Problem:
Score : 600 points

Problem Statement

There are N points in a two-dimensional plane. The initial coordinates of the i-th point are (x_i, y_i). Now, each point starts moving at a speed of 1 per second, in a direction parallel to the x- or y- axis. You are given a character d_i that represents the specific direction in which the i-th point moves, as follows:

If d_i = R, the i-th point moves in the positive x direction;

If d_i = L, the i-th point moves in the negative x direction;

If d_i = U, the i-th point moves in the positive y direction;

If d_i = D, the i-th point moves in the negative y direction.

You can stop all the points at some moment of your choice after they start moving (including the moment they start moving).
Then, let x_{max} and x_{min} be the maximum and minimum among the x-coordinates of the N points, respectively. Similarly, let y_{max} and y_{min} be the maximum and minimum among the y-coordinates of the N points, respectively.

Find the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}) and print it.

Constraints

1 \leq N \leq 10^5

-10^8 \leq x_i,\ y_i \leq 10^8

x_i and y_i are integers.

d_i is R, L, U, or D.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 d_1
x_2 y_2 d_2
.
.
.
x_N y_N d_N

Output

Print the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}).

The output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-9}.

Sample Input 1

2
0 3 D
3 0 L

Sample Output 1

0

After three seconds, the two points will meet at the origin. The value in question will be 0 at that moment.

Sample Input 2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Sample Output 2

97.5

The answer may not be an integer.

Sample Input 3

20
6 -10 R
-4 -9 U
9 6 D
-3 -2 R
0 7 D
4 5 D
10 -10 U
-1 -8 U
10 -6 D
8 -5 U
6 4 D
0 3 D
7 9 R
9 -4 R
3 10 D
1 9 U
1 -6 U
9 -8 R
6 7 D
7 -3 D

Sample Output 3

273

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>

using namespace std;

using int64 = long long;

int64 Min(const vector<int64>& xs) {
  int64 value = INT64_MAX;
  for (int64 x : xs) value = min(x, value);
  return value;
}

int64 Max(const vector<int64>& xs) {
  int64 value = INT64_MIN;
  for (int64 x : xs) value = max(x, value);
  return value;
}

void Process(
    const vector<int64>& xs_pos, const vector<int64>& xs_neg,
    const vector<int64>& xs_still, vector<int64>* check_points) {
  if (xs_pos.size() > 0 && xs_neg.size() > 0) {
    int64 pos_max = Max(xs_pos);
    int64 neg_max = Max(xs_neg);
    if (neg_max > pos_max) {
      check_points->push_back((neg_max - pos_max) / 2);
    }

    int64 pos_min= Min(xs_pos);
    int64 neg_min = Min(xs_neg);
    if (neg_min > pos_min) {
      check_points->push_back((neg_min - pos_min) / 2);
    }
  }
  if (xs_pos.size() > 0 && xs_still.size() > 0) {
    int64 pos_max = Max(xs_pos);
    int64 still_max = Max(xs_still);
    if (still_max > pos_max) {
      check_points->push_back(still_max - pos_max);
    }

    int64 pos_min = Min(xs_pos);
    int64 still_min = Min(xs_still);
    if (still_min > pos_min) {
      check_points->push_back(still_min - pos_min);
    }
  }
  if (xs_neg.size() > 0 && xs_still.size() > 0) {
    int64 neg_min = Min(xs_neg);
    int64 still_min = Min(xs_still);
    if (neg_min > still_min) {
      check_points->push_back(neg_min - still_min);
    }
  }
}

int64 Range(
    const vector<int64>& xs_pos, const vector<int64>& xs_neg,
    const vector<int64>& xs_still, int64 t) {
  vector<int64> xs;
  for (int64 x : xs_pos) {
    xs.push_back(x + t);
  }
  for (int64 x : xs_neg) {
    xs.push_back(x - t);
  }
  for (int64 x : xs_still) {
    xs.push_back(x);
  }
  return Max(xs) - Min(xs);
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);

  int n;
  cin >> n;
  vector<int64> xs_pos, xs_neg, xs_still;
  vector<int64> ys_pos, ys_neg, ys_still;
  for (int i = 0; i < n; i++) {
    int64 x, y;
    string c;
    cin >> x >> y >> c;
    x *= 2;
    y *= 2;
    if (c == "R") {
      xs_pos.push_back(x);
      ys_still.push_back(y);
    } else if (c == "L") {
      xs_neg.push_back(x);
      ys_still.push_back(y);
    } else if (c == "U") {
      xs_still.push_back(x);
      ys_pos.push_back(y);
    } else if (c == "D") {
      xs_still.push_back(x);
      ys_neg.push_back(y);
    }
  }

  vector<int64> check_points;
  check_points.push_back(0);

  Process(xs_pos, xs_neg, xs_still, &check_points);
  Process(ys_pos, ys_neg, ys_still, &check_points);
  int64 solution = INT64_MAX;

  vector<int64> new_check_points;

  for (int64 t : check_points) {
    int64 x_range = Range(xs_pos, xs_neg, xs_still, t);
    int64 y_range = Range(ys_pos, ys_neg, ys_still, t);
    solution = min(x_range * y_range, solution);

    new_check_points.push_back(t + abs(x_range - y_range));
    new_check_points.push_back(t + abs(x_range - y_range) / 2);
  }

  for (int64 t : new_check_points) {
    int64 x_range = Range(xs_pos, xs_neg, xs_still, t);
    int64 y_range = Range(ys_pos, ys_neg, ys_still, t);
    solution = min(x_range * y_range, solution);
  }

  cout << solution * 0.25 << endl;
}