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 Julia (0.5.0) (judge verdict: Wrong Answer)

parseInt(x) = parse(Int, x)
parseMap(x::Array{SubString{String},1}) = map(parseInt, x)
function f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,s)
    if s<0
        10^16
    else
        (max(rm,rr+s,lr-s)-min(lm,ll-s,rl+s)+0.0)*(max(um,uu+s,du-s)-min(dm,dd-s,ud+s)+0.0)
    end
end

function main()
    n = readline() |> parseInt
    r = Tuple{Int,Int}[]
    l = Tuple{Int,Int}[]
    u = Tuple{Int,Int}[]
    d = Tuple{Int,Int}[]
    for i in 1:n
        x,y,w = readline() |> split
        if w=="R"
            push!(r,(parseInt(x),parseInt(y)))
        elseif w=="L"
            push!(l,(parseInt(x),parseInt(y)))
        elseif w=="U"
            push!(u,(parseInt(x),parseInt(y)))
        elseif w=="D"
            push!(d,(parseInt(x),parseInt(y)))
        end
    end
    rm = -10^8
    lm = 10^8
    um = -10^8
    dm = 10^8
    rr = -10^8
    rl = 10^8
    lr = -10^8
    ll = 10^8
    uu = -10^8
    ud = 10^8
    du = -10^8
    dd = 10^8
    for i in 1:length(r)
        um = max(um,r[i][2])
        dm = min(dm,r[i][2])
        rr = max(rr,r[i][1])
        rl = min(rl,r[i][1])
    end
    for i in 1:length(l)
        um = max(um,l[i][2])
        dm = min(dm,l[i][2])
        lr = max(lr,l[i][1])
        ll = min(ll,l[i][1])
    end
    for i in 1:length(u)
        rm = max(rm,u[i][1])
        lm = min(lm,u[i][1])
        uu = max(uu,u[i][2])
        ud = min(ud,u[i][2])
    end
    for i in 1:length(d)
        rm = max(rm,d[i][1])
        lm = min(lm,d[i][1])
        du = max(du,d[i][2])
        dd = min(dd,d[i][2])
    end
    println(min(f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,(lr-rr)/2),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,rm-rr),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,lr-rm),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,(ll-rl)/2),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,ll-lm),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,lm-rl),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,(du-uu)/2),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,um-uu),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,du-um),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,(dd-ud)/2),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,dd-dm),f(rm,lm,um,dm,rr,rl,lr,ll,uu,ud,du,dd,dm-ud)))
end

main()