#include "prison.h"
#include <vector>
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using db = long double; // or double, if TL is tight
using str = string; // yay python!
// pairs
using pi = pair<int,int>;
using pl = pair<ll,ll>;
using pd = pair<db,db>;
#define mp make_pair
#define f first
#define s second
#define tcT template<class T
#define tcTU tcT, class U
// ^ lol this makes everything look weird but I'll try it
tcT> using V = vector<T>;
tcT, size_t SZ> using AR = array<T,SZ>;
using vi = V<int>;
using vb = V<bool>;
using vl = V<ll>;
using vd = V<db>;
using vs = V<str>;
using vpi = V<pi>;
using vpl = V<pl>;
using vpd = V<pd>;
// vectors
// oops size(x), rbegin(x), rend(x) need C++17
#define sz(x) int((x).size())
#define bg(x) begin(x)
#define all(x) bg(x), end(x)
#define rall(x) x.rbegin(), x.rend()
#define sor(x) sort(all(x))
#define rsz resize
#define ins insert
#define pb push_back
#define eb emplace_back
#define ft front()
#define bk back()
#define lb lower_bound
#define ub upper_bound
tcT> int lwb(V<T>& a, const T& b) { return int(lb(all(a),b)-bg(a)); }
tcT> int upb(V<T>& a, const T& b) { return int(ub(all(a),b)-bg(a)); }
// loops
#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define rep(a) F0R(_,a)
#define each(a,x) for (auto& a: x)
const int MOD = 1e9+7; // 998244353;
const int MX = 2e5+5;
const ll BIG = 1e18; // not too close to LLONG_MAX
const db PI = acos((db)-1);
const int dx[4]{1,0,-1,0}, dy[4]{0,1,0,-1}; // for every grid problem!!
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());
template<class T> using pqg = priority_queue<T,vector<T>,greater<T>>;
// bitwise ops
// also see https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html
constexpr int pct(int x) { return __builtin_popcount(x); } // # of bits set
constexpr int bits(int x) { // assert(x >= 0); // make C++11 compatible until USACO updates ...
return x == 0 ? 0 : 31-__builtin_clz(x); } // floor(log2(x))
constexpr int p2(int x) { return 1<<x; }
constexpr int msk2(int x) { return p2(x)-1; }
ll cdiv(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up
ll fdiv(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down
tcT> bool ckmin(T& a, const T& b) {
return b < a ? a = b, 1 : 0; } // set a = min(a,b)
tcT> bool ckmax(T& a, const T& b) {
return a < b ? a = b, 1 : 0; } // set a = max(a,b)
tcTU> T fstTrue(T lo, T hi, U f) {
++hi; assert(lo <= hi); // assuming f is increasing
while (lo < hi) { // find first index such that f is true
T mid = lo+(hi-lo)/2;
f(mid) ? hi = mid : lo = mid+1;
}
return lo;
}
tcTU> T lstTrue(T lo, T hi, U f) {
--lo; assert(lo <= hi); // assuming f is decreasing
while (lo < hi) { // find first index such that f is true
T mid = lo+(hi-lo+1)/2;
f(mid) ? lo = mid : hi = mid-1;
}
return lo;
}
tcT> void remDup(vector<T>& v) { // sort and remove duplicates
sort(all(v)); v.erase(unique(all(v)),end(v)); }
tcTU> void erase(T& t, const U& u) { // don't erase
auto it = t.find(u); assert(it != end(t));
t.erase(it); } // element that doesn't exist from (multi)set
#define tcTUU tcT, class ...U
inline namespace Helpers {
//////////// is_iterable
// https://stackoverflow.com/questions/13830158/check-if-a-variable-type-is-iterable
// this gets used only when we can call begin() and end() on that type
tcT, class = void> struct is_iterable : false_type {};
tcT> struct is_iterable<T, void_t<decltype(begin(declval<T>())),
decltype(end(declval<T>()))
>
> : true_type {};
tcT> constexpr bool is_iterable_v = is_iterable<T>::value;
//////////// is_readable
tcT, class = void> struct is_readable : false_type {};
tcT> struct is_readable<T,
typename std::enable_if_t<
is_same_v<decltype(cin >> declval<T&>()), istream&>
>
> : true_type {};
tcT> constexpr bool is_readable_v = is_readable<T>::value;
//////////// is_printable
// // https://nafe.es/posts/2020-02-29-is-printable/
tcT, class = void> struct is_printable : false_type {};
tcT> struct is_printable<T,
typename std::enable_if_t<
is_same_v<decltype(cout << declval<T>()), ostream&>
>
> : true_type {};
tcT> constexpr bool is_printable_v = is_printable<T>::value;
}
inline namespace Input {
tcT> constexpr bool needs_input_v = !is_readable_v<T> && is_iterable_v<T>;
tcTUU> void re(T& t, U&... u);
tcTU> void re(pair<T,U>& p); // pairs
// re: read
tcT> typename enable_if<is_readable_v<T>,void>::type re(T& x) { cin >> x; } // default
tcT> void re(complex<T>& c) { T a,b; re(a,b); c = {a,b}; } // complex
tcT> typename enable_if<needs_input_v<T>,void>::type re(T& i); // ex. vectors, arrays
tcTU> void re(pair<T,U>& p) { re(p.f,p.s); }
tcT> typename enable_if<needs_input_v<T>,void>::type re(T& i) {
each(x,i) re(x); }
tcTUU> void re(T& t, U&... u) { re(t); re(u...); } // read multiple
// rv: resize and read vectors
void rv(size_t) {}
tcTUU> void rv(size_t N, V<T>& t, U&... u);
template<class...U> void rv(size_t, size_t N2, U&... u);
tcTUU> void rv(size_t N, V<T>& t, U&... u) {
t.rsz(N); re(t);
rv(N,u...); }
template<class...U> void rv(size_t, size_t N2, U&... u) {
rv(N2,u...); }
// dumb shortcuts to read in ints
void decrement() {} // subtract one from each
tcTUU> void decrement(T& t, U&... u) { --t; decrement(u...); }
#define ints(...) int __VA_ARGS__; re(__VA_ARGS__);
#define int1(...) ints(__VA_ARGS__); decrement(__VA_ARGS__);
}
inline namespace ToString {
tcT> constexpr bool needs_output_v = !is_printable_v<T> && is_iterable_v<T>;
// ts: string representation to print
tcT> typename enable_if<is_printable_v<T>,str>::type ts(T v) {
stringstream ss; ss << fixed << setprecision(15) << v;
return ss.str(); } // default
tcT> str bit_vec(T t) { // bit vector to string
str res = "{"; F0R(i,sz(t)) res += ts(t[i]);
res += "}"; return res; }
str ts(V<bool> v) { return bit_vec(v); }
template<size_t SZ> str ts(bitset<SZ> b) { return bit_vec(b); } // bit vector
tcTU> str ts(pair<T,U> p); // pairs
tcT> typename enable_if<needs_output_v<T>,str>::type ts(T v); // vectors, arrays
tcTU> str ts(pair<T,U> p) { return "("+ts(p.f)+", "+ts(p.s)+")"; }
tcT> typename enable_if<is_iterable_v<T>,str>::type ts_sep(T v, str sep) {
// convert container to string w/ separator sep
bool fst = 1; str res = "";
for (const auto& x: v) {
if (!fst) res += sep;
fst = 0; res += ts(x);
}
return res;
}
tcT> typename enable_if<needs_output_v<T>,str>::type ts(T v) {
return "{"+ts_sep(v,", ")+"}"; }
// for nested DS
template<int, class T> typename enable_if<!needs_output_v<T>,vs>::type
ts_lev(const T& v) { return {ts(v)}; }
template<int lev, class T> typename enable_if<needs_output_v<T>,vs>::type
ts_lev(const T& v) {
if (lev == 0 || !sz(v)) return {ts(v)};
vs res;
for (const auto& t: v) {
if (sz(res)) res.bk += ",";
vs tmp = ts_lev<lev-1>(t);
res.ins(end(res),all(tmp));
}
F0R(i,sz(res)) {
str bef = " "; if (i == 0) bef = "{";
res[i] = bef+res[i];
}
res.bk += "}";
return res;
}
}
inline namespace Output {
template<class T> void pr_sep(ostream& os, str, const T& t) { os << ts(t); }
template<class T, class... U> void pr_sep(ostream& os, str sep, const T& t, const U&... u) {
pr_sep(os,sep,t); os << sep; pr_sep(os,sep,u...); }
// print w/ no spaces
template<class ...T> void pr(const T&... t) { pr_sep(cout,"",t...); }
// print w/ spaces, end with newline
void ps() { cout << "\n"; }
template<class ...T> void ps(const T&... t) { pr_sep(cout," ",t...); ps(); }
// debug to cerr
template<class ...T> void dbg_out(const T&... t) {
pr_sep(cerr," | ",t...); cerr << endl; }
void loc_info(int line, str names) {
cerr << "Line(" << line << ") -> [" << names << "]: "; }
template<int lev, class T> void dbgl_out(const T& t) {
cerr << "\n\n" << ts_sep(ts_lev<lev>(t),"\n") << "\n" << endl; }
#ifdef LOCAL
#define dbg(...) loc_info(__LINE__,#__VA_ARGS__), dbg_out(__VA_ARGS__)
#define dbgl(lev,x) loc_info(__LINE__,#x), dbgl_out<lev>(x)
#else // don't actually submit with this
#define dbg(...) 0
#define dbgl(lev,x) 0
#endif
const clock_t beg = clock();
#define dbg_time() dbg((db)(clock()-beg)/CLOCKS_PER_SEC)
}
inline namespace FileIO {
void setIn(str s) { freopen(s.c_str(),"r",stdin); }
void setOut(str s) { freopen(s.c_str(),"w",stdout); }
void setIO(str s = "") {
cin.tie(0)->sync_with_stdio(0); // unsync C / C++ I/O streams
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) setIn(s+".in"), setOut(s+".out"); // for old USACO
}
}
/* stuff you should look for
* int overflow, array bounds
* special cases (n=1?)
* do smth instead of nothing and stay organized
* WRITE STUFF DOWN
* DON'T GET STUCK ON ONE APPROACH
*/
std::vector<std::vector<int>> devise_strategy(int N) {
auto repr = [&](int x) {
vi seq;
int up = 5102;
while (true) {
assert(1 <= x && x <= up);
if (x == 1) {
seq.pb(-10);
return seq;
}
if (x == up) {
seq.pb(10);
return seq;
}
if (up > 6) {
--x;
int cnt = 0;
while (x > up/3) x -= up/3, ++cnt;
assert(0 <= cnt && cnt <= 2);
seq.pb(cnt);
up /= 3;
} else {
assert(up == 6);
--x;
int cnt = 0;
while (x > 2) x -= 2, ++cnt;
assert(0 <= cnt && cnt <= 1);
seq.pb(cnt);
seq.pb(x-1);
return seq;
}
}
};
// FOR(i,1,5103) dbg(i, repr(i));
// exit(0);
// 0
// {1, 0}
// {1, 1}
V<vi> nex(21, vi(N+1));
auto hsh = [&](int steps, int val) {
// dbg("HSH", steps, val);
assert(0 <= val && val < 3);
assert(steps > 0);
return 3*steps + val - 2; // 1 ... 18
};
F0R(steps, 8) {
const int cur = (8-steps)&1;
assert(cur == 0 || cur == 1);
auto found_cur = [&](int x) {
assert(x);
if (x < 0) return -cur-1;
return -(cur^1)-1;
};
if (steps == 0) {
const int id = 0;
nex.at(id).at(0) = cur;
FOR(k,1,N+1) {
vi r = repr(k);
if (abs(r.at(steps)) == 10) {
nex.at(id).at(k) = found_cur(r.at(steps));
} else {
nex.at(id).at(k) = hsh(steps+1, r.at(steps));
}
}
} else if (steps <= 6) {
F0R(j,3) {
const int id = hsh(steps, j);
nex.at(id).at(0) = cur;
FOR(k,1,N+1) {
vi r = repr(k);
if (steps-1 >= sz(r)) {
} else if (r.at(steps-1) != j) {
nex.at(id).at(k) = found_cur(r.at(steps-1)-j);
} else if (abs(r.at(steps)) == 10) {
nex.at(id).at(k) = found_cur(r.at(steps));
} else {
nex.at(id).at(k) = hsh(steps+1, r.at(steps));
}
}
}
} else {
F0R(j,2) {
const int id = hsh(steps, j);
nex.at(id).at(0) = cur;
FOR(k,1,N+1) {
vi r = repr(k);
if (steps-1 >= sz(r)) {
} else if (r.at(steps-1) != j) {
nex.at(id).at(k) = found_cur(r.at(steps-1)-j);
} else {
if (r.at(steps) == 1) {
nex.at(id).at(k) = found_cur(1);
} else {
nex.at(id).at(k) = found_cur(-1);
}
}
}
}
}
}
// dbg(nex);
return nex;
// 2, 3, 3, 3, 3, 3, 3
// start: 0
// V<vi> nex(23, vi(N+1));
// nex.at(0).at(0) = 0;
// auto hsh = [&](int i, int j) {
// if (i == 0) {
// assert(j == 1);
// return 22;
// }
// return 3*(7-i)+j+1;
// };
// // ROF(i, 1, 8) F0R(j, 3) dbg(i, j, hsh(i,j));
// auto get_tri = [&](int x, int tri) {
// rep(tri) x /= 3;
// return x%3;
// };
// F0R(z,10) dbg(z, get_tri(2187, 7), get_tri(2181, 7));
// FOR(i,1,N+1) {
// nex.at(0).at(i) = hsh(7, get_tri(i, 7));
// }
// // m = 24
// ROF(i, 0, 8) F0R(j, 3) {
// if (i == 0) {
// if (j != 1) continue;
// }
// const int pre = (7-i)&1;
// const int nxt = pre^1;
// nex.at(hsh(i, j)).at(0) = nxt;
// FOR(k,1,N+1) {
// if (get_tri(k, i) != j) {
// nex.at(hsh(i,j)).at(k) = (j < get_tri(k, i) ? -pre-1 : -nxt-1);
// } else {
// int bit = get_tri(k, i-1);
// if (i == 1 && bit == 0) {
// nex.at(hsh(i,j)).at(k) = -nxt-1;
// } else if (i == 1 && bit == 2) {
// nex.at(hsh(i,j)).at(k) = -pre-1;
// } else {
// if (i == 1) assert(bit == 1);
// if (i != 0) nex.at(hsh(i,j)).at(k) = hsh(i-1, bit);
// }
// }
// }
// }
// return nex;
// auto hsh = [&](int i, int j) {
// dbg("AH", i, j, 3*(12-i)+j+1);
// return 3*(12-i)+j+1;
// };
// R0F(i,13) {
// FOR(j,-1,2) {
// if (j == -1) {
// nex.at(hsh(i,j)).at(0) = 0;
// } else {
// nex.at(hsh(i,j)).at(0) = 1;
// }
// FOR(k,1,N+1) {
// if (j == -1) {
// nex.at(hsh(i,j)).at(k) = hsh(i,(k>>i)&1);
// } else {
// if (((k>>i)&1) != j) {
// nex.at(hsh(i,j)).at(k) = (j == 0 ? -1 : -2);
// } else if (i > 0) {
// nex.at(hsh(i,j)).at(k) = hsh(i-1,-1);
// }
// }
// }
// // {i, nothing}
// // {i, 1}
// // {i, 0}
// }
// }
// return nex;
}
