Problem: 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}.

Examples

Input

2
0 3 D
3 0 L


Output

0


Input

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


Output

97.5


Input

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


Output

273

Solution (in PYTHON3):

Solution:
n = int(input())
info = [list(input().split()) for i in range(n)]
for i in range(n):
    x, y, dire = info[i]
    info[i] = (int(x), int(y), dire)
INF = 10 ** 18


def solve_x(mid, op=max, e=-INF):
    # mid秒後のxの位置: x0
    # mid+1秒後のxの位置: x1
    x0, x1 = e, e
    for x, _, dire in info:
        if dire == "U" or dire == "D":
            x0, x1 = op(x, x0), op(x, x1)
        elif dire == "R":
            x0, x1 = op(x + mid, x0), op(x + mid + 1, x1)
        else:
            x0, x1 = op(x - mid, x0), op(x - mid - 1, x1)
    return x0, x1


def solve_y(mid, op=max, e=-INF):
    # mid秒後のyの位置: y0
    # mid+1秒後のyの位置: y1
    y0, y1 = e, e
    for _, y, dire in info:
        if dire == "L" or dire == "R":
            y0, y1 = op(y, y0), op(y, y1)
        elif dire == "U":
            y0, y1 = op(y + mid, y0), op(y + mid + 1, y1)
        else:
            y0, y1 = op(y - mid, y0), op(y - mid - 1, y1)
    return y0, y1


def solve_s(t):
    """時刻tにおける面積sを求める"""
    x_max = -INF * INF
    x_min = INF * INF
    y_max = -INF * INF
    y_min = INF * INF
    for x, y, dire in info:
        if dire == "L":
            x_max = max(x - t, x_max)
            x_min = min(x - t, x_min)
            y_max = max(y, y_max)
            y_min = min(y, y_min)
        if dire == "R":
            x_max = max(x + t, x_max)
            x_min = min(x + t, x_min)
            y_max = max(y, y_max)
            y_min = min(y, y_min)
        if dire == "D":
            x_max = max(x, x_max)
            x_min = min(x, x_min)
            y_max = max(y - t, y_max)
            y_min = min(y - t, y_min)
        if dire == "U":
            x_max = max(x, x_max)
            x_min = min(x, x_min)
            y_max = max(y + t, y_max)
            y_min = min(y + t, y_min)
    return (x_max - x_min) * (y_max - y_min)
            
            
ts = set([])
# xmaxについて
# 単調減少 → 不変
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    x0, x1 = solve_x(mid, op=max, e=-INF)
    if x0 > x1:
        ok = mid
    else:
        ng = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)
# 不変 → 単調増加
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    x0, x1 = solve_x(mid, op=max, e=-INF)
    if x0 < x1:
        ng = mid
    else:
        ok = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)

# xminについて
# 単調増加 → 不変
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    x0, x1 = solve_x(mid, op=min, e=INF)
    if x0 < x1:
        ok = mid
    else:
        ng = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)
# 不変 → 単調減少
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    x0, x1 = solve_x(mid, op=min, e=INF)
    if x0 > x1:
        ng = mid
    else:
        ok = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)


# ymaxについて
# 単調減少 → 不変
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    y0, y1 = solve_y(mid, op=max, e=-INF)
    if y0 > y1:
        ok = mid
    else:
        ng = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)
# 不変 → 単調増加
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    y0, y1 = solve_y(mid, op=max, e=-INF)
    if y0 < y1:
        ng = mid
    else:
        ok = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)

# yminについて
# 単調増加 → 不変
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    y0, y1 = solve_y(mid, op=min, e=INF)
    if y0 < y1:
        ok = mid
    else:
        ng = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)
# 不変 → 単調減少
ok = -1
ng = INF
while abs(ok - ng) > 1:
    mid = (ok + ng) // 2
    y0, y1 = solve_y(mid, op=min, e=INF)
    if y0 > y1:
        ng = mid
    else:
        ok = mid
ts.add(ok - 2), ts.add(ok - 1), ts.add(ok), ts.add(ng), ts.add(ng + 1), ts.add(ng + 2)


ans = INF * INF
for t in ts:
    if t >= INF - 10:
        continue
    if t - 0.5 >= 0:
        ans = min(solve_s(t - 0.5), ans)
    if t >= 0:
        ans = min(solve_s(t), ans)
    if t + 0.5 >= 0:
        ans = min(solve_s(t + 0.5), ans)
print(ans)