#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 1 "library/other/io.hpp"
#define FASTIO
#include <unistd.h>

// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
  pil = 0;
  if (pir < SZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}

void rd(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}

template <typename T>
void rd_real(T &x) {
  string s;
  rd(s);
  x = stod(s);
}

template <typename T>
void rd_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}

void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }

template <class T, class U>
void rd(pair<T, U> &p) {
  return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd(x);
    rd_tuple<N + 1>(t);
  }
}
template <class... T>
void rd(tuple<T...> &tpl) {
  rd_tuple(tpl);
}

template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
  for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
  for (auto &d: x) rd(d);
}

void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd(h), read(t...);
}

void wt(const char c) {
  if (por == SZ) flush();
  obuf[por++] = c;
}
void wt(const string s) {
  for (char c: s) wt(c);
}
void wt(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt(s[i]);
}

template <typename T>
void wt_integer(T x) {
  if (por > SZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt(s);
}

void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }

template <class T, class U>
void wt(const pair<T, U> val) {
  wt(val.first);
  wt(' ');
  wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt(' '); }
    const auto x = std::get<N>(t);
    wt(x);
    wt_tuple<N + 1>(t);
  }
}
template <class... T>
void wt(tuple<T...> tpl) {
  wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}
template <class T>
void wt(const vector<T> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}

void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt(head);
  if (sizeof...(Tail)) wt(' ');
  print(forward<Tail>(tail)...);
}

// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define U32(...)   \
  u32 __VA_ARGS__; \
  read(__VA_ARGS__)
#define U64(...)   \
  u64 __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

using Re = double;
const Re INF = 4'000'000'000'000'000'000;

// a_i-sqrt(b_i-x)
template <typename T, bool COMPRESS, bool MINIMIZE>
struct LiChao_Tree_1 {
  using FUNC = pair<T, T>;
  vc<FUNC> funcs;

  static inline T evaluate(FUNC& f, ll x) {
    auto [a, b] = f;
    if (x > b) return INF;
    return a - sqrt(b - x);
  }

  vc<ll> X;
  ll lo, hi;
  vc<int> FID;
  int n, log, size;

  inline int get_idx(ll x) {
    if constexpr (COMPRESS) { return LB(X, x); }
    assert(lo <= x && x <= hi);
    return x - lo;
  }

  template <typename XY>
  LiChao_Tree_1(const vc<XY>& pts) {
    static_assert(COMPRESS);
    for (auto&& x: pts) X.eb(x);
    UNIQUE(X);
    n = len(X), log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    FID.assign(size << 1, -1);
  }

  LiChao_Tree_1(ll lo, ll hi) : lo(lo), hi(hi) {
    static_assert(!COMPRESS);
    n = hi - lo, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    FID.assign(size << 1, -1);
  }

  void add_line(FUNC f) {
    int fid = len(funcs);
    funcs.eb(f);
    return add_line_at(1, fid);
  }
  void add_segment(ll xl, ll xr, FUNC f) {
    int fid = len(funcs);
    funcs.eb(f);
    xl = get_idx(xl), xr = get_idx(xr);
    xl += size, xr += size;
    while (xl < xr) {
      if (xl & 1) add_line_at(xl++, fid);
      if (xr & 1) add_line_at(--xr, fid);
      xl >>= 1, xr >>= 1;
    }
  }

  // [fx, fid]
  pair<T, int> query(ll x) {
    x = get_idx(x);
    int i = x + size;
    pair<T, int> res;
    if (!MINIMIZE) res = {-INF, -1};
    if (MINIMIZE) res = {INF, -1};
    while (i) {
      if (FID[i] != -1 && FID[i] != res.se) {
        pair<T, int> res1 = {evaluate_inner(FID[i], x), FID[i]};
        res = (MINIMIZE ? min(res, res1) : max(res, res1));
      }
      i >>= 1;
    }
    return res;
  }

  void add_line_at(int i, int fid) {
    int upper_bit = 31 - __builtin_clz(i);
    int l = (size >> upper_bit) * (i - (1 << upper_bit));
    int r = l + (size >> upper_bit);
    while (l < r) {
      int gid = FID[i];
      T fl = evaluate_inner(fid, l), fr = evaluate_inner(fid, r - 1);
      T gl = evaluate_inner(gid, l), gr = evaluate_inner(gid, r - 1);
      bool bl = (MINIMIZE ? fl < gl : fl > gl);
      bool br = (MINIMIZE ? fr < gr : fr > gr);
      if (bl && br) {
        FID[i] = fid;
        return;
      }
      if (!bl && !br) return;
      int m = (l + r) / 2;
      T fm = evaluate_inner(fid, m), gm = evaluate_inner(gid, m);
      bool bm = (MINIMIZE ? fm < gm : fm > gm);
      if (bm) {
        FID[i] = fid;
        fid = gid;
        if (!bl) { i = 2 * i + 0, r = m; }
        if (bl) { i = 2 * i + 1, l = m; }
      }
      if (!bm) {
        if (bl) { i = 2 * i + 0, r = m; }
        if (!bl) { i = 2 * i + 1, l = m; }
      }
    }
  }

private:
  T evaluate_inner(int fid, ll x) {
    if (fid == -1) return (MINIMIZE ? INF : -INF);
    return evaluate(funcs[fid], (COMPRESS ? X[min<int>(x, n - 1)] : x + lo));
  }
};

// a_i+sqrt(x-b_i)
template <typename T, bool COMPRESS, bool MINIMIZE>
struct LiChao_Tree_2 {
  using FUNC = pair<T, T>;
  vc<FUNC> funcs;

  static inline T evaluate(FUNC& f, ll x) {
    auto [a, b] = f;
    assert(x >= b);
    return a + sqrt(x - b);
  }

  vc<ll> X;
  ll lo, hi;
  vc<int> FID;
  int n, log, size;

  inline int get_idx(ll x) {
    if constexpr (COMPRESS) { return LB(X, x); }
    assert(lo <= x && x <= hi);
    return x - lo;
  }

  template <typename XY>
  LiChao_Tree_2(const vc<XY>& pts) {
    static_assert(COMPRESS);
    for (auto&& x: pts) X.eb(x);
    UNIQUE(X);
    n = len(X), log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    FID.assign(size << 1, -1);
  }

  LiChao_Tree_2(ll lo, ll hi) : lo(lo), hi(hi) {
    static_assert(!COMPRESS);
    n = hi - lo, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    FID.assign(size << 1, -1);
  }

  void add_line(FUNC f) {
    int fid = len(funcs);
    funcs.eb(f);
    return add_line_at(1, fid);
  }
  void add_segment(ll xl, ll xr, FUNC f) {
    int fid = len(funcs);
    funcs.eb(f);
    xl = get_idx(xl), xr = get_idx(xr);
    xl += size, xr += size;
    while (xl < xr) {
      if (xl & 1) add_line_at(xl++, fid);
      if (xr & 1) add_line_at(--xr, fid);
      xl >>= 1, xr >>= 1;
    }
  }

  // [fx, fid]
  pair<T, int> query(ll x) {
    x = get_idx(x);
    int i = x + size;
    pair<T, int> res;
    if (!MINIMIZE) res = {-INF, -1};
    if (MINIMIZE) res = {INF, -1};
    while (i) {
      if (FID[i] != -1 && FID[i] != res.se) {
        pair<T, int> res1 = {evaluate_inner(FID[i], x), FID[i]};
        res = (MINIMIZE ? min(res, res1) : max(res, res1));
      }
      i >>= 1;
    }
    return res;
  }

  void add_line_at(int i, int fid) {
    int upper_bit = 31 - __builtin_clz(i);
    int l = (size >> upper_bit) * (i - (1 << upper_bit));
    int r = l + (size >> upper_bit);
    while (l < r) {
      int gid = FID[i];
      T fl = evaluate_inner(fid, l), fr = evaluate_inner(fid, r - 1);
      T gl = evaluate_inner(gid, l), gr = evaluate_inner(gid, r - 1);
      bool bl = (MINIMIZE ? fl < gl : fl > gl);
      bool br = (MINIMIZE ? fr < gr : fr > gr);
      if (bl && br) {
        FID[i] = fid;
        return;
      }
      if (!bl && !br) return;
      int m = (l + r) / 2;
      T fm = evaluate_inner(fid, m), gm = evaluate_inner(gid, m);
      bool bm = (MINIMIZE ? fm < gm : fm > gm);
      if (bm) {
        FID[i] = fid;
        fid = gid;
        if (!bl) { i = 2 * i + 0, r = m; }
        if (bl) { i = 2 * i + 1, l = m; }
      }
      if (!bm) {
        if (bl) { i = 2 * i + 0, r = m; }
        if (!bl) { i = 2 * i + 1, l = m; }
      }
    }
  }

private:
  T evaluate_inner(int fid, ll x) {
    if (fid == -1) return (MINIMIZE ? INF : -INF);
    return evaluate(funcs[fid], (COMPRESS ? X[min<int>(x, n - 1)] : x + lo));
  }
};

void solve() {
  LL(N);
  vi H(N), U(N), V(N);

  FOR(i, N) read(H[i], V[i], U[i]);

  /*
  auto f = [&](int i, int j) -> Re {
    ll d = H[j] - H[i];
    ll u = U[i], v = V[j];
    if (u * u < 2 * d) return INF;
    if (u * u + 2 * H[i] <= v * v + 2 * H[j]) {
      return (2 * d) / (u + sqrtl(u * u - 2 * d));
      // return u - sqrtl(u * u - 2 * d);
    }
    // return sqrtl(v * v + 2 * d) - v;
    return (2 * d) / (v + sqrtl(v * v + 2 * d));
  };

  vc<Re> dp(N, INF);
  dp[0] = 0;
  FOR(j, N) FOR(i, j) { chmin(dp[j], dp[i] + f(i, j)); }
  Re ANS = dp[N - 1];
  if (ANS == INF) return print(-1);
  print(ANS);
  */

  vi VH(N), UH(N);
  FOR(i, N) VH[i] = V[i] * V[i] + 2 * H[i];
  FOR(i, N) UH[i] = U[i] * U[i] + 2 * H[i];

  vc<Re> dp(N, INF);
  dp[0] = 0;
  LiChao_Tree_2<Re, true, true> LCT2(VH);

  auto dfs = [&](auto& dfs, int L, int R) -> void {
    if (R == L + 1) {
      // 計算
      if (L > 0) { chmin(dp[L], LCT2.query(VH[L]).fi - V[L]); };
      // LCT2 に線分として追加
      // a+sqrt(x-b)
      LCT2.add_segment(2 * H[L], UH[L], {dp[L], 2 * H[L]});
      return;
    }
    int M = (L + R) / 2;
    dfs(dfs, L, M);
    // [L,M) to [M,R)
    // evaluate する点は 2h[j]
    vi point;
    FOR(j, M, R) point.eb(2 * H[j]);
    LiChao_Tree_1<Re, true, true> LCT1(point);

    // 必要なら高速化可能
    vc<int> I, J;
    FOR(i, L, M) I.eb(i);
    FOR(j, M, R) J.eb(j);
    sort(all(I), [&](auto& a, auto& b) -> bool { return UH[a] > UH[b]; });
    sort(all(J), [&](auto& a, auto& b) -> bool { return VH[a] < VH[b]; });

    // VH[j] が小さな j から順に計算を行う
    for (auto& j: J) {
      // UH[i] <= VH[j] となる i からの遷移を追加
      while (len(I) && UH[I.back()] <= VH[j]) {
        // LCT1: a-sqrt(b-x)
        // dp[i] + U[i] - sqrt(UH[i]-2*H[j])
        int i = POP(I);
        LCT1.add_line({dp[i] + U[i], UH[i]});
      }
      // dp[j] を計算
      chmin(dp[j], LCT1.query(2 * H[j]).fi);
    }
    dfs(dfs, M, R);
  };
  dfs(dfs, 0, N);

  Re ANS = dp[N - 1];
  if (ANS == INF) { return print(-1); }
  print(ANS);
}

signed main() {
  solve();
  return 0;
}