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 Python (3.4.3) (judge verdict: Accepted)

dirs = {'R': (1, 0), 'L': (-1, 0), 'U': (0, 1), 'D': (0, -1)}

N = int(input())
P = {d: [] for d in dirs}
for i in range(N):
    x, y, d = input().split()
    x, y = map(int, (x, y))
    P[d].append((x, y))

def fst(x): return x[0]
def snd(x): return x[1]
inf = float('inf')

# 方向別のX・Yの最大最小
S = {d: {'xmin': min(map(fst, P[d]), default= inf),
         'xmax': max(map(fst, P[d]), default=-inf),
         'ymin': min(map(snd, P[d]), default= inf),
         'ymax': max(map(snd, P[d]), default=-inf) }
     for d in dirs}

vars = ('xmax', 'xmin', 'ymax', 'ymin')
maxmin = {'max': max, 'min': min}
dirs = {d: {'x': dirs[d][0], 'y': dirs[d][1]} for d in dirs}

# S[L][xmax] - t1 == max(S[U][xmax], S[D][xmax])  t < t1 -> S[UD] < S[L]
# S[R][xmax] + t2 == max(S[U][xmax], S[D][xmax])  t < t2 -> S[R]  < S[UD]
# S[L][xmax] - t3 == S[R][xmax] + t3              t < t3 -> S[R]  < S[L]

# 変化がある時刻の集合
T = set()
T.add(0)
for v in vars:
    u, d, c1, c2 = ('R', 'L', 'U', 'D') if v.startswith('x') else \
                   ('U', 'D', 'R', 'L')
    C  = maxmin[v[-3:]](S[c1][v], S[c2][v])
    t1 =  S[d][v] - C
    t2 = -S[u][v] + C
    t3 = (S[d][v] - S[u][v]) / 2
    T.update([t1, t2, t3])

q = inf
for t in (t for t in T if 0 <= t and t != inf):
    Q = {v: maxmin[v[-3:]](S[d][v] + t * dirs[d][v[0]] for d in dirs)
         for v in vars}
    q = min(q, (Q['xmax'] - Q['xmin']) * (Q['ymax'] - Q['ymin']))

print(q)