#include<cmath>
#include<math.h>
#include<ctype.h>
#include<algorithm>
#include<bitset>
#include<cassert>
#include<cctype>
#include<cerrno>
#include<cfloat>
#include<ciso646>
#include<climits>
#include<clocale>
#include<complex>
#include<csetjmp>
#include<csignal>
#include<cstdarg>
#include<cstddef>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<ctime>
#include<cwchar>
#include<cwctype>
#include<deque>
#include<exception>
#include<fstream>
#include<functional>
#include<iomanip>
#include<ios>
#include<iosfwd>
#include<iostream>
#include<istream>
#include<iterator>
#include<limits>
#include<list>
#include<locale>
#include<map>
#include<memory>
#include<new>
#include<numeric>
#include<ostream>
#include<queue>
#include<set>
#include<sstream>
#include<stack>
#include<stdexcept>
#include<streambuf>
#include<string>
#include<typeinfo>
#include<utility>
#include<valarray>
#include<vector>
#include<string.h>
#include<stdlib.h>
#include<stdio.h>
#define LL long long
#define ll   long long
#define ull unsigned long long
#define ULL ull
#define mp make_pair
#define pii pair<int,int>
#define piii pair<int, pii >
#define pll pair <ll,ll>
#define pb push_back
#define PB pop_back
#define p push
#define P pop
#define INF 2147483647
#define ull unsigned long long
#define vi vector < int >
#define vpii vector < pii >
#define pq priority_queue
#define rep(i,n) for (int (i)=0;(i)<n;(i)++)
#define End return 0
#define mi multiset <int>
#define si set <int>
#define sll set<ll>
#define mll multiset<ll>
#define vll vector <ll>
#define Rep(i,a,b) for (int (i)=(a);(i)<=(b);(i)++)
/*
JLjjuYjuujUuuYJLjJjjuuuu22S5SF5u1u2U2u2U5U2U1U1U1u1u2U52F21u1U2u1U1U525U5151F11U2U2uUu2U5UF5SSkSX55U122252522U521UUjuuUJUuuJuuFSPqN0OM@MMMMOMMMMMMMOOG8Z8ZEXkUUYjLuu515251S1FFF1SFkFXSPFSFkFXSF5S5SSF1S5F155k1F1F1F5S5F151F1S1F2F5F15155S151F1521u2U2uuuUuuuujuuuuUuuuUuuuuuuuUuuuUuuuUuUuSP8GGE8Z8GGGO8OG8G
YvvYvYLYLJLYvLvLvvvYLYYuuuJjLYLYLjuujjYujuYujuJujuYuJuuujUJuJuJuJujUuuu2u1U522u2uuJjYjJuuUu2151F22u2jUuUuuuuJJLjLYvLvJYjJuuFX8M@BBMBB@MBOMOOOOOMOOOOOOGMOMOMOM80XXF2LJjUjuu5S5U1U1UF152F1F1F15U1UUUF21U1u2UUU1u1u2u2u2uUu2u2uUuUuUu2u2U2uUuUU2uUuujuJYLJYJYJYJYjYjLjYJYjLJYYLjYJLJYJLJLYLjuP08N00ZEZZGEGEGEG
YvJYjYjYuYJLJLYLJYYYuJuuuYvr;iii7vjj2uujuuUU2u2uUuUuUu222UUu2uuj2U2u2U1U11k5SFk5F21U1u2U5251S5S1F2121U522JuYjYu1PqZGMOMMBBBB@BBBBBMMMO8OMOMOMOMOM8MOMOMMMOMMBM@B@@@MO0NXS21U15SFFFS5kFSSSFkSkFS5F555S5S151F151F1F25151F1F15152F251F111F252F1F2521u2uuuuuUuUuuuuuUjuuUjUuuuujuuuuUjUuuuUuuu5P8GZEGEGEGZ8G8G88
LLYYLJJjYJLYLYLYLJLJYJJjv7i:,..:,:;vJuYujuu2U2uUuujuuuu1U2UUuujuuUuUu2u1U12F1S5F15U2U1U51F1F5F5522U12UjuJU1PqOOMB@@@B@B@B@B@MMOO8OEGZO8MOMOO8OGMGOGO8M8MOMOMMMOMOMMMBBMBMMZq5515US5SFXFkSkFkFS1S555S5F5F151F5F1F151F2F2F25151F152525U525212F15U1U1u2uUuuuujUjuuujujujUjujUuujujujUjuuuuuJu5qEO0EZZE8ZGGO88GO
YvJYjYJLJLJLYLjYJLJYJJuv7i:.. ...,:rYLJLuYujUuujuJujujUu2uUuUjujUuUu2u2u1U1UF251S5515155S1521U2U1uuJjYU5E8MB@BM8MMMOMMBMMOM0E0EqENE0ZZOOO8OGOOMOMOMOOOM8OOMOMOMMMOMOMMMOMMBMBOES5U2U15SFXSk5S1S1S5FFF1F5515152F25211F11112525252F2511152F2515U2U2uUuujujUuuuujujujujujujujujuJuuujujuJujuuFPZZZ08Z8EGZ8Z8G8G
LLYJYjYJLjJJYjYJYJYuYjJLri,.   ..,:rvLLYLuJuuuJujujuJuuUuUuujuJujuu2u2UUU511U12F1F152F12uU22uuLLYju5XZM@B@MMMMOOM@MMOMO8EZqNq0qZZOOMOMMMMBMBMMMBMMMMMMOMOMOMOMOMMMMMMMOMOMOMMBM@MOq1u21SSk5SSS5S551S5F1F1F252F112521252F21252F152525U511252115U2UUUUuujuuuuuuujujuJuuujUuuuuuujujUjujujuJU5q0Z0GEGZGZ8ZGZGZ8
L7LvYLYvLLJvYLYLYLYLLvvri::.. . .,irLLLvJYujUuUuuJuJuuuuUuUuujuJuuUuUu2u2u2U2U121u1U1U2U1jJLjYu5EOBBBBBMMOO8MMBB@BBMMZZ00NG8MOMOMMMMBMBMMMBMMMMOMOMOMOMMMOMMMOMOMOMOMMMOMOMOMOMMBM@BMqP1U1SFS1F5S2S5F1F1F151F251F25152515252515211521151F252F22U1u2uuuujujUjujujUjUjujujuuuJujujuuujujujuuSXZEZEZ08ZGZ8Z8ZGG
rr7v7v7v7vvv7v7L77r7;;ii::::,,,::ir7vLvLLJJuuUuujujuJUuUuUuUuuJujUu2u1u1UUuuu2U1u2jUJuJYLJJ5kEOBMMOMB@MM8MOM8BB@M80ZEZGMOMMBMBMBOO8O8O8OOM8MOOOMOMOMOMOMOMOMOMOMOMMMOMOMMMOMOMOMMBMMB@OZF52FFS1S5S1k5F552F251525112525252525151525252F1F252F25u1UUuuuujujuuUuuJujujuJujuJujUJujUjuJuuujuj2Fq0G0GEGZGZGZGZGZ8
rir;rr777777777rrrrii::,::i:iiiirrvvLvLLuJuuUuUuUJjJujUuuuUuuJuuuuUuUu2u5uUuUu2uUuuLuLLJSNMB@BBMMMMMMMMOMB@B@B@MGPNNZZOOOZZ0EE8Z8ZG0GZOOM8O8MOMOMOMOMMMOMOMOMOMOMOMOMOMOMMMOMMMOMOMMBB@B@ZXUU5S251S5F2F151515252F21252521251125U12F2121U5U5U512U2u2uuuUuUuuuuuujujUjuJUuujujujuuuuuuUjuuuuSPEEZEGE8EGE8Z8Z8Z
:::i:iirir;rr7;ririi::.,::irr77v77r7i7vujuj1uuJuuuJjjuuuu2uUjuJujuuujUu2U2UUu2juuuvLLFEOOMMBMMMMOMMMOMMBB@B@MMZENG8OZZEZNNXq0ZN0PZ0Z0GGO8O8O8MOMOMOMOMOMOOOMOMOMOMOMMMOMOMOMOMMMMMMMMM8BB@ME11U1152F55252F15112125252525252525U1U5U1U12525U125U2U2U2uujuuUuUJujujujuuuJuuUJujujuJuJUjuuUu25NEG0GEZEGE8E8ZGZ8
::::i:iiiiiiri;iiii::::::irr77v7r::,::;r77LYjJujujuJuuuuUuUjuJujujUjujuu2U121uJLvL5XOM@B@MMOMMMMMMMOMOBBBMMZEqEZO8M8OGO8OZGZ88G0E0Z0ZZOGO8M8OOMOMOOOMOOOMOMOMMMOMOMMMMMOMOMOMMMMMOMOMOOOMMBBMNk21U15S1S5F251F11U525211F2F25111521U1U12121U52121u1U2uujUuUuUuujujuuuuUjuuujuuujujujuuuuUuuuFPGEGEG0ZZ8ZGZ8ZGZ
:::::::iii:iiiii:i:::i:ii;irr7rr:,.....,,ii7vJLJjujujuJuJujuuujuuujujuuuu512u2u1X88MMBMMMMOBM@MMOOOOOMMMMOZZE8OOG80Z8OEEq0NZZOGGEG8OOMOMOMOMOMOMMMOMMMOMOMOMOMOMMMOMOMMMOMOMOMOMOMOMMBMMOMOBB@M0222F1S5F5F2525U12525152515252521U121U5U121U5U1U1U2u2uuuUuUuUuujujujuuujuJujujuJujuuujuuUjU50EG0GEZZ8EGZ8E8E8
,.:,::::::::::::::::::iirirrr;r;i,.     .,:i7LjYujuJjJujuJuJuuujujuYjJuJuuUjUkGOBMMMMMMOOMBMMOO8OGMMBMM88EZZO8O0E0GEG0qP0NGE88O88GMOMMMOMOMOMMMOMOBMMMMMMMMMMOMOMOMOMOMOMOMOMOMOMMMMMOBMMMMOOMBBMX5uF151SF1U515211511252525251125U121212121U5U2u2U2uUjUuUuUuujuuuJuuuJujuuuuujujuuujUjUuuUSPZZZ0GEZEGE8GGE8Z
...:,:,::::::::::::::::ii;irrrrr:,.      ,i7LuJjJJYjJujuJujuJuJuuUjuYjjuJuuXEOOOOMOMOMOMOMMMOOZN0OOMMM8OGO8O8O8OOMOMMMOMOMMM8OGOGOGO8MOMMO8MMBMMOMOMOMOMOMMMMMMMMMMMMMMMOMOMMMOMOMOMMMMMMMMMOOM@BBGX51u11S2511251F252521151525U1U1252121U52121U2UUuUuUuujuJuJuJujuJuuUjUjujujujujUuuuUjuu2S0ZG0Z0GEGZ8ZGE8EO
..,.,.:,:,:,:,:::,::::::iiriririi:,.. ..:iLJujjYjYjYjJuJujujuJujUjUJJvYu1SGMB8G8MOMOMOMMMMM8OZE0G8MOOOMOMOMOM8OG8EENNqNq0P0E8ZZ08G8ZGZGZ8ZMOMMMOMOMOMOMMMMMMMMMMBMMMBMMOMOMOMOMOMOMMMMMMMMMM@MMB@B@B05Uu1uF21252F211F2125151521U1U5U1U5U521U5U2U2UUuUu2uujuJUjujuuujujuuUjuuUjujUjUuUuUjUUkP8EZ0Z0ZEGEGE8ZGG
.......,.,...,.,,,,::::iiiiri;;riiii:::rvuuuYuYuJuYuYJYjjujuYuJujuUuLj208MMMOZEMMMOMOMMMMMOOZZEOOMOMOMOO8OOMMMMMOMMMMMOMOOMMMMMMM@MBOM8OZ8OMOMOMOMOMOMOMOMOMOMMMMMMMMMOMOMOMMMMMMMOMMMMMMMOBMMOMMBMBBMk2u125212125122F1525151525U125U1212121U1U1U2uUuuuuuUuujujujujUJujujuJujujuuuuujujuu2S0Z80ZEZ0GE8EGZGEO
. ..............,.,,::::iiiirirrri7rvLuJujuYuYjYuJjYjLJYjJjYJjuJuYLLFEMMMO8GOOMOOOOOMOMMMOOG80GOOOO8GZ8ZOGOOO8OG8GGZMOMOOOMOMOMOMMMMBMBB@MMOMOMMMOMOMOMMMOMOMOMOOOMOOOMMMOMOMOMMMMMMMMMOMMMOMOMMBMMMBBBZkuUU51525252F1FUF152F1521U1U5252521U521U2uUuuuUjuuuJuuuuuuUuUJUuuuuJuuuJujUuUjujUUkqGEZ0GEGEZZGZ8G8G
   ... ..........,.,,::::iiiir;rr7rvvuuUJuYuYjJuJjLJYjYJLuYjYuLYvvvFZMO8EZGMMMOO8MMMMMOM8OGGEGE8G8GGEZZ8G8GO8M8OEG0ZEGG8Z8GM8OG8GOGOGOOMMMMMMMMMOMOMMMOMOMOMMBMMMBMBMMMMMMOMOMOMOMMMMMOMOMMBMBMBMBMMB@BBOEu2251F1F2F2522U1U521U1U2U1U5u2U12121U122u2j2UUuuuujujujuJUjuJuuujUuuuuuujUuujujUF0ZGEGE8ZGZ8EGEGEG
. . ... ........,.:,::::::ii;irirr77LJjYujuYuJuJJYjYjYJYJYJYjvYLLvNOMOOEZZOMMOOOMMM8O888MMMOOE0kN0GGO8OOOGOEO88G8GOG8EZ0NXPX00GGO8OZ0kq0MOMOMOMMMMBMBMBMBMBOMMMOMMMMMOMMMOMOMOMMMOMMMOBMMMBMMMMMBMBMBMBMBBG5uJ2151F1521252121U121212121u221u221U1u2jUu2uUjuuujujUuujuuuuujuuUuuuUjUuUuUuuuSq8ZGEGEGEGEGEGEZZ
   . . . ........,.,,::::iiiirirr7rvLuJjjUjuJuJjLjYjYJLJYJYjJY7vUZ8OEGEOOO8MO8GMMM88ZGZOOMOGqSF0qEEZEGGOZZEZ0EPqq0qENZNE0ZNNXNN8GO88EZ00kEOOOMOMMBMBMBMBMBMMMMOMMBMMOMOMMMMMOMOMMMOMMMMMOBMMMMMBMMMBMBMBB@B8F2u52F2F11U1U121U1U521U121U1U1U2u5U1U2uuu2u2uUuujuuujuJujuuuuuuUuUuUjUuujuuUJ2F0ZGEGEGEGZGEGE8Z8
          ......,.,,::::::i:;i;;rr77JJUJujUuuJjJjYJYjYJYjYuLvrJqMMGEEN8ZO8O8ONZGM8GNZEOOOGGN0XN0ZEGGGGO8OOMOMOOZGNE0Eq0EGZ88OGGZ8G8GMMMOO8MOMOMOMMMMMOMOMOMMBMBMMMMMMOMMMOMOMOOMMOMMMOMOMMBMMMMMBMMMBMBMBM@BMP1uS5525112122U12121U1U121u2u1u2U521u2uUuUu2uUjujuuUjuJujujujujUuUuujuuUjUjuuSqGZGEZEEEZ0GE8GGZ
         ......,.,,,,::::::iiri;;rrLLjYjYjJuJjJuYJLJLYYJLJv77UZBM80ZEGGOOOGGEGE88G0EGOGGNPS00NkqPGZMOMOM8OZGZ8EZZOOM8O8OGOGO8M8OOOOMMMOMMMMMOMOMOMOMOMOMOMOMOMMMOMMMMMMBMMMBMMOMMMMMMMMMMMMBMMMBMMMBMBMBMMM@BBNFuU1F211F1FU51F2521U521U525U52121U2uUuujUuUuujuuUuujujuuUjUjuuUjujujUjuuuu2SZZGZG0E0G0Z0GEG0G
      . ........,.,,::::::iiiiiirr7vJJJYjJjJjYuLjYJYuJLLYv7YEB@OGEOOMOMOMOOGGEG0G0EZM88P5FXXqFSXNq8GO88GO8OOOZG0ZNNNGZGZ8Z88MOMOMOMMMMMOMMMOMOMOMOMMMOMOM8MOO8MOMOMOMMMMMMMMMMMMMMMOMMBMBMBMMMBMBMBMBMBMBMMOBBB0Su52F11u52F2525U1252125U1U521U1U2uUUUuuuUuujujuJujuuujujujUuUuujUuuuuuuuUUXN8ZGEZ0G0ZEGEZ0ZE
       . ......,.,.,,::::::iiiirr77LYJJJLuYuJuYjYJLjLjv77uXBMB8ZZO8OGO8O8OZZ000ZEZ0G88XXXqqPXkXZEG8M8OZGE8ZZGMOM8MOOGO8O8OOO8O8OOOOMOMMMOMOMMMOMMMOMMMMM8MOOOOOMOMOMOMOMOMOMOBMBMMOMMMMMMBMBMBMBMBMBMBMBMBMMO@@BZ5U12k52U115U52125U5U521U5U5U122U2u2uuuujUuujuuuuuuUjuuujujUuuuuuuuUuUjUj2SEE8EG0Z0ZEZ0G0G0Z
        ......,.,.,.,,::::::i:iir7LLYYjLJYujuJuJuYYYL7LYPO@OM8O8GEO8OG8Z80ZP0qEZOGGNNk55XSk5kEOZGGO8O8O88ZGEZ0Z0GEO8O8OOO8MOOOMOMOMOMOMOMOMOMMMOMOMOMMMOMOMOMMMOMOMOMOMOMOMOMMBMMMBMBOMMBMBMBMMMMMMMBMBMBMMMBMBMB0Xuu1F2125U5152125U1U1U5212125U2u2uuuUjuuUjujuJujuuuJujuuUuujUuUjujuuuuuuSPZ0E0ZEG0ZEZEG0ZE
       ..........,.,,:,::::::ii;rvvLLJLuJuJuJjYJLjLJLrLZB@BOGZ0ZGMOMO8ZGq0NEE0EOOOPk51J7rYuuuP08ZOZ8Z88O0Z0ZEZ0GZZN0EOZ8G8GOOMOMMBMMOMOMOMOMOMOMMMOMMMOMMMMMOMOO8O8OOMOMMMOMOMOMMMMBMMMMMBMMMBMBMMMBMBMBMBMBMBMBBBZ1uF212F2511U5U5U5U1U521252121UUuUuUuuuUuuJujujUjujuuuuujuuuuUuujuuUuuu1F0EGqEEZ0ZEZEZEZEG
  .   ..............::::::::iiir7vLvYLJYuYuLJvJYjJuv7LqMMZOEZ0GGO8MOOEZqNPEZ8EG8MGkLLvvvu2XFkXEG8GO8OGO88EZ0ZEGEGZO8OOMOMMMOMOMMMMMMMOMOMOMOMOMOMMMMMMMOMOMOMOMOMOMOMOMOMOMOMOMMMMBMMMBMBMMMBMBMBMBMBMBMBMBMBMBMBBMk1255F252FU1212121U52521U121U1U2uUuUuuuUJuuujuuujujujujujUuUjUuUjuuujuUSqZEENGEZEZ0ZEE0Z0
   . . . .   . ......:,::i:iirr77LvLLJLJYjJjLYLuLLvLYGB@8GG8ZO8OOMOM88EZN0N8ZG0ZOGUvrLJ21PP0qGZ8ZO8GZ8EGOMOM8OEZE8OMOMOMMMMMOMMMOMMMMMMMOMOMOMOMOMMMOMMMOMOMMBMMMBMMMMOOOMMMMMMMMBMMMMMBMBOBMMMMMBMBMBMBMMMBMBMMM@B8k2uF151522U515U1U525U1U5U521uUuUuUuUuuuuJUjuuujUuujujuuUuuuUuuuuJujuJu5q0Z0E0G0GEZ0ZNZNZ
                    ....::::iiir7vLLjYLLjLjLJYuLvvJ5OBB8O8EE88MMOOMGGq0NENZZ80PXPF277rLJu2k0O8OZGEZ00qNPqP0EZEG8MOMOMOMOMOMMMMMMMOMMMMMMMMMMMOMMMOMOMMMMMMMMMMMMMMMOMMMOMOMOMOMMMMMMMMBMBMMMBMBMBMBMBMBMBMBBBMMMMM@B8125S1F11U521U1U12521U1U1U122u2u2uUuuuujuuujuuujuuujuuujUuuuujUuujuJjJFPZEZ0GZZ0GEG0E0Z0
                     . ..,,::ii;rvvYLJLJYJYJLYuJ7vFMMM00GONZZO8MOO00kPkqPPqOZNFFU2JLr77ju25PXPPN0ZEG0ZEEPPXPkNNG8MOMOMMMMBMMMMMMMMOMMMOMOMOMOMMMMMMMMBMBMBMMMMOMOMOMOMMMOMOMOMOMOMMBMBMBMMMMMMMBMBMMMMMBMBMBMBMBMBM@MPu52F2511U5U1U5U1U125U52121UUUUuuuuuUjUuuuUjUuujujujujuuUjUuUuuJuYjLuUP0EE8ZGEZ0G0Z0ZNG
                         ...,::ir7v7LLYLuJuujLYLJ28MM8O8OEOGOOO8OOGPPkPXXkZq11FvvLYLYu5SNPZEZPE08G8G8EZEGZGZZN008GMOMOMMMMBMMMMMMMMOMMMMMOMOMMMOMMMOMMBMMMMOMMMOMOMOMOMMMMMMBMMMMMBMMMBMMMBMBMBMBMBMBMBMBMBMBMBMBMMB@E1u221212FU5U5U5U125U12121U2uUuuuUuuuuuUuUjuuUjuuUuUjUjujUuUu2jjYjLYLuPGNEEGEZEZ0Z0E0ZE
               . .          .,,::rr77YLJYuJjjJ7LSMBBOMGOGZGO8M8OGOZZSFFSF5q0LY2viii7vuL2q0ZOGOZGG8ZO8OZ8ZO8O8O8O8OOMMMOMMBMMOMMMMMMMMMOMMMOMMMOMOMOMOMOMOMMMOMOMOMOMOMOMMMMMMMMMMBMBMMMMMBMBMBMBMBMBMBMBMBMBMBMBMMM@BMP2u5212521U121U1U52521U122U2u2uUuUuUjujujUuujuuuuUuUuUuuuUjUuuLYLLvLu0EEqGZG0Z0E0GNE0G
            ........... .   ..::::;;7vJYjYuJLvjkMBB8GGOGOZ8GO8GEGNNXPkP55uSXF7r:iirv22uuF5q0Z0Z00NEGMOMOMOMOMOMOMOMMMMMMMMMMMMMMMMMMMMMOMOMMMOMOMOMOMMMOMMMOMMMOMOMOMMMOMMMMMMMMBMBMBMMMBMMMMMBMMMBMMMBMMMBMBMBMBBBOBBM2u252515U1U5U52121212121U2u2uUuUjuuuuuuUuuuuuUuUuuuuuuuUjuuuYYvv7v7UPONEZGEZEG0GEENEE
         . . ............. ..,,::iirrvJjYJJJvuE@BMMOZ8GO8OGMGGEZN0kk5SkX1kuUFUrri:iL1qqPk008E0E8ZE0Z0ZGGGOOMMMOMOMOMMMMMMMMMMMOMOMOMOMOMOMOMOMOMOMOMOMOMOMOMOMOMMMMMOMMMMMMBMMMBMMMBMBMMMBMBMMMBMMMBMBMMMMMMMMMMM@MMM@MXuUuF252121U52121U521U122UUuUuUuujUjuuuuuuujuuuuUuUjuuuuUjUJJvv777vj0ZZNGEZEZ0ZEG0ENZ
      .   . ........,,,.,...,::iirrr77jYuLLvu8@BMOMOZZOGOZO8O8O8ZkF2UukkNkSYLv7:ivu1SFkXZNGEGZO8M8OGOGM8OGOOMOMOMOMMMMMMMMMOMOMOMOMOMMMMMOMOMOMMMOMOMOM8MOMOMOBMBMMOMOBBBMBMMMMMBMMMBMBMBMMMBMMMBMBMBMMMBMBMBMBMBMBO@BBMFuU1FU125U555212121U521U2u2uuuuuuuUjuuujuuujujuuUjujUuUjuuuLv7v777Uq8NZZZEEEZEZNE0EZ
     . . ........,.,,:,:::.::;rvvL7LLuJLJL7UO@MMOMZEZ8OMOM8OGO88kFu5155XFFuuju7i72kZZGNZE0qN08ZOZ8GOZOOO8M8MOMOMMMMMMMMMMMMMOMOMMMOMOMMMMMOMOMMMOMOMOMOMOMMMMMMMMBMMMMMBMBMBMBMBMBMBMBMBMMMBMBMBMBMBMBMBMMMBMBMMMBMMMBB@P1U551U52F252121U121U5U1U2u2uUuUjuuuuuuUuUuUuUuuuujuuUuUuuJYvv7777u0Z0NGEG0ZEGEE0Z0G
  . ..........,..,,,:,:::::ir7LYuYJLuuUL7vFO@88GMZ8GO8OOOOO880ENX5SSFu2uS5kqXJj2F5kSkPZZZ0ENZ0Eq0qE0E0OGOOMMMMMOMOMMMMMMMMMMBMBMMMMOMMMOMMMMMMMMMOMOMOMOMMMOMMMMMMBMBMBMMMMMBMBMBMBMMMBMBMBMBMBMBMMMBMMMBMBMBMBMMMBMMMBBMS22S15251121U1U125U521U2UUuUuUjujUuUjuuuuUuUuuuujujUuUuuYJvL77777UNG0ZEZEZEZEGNE0ZZ
   . ........,,,,::::::::i;77YYuYuvYYLLJLSB@MO8BBOZMOMMM8OZOEqP0PPSPqX2UjuuP0OE0kPXP5F2PEZZG0EqENZNZZ8OMOMOMOMOMOMMMMMMMOMMMMMMBMMMMMMMMOMOMOMOMMMOMMMMMOMOMOMMMMMMBMBMBMBMBMBMBMBMMMBMBMMMBMBMBMMMBMBMBMBMMMBMBMMB@MBM@M0u15F15151F25U52121U1U1U1u2uUjujuuUjuuuuUuuuuuujuuuu2uUjjvL777v7vU0ZZNGEZ0GEGEE0E0G
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*/
//orz tjw
//orz yjz
//author: FST_master ST_master T_master _master master aster ster ter er r
using namespace std;
inline void cmax(int &a,int b){
	a=max(a,b);
}
inline void cmin(int &a,int b){
	a=min(a,b);
}
inline int read()
{
	int x = 0, f = 1; char ch = getchar();
	while (ch < '0' || ch > '9') { if (ch == '-') f = -1; ch = getchar(); }
	while (ch >= '0' && ch <= '9') { x = x * 10 + ch - '0'; ch = getchar(); }
	return x * f;
}
struct edge{
	int to,cap,rev;
	edge(int x,int y,int z){
		to=x,cap=y,rev=z;
	}
};
struct Max_flow{
	int Siz;
	int src,snk;
	vector<vector<edge> >G;
	vector<int> dist,seen;
	Max_flow(){
		Siz=0;
		src=-1;snk=-1;
		G.clear();
		dist.clear();seen.clear();
	}
	void Set(int x,int y){
		src=x,snk=y;
	}
	void resize(int s){
		Siz=s;
		G.resize(s+1);
		dist.resize(s+1);
		seen.resize(s+1);
	}
	void add_edge(int u,int v,int c){
		edge t1(v,c,G[v].size()),t2(u,0,G[u].size());
		G[u].push_back(t1);
		G[v].push_back(t2);
	}
	void bfs(int s){
		for (int i=0;i<dist.size();i++) dist[i]=-1;
		queue<int> Q;
		dist[s]=0;
		Q.push(s);
		while (!Q.empty()){
			int x=Q.front();
			Q.pop();
			for (int i=0;i<G[x].size();i++){
				int v=G[x][i].to;
				if (G[x][i].cap && dist[v]==-1){
					dist[v]=dist[x]+1;
					Q.push(v);
				}
			}
		}
	}
	int dfs(int x,int dest,int flow){
		if (x==dest) return flow;
		for (int &i=seen[x];i<G[x].size();i++){
			edge &e=G[x][i];
			if (e.cap && dist[e.to]==dist[x]+1){
				int ff=dfs(e.to,dest,min(flow,e.cap));
				if (ff){
					e.cap-=ff;
					G[e.to][e.rev].cap+=ff;
					return ff;
				}
			}
		}
		return 0;
	}
	int dinic(int u,int v){
		int ans=0;
		while (1){
			for (int i=0;i<seen.size();i++) seen[i]=0;
			bfs(u);
			if (dist[v]==-1) break;
			int f;
			while (f=dfs(u,v,INF)){
				ans+=f;
			}
		}
		return ans;
	}
	int Maxflow(){
		return dinic(src,snk);
	}
};
const int Maxn=100005;
struct node{
	node*l,*r;
	node(){
		l=NULL;
		r=NULL;
	}
};
node* root;
char c[Maxn];
vector<LL> v;
LL L;
void go(node* now,int dep){
	//cout<<dep<<endl;
	if (!now) return;
	if (!now->l && !now->r) return;
	if (!now->l || !now->r){
		v.push_back(L-(LL)dep);
	}
	if (now->l){
		go(now->l,dep+1);
	}
	if (now->r){
		go(now->r,dep+1);
	}
}
int main(){
	int n;
	scanf("%d %lld",&n,&L);
	root=NULL;
	root=new node;
	for (int i=0;i<n;i++){
		scanf("%s",c);
		int l=strlen(c);
		node* now=root;
		for (int j=0;j<l;j++){
			if (c[j]=='0'){
				if (!now->l){
					now->l=new node; 
				}
				now=now->l;
			}
			else{
				if (!now->r){
					now->r=new node; 
				}
				now=now->r;
			}
		}
	}
	go(root,0);
	LL fin=0;
	for (int i=0;i<v.size();i++){
		LL cc=1;
		while (v[i]%2==0){
			v[i]/=2;
			cc*=2;
		}
		fin^=cc;
	}
	if (fin>0){
		printf("Alice\n");
	}
	else{
		printf("Bob\n");
	}
	return 0;
}

Measured memory_bytes: 9076