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 PyPy3 (2.4.0) (judge verdict: Wrong Answer)

N = int(input())

x = [0] * N
y = [0] * N
d = [0] * N
Rx = []
Lx = []
Ux = []
Dx = []
Ry = []
Ly = []
Uy = []
Dy = []
xsort = []
ysort = []
for i in range(N):
    tx, ty, td = input().split()
    x[i], y[i], d[i] = int(tx), int(ty), td
    if td == 'R':
        Rx.append(int(tx))
        Ry.append(int(ty))
    if td == 'L':
        Lx.append(int(tx))
        Ly.append(int(ty))
    if td == 'U':
        Ux.append(int(tx))
        Uy.append(int(ty))
    if td == 'D':
        Dx.append(int(tx))
        Dy.append(int(ty))
    xsort.append([int(tx), td])
    ysort.append([int(ty), td])
Rx.sort()
Lx.sort()
Ux.sort()
Dx.sort()
Ry.sort()
Ly.sort()
Uy.sort()
Dy.sort()

xsort.sort()
ysort.sort()

t = [0]
if xsort[0][1] == 'R' and (Lx or Ux or Dx):
    xlside = xsort[0][0]
    lt = 10**8 + 1
    if Lx:
        lt = (Lx[0] - xlside) / 2
    if Ux:
        lt = min(lt, Ux[0] - xlside)
    if Dx:
        lt = min(lt, Dx[0] - xlside)
    t.append(lt)

# rside
if xsort[-1][1] == 'L' and (Rx or Ux or Dx):
    xrside = xsort[-1][0]
    lt = 10**8 + 1
    if Rx:
        lt = (xrside - Rx[-1]) / 2
    if Ux:
        lt = min(lt, xrside - Ux[-1])
    if Dx:
        lt = min(lt, xrside - Dx[-1])
    t.append(lt)

# tail
if ysort[0][1] == 'U' and (Dy or Ly or Ry):
    ytail = ysort[0][0]
    lt = 10**8 + 1
    if Dy:
        lt = (Dy[0] - ytail) / 2
    if Ly:
        lt = min(lt, Ly[0] - ytail)
    if Ry:
        lt = min(lt, Ry[0] - ytail)
    t.append(lt)
# top
if ysort[-1][1] == 'D' and (Uy or Ly or Ry):
    ytop = ysort[-1][0]
    lt = 10**8 + 1
    if Uy:
        lt = (ytop - Uy[-1]) / 2
    if Ly:
        lt = min(lt, ytop - Ly[-1])
    if Ry:
        lt = min(lt, ytop - Ry[-1])
    t.append(lt)

ans = 10**32+1
for time in t:
    xl = 10**8 + 1
    if Lx:
        xl = min(xl, Lx[0] - time)
    if Rx:
        xl = min(xl, Rx[0] + time)
    if Ux:
        xl = min(xl, Ux[0])
    if Dx:
        xl = min(xl, Dx[0])

    xr = -(10**8 + 1)
    if Lx:
        xr = max(xr, Lx[-1] - time)
    if Rx:
        xr = max(xr, Rx[-1] + time)
    if Ux:
        xr = max(xr, Ux[-1])
    if Dx:
        xr = max(xr, Dx[-1])

    yd = 10**8 + 1
    if Dy:
        yd = min(yd, Dy[0] - time)
    if Uy:
        yd = min(yd, Uy[0] + time)
    if Ly:
        yd = min(yd, Ly[0])
    if Ry:
        yd = min(yd, Ry[0])

    yu = -(10**8 + 1)
    if Dy:
        yu = max(yu, Dy[-1] - time)
    if Uy:
        yu = max(yu, Uy[-1] + time)
    if Ly:
        yu = max(yu, Ly[-1])
    if Ry:
        yu = max(yu, Ry[-1])

    rec = (xr-xl)*(yu-yd)
    ans = min(ans, rec)


print(ans)