⢀⡆⠀⠚⠢⡠⢔⡒⠳⠦⣤⢬⣽⣿⣶⣶⣿⣿⣿⣷⣶⣾⣿⣭⢤⡤⠶⠓⣒⠤⠠⠚⠂⠀⣄
⠀⠀⠀⠀⠈⠱⠶⡄⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠶⠾⠿⠿⠿⠿⠿⠷⠶⠦⣤⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠈⣇⠀⠀⡅⠠⠄⠀⠆⠀⠀⠀⠀⢆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣆⡴⠃⠀⠀⠎⠀⠀⠀⠠⠀⠀⡏⠀⡠⢇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡸⠀⠁⠀⠀⠀⠈⠒⢇⠀⠀⠈⠢⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣈⡇⠀⠀⠀⠀⠀⠀⢄⢸⡇⢸⣰⠃⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⠀⠀⠀⢸⡇⠀⣼⠅⠀⠀⠀⢀⠀⠀⠀⢸⡇⣸⠏⣀⡆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⢻⣧⠞⠁⠀⠀⠀⠀⠀⢣⡀⠀⢸⣧⠟⠉⠀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⣿⠀⠀⠈⠠⣀⠀⠀⣼⡇⠀⣼⡟⠛⠛⠫⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠄⠀⠀⣸⠀⣠⣛⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠈⠙⠓⣶⠶⠶⣷⣄⣸⣿⠀⠈⢢⡴⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠑⠦⣀⡨⠶⠋⠁⢀⠠⠈⠉⠻⣿⣆⢠⡿⠀⠀⠀⣠⣿⣿⣶⢶⣄⣀⠠⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⣷⣿⠏⠀⠀⠀⠉⠛⠛⢶⣒⣒⢛⠓⠒⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⣼⣿⠏⠀⠀⢠⠀⠤⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡥⠖⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢀⣀⣸⣄⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣭⡭⠥⠤⠤⠤⠤⠬⠤⣤⣤⣤⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢰⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠤⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠞⠂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣀⣀⠀⠠⠄⠀⠀⠼⠇⣀⣸⡇⡄⢠⠀⢸⠀⢀⡄⠄⠄⢀⡟⠂⢀⠀⠈⣢⠀⠀⢀⣯⡀⠀⡸⡿
⠀⠀⠠⢘⡩⣐⣀⢁⡤⣄⢀⣉⣀⠂⠌⡑⠍⡈⠢⡉⠀⠺⠐⢠⠉⠀⡇⠀⠀⢠⡏⠁⠀⠀⠀⠉⠀⠀⠁⠈⠁⠐⢠⡀
⠀⠀⠈⢁⠴⠋⠁⠜⠠⠀⠄⠡⠈⢑⠤⣌⠐⠈⠢⠈⢟⠋⠀⠀⠐⠒⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⢹⣹⠁
⠀⠀⡞⠃⠐⣊⣤⣶⠶⠶⢶⣦⣄⠑⠂⡉⢢⡀⠂⠑⣞⠒⢒⣒⡒⠒⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿
⢠⢎⠈⢊⣾⠟⡁⢐⠆⠠⣒⣌⡙⣷⡄⠑⠀⠙⡏⢱⠷⠶⢾⠅⠀⠀⡷⡶⠰⢾⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣼⣿⡀
⠘⠢⡅⣼⡏⡈⡚⠡⠒⡖⢀⠹⠵⠘⣿⡀⢈⡐⣯⠫⠭⠭⠥⠝⣯⡍⠉⢠⡤⠬⢽⠁⠀⡠⠤⠀⠀⠤⢄⡘⠳⡄⠙⠃
⢟⠂⠀⢻⡇⠇⣇⠰⡊⠓⡢⠠⢑⠀⣿⠇⠠⡈⡋⠀⠀⠀⠀⠀⠹⡋⠋⠋⠀⠀⢸⡯⠉⠀⡠⠤⠤⣄⡀⠈⢻⡅⠀⡀⢔
⠀⣀⠀⠨⣷⡌⠲⡳⠄⠀⣴⠯⢂⣼⠟⢀⠐⢌⡇⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠸⠁⢠⠎⢄⠄⠀⡀⡹⡄⠀⠷⠈⠂
⠀⠳⡄⠐⠈⠻⢷⣤⣤⣤⣤⣶⠟⠋⠀⠎⢢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⡄⡐⠀⠀⢀⢃⡇⠀⡀
⠀⠀⠈⢦⣉⠑⠀⠠⢉⠉⡀⠠⠀⠊⣁⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠙⠦⢥⡤⠵⠋⠀⡐
⠀⠀⠀⠀⠀⠑⠣⠶⠅⣴⣤⠇⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⣀⣀⣀⠄⠊
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⡀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠸⠿⠀⢿⣿⠺⢿⡶⢀⡄⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣿⣾⢛⠹⠷⠞⠀⢰⣦⡀⣴⡇⢿⠇⠀⢸⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣤⡉⣿⠇⠀⣷⣦⢠⣝⠃⡙⠁⣠⣴⠆⣀⢼⡿⠇⢀⡄⢠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣜⠃⣶⡠⢰⣬⣄⠈⣷⠄⡙⢀⡟⢣⣾⣶⠙⠋⣾⡿⢀⣴⣇⢻⠟⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡉⠉⠀⠉⠁⠳⣍⠙⣇⣠⠊⣴⡄⠶⣶⢧⣸⣴⣦⠉⠀⠘⢿⡣⠔⠀⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⣙⣿⠗⢾⡛⠋⠻⢾⣿⡄⠛⠀⠀⣤⣤⣹⣇⣹⣴⠗⠫⣿⣶⠆⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡜⠛⣋⡀⢈⣭⣝⠲⣾⣿⣋⢻⣇⠙⠛⠢⣉⣹⡿⢛⣭⣽⣓⠒⢰⡾⠀⣶⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⢀⡘⢿⣆⡭⠴⣾⣿⣍⠽⠿⣿⣷⣤⣤⣾⣿⣷⣮⣭⡭⠁⠀⣶⡿⠃⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠏⠠⡶⠀⢀⡀⢺⣷⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠙⠍⠛⣻⣯⣅⠀⠘⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣌⠙⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻⣿⢿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠖⣛⠿⢛⢿⡥⡴⠋⣹⡈⢯⡛⢤⣘⠿⣛⡛⠛⠛⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠸⡀⠀⠃⠈⠂⠉⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢤⡀⠀⠀⠀⢀⠀⣠⣾⠀⠆⢀⡶⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠉⠂⣀⡀⢹⠦⣀⠸⠄⢩⠉⠃⢦⣔⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠁⠈⠙⢱⣦⣈⠑⢦⡔⠀⢠⣁⠘⠀⢀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢻⣦⣀⣀⡀⢀⠈⠻⠃⠸⠃⠀⠸⠻⡄⠘⠈⣠⡁⢠⠇⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠴⠚⠉⢙⠋⢁⠘⣶⡃⠀⠀⠀⠀⠀⠰⠁⠀⠀⢿⢰⡜⠀⠀⠀⡀⣼⣸⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠘⠇⠀⡄⡼⣷⠀⠀⠀⠀⡀⢀⡆⠀⠀⠀⠀⠀⢃⢀⣼⡇⢱⣭⠀⡆
⠀⠀⠀⠀⠀⠘⢦⡀⢿⣝⠢⣄⠦⢄⠐⠛⢧⣌⠀⠀⠀⠀⠀⠀⠀⡴⣷⠆⠀⢀⠛⢯⡋⣇⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⠷⡄⠁⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⣈⠻⠱⠆⠀⡾⢷⣄⡉⠋⠁⢠⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠓⣂⡀⢀⠀⠀⠀⠀⠁⠄⠼⠛⢿⡀⠀⠀⠀⡓⠦⣈⠻⠆⠠⠋⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⠳⢝⡷⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣧⣦⠉⠀⠀⠀⡀⣠⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣑⣮⠉⠀⣄⠀⠀⠀⠀⠁⠰⢏⠙⠺⢙⠿⣀⠀⠀⣹⡉⢻⡍⠸⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠉⠑⢢⡀⠀⠀⠀⠢⣄⠀⠀⠈⠈⠑⠾⣿⣃⢀⡁⠀⢀⡆⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠌⢢⠀⠄⠀⠈⠓⢤⡀⠀⢂⠄⡂⣹⡯⣷⡀⠉⣤⠤⢄⠀⠠⠐⠙⢒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠙⢦⡑⠄⣀⠀⠙⢶⣧⡈⠀⠈⠀⠈⡷⠈⠙⢆⡉⠉⠉⠉⠒⠭⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢵⠂⣠⣈⠲⢾⣶⣤⠁⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠹⡄⠙⠂⠀⠀⠙⢦⣃⠌⠀⢀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠈⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣎⣁⣙⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⣤⣤⣤⢤⣤⣤⣤⠤⣼⣠⢄⠀⢠⢔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢸⠠⠀⠀⠀⢃⢃⢀⢸⠀⣇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠀⠀⢸⠸⠤⠤⠤⠼⠸⡌⣁⣠⠗⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⣻⣿⠦⣾⢞⠭⡉⠉⢹⢠⠀⠀⢸⠀⠀⡇⠭⠍⠉⠓⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣵⠙⠃⠁⠈⠁⠚⢩⢮⣢⡸⢸⠀⠀⠈⠁⡨⣲⣭⣿⣽⣁⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠄⠀⠔⠉⠉⠉⢢⠀⠒⢧⡏⠉⠉⠉⢛⢮⡾⡏⠃⣁⣈⠘⠩⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠸⡀⠀⠀⠀⢠⠀⢈⢻⣿⣉⣉⣹⣾⣾⣛⠀⡎⠀⠀⠙⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⡀⠙⠢⠤⠔⠃⠠⣌⠏⠿⠤⠤⠽⢰⡻⡧⠀⠣⣀⣀⠴⢁⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣆⣠⡀⣆⣸⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠫⣖⣠⢠⣄⠦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢸⣷⠀⠀⠀⣤⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣬⣿⣤⠉⣾⢰⣆⠻⣚⣛⠁⠀⢴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⣀⣻⣿⡆⣶⠉⠓⣶⢣⡘⢿⢢⡭⢁⡀⣿⣌⢱⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⣿⡇⢉⠩⣤⣤⡈⠀⣧⢉⡋⢻⡞⣰⣾⡿⢀⢿⣿⢈⡁⣰⡄⣶⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣬⡀⠟⣶⠀⢭⣬⡶⣤⡸⢳⡎⣷⢛⡋⠀⢸⣷⠀⢛⡁⣿⣇⡉⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⣦⡀⠠⣭⣤⡤⡉⣾⠀⢹⡇⠨⣤⣪⡁⣿⠛⢡⣦⠷⢈⣀⡍⣁⣼⣋⣛⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠙⢹⡇⡀⠉⠁⢘⢦⡼⢻⣰⣷⠨⣥⡙⣿⢠⡞⢿⣴⣣⠴⢺⡅⠈⠛⠿⠓⠀⢸⣿⣀⣴⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⠃⠠⡼⠇⠸⠆⣿⡆⡙⢾⣜⠁⢶⣌⠁⣿⢈⣷⠿⣷⣤⡛⢂⠀⣜⢣⡄⢰⣦⡘⠟⣩⣵⠆
⠀⠀⠀⠀⠀⠸⣷⡄⣦⠀⣌⠛⠃⢹⡍⣛⣇⡪⣹⠇⣶⡙⣷⡜⠿⢶⣿⣿⠃⢠⡌⢉⠉⢸⣷⣉⠴⢛⢈⠿⡷⠘⣛⡋
⠀⠀⠀⠀⠀⠰⢾⡄⢉⠰⣿⡆⣷⣄⢷⡈⠉⢤⡬⠓⠾⣤⣘⣿⣆⣼⣿⡏⣀⠳⠕⣫⡄⣦⠙⡿⢚⣋⡘⢇⡜⢙⣋⠁
⠀⠀⠀⠀⠀⠰⠾⠿⠗⠑⢮⣄⣙⠿⠈⢳⣄⣀⣚⣛⡛⠀⠹⣿⣿⣿⣿⢣⣼⠷⡞⣩⡤⣉⡾⢳⡿⢟⣥⠟⣴⣿⢏⣾⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣃⣤⣭⡝⢛⣛⣛⢋⣹⠿⠿⠿⣶⣼⣿⣿⣿⠈⠿⣴⣧⣽⣿⣿⠷⠶⠶⠿⠧⠤⠽⢥⢠⣅⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣠⣤⣉⠝⣭⠸⡆⠀⠀⠘⣿⣿⣿⣿⠀⣴⠟⠛⠉⢩⡍⣙⠲⣟⣧⡄⢾⣷⡌⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⡆⢿⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⢸⣿⢀⡓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠈⠐⠠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢇⡀⠀⠀⠀⠀⠘⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢑⢢⠀⠀⠀⠀⠀⡀⠀⠛⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⣷⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠉⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠸⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⢠⣄⠈⠀⡈⠀⠀⡀⠀⠀⠈⠓⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠈⠳⠀⠄⡀⢀⣍⢠⡾⢃⡘⠻⠃⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠘⠀⠀⠸⠀⠀⠈⠛⠀⠙⠁⢠⠀⢠⡀⢀⠇⡚⠂⢻⠀⠘⢂⠈⠀⢀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠈⠀⠀⡄⠳⠀⠆⢠⣄⡄⠈⢆⢈⠀⣤⠀⠈⡀⠋⣠⠀⠶⠉⠉⠠⠀⢯⠀⡠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⡀⠀⡀⠁⠢⠀⠈⠀⠰⠮⣀⠀⠈⠀⠀⠸⣤⠊⠀⠀⠋⣠⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⢠⡙⠀⣠⠈⠀⢧⠀⠀⠘⡀⠀⢳⡀⢸⡇⠀⢠⡴⠈⠀⠀⢸⠆⢀⣿⡤⠀⠠⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣶⣀⠀⠛⠠⣀⠀⡃⠀⠀⡃⠀⠀⠙⣿⠀⠀⡟⠀⣠⠤⠔⠃⠈⠁⠀⠀⠘⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⣠⠀⠀⠀⠀⠠⢄⠉⠙⢦⢀⠇⠀⢦⣸⣏⢀⡴⠃⣾⣠⡤⠄⠀⠐⠀⠀⡀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠉⠀⠰⠖⢀⠀⠀⠈⠉⠉⠙⢻⣄⡄⠀⣿⡿⠋⠀⠀⡞⠁⠀⡀⠤⠤⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠚⠋⠀⠀⣀⡀⠐⠢⠀⢿⣷⣀⣿⠀⠀⢀⡼⠁⠀⠠⠀⣀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⡄⣨⠆⠀⠀⠀⠙⣯⣿⣧⡞⠋⢀⠒⠂⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠢⢤⣤⣤⣄⣀⣿⣿⡏⠤⣠⠼⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣴⣶⣯⣾⣿⣿⣯⣽⣶⣶⣿⣿⣷⣿⡶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣏⣽⣶⣿⣷⣍⡻⣿⣿⣿⣿⣿⣴⣷⣌⣭⣿⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣔⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣮⣽⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣵⣾⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣷⣌⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠋⠉⠉⠟⠛⠛⠛⠉⠋⠉⠉⠉⠁⠀⠈⠉⠉⠙⠛⠉⠈⠉⠉⠀⠈⠛⠛⠛⠿⠟⠿⣿⢿⠿⠛⠷⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡿⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢆⠀⠀⠀⠀⠀⣠⠋⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⠀⠀⢀⡴⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⢀⡷⠊⠉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⡀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠙⢆⠀⢸⡇⠀⠀⠀⠀⠀⡴⠁⠀⢠⡆⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠑⣾⡇⠀⠀⠀⠀⡼⠀⠀⢀⣾⠀⠀⠇⠀⠀⠀⢀⣀⣀⣀⢀⣀⠤⠚⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⢇⠀⠀⠀⠠⡇⠀⡰⠋⡏⠀⢸⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠸⡆⠀⠀⠀⠳⡼⠁⢸⠁⠀⢸⠟⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠉⠒⢦⡀⠀⠈⣧⠀⠀⢀⠀⢷⠀⠀⠀⠀⡇⠀⠸⡄⢠⠃⠀⠠⡀⡀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⢆⠀⢸⡆⠀⢸⡄⢸⠀⠀⠀⠀⡇⠀⠀⢳⣜⠠⡀⢠⡀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⡆⢠⠇⠀⢨⡟⣎⠀⠀⠀⠀⢣⠀⠀⠀⠙⣄⢱⠐⡇⢾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡏⠀⠀⢈⡇⠀⠙⠢⡀⠀⣸⠀⠀⠀⠀⠘⡄⡆⣷⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠈⢡⡄⠀⣸⠁⠀⣀⠀⢸⡆⠁⠀⡜⡅⠀⠀⡇⡇⣾⣸⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣇⠀⣿⠀⠀⠘⡇⡰⢻⠀⣰⠃⠀⢸⠇⠀⠀⡇⢸⠀⢀⠇⡏⢁⡤⠒⣉⠟⣂⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠹⡄⣯⡜⠀⢠⡷⠁⢸⠀⢟⠀⠀⡟⠀⣀⠀⢃⢸⠃⢸⠀⢡⠋⠀⡞⠁⠈⠀⠀⢀⡠⠟⠓⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠹⡄⠙⢷⣱⡀⢸⣇⢰⠈⣇⢸⢰⡀⢿⠀⠈⢷⡘⣼⢾⣿⡀⡏⠀⠀⡆⢀⣸⡀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⢦⣀⠙⠳⣄⢿⡘⠂⢸⡎⢸⣹⠘⠂⠀⢸⢳⠀⣾⣿⣷⡇⠀⠀⡇⠘⣿⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢧⡈⠉⡳⢌⠙⢿⠀⠀⠁⢸⠀⢀⡠⠴⠋⡸⣤⠛⣣⣿⠀⠀⠀⢸⢰⠿⠇⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠈⠻⣤⡙⣄⢳⠀⠓⢀⣾⣿⢀⡞⠀⠀⠀⡇⡿⡄⠈⠁⠀⣴⡄⢸⢀⣶⣾⣿⣤⣿⣒⢩⠟⣦⣶⡀
⠀⠀⣀⣤⣴⡰⢶⣶⣶⡟⣈⢩⣍⢩⣭⣈⣓⣄⣈⠙⠊⢷⠃⠀⠸⡇⣿⡀⡇⠀⠀⠀⣟⡱⢃⣾⣾⣾⠟⣠⣾⣿⣿⣿⣟⣻⡧⠤⠼⢾⣽⣿⡝
⠀⢼⠁⢸⣿⣻⣦⠿⠿⠓⠐⠋⠉⠁⠀⣩⠝⠋⠉⣒⡒⡾⠀⠀⠀⠳⠘⠏⠓⠂⠀⢘⣭⢶⣾⣷⣤⣶⠿⣯⣤⡤⢤⣠⣍⠁⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠉⠛⠤⠴⠶⠶⢦⣤⣶⣶⡶⠖⠋⢀⣤⣴⣾⡿⠻⠅⢀⣴⠋⠙⠦⣐⠉⠉⠛⠦⠭⠽⣭⣭⢿⣟⣛⣛⡺⠔⣫⠀⣼⣶⡶⣶⣾⡋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣭⣄⣠⣴⣿⣾⣋⣭⣴⣯⣿⠶⣦⣟⡲⢄⡈⠙⠒⢤⣀⣀⠀⠀⠀⠀⠈⠁⠈⠙⠋⠙⣄⡟⠓⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⣯⠶⣿⣿⣿⡾⠟⣯⣴⣾⣿⣿⣊⢿⣿⣿⣛⡿⣦⡀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠍⠉⠉⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠻⣿⠰⠿⠋⠉⠀⠀⠀⠀⠈⠁⠀⠙⠄⠙⢿⣿⣿⣿⣿⣦⣄⣀⠀⠉⢻⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡿⢻⣮⣳⣄⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣯⣄⣠⡼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠃⠀⠀⠀⣀⣀⣀⣠⠟⠉⠉⢀⠙⡄⠀⣀⣠⡤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⢀⣴⢊⣵⡒⣶⣾⡀⠀⠀⢸⢸⠷⠋⠁⠀⠀⠀⣠⡤⣾⡷⣄
⠀⠀⠀⠀⠀⣠⡴⣒⣚⣛⣉⡉⢉⣉⣉⠉⠭⠭⠭⠭⠹⣶⢥⡷⢷⢰⣸⢃⣀⣀⠀⠀⢀⡼⠑⠃⠈⢻⡈
⠀⠀⠀⣀⣼⠉⠀⢴⠆⠀⠀⢸⠀⠀⣶⠀⠀⠀⠀⣋⡱⢸⠿⠭⠭⢽⡟⣽⡙⠘⢉⠝⠓⡢⠀⣀⡀⠀⢻
⠀⢠⣮⠔⡏⣾⠭⢱⣸⠩⢱⠘⣴⠟⢯⠉⢲⡤⠒⢲⢾⡿⠤⣤⠤⠼⣾⣇⠙⡄⠈⠉⠁⠀⢰⡟⢧⠀⠘
⠀⠈⢯⣤⡇⡏⠰⢸⣽⣒⢸⠀⠏⢆⡸⢀⣮⣬⡟⢋⣿⣆⠀⣉⣈⢹⣽⡇⠉⢁⠔⠂⢱⠀⡞⠀⣸
⠀⠀⣀⣀⣧⣿⢩⢴⢹⠶⠬⠀⠀⣿⠿⠿⠋⠛⠧⡬⡿⠰⠹⢷⣼⠸⢹⣌⣢⠈⠉⠉⠀⠀⡇⠀⢹
⢠⠞⠁⡜⢻⢸⣒⢸⢸⣏⠀⠀⡇⢸⣠⠖⠒⡲⠾⣟⣻⠍⣩⠭⣭⣤⢼⡄⠀⢀⠔⠒⢲⠀⢷⠀⡞⠀⢰
⣾⠀⢸⢠⣻⣎⣑⣊⠈⠛⠒⣁⡇⢸⠁⠀⡐⢀⣄⢸⣐⣟⠝⡄⠀⠈⡿⣄⣱⠈⠈⠉⣀⠀⠘⢟⠀⠀⣾
⡷⠀⢸⢸⠣⢷⣄⣀⣦⠀⣀⣀⣀⡏⠀⠀⡇⡆⣸⢸⣿⠿⠚⠛⠋⠉⠁⣟⠦⡀⠯⠉⠤⠃⠀⠀⠀⢰⠋
⢻⣶⡼⡄⠒⣹⠁⠈⠉⠉⠉⠁⠀⣇⣀⠀⡇⢃⠸⢸⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⣀⣰⣖⣩⣝⡠⠶⠋
⠀⠙⠺⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠻⣇⣦⡧⠈⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⡄⠀⠀⠀⠀⠀⣴⠀⠘⠃⠀⠜⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠸⡆⠀⠀⠀⠀⣟⢧⡀⠙⠤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⣌⠇⠀⠀⠀⠀⢳⡨⢘⠦⣄⠀⠑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⠀⠀⠀⠀⠀⠀⠈⢖⡡⢂⠠⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡡⠹⡄⠀⣠⠀⠀⠀⠀⠀⠳⣅⠪⢀⠤⠛⢆⠀⠀⠀⠀⠀⠀⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠑⠪⢒⠩⡇⢰⢳⠀⠀⠀⠀⠀⠀⠈⢲⠁⡊⢔⠨⢣⠀⠀⠀⢀⠔⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡐⠅⣊⠑⣺⠃⠘⢌⠳⣄⢲⣤⣀⠀⠀⠀⢹⡔⠅⣢⡜⡇⠀⠀⡎⡰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠐⡩⠐⠥⣻⠀⠀⠀⠑⠺⠄⢳⠍⢣⡄⠀⠀⠇⢊⡀⢀⢽⠀⠀⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢕⠀⠷⡀⠐⢧⡀⠀⠀⠀⠀⣜⠠⠲⣷⠀⣜⠁⠸⡁⢀⡸⠀⠀⡴⣷
⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⠀⠀⠀⢹⡡⠀⢇⠈⠢⡀⠙⢆⠀⠀⡼⠑⠀⠀⣷⠞⠁⡠⠁⡆⢀⡇⠀⠀⡏⢸⡆
⠀⠀⠀⠀⠀⠀⠹⡀⠹⠲⣄⠀⠀⠀⢫⠄⠘⣄⠀⠙⢢⡈⢳⣞⠉⡰⢣⠀⠋⢀⠜⠁⠰⠁⡼⠀⠀⣸⡡⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡘⢆⠀⠀⠀⠳⣄⠈⢆⠀⠀⠱⠀⠁⡜⠀⠈⠄⡴⠃⠀⢠⠃⣼⠁⢀⡴⠏⠀⠡⢸⠀⠀⠀⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣌⡙⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⢣⡘⠀⠀⢰⠋⠀⠀⢠⠃⣴⣧⠶⠛⠀⣀⠀⠨⣼⠀⠀⠀⡰⠋⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡉⠒⢤⡀⠀⠀⢸⡄⠈⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠇⡘⠋⢀⡠⠔⢊⠃⠀⣠⠇⠀⠀⢎⡠⠖⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠕⠈⠹⡄⠀⢸⡃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⡠⠊⠁⠀⡰⠁⢀⡰⠃⠀⠀⠘⢁⡤⠞⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠗⡄⢹⣄⣾⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡜⠀⠠⡞⠁⣠⠤⢀⡴⣃⢢⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⡆⠘⡄⠿⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡟⢠⠾⣟⢀⣞⠁⠒⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢠⠁⠀⠶⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⢀⣷⠝⠂⢟⣼⢃⡾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢸⠀⠀⠘⣦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢇⠈⡇⡼⡇⢻⠟⡎⢠⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠣⣠⠀⠱⡈⢰⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠒⠀⡘⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠌⢲⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⢚⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠉⠐⠂⠤⠤⠀⠀⢀⣀⣀⣀⣀⠀⠠⠌⠑⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠒⠈⠄⠂⠡⠐⡈⠔⢊⠡⠐⡀⠄⣀⠤⢐⠀⢀⡠⠄⡂⠄⣂⠀⢂⠠⢀⡀⢤⡒⢄⠂⠐⡂⠤⡰⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠈⠀⠈⠁⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠐⢀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣄⠀⠉⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢫⠀⡤⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⡄⠶⠇⢀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠋⠚⡃⢀⡀⠈⠃⠀⠀⡀⠀⠐⠃⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠙⢎⠁⣿⠁⡀⠀⠁⠉⠀⠀⡄⠉⣭⡀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠷⠰⠋⠑⢧⡀⠋⠀⠀⠀⠀⢠⣄⢰⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠂⠀⡀⠀⢀⠄⠀⠘⠢⠤⠀⠁⢦⠀⠀⠀⠐⣯⡉⣿⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠦⠈⡃⠀⠸⠃⢠⡓⠀⠀⠆⠀⢀⡃⠀⣻⡄⠈⣧⠉⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢠⡤⠁⠀⠈⣁⠀⡟⠆⠀⠛⠀⠀⢻⠃⠈⠛⢿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣙⠀⠀⠰⠎⠁⠘⠋⠁⠴⢀⡙⠀⠶⠀⢻⠤⣀⡀⢠⡤⠴⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠀⠠⠠⠆⠐⢀⠀⠠⠙⠰⠆⣀⣼⠀⠉⠷⠀⠀⠀⠀⡠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣀⠰⣌⣀⣧⡄⠀⠀⠀⠁⠈⠱⠋⠀⠀⠀⠈⢻⣆⡀⠶⠛⠚
⠀⠀⠀⠀⠀⠀⠈⠁⠋⠙⠿⢁⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡏⠠⠆⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠒⣰⠃⣠⣄⠀⠀⠀⠀⠐⡃⠀⠀⢸⣬⡅⠈⠓⢀⠀⢰⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠶⣧⣄⣻⠀⠀⢠⡶⠀⠈⠁⠀⠀⣸⠉⠀⢮⠁⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡏⠶⠚⡻⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠁⠸⠐⢶⣋
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠶⢄⠶⣀⠤⢨⠗⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⠀⠀⣿⠛⢁⡀
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠃⠒⠀⠀⠟⠀⠀⠀⠹⠥⡶⠄⢀⣖⣠⠟⠛⠁
⠀⠀⠰⣿⡃⠀⠀⠀⠀⠀⠀⢒⡀⠠⡐⠆⠀⡀⣾⠃⠀⠠⠀⠲⠀⠀⠠⠈⢙⣁⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠘⢣⡄⠀⠀⠀⠀⠀⠀⠀⢂⣤⠘⠢⣤⠀⠀⠀⠄⠀⢀⠀⠀⠲⡶⠆⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⢀⣄⠙⠀⠀⠀⠀⠀⠀⠐⣿⣿⢀⣤⠘⣀⠰⠂⡄⠀⣼⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠐⢶⠀⢠⣦⠀⠠⣌⠛⣤⡿⣀⡽⢃⡀⠀⢰⠇⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣅⠀⠀⠙⠻⣿⣾⡏⠀⠁⢄⠀⡜⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⣴⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⣠⡀⠘⡿⠀⠈⠀⠀⠰⡇⢀⠃⠀⠀⠀⣀⣦⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢺⣷⠀⠀⠀⠀⢰⣦⠀⠠⠐⠚⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣴⠀⠀⠀⢈⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠘⠻⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠈⢰⡞⠀⠀⠀⠐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢄⢸⡇⢀⣾⠇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡆⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣶⣶⣶⣖⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢤⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣆⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⠼⣿⣾⣿⣟⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⠀⠀⠀⢰⣾⠿⣿⢿⣿⣿⣞⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣛⠿⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣽⣿⢯⣯⣯⣿⣿⣩⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⢹⣾⣿⣿⢿⣿⣿⣿⡿⠿⣧⣿⡏⢻⣽⣿⣹⣿⡿⠋⢉⡽⣿⣟⣿⣿⣿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣿⣿⠝⠒⠚⠛⢷⣾⣻⡿⣹⣿⣽⠟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠏⠫⣷⣿⣿⡟⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠿⠿⠧⠿⠿⠿⠿⠿⠶⠶⠦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⢀⣿⣿⡀⣸⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠇⣸⠛⠀⡟⠸⣧⠀⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡎⠁⠀⡔⠁⠀⠀⠀⠀⡿⠘⠃⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡧⢤⡆⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢻⣯⢹⣴⠀⠀⠀⠀⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠋⢣⠛⣿⡆⢿⠀⠀⡀⠀⠀⠸⣿⣇⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣶⡌⢈⡝⢋⡄⢀⠀⢀⡌⢳⠻⣆⠀⡇⠀⠀⢀⣍⠣⠙⢶⣿⡀⣠⣏⠷⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⣁⣀⡀⠀⠀⢿⡿⠊⣰⣿⣿⠂⠀⣾⠃⡾⠀⢸⡷⣧⡀⣠⣾⡯⣀⠀⠀⡙⢷⡿⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣸⠟⠻⠶⣽⣤⣶⡟⣡⡾⣿⢣⠇⠀⣼⣿⠀⡇⠀⢾⣧⣧⢹⠙⢿⣿⠸⣦⡀⠈⢮⣥⠤⡀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠉⢟⣻⡿⠼⢧⣿⣀⣸⣿⣾⣼⣥⠀⢸⣿⣿⣎⣆⣀⣾⣧⡽⠟⠋⠉⠙⠲⣤⠀⠀⠌
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠎⢸⡟⠀⠁⠁⠀⠀⠀⠀⠀⣾⢸⡗⠁⠀⠉⠉⠁⠀⠀⠹⡇⠀⠖⠊⠁⠀⢿⣦⡔
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢰⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣭⢺⡇⠼⠉⠁⠀⠀⠀⠀⠀⣻⣀⣀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⣗⡤⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢽⠇⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⠞⠉⡦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠐⠒⠂⠒⠒⠚⠒⠈⠓⠦⠶⠶⠖⢒⣒⠇⡀⢠⠞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⣤⡷⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢠⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠰⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡤⠆⠒⢄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠠⠤⠀⠀⠤⠿⣀⠀⣿⠀⠀⠀⠀⠠⠀⢀⠄⠤⠀⡀⣿⠢⢀⠀⠀⢡⡀⠀⠀⣸⡇⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠄⣊⠁⠐⠀⢁⣀⡀⠈⣀⠂⠤⢙⠪⢁⠘⢌⠉⠒⠺⠀⠤⠀⠀⣿⠀⠀⠀⡞⠉⠀⠀⠀⠉⠉⠀⠉⠀⠃⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠊⡩⡟⡏⢁⡁⠌⢠⠈⡩⠛⠢⣀⡑⢄⠑⢄⠑⡖⠚⠓⠚⠒⠒⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣻
⠀⠀⠀⠀⠀⠀⠀⣴⢋⠀⠄⣁⣤⣤⣶⣦⣤⣀⡘⠀⠀⠛⠄⠑⠄⢢⡼⠤⠤⠤⠤⠤⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣔⠉⠄⣠⣾⠛⡩⠄⣐⡠⠌⡙⢿⣦⠈⠄⠈⣷⡞⢶⣯⣤⣼⣃⠁⠀⢁⣶⣴⣦⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢀⣿⣿
⠀⠀⠀⠀⠀⠸⢄⢢⢰⡿⢁⣚⠗⡩⢄⢌⠁⡭⠄⢻⡇⠀⠆⠌⡇⡞⠀⠀⠈⢲⣶⣀⣃⣙⣙⣉⣓⡄⠀⠀⣀⣀⠀⣀⣀⠀⠰⢄⠙⠿⠆⠐⢢
⠀⠀⠀⠀⠀⣴⠄⡁⢸⡇⢸⠰⠠⡤⠾⣄⡁⠰⡱⢸⣿⠀⡈⠐⡇⠀⠀⠀⠀⠐⠻⠷⠶⠶⠍⠀⠉⣗⡶⠊⠀⣀⣀⣀⠀⠐⠲⣾⠀⠀⠀⡠⡼
⠀⠀⠀⠀⠀⠙⠂⠼⠸⣷⡀⠅⡦⡀⠀⢊⡠⣅⠃⣼⡇⠀⢍⢢⡅⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠰⠎⠀⣠⢞⠍⠀⠩⡑⣆⠀⠹⡀⢀⠅⠊
⠀⠀⠀⠀⠀⠀⢷⡒⢂⡙⢿⣦⣁⠛⠋⠃⣊⣠⣾⠏⠀⠔⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⡀⠀⢈⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠑⣄⠤⠠⡉⠛⠟⠛⠻⠛⠉⢀⢤⠀⣢⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠱⣄⠀⠀⠁⣢⠞⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠡⣠⢀⡱⠂⡐⢀⠃⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠙⠛⠋⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⢸⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠙⠁⠀⢇⠀⠀⠀⠘⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢆⠀⠀⠀⠘⣧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢦⠂⠀⠀⣆⣹⡄⠀⠀⠀⣰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡇⠀⠀⢿⣿⠇⠀⠀⢰⣿⡇⠈⠀⡄⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢸⣿⡇⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡠⠒⢄⡤⣂⢀⡀⠀⠈⠉⠀⣠⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⠆⠀⠀⣀⣀⠔⠁⠀⣀⡀⡁⢘⡄⠀⠀⠀⠐⠋⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢼⣿⡏⠀⣾⣿⠟⡿⣽⠟⠁⠴⢦⡀⠀⠀⢃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⡧⠛⠂⠈⢉⣷⣤⣿⣿⣿⣿⣿⣿⡅⠀⠀⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡔⠀⠀⡀⣀⢠⣼⣿⣿⣿⣿⣿⡇⠉⢿⣿⠀⠈⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡅⣀⣆⣣⣽⠾⢻⣿⣿⣿⣿⣿⣧⣬⣿⣟⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⢸⣿⣿⣿⣟⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢸⣿⣿⣿⣿⠁⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣃⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡻⡟⠢⠚⢿⣿⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣗⣷⠀⣈⣹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢡⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⢾⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⣲⢤⡀⠹⣿⣧⣿⣿⢃⣤⡖⣲⣶⣔⣦⣄⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣾⣷⣤⣿⣆⣿⡦⡈⠻⠛⢡⣾⣻⣷⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⣿⣧⣤⣴⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠓⢦⣤⣝⠙⡻⣿⣿⣷⠀⠂⢸⣾⡟⣡⠇⠀⠀⠀⠀⠀⠀⠐⡄⢀⡆⢾⣧⣴⠛⠋⠀⠙⠻⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠤⠽⢿⠀⡔⠐⡯⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡴⠏⠓⠀⠀⠀⠀⠀⢀⣠⠆⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⢀⣀⠀⠀⠀⣀⢈⣡⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣄⣀⣀⡀⠀⢄⠀⠀⢀⠂⠀⠀⢀⣸⢉⣉⣠⣼⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣲⣦⣽⣦⡐⣤⣄⣠⣤⣄⠀⠀⠀⠈⣷⡿⣿⣿⣿⣷⡶⠦⢱⣦⣼⣿⣍⣶⡾⢿⣿⣿⣿⣗⣾⠗⠲⣤⡀
⠀⠀⠀⠀⠀⢀⢐⣿⠟⠩⠤⠤⠈⣉⡛⢿⣼⣿⣿⣿⣿⣿⣆⡀⠀⣼⣿⠟⠻⣿⣿⣧⡀⡟⠓⡛⢉⣠⣤⣴⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣦⡤⠄
⠀⠀⠀⠠⠖⠋⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡉⠋⢀⣰⣾⣼⠏⠀⠀⠉⢹⣿⣾⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠉⠉⠁⠀⠀⠈⢹⣽⠏⠁⠀⠀⠀⠀⠀⢻⡿⠟⠋⠋⠩⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠷⣄⠀⠀⠢⢤⣬⣵⣏⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣄⣀⠉⠉⢉⢉⣁⣀⣹⣧⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠿⠛⠛⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠙⠿⡿⢻⡿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣿⡇⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣱⣶⣿⣿⣿⣿⡇⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡠⠀⣀⢀⣀⠀⣀⣠⠔⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠛⠿⣿⣿⣿⣿⣇⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣿⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡤⠤⠴⠶⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⢀⠀⠠⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠏⢢⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠤⣉⠒⠒⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠑⣮⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡒⠤⣍⣉⠀⠀⠼⡆
⠀⠀⠀⢀⣀⣀⣀⠀⠀⡀⠀⢀⠀⢸⠀⢠⠀⠀⢂⠀⠀⣡⡇⠀⡀⠀⢈⠓⠂⠤⠀⠀⢚⡇
⠀⢀⡾⠿⠿⣯⣿⣷⣀⣷⣤⣾⣀⢸⠀⢸⠀⠀⠉⠾⠀⠋⠀⠙⢄⠀⠀⠉⠁⠀⠀⠀⠉⡇
⠀⡟⠀⢁⡀⠘⣮⣙⣿⣿⣿⣿⠉⣸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⣤⠤⠤⠃
⢸⠃⠀⠎⠀⠀⢹⡙⢿⣿⣿⣿⣿⡇⣀⣀⣀⣠⠤⠴⣰⡖⠶⢿⣟⣛⣛⣿⣿⣿⣯⣤⣤⣶⣾⣷⣤⠀⣀
⢿⠀⢸⣠⠴⣆⢸⠿⣶⣿⣿⣿⣿⣿⣴⢊⠩⣙⢿⣞⢿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⣼⣿⣿⣷⣄⠑⡄
⢾⠀⠸⠘⠉⠛⢸⣷⣌⣻⣿⣿⣿⣿⡇⠃⢔⡝⢷⣿⣏⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣸⡟⢸⠉⠹⣿⡀⠐
⠸⡇⡀⢇⠢⡈⣸⠻⣿⡿⠿⠿⠿⠿⣇⠀⠠⣷⠈⣿⣿⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⠁⣾⢲⢻⣿⡇
⠀⢻⡀⢀⠀⢠⣿⣷⣬⠇⠀⠀⠀⠀⢻⡀⢴⡟⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⡄⠉⠈⣺⣿
⠀⠀⠻⣶⣶⣿⣌⡻⠛⠀⠀⠀⠀⠀⠈⠳⡀⠁⢠⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⠏⠀⠈
⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠩⠞⠋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⡅⢶⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣇⠀⠀⠟⠁⣸⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣿⣿⣿⠀⣿⠀⣆⣀⡤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢿⣿⣧⢰⠦⠟⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⠆⢸⣿⣧⠈⣠⠚⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⡟⣋⣨⡽⠿⠛⣿⣿⠗⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣏⣻⠟⣿⠀⠶⠂⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠈⢿⣷⡶⡄⣄⠒⣘⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⣏⣤⣤⢰⣂⣭⣿⣿⣿⣷⠶
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠏⠀⠈⠙⠋⠛⠛⠋⠉⠁
⢀⠙⠉⣠⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⣿⣰⣿⠄⠀⠀⢀
⠀⢱⡶⢼⡧⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⢙⣟⣁⡤⠂⣠⡎
⠀⠈⠃⡤⠛⠀⠈⠁⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⡘⢿⣿⣿⡿⠛⠁⡤⣨⠏
⠀⠀⠀⠀⠀⣤⡄⢳⢀⡟⠀⠀⠀⠀⠀⠀⢠⡌⢻⣄⠙⣿⣿⣧⡾⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡈⠃⠎⠀⢠⠀⠀⠀⠀⠀⣿⣿⣄⢿⠀⡞⠉⠀⣀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⡞⠀⠀⠀⠀⠘⣿⣿⣿⣧⣷⢁⣶⠈⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⢀⡰⣤⡀⣿⡿⠙⢿⡿⠈⣳⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⣠⠛⢱⡍⢿⣾⠃⡀⠀⢀⡾⣥⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡆⣿⣀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣯⣄⠀⣤⡈⣿⡀⢲⠀⢸⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢿⡍⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣉⢈⡉⠀⠘⠿⠻⣷⢸⣦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢿⣿⣦⡀⠀⢿⣿⠋⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣧⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡇⢸⠟⣿⣿⠇⠀⡏⣀⢃⡼⠃⠀⠀⠀⠠⣀⠀⠀⠀⠐⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠊⠀⠈⠻⠀⠆⠘⣿⠚⠁⠀⠀⠀⠀⠀⠙⢷⡌⠀⠰⡀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⢠⡇⠀⢺⣇⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⣤⣹⣦⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⣿⡄⠄⠀⠀⠀⠀⠀⠀⠈⢻⡉⣿⣿⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢰⠀⢠⡆⢸⣇⠀⠀⠀⠀⠀⢀⡴⠀⢾⣇⣡⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢲⣤⣄⡀⠀⠙⠷⣦⠸⡇⠘⡟⠀⠀⢀⡦⢀⠈⠁⠀⣬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣌⡳⢦⡄⠀⠘⠇⠀⠀⠀⢐⣈⢰⣷⠐⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠻⢿⣿⣷⡆⢰⡇⠀⠀⣴⠿⣿⠀⢻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⢿⣇⣘⡁⢉⡼⢁⣤⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢀⠙⢿⠁⢩⡄⢸⣏⢉⢡⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠈⡀⣼⡇⢸⣿⡤⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⢘⣷⠘⡇⢠⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⣿⢀⡟⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡆⠘⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢎⣿⡄⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⢰⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⠛⠃⠘⡿⠀⣾⣿⠃⠀⢀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠁⠀⣶⡄⠈⢢⡉⠀⠀⠀⣟⡰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⡀⠀⠸⡇⠀⠀⠀⠁⢢⠀⣧⡴⠚⠋⠀⠀⠰⣷⣞⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⢤⡃⠀⣴⡄⠀⠀⣷⠏⢠⡶⠀⠀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢧⡀⢿⠇⠀⢸⡿⠊⠀⠀⢀⠰⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠗⠐⠊⣩⡍⠉⠛⣶⣤⡀⢸⡇⠀⢀⣠⠾⠒⢒⣀⠀⠴⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⣀⣉⣀⣀⠞⠁⠈⢻⣾⣿⣴⣯⠥⣤⠄⠈⠻⠷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠁⠀⠀⢠⣄⠀⠀⢻⣿⣿⢀⣀⠀⠀⠀⣤⠲⠦⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⢸⣿⡇⠈⠉⣷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣞⡽⠃⣀⣀⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⠀⠀⢠⠀⢀⣭⠶⠛⢉⡵⠟⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⠀⢠⣾⢠⡞⠁⠀⢠⡟⣠⠶⢲⡿⠃⢀⡤⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⢁⡴⢋⡏⣾⠀⠀⠀⢸⡞⠁⢠⡟⠀⠰⠾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢠⣏⠄⢸⣇⣿⡕⠀⠀⠈⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣇⢸⣿⣶⠀⠻⠏⠀⠀⠀⠀⣠⠞⢡⣶⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⢦⣏⠀⠀⠀⢀⡴⡖⠀⢰⠏⣠⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⢠⡄⡾⠰⢁⡀⢸⡟⠁⣸⠁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢰⠃⢷⠇⠀⠛⡆⢀⠀⣠⡿⠞⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⠀⠈⠀⠀⠀⠷⡿⠀⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⢣⡀⠀⠀⣀⣴⣃⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⣿⡆⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣿⠀⠀⠀⠀⠀⢀⡏⡇⠰⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠀⠀⠀⠀⠀⡼⢠⡇⠀⡇⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠁⢹⡄⠀⠀⢀⡼⠁⡄⡀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⡀⠛⠶⠶⠋⢠⠞⢠⠃⠀⠟⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠸⣿⠀⡇⠀⢇⠀⢰⠃⢠⠏⠀⠀⡇⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⢦⡀⠀⠀⡏⠇⢡⡀⢸⠀⠂⢠⢃⣴⡖⠀⡇⣸⠁⠀⠀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡆⠀⡀⢻⡀⢀⡇⢸⠀⠃⠘⢀⠀⣧⡞⢹⠁⠀⡇⣿⠀⠀⠀⡟⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠃⡇⠀⡃⠈⠷⠼⠃⡘⢀⡴⢻⡟⠙⣿⠁⠸⡆⠀⡇⢻⡄⣀⣼⠃⡇⠀⢹
⠀⠀⠀⠀⠀⠀⢠⠏⢰⠁⠀⢁⠀⠀⠀⢠⢣⡟⢠⡏⠀⠀⢹⠀⠀⢻⡄⠰⠀⠉⠉⣁⠀⣃⠀⢸
⠀⠀⠀⠀⠀⠀⡞⠀⡦⠀⡆⠀⠱⣄⠀⠈⠘⡇⢸⡄⠀⠀⣾⠀⠀⠀⢻⠀⡆⡰⣱⣿⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⡇⠀⣀⠈⢣⠀⠀⢻⡀⠙⠶⠞⠁⠀⠀⠀⡾⢀⠔⣱⠏⡿⠀⡛⣰⠃
⠀⠀⠀⠀⠀⠀⢹⡀⢳⠀⢰⠀⣿⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢼⣁⣤⠞⠁⢰⡇⢠⠃⢿⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠈⢷⡈⢆⠈⢧⣿⠹⣤⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡾⢁⠆⠀⠈⢛⠋⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⢢⡈⠻⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣡⣤⡄⢀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠦⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⡴⠛⠐⣩⠞⠁
⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠉⠳⢦⣌⡳⣦⣄⣀⣀⣀⣀⣀⣠⣤⠴⠖⠚⣋⣡⡤⠖⠋⣁⣀⣤⡤⠶⢶⡄
⠀⠀⠀⠀⣿⠀⢄⣀⠉⠛⠲⢦⣄⣀⠀⠉⠛⠲⠿⠿⠭⠭⠴⠶⠶⠖⢚⣋⣭⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⢀⠀⠉⠒⠀⠀⠤⣈⡙⠲⠦⣤⣀⣀⢀⣀⣤⠶⠚⠋⠉⠀⡀⠀⠀⠐⢀⣀⠤⠒⠂⠀⣀⣼⠁
⠀⠀⠀⠀⠻⢦⣀⠀⠉⠐⠂⠤⠀⠀⠀⢉⣠⡤⠞⠛⣉⠡⠄⠐⠂⠀⠀⠀⠀⠀⠀⠈⠁⣀⣠⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣤⡤⠶⠛⠉⠀⠀⠀⠉⠀⠀⣀⡠⠤⠖⠀⠀⣀⣤⣴⣖⡋⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⢁⡠⠄⠂⢀⡠⠔⠀⠤⠂⠉⣀⣠⡤⠶⠚⠋⠁⠠⢀⡉⠉⠛⠲⠦⣄⣀⡀
⠀⠀⢠⡶⠛⠻⣍⡀⠀⢀⡈⠄⠒⠂⠉⠀⠀⣀⣠⠴⠶⢯⣉⣀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠂⠀⠀⠉⣹⠿⠶⠦⣄
⠀⢠⡟⡼⢬⡑⡌⣷⠀⢀⡀⠤⣒⣮⠽⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣀⡀⠀⠀⢀⠀⠀⠀⣼⠡⢊⣑⡀⠈⣧
⠀⡸⢧⣳⣬⢔⣱⣧⡤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⣄⣀⠰⡇⠠⣘⢀⡱⣠⠏
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠚⠁

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢀⣿⡄⠀⠀⠐⣄
⠀⠀⠀⠄⡀⠀⠀⠂⣤⡀⠘⣿⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠫⢝⣶⣄⠀⠀⠀⠀⠀⡖⠀⢹⣿⠀⢠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⡀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠙⣆⠀⢀⠀⣱⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠖⠀⠀⣠⠀⠀⠀⠀⠁⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠙⠇⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⣀⣤⣤⣀⡀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⣤⢀⡀⠉⠤⠈⡿⠴⢤⠀⠀⢄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢐⣛⡣⠀⢋⠛⠈⣡⡄⡔⠒⠊⢠⣐⢎⡉⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠀⣭⣉⢀⠣⠆⢀⣰⠀⠒⠁⠂⣦⡀⣚⣁⠜⠃⠾⠃⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠆⡂⠩⢗⡉⠛⢠⡌⠻⢼⡩⠧⣽⠀⠘⠀⠿⠀⢤⡉⠈⡆⣙⠁⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢢⣅⢠⢨⠛⠖⡄⠰⠆⢺⠛⣶⣭⡓⣤⡃⣦⣻⠞⣁⠨⠶⢈⠷⠀⣄⣇⠤
⠀⠀⠀⠀⠀⠀⣤⡴⠘⢂⣑⠚⣄⠥⣤⢀⣤⢖⡻⣟⠰⡀⠀⣶⢶⡞⢱⣄⣿⣘⣦⣊⣒⠇⢠⡌⠔⠃
⠀⠀⠀⠀⠀⠠⠾⣤⠘⠿⢧⠇⢈⡱⢦⣌⡬⢻⡁⢻⣄⣴⠊⠷⣼⢠⡞⢭⢀⠖⣤⠐⢀⣵⡊⠁⡆⡒⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠋⠴⠀⣂⡘⠁⣛⣉⢷⡖⠘⡙⣿⡑⠄⡽⣣⡟⡰⠦⣼⠳⠀⣶⢀⡘⣥⠰⠇
⠀⠀⠀⠀⠀⠀⠠⠝⢡⠤⣃⠤⣠⠴⠶⢤⣩⡟⢾⣅⢸⣧⣾⣵⠟⢸⣷⣎⡬⣕⠂⣽⣦⠉⡀⠠⠜⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠇⣲⣀⠘⠂⣈⠋⣰⢿⣮⠳⣦⡙⣿⣿⣿⣷⡿⣿⡉⣱⡦⠝⢃⡄⢍⡚⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠨⠌⢁⠱⠀⣧⠡⠀⣀⠈⢻⣾⣿⣿⡏⠀⠈⠃⢀⡰⠒⠀⠠⡌⠑⢇⡐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠃⠀⣅⠷⠀⠰⡶⡉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠨⠤⢀⣄⠃⠀⠠⠾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠘⠁⠙⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣴⣶⣿⣧⣿⣿⠁⠉⠙⠋⠻⠓⠶⣦⣼⣦
⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⢤⣤⣹⣵⠿⢿⣿⣿⣿⣿⣤⣤⣤⣰⣥⣴⣶⠾⢶⣷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢘⣿⢿⠇⢰⡇⠀⡀⣿⣹⣿⠻⢏⢉⡁⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⡘⠻⠀⠀⣸⡇⠀⢱⡟⡿⣿⡤⠠⠄⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠉⠿⠯⠓⠚⠛⠻⠷⠐⠀⣤⠀⣀⣀⣀⣘⣿⣧⣤⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡿⣉⣤⣤⣄⡀⠀⠀⠀⡀⠀⠀⠣⠾⠿⠿⠷⠞⣁⣀⡀⠀⠈⠁⠀⠄
⠀⠀⠀⠀⢀⠀⣠⠶⠛⠉⠋⣿⣾⣿⣿⣿⣿⣿⣷⣦⣄⡈⠩⠭⠀⠀⠀⠀⠀⣠⣿⣿⡁
⠀⠀⠀⠀⣸⣆⣠⣤⣤⣀⣶⣾⣿⣧⡾⠛⢛⣩⣿⣿⣿⣿⣿⣷⣖⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠿⢦
⠀⠀⣀⣶⣭⢭⣿⣿⣟⣿⠿⠛⢻⣿⣶⣿⠛⠉⠀⢋⣴⣿⣿⣿⣿⣿⣿⣿⣛⣭⣿⣿⣿⣿⣿⠏⡠⠊⢙⠇
⠀⣾⢿⣷⣿⣿⣿⣿⡿⢁⡾⢛⡛⢻⣿⣿⣷⣤⠖⠋⠀⡉⠐⠮⣝⣻⣿⣿⣿⣿⣿⡿⠿⠯⣿⣿⠟⠀⣠⢳⣦
⠀⣷⣿⣿⣿⣻⣿⣿⠃⡞⣰⢏⣾⣟⣽⣿⣿⣧⣠⠤⠊⡁⢠⡀⠈⢻⣿⣿⡯⣿⣿⣶⣄⢸⣿⣿⣶⣿⡵⠛⣅
⠀⡏⣿⣿⣿⢿⣻⣿⢠⡇⣿⣸⣿⣸⣿⣿⡟⣿⣿⡶⠋⠀⢀⠌⠲⣤⣙⣿⣿⣹⣿⣿⡿⠛⠻⣿⣿⣿⡇⢠⠌
⠀⠸⡹⣿⣿⢿⣻⣿⢸⡇⣿⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣼⠀⠹⣿⣿⣿⢺⣽⣿⣿⢚⣿⣿⣿⣿⣿⢀⡓
⠀⠘⢯⣵⣟⣻⣿⣿⡜⡇⢻⡜⢻⣿⣶⣿⣿⣿⣿⣿⡤⠚⠉⠨⢶⣤⣈⣧⣿⢻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠉⠉⠉⠈⢧⢻⡈⢷⡀⢻⣟⠿⣿⣿⣿⡛⠓⡊⠁⣠⠆⠈⢯⣛⣿⣾⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣷⡀⠳⣄⣈⢁⡞⣿⣿⣿⣾⣵⡿⠿⣶⠄⠘⠉⠻⣿⣿⣿⣷⡿⣿⣿⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣴⡿⣱⣿⣯⠉⣉⣥⣤⣬⣩⠆⠀⠀⠀⠘⠿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣥⣤⣾⣿⣿⣿⠿⢛⣫⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⡿⠿⠛⠋⠁

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠰⡀⠀⠀⠀⠀⠀⣠⢎⡕⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠳⡀⠀⠀⠀⡴⢡⠎⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠻⡲⢄⠀⠀⠀⡠⠜⠀⡿⠀⠀⠀⢱⠖⠑⠋⣰⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠈⠢⡱⣄⠀⡇⠀⠀⣇⠀⠀⠀⢘⠀⠀⣰⠏⢀⣄⠀⣙⠀⠀⠀⠀⣀⣤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⢮⡑⠋⠳⠀⣿⠀⠀⣀⠜⠀⢀⡟⡰⡟⢠⠀⢧⠀⢀⢔⡫⠋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠳⣄⠀⢇⢹⡀⢸⠀⠀⠀⣸⠏⡜⢀⡜⠀⡰⢟⡵⠋⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢦⣀⠀⠙⣆⠈⠛⠧⡀⣇⠀⢀⡏⡼⠀⡜⡔⠉⣴⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⡳⡄⠘⣧⠀⠀⢃⡸⠓⠊⡼⠁⠀⢧⣹⡾⠁⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⡀⠀⠀⠀⠘⢆⠘⢮⢢⣘⣧⠀⢘⡂⠀⣸⠃⠀⠀⣸⡟⠁⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠛⢕⠢⣀⠀⣸⠀⠈⢳⡄⠈⣧⣿⠇⢀⡏⠀⠀⠀⢻⠀⠀⠰⣋⡠⢄⡶⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠙⢮⡉⠉⢄⠀⠀⠹⡄⠸⡉⠀⣸⠀⠀⠀⢀⡼⠄⠒⣋⡵⠚⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠙⢦⡎⠀⠀⠀⠹⡄⠈⠱⡇⠀⠀⣰⠛⢁⡴⠞⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⢻⡖⢤⡀⠀⠹⡄⠀⡄⠀⠀⢱⣴⠋⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠆⠀⠀⠹⣤⡃⠀⠀⢳⣸⠃⢀⣴⠟⠁⠀⠀⡔⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠒⠀⠛⠘⠒⠀⠘⠛⠚⠃⠛⠂⠐⠒⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⡀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠻⣝⠮⣝⡲⢤⣀
⠀⠀⠀⠀⠈⠳⣄⠙⠲⢬⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⠦⣝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠲⢭⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠉⠓⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠂⠀⠈⠙⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡇⠀⠉⠙⠒⠦⠤⠤⠤⠴⠒⠋⢀⡠⠖⠚⠛⠓⠲⣄⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠤⠒⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡞⠀⠀⠀⠀⠀⠀⠈⢳⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠛⢷⣤⡀⠀⠡⠘⣇⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠻⣷⢤⡀⠈⠳⢤⣀⣀⣀⠴⠚⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡓⠦⣤⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠒⠛⠓⠲⢤⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⡀⢷⡀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠑⠦⣄⣀⣀⡤⠖⠁⢐⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠰⠃⠚⣶⢶⣶⣒⣒⣂⣒⡶⡶⠶⠶⠶⠶⠶⠾⠼⠤⠤⣤⣤⣦⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⢴⡶⠦⢀⣀⣀⣀⣈⡉⢉⠉⠉⠉⠙⡒⢒⡒⣶⠖⢶⠀⠀⢹⣇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⣹⣭⣭⣭⣶⣦⡄⠐⡀⠀⠀⠀⢨⠏⠉⠉⢻⣿⣿⣿⡷⣶⣶⣮⠻⡀⢶⣿⡇⣿⡿⠻⢿
⠀⠀⠀⠀⠀⣀⣴⣶⣶⣷⣴⣦⣰⢟⢿⣿⣿⣿⣿⠋⢰⠀⠀⠀⠀⠸⣧⢤⣀⣸⠿⠿⢿⢇⣞⣸⣿⠀⠀⠸⣧⣷⣿⡷⣦⣾⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣍⡸⣨⣿⡿⣿⣶⣶⡀⠀⠀⠀⠀⢸⢘⣉⡙⠃⣾⡃⣏⠉⠉⣿⠀⠀⠠⣿⣿⡏⡏⣿⣿⡇
⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⣾⣿⣿⡿⣷⠏⣸⡧⣿⣿⣿⣇⣤⣤⣤⣀⡈⠈⠛⢻⣿⠼⠆⣄⣤⣠⣿⠀⠀⢀⣿⣿⡷⡇⣿⣿⡇
⠀⠀⠀⣹⣿⢽⣿⠏⠊⢉⠘⢿⣿⣿⣮⡸⣻⣷⣶⡶⣶⠦⣀⣑⣓⡃⠉⠿⣄⠈⠀⠀⠀⠀⠃⠽⣿⠀⠀⠀⣿⣿⣇⡇⣿⣿⡇⢀
⠀⠀⠰⣿⣿⢸⣿⠀⠀⠀⣀⢰⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣟⢙⡿⣿⣿⣿⣦⣔⣀⡀⠀⠀⠠⣿⡀⠀⢀⣿⣛⡋⣁⡀⡤⡦⣿⣿⣦⡀
⠀⠀⢀⣿⣿⠈⠻⡀⡀⠤⠋⣸⣿⣿⣿⣿⣿⣿⠟⠛⠓⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⣇⣰⣿⣿⣿⣶⣶⠙⡏⠁⠀⠀⡇⣿⣿⣿⣷
⠀⠀⠘⠻⣿⣇⠀⠕⠈⡈⢀⣿⣿⣿⣿⣿⣿⠤⠤⠾⠑⠉⠙⠛⠛⠿⠛⠛⠉⠁⢀⣤⣶⣿⣿⣦⡙⢿⡿⣿⣿⢀⡇⠀⠀⢀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣷⣄⣀⣰⣿⣿⣿⣿⣿⣿⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣾⡿⢫⠟⠛⢻⣿⣴⢿⡿⠽⣿⣵⡖⢿⠏⠀⠉⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⠀⠀⠀⠤⠒⠀⠤⠄⠀⠤⠄⣐⣢⣾⣿⠁⠏⠀⠄⠀⣿⣿⣼⡷⢦⣿⣿⣿⣦⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠃⠺⡀⠀⣿⣿⣿⡇⠀⠀⠀⠉⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠤⢄⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠂⠚⠉⠘⠢⠔⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⠀⠀⠀⠀⠀⣀⣴⣄⣀⡀⠀⣀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⢀⣸⣯⢉⡙⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂
⠀⠀⠀⢀⡀⠤⠀⠒⠂⠚⠁⠹⠛⠆⢴⣏⣲⣿⣿⣿⣯⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣲⣊⣄⢰⣿⡛⠁⠈⠉⠙⠐⢣⣤⡀⣄⢚⡃⠀⢠⠌⡧⠴⠛⠀⠀⠀⣤⠀⠀⡄
⠀⢀⣀⠀⣠⣀⣠⡀⢀⠀⠲⣼⣿⣿⣿⣿⡿⢆⣙⢶⣤⣆⡸⣷⣿⣯⣿⣦⠿⠷⣦⡄⣶⣬⣥⣀⣀⣾⣷⣊⡀⠂
⠀⠀⠁⠚⠒⠛⠿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡶⠿⠿⣿⣿⣿⡿⢻⣿⣽⣥⣔⣒⡋⠁⠐⠋⠉⠋⠁⢛⣿⣿⠛⠁
⠀⠀⠀⠄⠠⠀⠂⠀⠒⠝⠛⢀⠛⠀⢜⢻⣅⢿⣯⣼⣿⣻⣷⣸⣿⢿⣷⣶⣷⣶⡆⢠⣄⣤⣄⣼⠛⠉⠁⠀⢀⠉⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢓⣦⣀⡚⠓⠙⠛⢛⣿⡿⢷⣇⡿⠋⠈⢿⡿⣿⠿⣿⡷⡿⠀⠉⠀⠀⠀⠀⣀⣈⡀⠀⠀⢀
⠀⢐⣂⡄⠀⢀⣀⡀⠀⠠⠉⢩⡭⢉⢽⡶⢦⣼⣿⡟⢁⡿⢋⡀⣸⣠⡏⢀⣤⣬⣟⣂⣄⠂⠀⠀⢠⣬⠣⠘⣫⠄⣤⣇⣩⡁
⠀⠈⢻⣶⢆⣍⠿⠖⣛⡉⣀⣮⡟⢥⣿⣧⣦⣿⣿⣧⣼⡧⣿⠛⢻⣿⢻⡏⠀⠸⠆⢴⠀⠀⠴⢡⣶⣷⠮⣿⣿⣾⣿⡶⣸⠿
⠀⠀⠀⠀⠉⠐⡿⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣎⣿⣿⣿⣇⣿⣴⣿⣧⣾⣯⣤⣾⣧⣶⡭⠾⢗⣋⣿⣿⣿⣿⡿⣛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠛⣫⠿⣿⣿⣿⣿⠟⢿⣿⣿⣾⣿⠻⣿⢟⣿⣿⣿⣿⣷⣎⢻⡁⢠⡾⠉⠘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⡿⠙⡃⠐⠀⠉⠛⠻⠟⠛⠛⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⢀⣀⣤⣤⣴⠶⠿⠿⠋⠉⠉

⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠤⠀⣀
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠲⢆
⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠐⠄⠀⠀⠀⡠⠁⠀⡠⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡐⠁⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠢⠀⠀⠀⠀⠸⠀⠀⠑⠠⡀⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠐⠣⡀⢡⠀⠀⠒⠀⠆⠀⠀⠀⠀⠀⠢⠀⠑⡀⠀⠄⠀⠡⠀⠀⠀⠈⠀⢄⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢢⠆⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠐⢄⠀⠁⠲⡀⠀⠀⠀⠁⠂⠀⠀⠁
⡇⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠲⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⠆⠀⠀⠀⠀⠞⠑⠂⠀⠠⠄⠀⠈
⡇⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠊⠐⠀⣀⠄⠁
⡇⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠣⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⡠⠉
⡇⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠀⠠⠊⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠈⠄⠈⠢⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠰⢀⠔⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠢⢀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠁⠢⣀⠄⠀⠑⠄⣀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡠⠂
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠄
⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠊
⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⢤⠤⠀⠀⠀⠈⠉⢹⠤⠐⣂⠀⠎⢓⠶⠶⠀⠠⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣟⡶⠅⢒⣶⡄⠤⢄⡀⣂⡁⠀⣤⢑⡠⣀⠂⠀⢻⣵⣧⣯⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⣤⣵⣿⡋⣠⣢⢸⢡⣃⣹⢹⣟⣂⣷⢴⣏⣵⡮⣺⣿⣿⣿⣜⣷⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⣿⣿⣿⣿⣵⣿⣿⡏⠱⡗⢡⡔⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣤⡧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣾⢹⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢻⣿⣿⣿⢹⣿⣿⡿⣿⣿⣿⣿⣿⣿⡴⢻⣿⣿⣧⠇⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡏⡇⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢙⢿⣿⢈⠘⢘⠝⢸⠧⠡⠝⠿⠿⠞⠈⠿⠃⡼⣤⠌⡟⡋⠛⡏⠀⢂⣹⡿⠏⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠿⠄⢀⡔⠒⡉⠁⣑⣦⣀⣀⣀⣀⣀⣀⣃⡁⣆⣙⣧⠀⣇⠀⢀⣀⢣⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⢊⣡⡤⠶⠁⡞⠉⢂⣿⡎⣉⣀⠏⠉⠀⠀⠀⠀⠀⢸⡗⠒⣢⠾⠽⢯⣉⠀⣸
⠀⠀⠀⣀⣤⠤⠖⠋⣁⣠⠴⠚⠉⠀⠀⠀⠀⢿⠉⣡⠛⡍⠢⢀⠆⠀⠀⠀⠀⠀⠀⠀⣷⡊⠁⡜⠈⣇⠉⣱⡟⣠⣀⣠⠂
⠶⠾⣯⣅⣤⣒⠛⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⣤⣷⣶⣯⣤⠤⠿⠆⠀⠀⠀⠀⠈⠛⠦⠧⣤⠾⠷⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⠋⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣍⠉⠉⠛⠛⠛⠛⠚⠛⠛⠛⠛⠉⠉⣩⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠶⣄⠀⠀⠀⠀⢀⡾⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢦⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠳⡄⠀⡴⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠘⢯⡉⠉⠉⠉⠛⠶⣤⡀⠀⣾⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠹⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠙⢦⡇⠀⠀⣀⠀⠀⢿⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠙⠀⠀⠀⠲⣄⠀⠀⢿⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⣇
⠀⠀⠀⠘⣏⠉⠙⠓⠦⣄⠀⠀⢿⠀⢠⠀⢸⠇⠀⢀⡴⠖⠛⠉⢉⡟⠀⠘⢧⡀⠘⣇⠀⣿⠀⢀⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠹⣆⠀⠀⢀⠈⠳⣄⠘⣆⢸⢀⡟⢀⡴⠋⣀⠀⠀⢀⡞⠁⠒⢤⣀⢳⣀⡿⣆⡿⢠⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠳⣄⡈⠛⢦⣹⡆⠙⣿⡟⠁⣾⣡⠞⠋⣀⡴⠋⠀⠀⠀⠀⠉⠛⢿⡁⢹⡟⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠻⣄⢸⢀⡼⠛⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⢧⣄⡀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠞⠁⡀⠀⠀⣀⠙⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠳⠦⠶⠃⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣀⡀
⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⢸⡆⠀⢀⣤⣄⠀⠀⣾⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠛⠋⠀⠈⠙⠳⣄
⠀⠀⠀⠀⢸⠃⠀⠀⠘⠓⠋⠀⠀⠛⠀⠈⠃⠀⠈⠓⠛⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠴⠟
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉
⡏⢘⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡃
⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⡎⠈⠆⢰⠀⡇⠀⡶⡀⢸⣰⠂⡆⠀⣆⡀⠀⠰⣁⠀⢺⠀⣞⣰⢠⠂⠑⡆⣦⠒⢰⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣥⣄⣧⣤⣃⣸⣀⣇⣜⣁⣣⣰⣼⣃⣧⣄⣧⣤⣀⣠⣼⣃⣸⣀⣯⣘⣌⣦⣴⣃⣇⣱⣌⣤⣄⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠼⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠸⡄⠸⣆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣾⠉⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢠⡇⣰⠇⢠
⠀⠀⠀⠀⠀⠘⡆⢸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⡻⡄⣰⡊⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟⢀⡞
⠀⠀⠀⠀⠀⢀⠇⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⢻⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⡀⢰⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⣇⠈⢷⣄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⢦⡀⠴⣫⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠳⠀⢠
⠀⠀⠀⠀⠈⠓⣦⠙⣷⢹⡄⠀⠀⠀⠀⠀⠀⠀⡏⠉⡳⡝⣦⡧⣀⡽⠀⠀⠀⠀⠀⠀⠀⢾⡁⢀⠀⣏
⠀⠀⠀⠀⠀⢰⢫⡞⢱⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠁⠈
⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣰⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠈⢳⣌⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢠
⠀⠀⠀⠀⠀⠱⡄⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣠⠞
⠀⠀⠀⠀⠀⠀⡿⠘⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢷⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⡜⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠇⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠛⠀⠀⡴⠀⠀⠀⠰⠋⠀⠀⠀⠠⠀⢀⡞⠀⠀⠀⠀⠐⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⢡⢧⠂⡆⠀⠀⠀⣴⣄⠀⠀⠀⢁⠀⣀⡆⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠃⠐⠀⣴⠀⠙⡀⠀⡀⠈⣶⡈⠙⠀⢸⠅⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⡅⠸⠁⠀⠉⠃⠀⠀⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠐⡈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠁⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡆⠳⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡸⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣙⢾⡿⠀⠀⠀⠀⣴⠞⣻⢧⡀⠀⠀⡀⠀⣿⠀⠀⠀⢀⡂⢀⡾⠀⠀⠀⠀⢸⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠁⠀⠀⢀⡴⠃⢀⣾⡿⠁⠠⠜⢻⡎⢹⣿⢄⠀⣸⠿⠽⠟⠶⣴⣦⣀⣀⣀⣂⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣧⣇⢀⡾⠋⠁⠀⡾⠉⠀⠌⠀⡠⣾⠂⢠⣷⠈⠛⢟⡀⠀⠀⠀⠀⠀⠀⠀⠩⠋⢿⠌⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠋⠀⠀⠀⣼⠃⠀⠀⠀⠀⡱⣿⠀⢸⣿⡆⠀⠹⣣⡀⠀⢠⡤⠀⠠⢰⠡⠚⢈⣵⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⡟⠁⠀⠀⣰⠇⠀⠀⠀⠀⢠⡇⣿⠀⠀⣏⣷⡀⠀⢻⢿⣿⠶⢶⣾⣤⣽⣥⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠬⡀⠀⢠⣾⡏⠀⠀⠀⠀⠠⠋⢠⢛⠀⠀⡏⢹⣧⠀⠈⡀⠸⣄⣈⡛⠚⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣆⠀⣈⢷⣾⡿⠀⠀⠀⠀⢠⠃⠁⠈⢸⠀⠀⡧⠋⠐⡄⠀⢱⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢻⠿⢳⠶⠖⠂⣰⣄⡀⠀⠀⢸⠀⣰⡇⠀⠀⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠋⠀⠓⠀⠀⠘⠋⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢉⠃⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⠈⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡘⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣀⣀⣀⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⢩⢸⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⠀⡇⠀⣀⣀⣀⣇⣸⣇⣀⣀⣀⣀⣀⣀⣸⠘⡀⠀⠀⠀⢠⡧⡇⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣹⡤⠼⠒⠒⠒⠓⠢⠤⢄⣸⡇⠀⠀⠀⠀⠀⠀⠘⡀⡧⢄⣀⣀⣸⠀⡆⠀⠀⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡤⢤⠤⠔⠙⠦⠤⡤⢄⠈⠙⠦⡀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠉⠉⠉⠒⠒⠣⠭⠦⣄⣀⡀
⠀⠀⠀⠀⠀⢀⠞⢁⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢤⠈⢢⡀⠀⠀⠀⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⢠⠃⠀⡴⠁⠀⢀⡠⠒⣉⠭⠭⢍⣑⠢⣄⠀⠀⠘⣄⠀⠹⡛⠛⠛⠛⣿⠀⢰⣶⣶⣶⣶⣶⠀⠀⠀⠀⢾⣿⣿⣿⡆
⠀⠀⠀⢀⠇⠐⡍⠀⠀⢠⢊⡔⠋⢀⡠⠤⣀⠈⠑⢄⠳⡀⠀⠈⡹⠀⢱⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⠀⠀⡠⠤⠾⠛⠿⠿⣇
⠀⠀⠀⢸⠀⢀⠇⠀⠀⡇⡜⠀⡰⢩⠒⠑⠢⡙⡄⠈⡇⢱⠀⠀⢹⡀⠈⡟⠻⠛⣿⠤⠼⠿⠿⠾⠿⢿⡖⠋⠀⣀⢤⠤⠤⣀⠈⠓⢦
⠀⠀⠀⠈⠉⠣⡀⠀⠀⡄⡇⠀⢇⢣⠀⠀⢀⠇⡆⠀⡄⢸⠀⠀⢠⡟⠈⠁⢲⣶⣿⠶⠶⠶⠶⠶⠶⡊⠀⡰⡫⢚⣭⣭⣭⡲⡕⡄⠈⢳
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⠹⡄⠈⠢⠭⠉⠥⠚⠀⡰⢃⠎⠀⠀⢾⣭⣭⣭⣭⡉⣿⣀⣀⣀⣀⣀⡰⠀⢠⢱⢱⡿⠁⠀⠙⣿⢸⢸⠀⠸⡄
⠀⠀⠀⠀⠀⠈⠒⣄⠀⠀⠱⢌⠓⠤⢄⣀⡠⠴⢊⡤⠋⠀⠀⡴⠾⠭⠭⠭⠭⠤⠿⠀⠀⠀⠀⠀⢘⡄⠀⣎⣌⢾⣦⣤⣔⠝⡼⡜⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠨⣂⣀⠀⠀⠉⠐⠒⠒⠒⠊⠁⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠪⣗⡨⠭⣕⡪⠞⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆⣠⣀⡀⠀⣀⣀⣄⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣄⣀⡈⠁⢁⣀⡠⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠚⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣶⣾⣷⣿⣿⠶⠁⡺⣿⣿⢿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⠿⢉⣉⣿⣿⣤⣬⢻⣿⡭⢄⣻⣿⣿⣿⣁⣤⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣥⣼⣿⣯⣷⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣿⣯⣛⣛⣩⣴⣤⣀⡀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠈⣹⠟⠚⠛⣿⣟⢻⣲⣷⣿⣿⣿⣷⣿⣿⣿⣿⠆⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⡀⣾⣿⣿⡟⠛⢉⠺⢛⡛⢻⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⢷⣼⡿
⠀⠀⠀⠀⠀⠚⠫⠋⣀⣠⣄⠴⣾⣿⣿⢀⡘⢯⣻⡟⢿⣿⣿⣿⣿⡿⢿⣿⡟⡫⣿⣿⡽⣿⣯⡳⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⢠⣽⣿⣿⣤⠀⠀⠙⢻⣦⣿⣿⡿⠉⠃⠀⠛⠀⠀⡓⠻⡆⠈⣿⠁⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠈⢻⠇⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣀⣀⣠⣠⣤⣤⣴⣶⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣤⡶
⠀⠀⠀⠀⠂⡞⠿⡿⢿⠟⠛⢿⣿⡿⡿⡿⠿⠋⠉⠁⠠⠬⠭⠿⠿⡿⠿⠿⢿⡿⠻⠟⠻⠻⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠤⠤⠤⠄⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⣠⠔⠋⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⢀⡤⠊⠁⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡷⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠖⢉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠴⠶⠞⠋⢸⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢁⡤⠶⠒⠲⣤⡀⠀⢸⡶⡏⠈⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⠋⠀⠀⠀⠀⠀⢿⠀⡟⠀⡗⠚⢣⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⢀⡟⢸⠃⠀⡟⠀⣈⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣧⠈⠣⡀⠀⢀⣀⡴⠋⣰⠏⠀⢰⢠⡞⠉⠉⠙⢷⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⣉⣉⣡⠴⠛⠁⠀⠀⡎⢸⡇⠀⠀⠀⠀⢷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⡇⠘⣧⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠙⣦⡀⠀⣀⡾⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠙⠛⠉⢀⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣲⣒⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⠘⠀⡰⠇⠀⠀⠃⠀⡠⠀⠀⠀⠀⢓⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⣄⠀⠀⠁⠀⣠⡀⠀⠀⠃⠀⢀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠈⠀⠂⣦⠉⠃⠀⠀⠠⣤⠉⢀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⢰⠋⠀⢠⠀⠇⠀⠉⠀⢀⠈⠀⠠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡤⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠦⠀⠀⠀⢁⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡫⢿⠇⠀⠀⢀⠖⢻⣧⠀⠀⣀⠀⡇⠀⠀⠀⣀⣠⡆⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⣐⣀⣀⣀⠀⢀⣠⠞⠀⣼⠞⢀⠐⢻⡇⢹⡿⣄⣼⠓⠛⠙⠲⠷⠤⠤⣴⣴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⢿⡤⠞⠀⠀⡼⠁⠀⠀⠠⣼⡇⢸⣧⠀⢩⡄⠀⠀⠀⠀⠀⢰⢳⠸⣥⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠅⠀⠀⣰⠃⠀⠀⠀⣸⢽⠃⠸⢻⡄⠀⢿⣶⣧⣯⣤⣄⣧⡠⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠥⢁⠀⢀⣼⡏⠀⠀⠀⠀⠃⡼⠁⠀⠏⣷⠀⠘⡈⢇⣀⡛⠻⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣹⣾⡟⠀⠀⠀⠰⠁⠈⠀⠀⢀⡞⠀⠆⠀⣁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠙⠙⢃⣴⠢⣀⠀⠀⣄⣼⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⢹⠋⠙⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⣀⣀⣀⣀⣀⡁⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢦⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⣹⠀⠀⠀⠀⠀⣠⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠳⣄⠀⠀⠀⠀⠉⢹⠋⠉⠉⠉⢹⡏⠁⠀⠀⠀⢠⠞⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⢠⡇⠀⠀⢸⡄⠀⠀⠀⣀⣠⣬⣄⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠸⡄⠀⠈⠙⠲⣄⠈⣇⢰⡆⣸⠁⣠⠖⠋⠁⠀⢰⠇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⢦⡀⠰⢦⡘⣆⠸⣼⣷⠇⣴⢃⡴⠂⢀⡤⠃⡀⠀⠀⢀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠓⠶⠽⠿⣄⢸⡇⣰⠿⠯⠶⠚⠋⠀⠀⡿⣆⠀⢸⠆⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡷⠃⠀⠀⢠⠏⠀⠀⠀⡇⠘⢦⣸⡄⡴⢶⡄⢻⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⢧⣄⡀⢠⠏⠀⠀⠀⠀⠇⠀⠈⠻⠀⣠⡞⠁⠀⢻⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠋⢀⠀⠀⣄⠙⠶⠦⣄⠀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⠼⠃⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⡞⠁⠀⣀⠀⠈⠓⠚⠁⠀⣀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠷⢤⣄
⠀⠀⠀⠀⣰⠋⠁⢠⣀⡼⠀⢠⠖⠲⡄⠀⢧⣀⡄⠈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠐⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠼⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠖⠒⠛⠋⠉⠁
⡄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⣆⠀⡆⠢⢰⠀⡖⠀⢶⠀⣆⡦⠠⠀⣐⡀⠀⢔⠀⠰⠂⢰⢰⢠⠂⢢⢰⡐⢠⣂
⣀⣀⣀⣀⣀⣀⣀⣥⣄⣧⣜⣸⣄⣇⣼⣉⣇⣧⣟⣴⣤⣤⣤⣀⣤⣇⣠⣄⣼⣸⣜⣤⣜⣸⣙⣬⣤⣄⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠋⠀⠀⠀⢀⠄⢀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠔⠡⠚⠁⠉⠉⠳⢶⡤⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢀⣼⣡⠔⢡⡐⠒⠢⠄⠀⠀⠀⠀⠀⢦⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡌⢧⠀⠀⣀⠤⢛⣽⠅⠀⠘⠈⠂⠀⠀⠀⠀⠠⣒⢄⡈⢣⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⠈⠀⢠⣴⡅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠢⠍⠛⠺⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧⠐⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢷⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢄⣀⣀⣠⣄⢫⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡟⠹⡜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡀⣉⣀⣈⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢳⠸⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣀⠀⠀⠀⠉⠙⢦⠤⣀
⠀⠀⠀⠀⠀⠀⠤⣶⠚⡉⠀⠀⣈⣧⢳⡎⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀
⠀⠀⠀⠀⠀⡠⠺⠛⣛⣵⠖⠋⠉⠙⣎⢇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠉⠻⣟⠒
⠀⠀⠀⠀⠀⠀⠀⣩⠞⠁⠀⠀⠀⠀⠸⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠦⠬⣲⣖⡚⠉⠙⠛⠯⣉⠁
⠀⠀⠀⠀⠀⣠⡾⠓⠀⠀⠀⠀⠀⠀⠀⠹⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠙⢦⠀⠀⠁⠉⠀⠀⠀⠈⠻⣗⠆
⠀⠀⠀⠚⡩⠋⠀⠀⠀⠀⠀⡠⠀⠀⣠⠔⢳⡵⠤⣄⡀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⠓
⠀⠀⢠⡾⡞⠁⠀⣀⠤⢊⡜⠁⢤⠊⠁⢲⣯⣄⢸⠳⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣖⠚⠓
⠀⡰⠋⣼⣷⠗⡺⠃⠀⠊⠀⠀⠀⠁⠀⠸⡼⡜⣎⢧⠙⣎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢭⣻⡓
⠀⠀⠀⠉⣠⣎⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡽⣌⢮⣳⡈⢫⣧⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠙⢯⡁
⠀⠀⠀⠰⠉⣬⡔⢃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢾⣦⡙⠿⣦⡙⡷⢜⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡖⠽⠦⠄
⠀⠀⠀⠀⢈⡽⡾⣡⣖⡴⠋⠀⠀⠀⠀⠀⠐⠒⠠⠭⠿⢧⣉⠛⢇⠰⢤⡁⠹⣵⠤⣀⡀⠀⠀⠀⠐⠒⠲⣼⠦⣄
⠀⠀⠀⠀⢨⡾⠋⣱⢻⡆⣀⠲⡄⢢⣄⡀⠀⢀⡀⠀⢀⡠⠔⡹⠬⣄⠀⢈⣣⢌⣦⣶⡜⠓⠂⠸⠮⢯⡑⠚⠦⡀⠁
⣀⣀⣀⣀⣀⣁⣈⣁⣀⣏⣁⣀⣈⣛⣀⣉⣑⣚⣉⣲⣭⣒⣉⣇⣀⣀⣉⣓⣚⣦⣀⣀⣈⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⠤⢤⣼⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣉⣭⠿⠿⠿⠷⠶⠶⠖⢿⣿⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠛⣿⠀⠀⠀⢠⣶⠀⠀⠀⠀⢸⣿⠀⣿⣿⡏⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⡇⠀⠀⠀⢸⣿⣧⣤⡀⠀⢸⣿⡴⠛⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢁⡄⢠⣤⣾⣿⣿⣿⡇⣀⢸⣿⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣬⣥⣼⣭⣭⣭⣭⣽⡇⣿⢸⣿⣿⠀⢠⣶⣶⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣼⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⢸⡟⠿⠛⢸⣿⣿⣶⣾⠟⣹⣿⣿⡿⢷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣯⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⡇⣾⣿⣿⣿⣿⣿⠋⣰⣿⣿⡿⢡⣾⣿⣭⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠭⠭⠭⠭⠭⠽⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⢸⣧⣿⣿⣿⣿⣿⠃⣰⣿⣿⡿⢡⣿⣯⣭⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣿⣽⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣿⣿⠇⣰⣿⣿⣿⠃⣿⣿⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣼⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⡿⣿⣯⣭⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⣿⣿⣼⣿⣿⣿⣿⠟⢷⡈⢿⣿
⠀⠀⣴⣿⣿⣭⣿⣿⣿⣿⣿⣿⣶⡶⠶⠶⠶⣶⣿⣿⡿⣿⣟⣩⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⢸⣇⢸⣿
⠀⢀⣿⣿⡿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⠿⢟⣿⣿⣿⣿⣿⠿⣦⠙⣿⣿⣿⣿⣏⣿⣿⣿⣿⣟⣭⣾⣿⣿⣿⣿⢣⡟⢸⣿⠃
⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣾⣿⣿⣿⣿⣿⣧⡀⢹⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠏⣼⠇⣸⡟
⠀⠈⣿⣿⣯⣿⣿⣇⠘⣿⣿⣿⣿⠋⢉⣿⡿⠿⢿⣿⣾⠿⣛⣿⣿⣿⣿⣿⣿⠇⣸⡇⢸⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⢏⣰⣿⠇
⠀⠀⠻⣿⣿⣯⣿⣿⣧⣌⡉⠉⣁⣤⣾⡿⠀⠀⠀⣌⢳⣿⣿⣿⣿⣿⣿⣿⠟⢀⣿⠁⣾⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⣿⢟⣩⣿⣿⣿⣿⣿⣤⡾⠃⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⠻⢿⣷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠻⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⢟⣿⣿⣿⣿⣿⣴⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⠟⣻⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⢀⣆⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⣾⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⢠⣿⣇
⠀⠀⠐⣿⡆⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⢹⠃⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⠟
⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⣸⠀⠀⠀⢸⣿⣿⣿⣆⣄⠀⠀⠀⠀⢠⠃⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⢰⣇⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠟⠀⠀⠀⣤⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢿⣷⣄
⢰⡇⣸⣿⡀⠺⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣼⡇⠀⠀⠀⣠⣤⠀⣀⣤⣀⡀⢀⣼⣿⣇
⠀⠻⣿⢿⣿⣦⣤⣾⣿⣿⣿⣿⣿⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⣿⣿⣿⡳⢤⣐⣿⠱⠿⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠓⠈⠻⠿⠿⠛⠛⠛⠉⠀⡆⣀⡸⡇⠀⠀⠀⢀⣾⣦⣤⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠉⠁⠐⢦⠀⠻⠿⠿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡖⢢⠤⢀⡠⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢡⡿⣇⠛⣛⣿⢿⣿⣿⠏⠻⢻⢷⣤⣀⣵⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣬⡫⠩⠝⠂⠍⣄⠘⣛⣦⣶⢶⣾⣿⣿⢻⠿⠿⢿⣾⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠐⠘⠀⠒⣶⠤⠤⣹⣾⣷⢾⡉⣾⡵⠻⢿⣤⢠⡄⢉⣽⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⣾⣿⠩⠉⢀⠀⠁⠐⠂⠳⡄⠠⠙⡃⠹⣿⡴⣦⡟⠻⡆⢳⣌⢽⣿⣿⣿⠥⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⠄⢿⣇⣆⢠⢈⠀⠃⠀⡀⣀⠀⡀⠤⠈⣧⠀⠀⢁⢰⣆⠞⣼⠼⣽⣿⣿⣷⣿⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠰⢇⣻⣶⠾⠛⠻⡶⠂⠀⠤⠀⣠⣙⡷⢂⠀⡠⢄⡐⣄⣸⣛⡿⡻⠁⠐⠢⢻⣿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⡂⠀⠀⣀⡆⠄⣒⣠⣾⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣟⠳⢈⣷⣾⣿⡛⠿⣟⣻⣿⡇⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣾⣯⣠⣾⣷⣿⣯⢿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⢿⢿⣿⣿⡿⠿⠷⣾⣽⣿⣿⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣛⠛⠻⣿⣄⢨⣿⣿⣥⣴⠟⣻⣿⣾⣿⣿⣿⣦⣧⣼⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢰⠾⡟⠛⠿⠟⠛⠛⠛⠻⣿⣾⠋⣿⡏⢁⣴⣿⠟⢩⣽⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢿⠟⣹⡿⠋⠀⢾⢿⣿⣿⣿⡿⣯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢀⠀⢰⣿⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⢺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣷⡜⠿⢿⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣒⠒⠂⠀⠀⠠⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢿⡻⢿⠿⣟⡛⢯⡿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢞⡩⠔⢚⣉⠍⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠱⢋⠔⢀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢾⡿⡿⢠⠃⡰⠋⠀⠖⡀⠀⠀⠀⠀⠀⢹⡇⢀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠘⠃⠁⠀⠀⢞⡽⠡⠀⠀⠀⠀⢸⢇⣸⣰⣤⣦⣴⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠁⣴⠀⢠⠂⢀⡔⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠀⠐⣄⡹⣿⡌⢦⠐⡖⡦⣱⠂⡠⠀⣊⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢤⡈⢦⡘⣿⣿⣿⣎⡱⣿⣼⣣⡞⡼⣿⡷⣿⡿⢫⠟⠏⠛⠋⡴⠁⠔⠁⣠
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠁⠀⠋⠁⠉⠘⠃⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣦⣿⣧⣿⠆⠀⠀⢸⣿⠉⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⠻⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣧⠀⠀⠀⠀⠀⠀⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢻⣿⣛⣋⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣛⣛⣛⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⣿⢿⣿⢸⣿⡿⢟⣋⣭⣭⢹⣿⣿⣏⢸⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⢻⢸⠏⣾⢋⣴⢿⣿⡿⢃⣾⣿⣿⣿⣦⢩⡁⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢬⢓⡀⠀⢀⣾⠃⠊⡵⠾⣋⣵⣿⣿⣿⣿⣿⡿⢘⡃⢿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠸⢋⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⢸⣿⣿⡆⠿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣿⣿⣿⣿⡟⣩⣤⣍⠴⢛⣨⣙⣫⣔⡳⠎⣩⣤⣍⢻⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣬⡻⠿⢿⣿⡄⢿⣿⠟⣸⣿⣿⣿⣿⣿⣿⣆⢿⣿⠟⣸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣜⣛⣡⣿⣿⣿⣶⡙⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⡿⢸⣿⠟⣛⢻⣿⡁⣿⣿⣷⣜⢻⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣡⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠷⠶⠿⠿⠾⠿⠷⠿⠷⠶⠾⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⢆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠄⣤⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⢀⣀⡄⠀⢀⠀⠀⠀⠀⠀⠒⡴⢠⠀⡀
⠀⢎⠵⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠪⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⢓⢀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠀⡤⠀⡀
⠈⠡⠂⠈⠉⠞⢀⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⡢⣀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢡⠒⢀⡀
⠨⠗⡀⠀⠀⠀⠀⠀⠇⠲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡂
⢐⠾⠀⠀⠀⠀⠀⠀⠈⠉⠼⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢺⡁
⠨⣴⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡕⠁
⠐⣲⠂⠀⠀⠀⠀⠀⠀⠀⠴⢊⡄⠀⠀⠀⠀⠀⠀⠐⠀⡠⠚⠻⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠣⡗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢃
⢀⡣⠂⠀⠀⠀⠀⠀⠀⠀⢰⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎
⠈⣜⠅⠀⠀⠀⠀⠀⠀⠀⢨⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉
⢀⣜⡃⠀⠀⠀⠀⠀⠀⠀⢐⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⡂
⢀⣪⠄⠀⠀⠀⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤
⠀⢑⠆⠀⠀⠀⠀⠀⠀⠀⠘⡳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠂
⠀⠈⠹⠐⠀⡀⠀⠀⠀⠀⡩⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡊⠄
⠀⠀⠀⠈⠨⠔⠂⡀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢕⡃
⠀⠀⠀⠀⠀⠈⠚⢡⡄⡀⠘⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠦⠤⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⣮

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⠶⠖⠒⠒⠒⠛⠛⠋⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⠶⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠀⢠⡀⠀⠀⠀⡍⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠞⠉⢹⡠⠞⢿⣦⡀⢠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣾⣷⡟⣠⣤⣸⠀⠀⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠺⡤⠟⣻⠀⢿⣷⠇⠀⠀⡇⡇⣀⣠⡤⠤⠶⠶⠶⠦⠤⣧⣅
⠀⠀⠀⢀⡤⠤⠤⠤⠤⠶⠶⠶⠦⠾⠤⠤⣤⣇⣀⣀⣀⣀⣀⣤⣤⣤⣼⣀⡀⠀⠀⠸⠶⠾⠅⠀⢀⡴⠒⠋⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠈⠓⢦⡀
⠀⠀⠀⡏⠰⠊⠀⠈⠉⠉⠉⠉⠉⠉⠑⠒⢦⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⢄⣀⡐⠳⢤⣄⣀⣀⣀⣀⡤⡦⠋⠉⠀⠀⠙⢻⣦⡀⠙⣄
⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢛⢻⠋⠑⠊⣈⠭⠁⡪⠀⢀⣀⣄⣀⠀⠀⢳⠙⣆⠘
⠀⠀⣀⣷⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠂⢠⡊⡠⠔⠎⠀⣠⠋⣀⡀⠈⢦⠀⠈⡇⢹⠞
⠀⡜⠁⠀⠘⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡘⡦⠰⠖⠲⠒⠀⠀⠀⠀⢤⡀⠀⠀⢀⠇⠱⠜⠀⠉⡠⠼⡀⢠⠇⡞⠑⢎⢆⠘⡆⠀⢻⢸
⠀⢣⡀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠹⡶⡄⢧⠀⡄⢀⡼⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠣⡀⠀⡠⠊⡠⡞⠁⣼⢸⠀⠀⠈⡼⠀⣧⠀⢸⢸⡆
⠀⠀⢙⣲⣞⢛⢁⠀⠀⠀⠀⠀⢠⣿⡄⢠⢙⠟⠿⢚⣁⠠⠤⠤⢤⡀⠀⠀⢀⢀⠀⡇⠀⠀⡞⢆⠘⣄⠑⠋⢠⡇⠀⡇⠈⠀⠀⠀⠡⡆⣿⠀⢸⢸⠁
⣠⠞⠁⠀⣸⠈⠈⢦⣀⠀⢀⣠⠞⢀⣧⢸⢸⠀⣾⣁⠀⠀⣠⠤⠤⡈⢢⠀⠀⠀⡀⡇⠀⢀⠆⠀⠓⠁⣠⠎⣠⠃⠀⡇⢸⠀⠀⠀⣴⠀⣿⠀⡿⢸
⢓⢶⢶⢾⠚⡇⡄⠀⠈⠉⢯⣄⣠⠞⣸⡘⡜⡰⢣⢫⣿⠛⢡⠤⣄⠈⢆⠣⠈⠒⠒⡏⠀⠈⣦⡑⢄⠐⠥⠊⠀⡆⠀⣇⠸⡄⠀⢠⡟⠀⡟⠀⡇⡼
⠈⠈⠀⠈⠀⣷⢡⠀⠀⠀⠀⠑⠲⠖⠃⢃⡇⠃⠂⠀⡇⢠⠃⠀⠈⡆⠈⡶⠀⠀⠀⡇⠀⠀⡇⠉⠲⠀⠀⠀⠀⠓⠄⢹⡀⠙⠒⠋⠀⣼⠃⢸⢁⡇
⠘⡄⠀⠸⡀⢸⣌⣉⣉⣉⣉⣉⣉⣛⣛⣸⣹⠀⠀⠀⡇⢸⡀⠀⢀⡇⠀⡟⠉⠉⠉⢇⠀⢰⡉⠢⡀⠀⢀⠔⢁⠴⠀⠀⠳⢤⣠⡤⠞⠁⠀⡞⣸
⠀⠙⣄⠀⠱⣄⠀⠀⠀⢀⣼⠃⠀⠀⠉⠉⠈⡇⠀⠀⢱⠀⠳⣤⠜⠀⢠⠇⠀⠀⠀⠈⠉⠉⠙⡦⡘⢦⠘⠖⠁⢠⠧⡀⠀⠀⠀⠀⠀⠀⡸⢱⠇
⠀⠀⠈⠛⠦⣤⣽⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠣⡀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠊⠀⠀⠔⣡⠞⠱⣄⠀⠀⠀⣀⠜⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⠿⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⣐⠠⢔⠁⠀⡨⠊⠉⠉⢉⡡⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣵⣿⣶⣶⣶⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣦⡘⢃⣤⠄⠀⢀⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣶⣄⠿⢃⡀⠀⠈⠛⠘⠉⢠⡄⠸⢟⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣙⣃⠀⡈⠉⠐⠛⠋⠀⠀⢠⠀⠀⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠁⠀⠀⠙⠲⡀⠀⠀⢀⣿⠖⠉⠀⠀⠀⠀⠀⢠⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠂⠀⠀⠀⢠⡼⣦⠀⣼⠁⠀⣿⣆⣴⠀⠀⠀⠈⠋⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠃⠀⠀⠸⠛⠀⠘⣷⡿⠘⠿⠮⠙⠋⠀⠀⠀⠀⠀⠀⢠⣶⠟
⠀⠀⠀⠀⠀⢀⣤⣀⠿⡆⠀⢸⡆⠀⠀⠀⢰⣿⢀⣀⠀⠀⣿⡇⠀⠀⢀⣀⠤⠒⠂⠈⠉⠁⠀⢴⡜⠿⠆
⠀⠀⠀⠀⡀⠈⠉⣽⠶⠂⠀⠀⠁⢀⡀⠻⢷⠅⠿⠃⠀⠀⣿⣷⠀⣠⠞⠁⠀⠀⣠⡄⠀⠀⠀⠈⢁⠀⣠⡄
⠀⢀⣤⣀⠿⡆⠀⣀⠀⠐⠈⠉⠉⠉⠛⠓⠶⢦⣤⡀⠀⠀⢻⣿⣼⠃⠀⠀⠀⢷⡝⡷⠆⠀⠀⠀⠻⢎⣯⡀⠀⠰⠿⣡⣤⡀
⠀⠀⠉⣩⡶⠀⣼⠟⠇⠀⠀⣀⣀⠀⠀⡀⠤⠶⠾⢿⣷⣄⠸⣿⡿⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠤⠤⠬⠅⠀⠀⠀⢶⡌⠉⠁
⠀⠀⠀⠉⠀⠀⠈⠀⠀⠺⢶⡌⠉⠃⣼⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣀⢾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⢀⣠⣤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⠀⢹⣇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶
⣤⣾⠋⢡⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠠⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⢀⣀⣤⣶⡶⠶⠞⣩⡿⠋
⠈⠉⠐⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠻⠶⣶⣄⡀⠀⠀⡙⢿⣄⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⣰⣤⣴⣾⠿⠉⠉⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⢻⣆⠀⠀⠀⢿⣷⠀⠀⠀⣰⡿⠋⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⣶⣄⢻⡄⠀⠀⢸⣿⠀⠀⣼⠟⠀⠀⣠⣾⢛⣁⣀⡤⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⢸⣿⠀⣼⠏⠀⣰⣿⠟⠓⠉⠉⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠺⢿⣦⡀⢸⡇⢠⡟⢠⣾⠟⠁⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⠘⣷⣿⣃⣤⣴⣶⡶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣰⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⡶⠶⠶⢾⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠟⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢶⣦⣤⣀
⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡀
⠀⠀⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢻⡇
⠀⠀⢸⡇⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣴⣶⣾⣛⣛⣛⠛⠉⠉⠉⠉⠉⠉⢹⣏⣿⠉⠉⠉⠉⢛⣛⣛⣛⣻⣷⣶⣦⣤⣤⣶⣶⠾⠟⠋⠀⠀⢸⡇
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠏
⠀⠀⠀⠀⠀⠉⠉⣿⠳⠶⢶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠶⠶⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣶⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠈⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠧⠤⢤⣄⠀⠀⠴⠖⠒⠂⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⠠⠶⡶⠀⠀⠀⠀⢶⡖⠆⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠐⠃⠤⠈⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠀⠀⠠⠊⠀⠔⠁⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠋⢀⣿⣿⣿⣎⢿⣿⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⣴⣿⣿⢻⣿⣿⣼⣿⡇⢀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣻⣿⣾⣿⣿⣣⣿⣿⣿⣿⣿⠃⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿⠇⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⣿⣼⣿⣿⣿⣿⡏⠋⢰⠸⣿⣿⣿⣿⣿⠇⣾⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣼⣿⣿⣿⡿⢿⣿⡇⣄⠸⣷⡙⢿⠟⣿⣿⣿⠟⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣎⣿⣿⣿⣿⣧⠈⡛⠳⠙⠀⠘⡇⠀⢰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣦⠳⣄⠈⠀⠀⠁⠰⠟⡛⢉⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠇⠀⡀⠀⠀⠀⢀⠀⠀⢀⠈⠁⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣾⣇⢀⣾⣆⣸⣷⣼⣷⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢫⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣷⡾⣿⣿⣷⣽⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣴⣿⣿⣟⣿⣏⣿⣿⣭⣿⣿⣿⣿⣿⣿⣷⣟⣽⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡿⠿⠿⠛⠛⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠻⠟⢻⣟⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣴⡦⠀⠈⠛⠃⠐⠿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⠀⢀⠖⢁⡀⠰⠂⠀⠀⠀⠰⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡆⠀⠀⠀⣀⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⣿⡿⠀⠀⢸⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢁⣤⣉⠉⢶⠋⣱⡄⠀⠀⠉⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⣙⠟⠉⠉⠙⠳⡄⠀⢉⣁⡶
⠀⠀⠀⢀⡄⠀⢀⡀⠉⠀⢰⣿⠗⠒⠒⢠⡄⠁⠀⠀⢁⣄⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣧⠀⠘⢧⣤⣤⣠⣿⣦⣄⠀⣸⣧⣤⠜⠓⠈⠉⠿⠀⠛⢠⣤⣤⣤⠤⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡿⠹⣿⢿⡟⠡⡔⠓⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠤⠀⠀⠠⣴⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⢶⣦⠀⠀⠊⠀⠀⠈⠙⠷⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠰⢆⡀⠀⠐⠲⣌⠉⠹⠶⠶⠦⣤⣤⣠⣤⠴⠶⠶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇⠸⣆⠘⢧⣄⠐⠂⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠛⣦⠈⣿⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡁⢀⡀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⣿⡌⠋⠉⡳⠄⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠉⢉⡁⠈⣻⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇
⠀⠀⠀⢠⡆⠀⠺⣅⣀⣉⣴⣦⣼⡏⠀⣿⡄⢰⡆⠀⠀⠀⠀⠀⠀⠀⢆⡘⠃⠸⡇
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⡀⠙⡿⢳⣶⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠚⠀⠀⠉⣿⣷⠈⢣⠀⠀⠀⠀⢀⠏⠀⡌⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡁⠀⣸⡇⠀⠀⢀⠟⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠃⠀⢸⡇⠀⠀⠿⠀⠀⡀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡀⢾⠀⠀⠠⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡆⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣈⣉⣉⣉⣉⣱⡿⠟⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⣀⣀⣀⣀⣀⣶⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠉⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⣧⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⠀⠀⠀⠀⣰⠇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⣆⠀⠀⢠⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠹⡄⢠⡏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢻⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡾⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣼⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣠⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠛⠋⢰⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡴⣿⡁⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⠁⠀⢀⡴⠋⠀⠈⠻⣆⠀⠀⢙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠛⢦⠟⠁⠀⠀⠀⠀⠈⢳⡴⠋⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣤⢞⣩⠤⠤⢤⣀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣀⣤⣤⣬⣛⢦⡀
⠀⠀⠀⠀⢠⡞⡵⠋⠀⠀⠀⠀⠈⢷⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⡞⠁⠀⠀⠀⠉⢳⡝⣦
⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⣼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⠀⣿⢸⡆
⠀⠀⠀⠀⠸⡆⢳⣄⠀⠀⠀⠀⢠⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⣠⠏⣼
⠀⠀⠀⠀⠀⠹⣦⡙⠳⢤⣤⠴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⣤⡤⠞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠴⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣴⢲⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣬⣦⣤⢤⣍⣻⣿⣿⡶⣿⠾⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠰⠦⣿⣉⠀⠐⣶⣶⣿⠳⠞⣿⣿⠿⠋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣻⣿⣧⡳⣾⣿⡞⢲⠛⡻⣿⣤⣶⡾⣤⣤⠖⠺⣿⣯⣣⣤⢠⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣶⣬⠵⢿⣿⣧⣠⣃⣽⣶⠀⠀⢘⠛⠛⢲⠿⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠻⠟⠁⡟⠻⣿⣿⣽⣷⣿⣿⣿⣿⣿⣷⣴⣾⣦⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⣤⣤⣥⣥⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣷⣿⣿⠛⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢀⣀⣤⡟⢿⡿⠷⡌⢤⡛⠻⢿⢿⣿⣿⣿⣿⣻⣿⣿⡿⢝⠻⠿⢣⣖⢾⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠛⣻⣿⣟⣿⣿⣷⣦⣾⣿⣶⣦⣼⣿⣿⡿⣷⣿⣿⠐⣀⣉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠇⠈⢋⣻⣿⠇⢿⡻⣿⣿⣿⢫⣾⣿⣿⣿⡟⠻⠉⠀⢴⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣬⣿⣿⡿⣧⢹⣿⣿⣿⣿⣵⡆⢀⣷⣼⣿⣶⣤⡐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄⡀⢀⢀⡛⣿⣿⣿⠹⣿⢠⣿⢿⣿⣿⣿⣇⠀⢿⣯⡍⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⢿⣿⣿⣮⣭⣯⣥⡉⠀⣿⣧⣶⣄⢋⣭⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠹⠷⠇⢡⣿⠉⠹⡟⠀⣿⣷⠛⠷⠻⣼⣿⣻⣿⣃⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⡉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠒⠢⠀⡈⠙⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠺⠛⠃⠃
⣿⣿⠿⠿⡿⢻⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣾⣿⡄⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⢀⣤⡄
⠀⠀⢸⣗⣛⡇⠀⠀⠀⠿⠿⠿⠿⠇⠀⠀⠀⡿⢽⠀⠀⠀⠀⢸⣗⣺⣧⠀⠀⠀⢸⣷⣺⡇⠀⠀⠀⠸⣿⡷⠉
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⢛⡛⠁⠀⠀⠀⠀⠾⢻⡆⠀⠀⠀⠀⢱⣮⠁⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠿⡼⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠠⠶⠶⠆⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⠰⠶⠶⠆⠀⠀⠀⠸⠿⠿⠇

⠀⠀⠀⣴⣧⠀⠀⠀⠀⠰⠶⠶⠆⠀⠀⠀⠀⣰⢦⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣏⣸⠃⠀⠀⠀⠀⡐⠇⡀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢰⣪⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠛⠫
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡟⡟⡇⠀⠀⠀⠘⣷⣿⡇⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⢀⣷⠶⡆⠀⠀⠀⠀⡟⢻⠁
⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⠧⠴⠇⠀⠀⠀⠰⠿⠿⠇⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠠⠿⠿⠆

⠀⠀⢰⣒⣾⣇⠀⠀⠀⢀⣶⣶⡆⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⢠⣀⣠⣤⠀⠀⠀⢠⣶⣶⡤⠀⠀⠀⢠⢤⣶⡄
⠀⠀⢸⡇⡀⣯⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⢸⣯⣹⡇⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⢸⣾⣿⡇
⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣄⢀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⡔⣿⠁⠀⠀⠀⠀⣚⢻⠀⠀⠀⠀⢸⠟⡙⣿⠀⠀⠀⢸⡽⠋⡇⠀⠀⠀⢸⣛⡛⡇⠀⠀⠀⢸⣿⣛⡇
⠀⠀⠀⠿⠿⠄⠀⠀⠀⠀⢛⣼⠀⠀⠀⠀⠼⠿⠿⠿⠀⠀⠀⠸⣧⣤⠇⠀⠀⠀⠸⠦⠥⠇⠀⠀⠀⠸⠿⠶⠇
⡀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡚⠛⠛⠿⠷⠾⠷⣦⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡵⠞⠋⠡⠀⠐⠉⠀⠀⠀⢀⡠⠀⠀⠉⠉⠛⠟⠲⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⠍⠀⡀⢀⠀⠜⠁⡀⠐⣉⣩⣥⡤⠴⠒⠒⠒⠒⠶⠤⢤⣉⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⢯⠀⠀⠚⠉⢁⡤⠀⠀⢉⡴⠟⠛⠉⠁⢀⡀⠀⠁⠤⠤⠄⡀⠀⠀⢈⠉⠛⢶⡄
⠀⠀⠀⠀⠀⢀⡾⠁⠥⠂⠐⠀⠀⠋⠀⣠⢞⡍⠊⠁⡀⠀⠐⢉⡀⠤⠖⠂⠀⠬⠤⢴⣒⠈⠙⠆⠈⢻⣦
⠀⠀⠀⠀⠀⣸⠃⠀⡀⢠⠀⡔⢀⠞⣴⢫⠏⡔⠀⠚⡁⠀⠊⠁⠐⠀⠈⠁⠀⠀⠤⠤⢤⡁⢤⡀⠀⢤⠙⣆
⠀⠀⠀⠀⢀⣿⢃⢿⣇⢏⡄⠁⠀⣼⢃⠏⠘⠀⠀⠊⠀⠀⡴⢀⡠⠴⠒⠚⠒⠲⠤⣀⠒⠲⠀⠈⢦⡀⠀⢹⡄
⠀⠀⠀⠀⢸⡇⡀⠘⠀⣼⠀⡀⢸⡇⠀⠀⠀⢰⢃⠆⠀⢀⡴⠋⠀⠐⠉⠀⠀⠀⢐⡊⠛⢆⠀⢠⠀⢇⠀⢸⡆
⠀⠀⠀⠀⣾⠁⠀⣰⠀⠁⠀⡇⣸⠀⢀⡀⠀⡎⢀⠀⠀⡞⠰⠁⠀⡴⠒⠀⠲⡂⠀⠁⠀⠈⡇⠸⠀⠈⡆⠀⣟⠄
⠀⠀⠀⠀⢻⡆⠀⢁⠞⠠⡆⠇⣿⠀⠀⣿⢸⣇⢱⠀⠀⣇⠀⣄⠀⡗⢄⡼⠀⠙⣦⢠⠀⢀⡇⠀⡀⢠⠀⠀⣸
⠀⠀⠀⠀⢸⡇⠀⡌⠀⠀⠀⠀⢿⡀⠀⠇⠀⢿⠘⠆⢀⢿⣆⠈⠂⠀⠀⠀⠀⢀⡟⠈⠀⣼⠁⠐⠁⡏⡄⢠⡏
⠀⠀⠀⠀⠀⡻⣄⠈⠏⢦⠀⢠⠘⡇⠣⠀⠀⠘⣆⠀⠈⠈⠙⠓⠲⢶⡤⠤⠖⠋⠐⠀⢀⡏⠀⠀⠰⠁⣠⡾
⠀⠀⠀⠀⠀⠀⠹⣗⠄⠀⠁⠀⠁⢧⠀⠀⠸⡄⠈⢷⡀⣀⠀⠂⠀⠀⡠⠄⣀⠀⢀⣴⠏⠐⠀⡴⠀⣨⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠳⣄⠰⣌⠲⠊⠳⣄⠀⠑⠀⠀⠉⠈⠓⠦⣤⣤⠤⠤⠤⠖⠋⠁⢀⠀⠈⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⠦⡀⠙⢶⣄⠀⠈⠂⠀⠀⣀⠀⠀⡉⠓⠂⠀⣀⠴⢋⣠⣴⠏
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠐⠈⠓⠦⣬⣀⠠⣙⠳⣤⣐⡒⠒⠀⠀⠘⠋⠴⠂⢉⣀⣴⠟⠋⢁⡀⣀⢀⣠⠄⡀
⠀⠀⠀⠀⠀⠑⠐⠂⠐⠝⠂⠐⠒⠀⠀⠍⠛⠛⠛⠶⠬⠽⢿⠿⠿⠿⠶⠶⠛⠋⠁⠋⠊⠁⠉⠋⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⡻⠶⡄⠀⠀⣴⢶⣞⣟⣛⣻⣛⢻⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⡴⠶⠿⢷⡖⢩⠀⠀⢹⠸⡏⠉⠉⠉⠉⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠋⠀⠀⠀⣼⠟⣿⠀⠀⢸⣧⣷⣀⣀⡀⢰⡏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣼⠇⠀⠀⠀⣸⣟⣩⡇⠀⠀⠸⣧⣍⣽⣹⣟⣻⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⣿⠀⠀⠀⠀⣿⣯⠼⢻⣆⠀⠀⣿⣹⠏⠉⠉⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣿⠀⠀⠀⠘⢯⣤⠖⠻⣧⠶⠛⠛⠋⠀⠀⠀⣿⠁⠀⠠⣴⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡇⠀⠀⠀⠀⠈⢳⣄⢹⡶⠃⠀⠀⠀⠀⠀⠘⠷⣤⣴⠿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣿⠀⠀⠀⠀⠀⣿⣽⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣹⡄⠀⠀⣼⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⣀⣤⠴⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡇⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠷⠶⠦⠄⠀⠀⠀⠀⣿
⣠⠞⣡⡴⠖⠛⠷⢾⣭⣟⡳⣤⡀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡇⠀⢀⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⣿
⣿⠀⣿⡀⠀⠀⠀⠀⠉⠛⢷⣮⡹⣦⡀⠀⠀⠀⠀⣼⠁⣾⠀⠀⢸⣿⣿⣛⡛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢠⡿
⠸⣦⡘⢷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣯⡙⠶⠤⡤⠞⢁⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠈⠛⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⠶⠟⠁⠀⠀⠀⣸⣿⣯⣭⣉⣉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠙⢦⡹⣄⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡜⣦⣀⠀⠀⢸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣤⣀⣀⠀⠀⠀⠈⠁⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢸⣿⡻⢟⣦⡀⢸⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠤⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⡹⢻⣦⣷⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡜⣹⠇⣳⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠒⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣾⣋⣷⣄⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣄⣀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⡳⣄⠀⠀⠀⠀⠛⠛⠛⠒⠒⠶⠶⠶⠶⠶⠶⠦⠤⠬⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢭⡓⢦⣄⣀⠀⠀⠀⣶⣦⣤⡀⠀⠘⠉⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⢭⣽⣿⣭⠽⢷⠞⠁⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢰⠃⢠⠂⠖⢀⠞⡥⠒⢍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸⠘⣸⣿⠀⠘⢦⡀⠈⠀⣼⢷⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⡘⣄⢧⢣⢿⣦⢳⡀⠙⣦⢣⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠀⢂⡄⢢⡻⣦⣷⣆⢻⣧⢧⣦⣿⠆⡇⡀⠈⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡺⢽⡛⢽⣿⣿⣶⡟⣸⣿⣏⠞⢁⠁⣠⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠓⠀⣽⠈⢿⣿⣿⣷⣿⣿⢡⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠤⣤⣀⣱⣀⣿⣜⣌⣿⣿⣿⣿⡿⠿⠿⠇⢄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⢀⡇⠁⠊⠈⢛⡿⠛⠛⠁⡀⠀⠀⢀⠀⠀⠀⢠⠟⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣤⣞⡁⣀⡴⣂⡁⠀⠀⠀⢀⣴⠀⢀⣤⣄⣬⣦⣻⡸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⣤⠄⠀⢸⠁⠉⣝⡇⠀⣰⠾⠿⣇⡌⠠⠀⢠⣿⡏⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠈⠻⢀⣠⣼⡧⣬⣽⡷⣿⡄⠐⠀⣽⠿⠲⡒⠛⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠋⢩⡀⠀⠝⠋⠉⠻⠟⠟⠉⠈⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⢠⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢶⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⣤⣤⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠩⠖⠂⠀⠀⣚⡧⠄⠀⠀⠀⠀⠈⠀⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⠦⢴⠀⠀⠀⠀⠀⠀⠈⢨⣖⡀⠀⠀⠠⠄⢀⠀⠤
⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⡀⠈⠡⠀⠀⠀⠀⠀⠈⠀⠀⡄⡄⣀⡠
⠀⠀⢧⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⢿⠀⢄⢢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠤⡀⠀⠀⠒⢝⠄⠁
⠀⠀⠀⠸⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⠀⠈⡀⠤⠀⡐⠀⠀⢀⠀⠈⡀⠈⢀⠀⡀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠈⢀⢂⠀⠃
⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠶⠀⣦⠀⠀⠀⠀⠀⠀⠾⣸⡜⣯⣗⢯⠵⡆⠀⠀⠀⠀⠠⠒⠤⡀⠀⠀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡄⣤⠡⣶⣦⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠼⣠⡎⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⢿⣿⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡁⢹⡂⡈⢀⠎⠀⠀⡁⠀⠂⠀⠐⠀⠀⠐⡘⡿⢰⠀⢀⠂⠀⠀⡄
⠀⠐⠲⣄⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠰⠁⠀⠀⠐⠀⠀⠀⢀⡙⢦⣾⡣⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢈⢳⡁⠀⠀⠀⠀⢈⡗⠈
⠀⠀⠀⠴⠆⠋⠀⠀⠀⠀⠀⠀⠄⠠⠀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠡⠒⠻⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠂

⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣀⣀⡀⢀⣾⣿⣧⣴⣶⣿⣷⡀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠉⠀⠈⢙⣿⣿⣿⣿⠏⠁
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠠⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⡏
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡷⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣧
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠠⣃⣼⣿⣿⣿⡁⠀⠀⢘⣿⣿⣿⣿⣿⣷⣄⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠿⠿⠿⠿⠿⠀⠀⠀⠀⢠⣿⣿⠟⣿⡿⢹⣿⣿⡆⠀⠀⠘⢿⣿⡿⠿⠛⠛⠻⠷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠓⢀⡟⣱⣤⣹⣿⡀⠀⠀⠀⠘⣀⣀⣤⣀⠀⠀⠀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⣶⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠋⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠛⠛⠻⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠔⠀⠀⣀⢡⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣠⣦⣀⣀⡀⠀⠂⠀⠀⠉⠀⠉⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⡿⠿⣷⢷⡂⠀⠰⠶⠶⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣶
⠀⠀⠀⣀⣀⣠⣀⣴⡴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⡛⢏⡋⠁⠀⠀⠈⠳⣽⣧
⠀⠀⠀⠉⣙⣿⣿⣿⣿⣷⣿⣽⣿⣟⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢋⠉⡙⠈⡁⠈⠀⠀⠀⠀⠘⠘⠇⠀⣤⣤⣀⣀
⠀⠀⠀⠀⠉⢻⣿⣿⣾⣿⣿⡿⣿⣿⣯⣭⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢏⠀⠀⠼⠉⢤⣤⡄⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤
⠀⠀⠀⢠⣤⣿⣧⣿⣿⣿⣿⣾⣽⣟⣷⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⠯⠂⠀⠀⠀⢸⣿⡿⠂⠀⠀⠀⠀⠀⠈⡀⣿⡿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⠁⠘⠉⠉⠈⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⣅⢀⠀⠀⠀⠀⠀⠀⡄⢀⡾⣿⡖⠁⠀⠀⠀⠘⡟⠰⠀⢀⡀⠔⢠⡐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢲⡀⠘⢂⢎⡣⣌⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠊⠆⠀⠁⢀⡀⠘⠀⣢⡀⠀⠐⠀⣠⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠐⡅⢸⠀⠀⠀⠀⢐⠀⠀⣿⣿⣿⣷⣯⣶⣶⣷⣤⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⡄⣨⡇⠀⠖⢿⡁⢠⣆⠀⢪
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠇⣀⡴⠀⠀⢸⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠐⠢⠀⠀⠀⠀⢀⣀⡊⡴⢋⡴⢠⣮⣷⡇⣾⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡇⠀⠀⠀⠈⠀⠀⠀⠒⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣤⠄⣠⡾⠉⠊⡬⠖⠖⣟⢱⣿⡿⢿⣧⠇⢨
⠀⠀⠀⠀⣀⡤⠀⠁⠒⠦⠀⠋⠈⠧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠟⠛⠛⠟⠶⠲⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠆⠐⢷⠈⠀⡀⠠⠀⢠
⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠏⢦⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⡿⠦⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠂⠀⠀⠀⠀⢐⠐⠒⠐⠀⠚⠃⠀⣀⠐⡀⡁⣯⢋⣤⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣫⡗⢨⠠⠀⠀⠦⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠷⠀⠀⠴⠾⠛⠺⠷⠁⠋⠿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣟⠉⠛⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠉⠉⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⢸
⣠⢖⠚⠋⠉⠛⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⣾
⠙⠦⣉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⣿
⠀⠀⠀⠉⠐⠂⢽⠟⠁⠒⣶⠒⠂⠈⠉⢳⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠀⠀⠀⠀⠈⠋⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠱⢤⡤⠂⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠒⠉⠀⠀⠀⡰⡷⠒⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⢶⠶⠾⠷⠶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣢⣀⡀⠀⢀⣠⣞⣡⠇⠀⠀⢄⠀⠈⠓⡄⠀⠀⠈⠙⡟⠛⢷⣦⠞⠉⡿⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠈⠉⠉⠀⠈⠁⠀⠀⠀⠀⠳⣄⡸⠀⣤⡀⠀⢠⣄⡴⠊⠁⠀⢸⡇⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢆⠀⠀⠀⣙⣦⣀⢀⣠⠴⠚⠁⠀⡜⣼⡑⣾⡿⠋⠀⠀⠀⠀⢸⠇⠀⢀⠇⠤⠤⠤⣤⣄⣈⣉⣿⣶⢤⣀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⣰⠋⠁⠀⠈⠉⠀⠀⢀⡠⢺⣇⣿⣿⠋⠀⠀⢹⠀⠀⠀⣿⠀⠀⢸⠀⠠⠤⣄⣀⣀⡈⢹⣿⡟⠢⠈⡆⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠻⠿⠒⢤⡤⠐⠒⠚⠛⢦⣸⡻⣼⡇⠀⠀⠀⠸⠀⠀⠀⡟⠀⠀⡇⠠⠤⢀⣀⣀⡉⠉⣿⣿⠷⠄⣠⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⢿⠉⢹⡇⠀⠀⠀⡆⠀⠀⢸⡇⠀⢠⠇⠀⡀⣀⣀⠈⠉⢹⣿⡿⠄⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠾⢷⠲⠦⣤⡀⠀⡄⠀⠀⠀⠀⠘⡇⢸⣇⣴⠖⠚⠷⣄⠀⣸⠁⠀⢸⠀⠀⠀⠈⠉⠙⠛⣿⣿⢧⣄⣰⠁⠀⡿
⠀⠀⠀⠀⡠⠊⢀⣠⣤⣄⠀⠀⠁⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⢻⠋⣿⠁⠀⠀⣷⡘⡆⣿⠀⠀⡞⠀⠉⠙⠒⠲⠦⢤⣿⣿⠀⠀⡏⠀⣸⣇
⠀⠀⠀⡜⠁⣴⠋⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠘⣄⢻⡄⠀⢠⡿⣰⢃⡇⠀⠀⡏⠈⠁⠒⠒⠶⠦⣼⣿⣃⠉⡸⠀⢠⣯⣭⢿⣦
⠀⠀⣸⠁⣼⠃⠀⠀⠀⠀⠐⣜⣆⠀⠀⠀⠀⠀⢹⡀⢀⣴⣶⣛⡛⠛⠻⢶⡶⠞⠁⢸⠇⠀⢠⠋⠐⠲⠶⠤⣤⣠⣿⡿⠉⢰⠃⠀⣼⠏⠀⠀⣈
⠀⠀⡇⢠⡏⢠⡾⢶⣄⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣷⣿⡛⠛⠙⠛⠿⣶⠾⠃⠀⠀⣸⠀⠀⠈⠁⠒⠒⠒⠤⠤⢼⣿⡏⢁⡏⠀⢰⠹⣆⢀⡴⢏
⠀⠸⣧⣿⠁⣼⠀⠀⠙⣧⠀⠀⠈⣿⡄⠀⠀⠀⣰⠋⠀⠙⢆⠀⠀⠀⠘⣆⢀⣀⣀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠷⠶⢾⣿⡛⠋
⠀⠀⠀⣿⠀⢿⠀⠀⠀⠘⣧⠀⠀⢸⣷⠀⠀⢠⠃⡞⠛⣆⠘⡆⠀⠀⠀⢸⡿⠿⠿⠿⠷⢶⣯⣥⣤⣤⣐⣀⣠⠄⠀⣀⠀⠀⡜⠀⢆⢀⡄⠹⣆
⠀⠀⠀⢹⡀⢸⡀⢦⡀⠀⢹⡄⠀⠀⣿⡇⠀⢸⢸⠀⠀⠘⡆⢹⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠶⠾⢯⣇⠀⡼⢸⠇⠀⢻
⠀⠀⠀⠸⡇⠈⣇⠈⣧⠀⠈⣇⠀⠀⣿⠻⠤⢾⢸⠀⢠⠀⢷⠘⡆⠀⠀⠀⢻⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠇⢸⠀⠀⣿
⠀⠀⠀⠀⢷⠀⢹⡄⠋⠀⠀⣿⠀⠀⣿⠀⠀⢸⢸⠀⠸⠀⢸⠀⡇⠀⠀⠀⢸⠂⠀⣸⢫⣆⣄⠈⠉⠙⣿⢲⡶⠦⡤⢤⡤⠊⢀⡎⢠⠃⠀⢰⠇
⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⣿⠀⠀⡿⠀⠀⠈⡇⠇⠀⠀⣼⢠⠇⠀⠀⠀⣾⠳⣤⠏⠀⢹⡟⠛⠛⠉⠁⣼⢠⠀⡇⠀⠀⢠⠎⢀⠎⠀⢠⠟
⠀⠀⠀⠀⠀⠘⢦⠀⠹⢤⣼⠏⠀⢠⠇⠀⠀⠀⣿⡘⣄⢠⡏⡸⠀⠀⠀⢠⡟⡄⠀⠀⢀⡞⠀⠀⠀⠀⡼⠃⢸⠀⢧⣀⡴⠋⡠⠃⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⡞⠀⢀⣀⣠⠋⠻⣬⣋⣴⠁⠀⣀⡴⠏⠀⠈⠶⣤⠎⠀⠀⠠⠔⠊⠀⠀⠀⠳⣄⣁⡠⢊⣀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠚⠟⢉⡄⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⡄⠀⣀⣀⡀⠀⠀⠐⠋⠍⣾⣟⣡⡞⠂⣶⣀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⣭⢬⡄⢠⣿⡿⢿⣿⢷⡲⢀⣺⣿⣿⡿⣷⢀⣭⠂⠠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡶⠀⠴⣧⢬⡼⣏⢸⣿⡧⣴⣄⢸⣿⢿⠁⣽⣷⢿⢘⣭⠄⡀⣈⢻⣿⡆
⠀⠀⣠⣠⣤⡄⠸⠀⠇⠀⠀⠉⠈⡇⢑⠚⠛⠓⠛⠛⠾⡿⠏⠛⠋⡙⠖⢋⡤⢸⠁⠘⡆⣿⣇⢴⢦⠄
⠀⠀⠈⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢳⢀⣀⣀⣀⣔⢴⣶⣶⣿⣦⡀⠛⠶⢋⣤⢸⡀⢀⠇⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⠀⢠⣤⢄⣚⣋⡤⢌⡲⣿⡟⢉⢿⣿⣇⣛⡒⢏⡵⡀⠳⣊⡴⠀⣰⣶⡶⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡺⢿⣶⣿⣿⣶⣶⣶⣶⣖⡻⣿⣇⣸⣿⣿⣿⣿⣿⣯⣽⢿⣦⣀⣠⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠋⠁⠀⠀⠉⠼⣿⣿⣾⠿⣿⣿⡿⠛⠋⠉⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⢻⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠈⢻⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣄⡀⠀⠀⢀⣿⡀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⡿⠃⠈⣿⣿⣿⣿⡿⠋⣀⡴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⣤⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡿⠋⣠⡾⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⠈⠻⣿⣿⡄⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⣰⠟⠉⠐⠚⣁⡤⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⡆⠀⠀⠉⠀⣴⣄⠸⣿⣿⣆⠀⡀⠀⣠⣾⣷⡶⣶⣶⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠻⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣷⠈⠉⠉⠀⣰⣿⣿⣿⠏⢠⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣧⠀⠀⢻⣦⣿⣿⣿⣿⣿⠟⢁⠀⠀⣠⣾⣿⡿⠟⢁⣴⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣶⣄⡙⠻⣷⣄⡘⢿⣿⣿⣿⣿⣿⡿⢏⣤⣾⡿⠟⢉⣠⣴⣟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣶⣄⡉⠛⠚⠻⠿⠿⠿⠟⠛⢛⣉⣥⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣄⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠰⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠾⠿⢿⣿⣶⣦⣬⣲⣦⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣴⡿⠟⠛⠛⢉⡭⠖⠀⣀⠀⠀⠀⠀⠉⠀⠈⠭⠿⠿⢷⣦⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠾⠋⠁⠀⠖⠁⠀⢁⠀⠀⠀⠀⠀⠀⣀⣠⠋⠀⣀⡀⠀⠀⠀⠬⠶⠈⠙⣻⣷⣔⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣫⠄⠀⠀⠀⠀⠀⣠⠝⠁⠀⠀⠒⠋⠉⣉⣁⣀⣀⡤⠶⠶⠶⠶⠶⠶⢦⣤⣄⣀⡈⠙⢿⣿⣦
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⣀⡤⠖⠒⠀⢑⠀⠀⠠⠖⠋⣠⣶⣿⣿⠿⠛⠉⠀⠀⠀⠤⠒⡀⠀⠀⠀⠀⠠⢌⠉⠙⠛⠻⢿⣧⡠⡄
⠀⠀⠀⢀⣴⠟⡴⠃⠀⠈⠁⠀⠀⢀⡴⠋⠀⠀⢀⣤⠾⢋⡁⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⠑⠛⠛⠒⣠⠄⠀⠀⠀⣤⣀⠀⠀⠹⣿⡀
⠀⠀⢠⣿⠃⢘⡵⠂⠀⠎⠀⠀⠐⠉⠀⠀⣠⣴⢛⡁⠊⠉⠀⢀⠀⠀⠐⠉⠀⠀⣠⡤⠶⠒⠀⠀⠤⠤⢤⣌⣛⡒⠀⠈⠳⡄⠀⠙⢿⣾
⠀⠀⣿⠇⠀⠀⠀⠀⡀⠀⢠⠀⠀⣰⠀⣴⡟⣱⠏⣠⠀⢀⡴⠋⠀⠀⠴⠞⠋⢡⠀⠀⠒⠒⠀⠀⠰⠀⠀⠀⠈⠙⠂⡀⠀⠀⠀⢀⠈⠹⣇
⠀⢠⡟⡄⢀⣄⡄⢰⠇⠀⡟⠀⠰⢃⣼⠏⣸⠏⣰⠃⠀⠈⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠦⠉⠳⢤⡀⠀⠙⠄⠙⣆
⠀⣼⡿⠁⡞⣼⢁⡟⢠⠀⠀⠀⢀⣾⠏⢰⠋⠀⠃⠀⠀⠘⠁⠀⠀⠀⡰⠃⢀⣤⠴⠒⠛⠉⠙⠛⠛⠲⢤⣀⠉⠛⠒⠢⠀⠀⠳⣄⠀⠀⠀⣿⢆
⢸⣿⠁⠀⠀⡏⠈⢠⡏⠀⠀⠀⢼⡟⠀⠁⠀⠀⠀⢠⡟⢀⠆⠀⠀⠀⣠⡶⠋⠀⠀⢰⠚⠁⠀⠀⠀⠀⠲⣍⠻⢦⡀⠀⠀⢀⠀⠘⡆⠀⠀⢻⡈
⢸⡿⢰⠃⠀⠀⠀⢸⠁⠀⡆⠀⣼⠃⠀⠂⠁⠀⠀⡿⢡⠏⠀⠀⠀⣴⠋⣠⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠙⠆⠀⠀⠳⣆⠀⢸⡄⠀⠱⡀⠀⠸⣧
⣿⡇⠀⠀⢠⡇⠀⠈⠀⠀⣇⠀⣿⠀⠠⣄⣀⠀⢠⡁⢀⣀⠀⠀⢰⠃⠸⠁⠀⠀⣰⠛⠉⠁⠉⠩⡍⠀⠀⠀⠀⠀⠀⢸⠀⠘⠁⠀⠀⡇⠀⠀⣧
⢺⣇⢠⠀⠘⢀⡔⠀⢸⠀⢸⢰⡟⠀⠀⠈⣿⠀⣾⡇⢸⡄⠀⠀⢸⠀⠀⢤⠀⠀⣇⢤⡀⣠⡇⠀⠈⢦⡄⠀⡄⠀⠀⣼⠀⠀⢀⠀⢀⠁⠀⠀⢸
⢸⣿⠀⠀⠀⠏⠀⠄⠸⠀⠘⢸⣧⠀⠀⠀⡟⠀⢹⣷⠀⣷⠀⠀⢸⣦⠀⠘⢧⡀⠉⠀⠙⠋⠀⠀⠀⢸⡇⠀⡇⠀⢠⡟⠀⠀⡞⢀⡟⠀⠀⠀⣾
⠈⣿⠀⠀⢳⠀⡀⠀⠀⠀⠀⠀⣿⡄⢀⠀⠃⠀⠘⢿⡆⠘⠀⢸⡈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣾⠁⠀⠈⠀⢸⠁⡜⠀⣸⡇
⠀⢘⣷⡀⠀⢱⠇⢳⡀⠀⢠⠀⢹⡇⠘⡆⠀⠀⠀⠘⣧⠀⠀⠀⠣⠘⠙⠛⠻⢶⣶⣤⣄⣀⣤⠴⠛⠁⠐⠁⠀⢀⡟⠀⠀⠀⠀⡟⠀⢀⣴⡟
⠀⠈⠈⢿⣦⣄⠀⠀⠻⠄⠈⢣⠀⡗⠀⠀⠀⠀⣆⠀⠘⢷⡄⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⡟⢁⡀⠀⢠⠌⠀⠠⢚⡿⠁
⠀⠀⠀⠈⠳⣌⠓⠀⠀⠀⠀⡀⠀⠹⣄⠀⠀⠀⠹⡆⠀⠀⠻⣆⠰⣤⡀⠈⠁⠀⠀⠀⠒⠒⣠⠴⠂⠀⣠⡾⠋⠀⠊⠀⣠⠟⠀⢀⣴⡟⠁
⠀⠀⠀⠈⠉⠙⠳⣄⡀⠰⣄⠙⠶⠲⠜⢷⡄⠀⠀⠹⠀⠀⠀⠀⠁⠀⠛⠷⣦⣤⣤⣤⡤⠤⠤⠤⠶⠛⠉⠀⠀⠀⠀⠀⠁⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠲⢤⡀⠀⡈⠻⣦⣄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠉⠀⠲⠦⠤⠀⠀⠀⠀⢀⣴⠟⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⣦⡀⠉⠢⣄⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⣠⣢⠄⠀⠀⣀⡤⠚⠉⣀⣤⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⣀⠉⠉⠀⠈⠛⠷⢦⣤⣀⠀⢤⣈⠛⢿⣤⣌⡉⠙⠉⠀⠀⠀⠀⠉⢁⠴⠚⠁⠀⣀⣀⣬⡽⠛⢉⢀⣀⠀⢀⡀⢀⢐⣤⠄⢀
⠀⠘⡦⠬⠥⠀⠴⣳⠶⢄⣠⠶⠤⠀⠀⣈⣙⠛⡳⠿⠿⣦⣤⣉⣙⣻⣿⣿⣿⣿⣿⣥⣄⣤⣴⡾⠿⠋⠩⠞⠰⠚⠁⠔⠫⠗⠋⠱⠚⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰
⠀⠀⠀⠀⢰⡀⠃⣴⠀⠀⠀⠀⠀⠀⠏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣥⠁⠀⠀⠀⠀⠀⠈⠓⠺⠃
⠀⠀⠀⠀⢸⢻⡾⠸⡠⢲⠃⠀⠀⣀⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⠓
⠀⢀⠀⠀⠸⠂⡟⡺⠁⡏⠀⠀⠀⡆⠉⠱⡇⠀⢸⠀⡀⢰⠀⠀⠀⠀⠀⢸⣏⡙⢹⡆
⠀⠀⣷⢄⠀⢠⡇⠃⠞⠀⠀⠀⠀⠙⠀⠚⠀⠀⠀⣼⡇⣾⣧⠀⠀⠀⠀⠸⣏⠀⣨⠃
⠀⠀⠸⣌⢳⡀⢡⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡗⣿⣼⡶⢠
⠀⠀⠀⠈⠑⠀⡄⢰⣦⣀⡈⢇⠀⠀⠀⠀⠀⠀⠓⣆⣿⣾⣿⢇⢈
⠀⠀⡠⠒⢀⠐⡇⠁⠀⡨⠙⠾⠀⠀⠀⠀⠀⠀⠸⣯⣿⡻⣿⣷⡏⠀⠀⢀
⠀⠴⠁⣴⠏⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣓⡻⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠻⡜⠁⠀⠈⣧⠏⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣧⢸⣅⡿⠀⠀⠸⣄⢀⠇⠀⠀⠀⠀⠀⠀⣆⣁
⠀⠀⠁⠀⣤⣴⡿⠦⢤⠀⠀⠀⠀⢂⡀⠐⠀⠀⠀⢀⢽⣿⣿⢧⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⢠⡄⣾⣿⣤⡀⠀⢀
⠀⠀⠀⠀⢿⠋⠀⠂⠒⠀⠀⠀⢠⠸⠷⢤⣄⠀⠀⠀⠀⠉⠉⢁⠀⠀⣠⣧⣤⡀⠀⠀⠀⠀⢸⡃⠀⠀⠈⠇⠀⠈⠃
⠀⠀⠀⠀⠸⣧⡀⣦⠆⠀⠀⠀⠸⣄⠀⣀⠌⠀⠀⠀⠀⠀⠀⠜⠀⠀⣿⣸⣽⠇⠀⠀⠀⠀⠈⢧⣀⣀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣟⠻⡿⣿⣿⣿⣿⡿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿⢾⣿⣿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣀⣤⣀
⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠁⣼⠁⠀⠙⣩⠿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣁⣶⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠽⠛⠚⠹⡇⠀⠰⣇⠀⠀⢰⠇⠀⠀⣤⣾⣿⣷⣝⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭
⠀⠀⢸⡿⢿⡿⠋⢻⢻⠿⢟⡟⣿⣿⣿⠿⠿⠿⣿⣉⡀⠀⢰⠏⠀⠀⠀⠀⡇⠀⠀⠙⢦⠀⡜⠀⠀⢸⡇⠀⠀⠀⠉⠻⠶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓
⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠙⠿⣷⣾⠀⠀⠀⠀⠀⣧⠀⠀⠀⣸⡗⠁⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠛⠛⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠳⣤⠀⣿⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⡀⠀⢀⣿⡿⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠙⣆⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠈⢧⠀⠀⠀⠀⣠⡿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⣂⠈⢳⡀⠀⣴⡟⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣔⣂⠀⠳⣾⠏⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣂⠀⠙⣆⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣂⠀⠘⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠹⣷⣂⣀⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧⢦⡄⢹⡍⢀⡴⢬⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⠿⡭⢿⠏⠹⡜⢯⡿⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢠⠋⠀⠀⠙⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣀⣰⠃⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⡴⡞⠉⢩⠇⠀⠀⠀⠀⠀⠀⠸⡏⠙⢳⢦⣀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠞⠁⢀⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡆⠈⠳⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⠀⠀⢸⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⣇⠀⠀⠈⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡞⠁⠀⠀⠀⢀⡟⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⡀⠀⠀⠀⠈⢳⡈⠳⢤⡀
⠀⠀⠀⠀⠀⢀⡤⢖⣛⣛⡛⠉⠀⢠⠏⠀⠀⠀⠀⠀⡼⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠙⡄⠀⠉⢛⣛⣓⡲⢤⡀
⠀⠀⠀⠀⡴⣫⡾⠋⠉⠉⠙⣧⠠⡏⠀⠀⠀⠀⠀⡼⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⠀⠀⠀⠀⠀⢹⡄⣴⠋⠉⠉⠙⢷⣝⢦
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⢸⠆⡇⠀⠀⠀⣀⡼⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢷⣀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠹⣯⣧
⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⢀⡾⣸⡃⢀⡴⣫⣥⣤⣤⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣤⣤⣬⣝⢦⡀⢘⣇⢷⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢿⢻⣄⠀⠀⢀⣠⠞⣹⣅⣹⣫⡾⠋⠀⠀⠀⠹⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠏⠀⠀⠀⠙⢷⣝⢏⣨⣏⠳⣄⡀⠀⠀⣠⡿⡿
⠀⠀⢀⡜⣳⢭⣟⣛⣭⠵⠋⠁⢸⣷⡟⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡀⠀⠀⠀⠀⠀⢻⣿⡇⠈⠑⠮⣽⣛⣛⡭⣞⢣⡀
⢀⡴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢦
⠻⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢿⣄⣀⣀⣴⢞⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⢦⣀⣀⣠⡾⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣮⣭⣵⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣮⣭⣵⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢭⣉⡓⠲⠦⣤⣀⡑⢄⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠉⠛⠳⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣶⣄⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠛⠋⠉⠉⠈⠙⠶⣄⠀⠀⠀⠀⠀⠈⢧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⣉⠥⠤⠤⠤⣄⠀⠀⠀⠀⠀⠙⢗⠺⠿⠦⠀⠀⠀⠀⠀⢳⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢁⡤⠚⠉⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠗⠒⠒⠲⠦⠄⠉⠲⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⠖⠦⡄⠘⣇⠀⠀⠀⠀⠰⣿⡷⠋⣠⠤⠒⠒⠙⠛⠁⠀⠀⠀⠀⠀⢀⣹⠦⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠸⡀⠀⠘⡄⠈⠙⠒⠶⠖⠚⠉⠀⠚⠓⠶⠤⢤⡀⠀⠀⠀⠀⠀⠀⣴⣋⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠋⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠋⠀⠀⠀⠀⢀⡀⣼⠀⠀⠀⠀⢠⠰⠋⢸⠄⠀⠀⠀⠀⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⢾⣡⠇⠀⠀⠀⠀⠘⢆⡷⠚⠒⣄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣤⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⠀⠀⣀⡴⠋⠈⣇⣿⣀⠀⢀⣀⣀⡜⢀⠞⢙⣞⠒⢢⡄
⠀⠀⣠⢄⠀⠀⠀⠀⠀⣀⣾⠉⠉⠈⠉⠉⠻⣍⠉⠉⠉⠳⣤⠴⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⠙⠻⣄⠀⠀⣇⣀⣀⣀⡀⠀⠀⢀⠘⠒⢉⣿⡑⢫
⣠⢄⣳⡈⢣⣀⠔⠂⣁⣀⣸⠀⠀⠀⠀⠀⢠⠈⠙⠢⢄⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣄⠀⠈⠓⠚⠳⠀⠈⣇⠀⣆⡇⠀⠀⠀⠀⠀⠀⢱⠶⣻⣥⠑⠓⣄
⠑⢦⣤⡁⠀⢀⠰⠯⠥⣄⠈⢧⠀⣀⣠⠤⡞⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠚⠉⠁⠈⠙⡶⠦⠤⠤⠴⠖⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢣⢻⢣⣔⣒⣛⡤
⠀⠫⠤⠴⠚⠉⠑⠒⠒⠁⠀⠀⠉⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡶⢯⠱⣅⡤⠋⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⠉⠒⠒⠒⠒⠒⡖⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⡼⡀⠀⠀⠀⠒⢻⠉⠉⠉⡟⠛⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡞⢦⡷⡼⠓⠲⠤⣄⣘⠀⠀⠀⣇⠀⠀⠀⣀⣀⠤⠧⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠭⣍⣐⣒⣒⣒⣒⣒⣒⣺⡄⠀⠀⢸⠉⠉⠉⠉⠀⠀⠀⠀⠉⢀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠄⡀⠀⡀⠀⠐⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠚⠁⠁⠀⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡅⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠐⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠁⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⡆⣿⠇⡲⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠦⢇⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⢀⣿⣸⣿⣛⣛⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠙⠒⠠⣄⣀⣀⣀⣀⣃⣰⣤⣦⡴⠶⠾⠿⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿