Problem:
Score: 600 points

Problem Statement

M-kun is a brilliant air traffic controller.

On the display of his radar, there are N airplanes numbered 1, 2, ..., N, all flying at the same altitude.

Each of the airplanes flies at a constant speed of 0.1 per second in a constant direction. The current coordinates of the airplane numbered i are (X_i, Y_i), and the direction of the airplane is as follows:

if U_i is U, it flies in the positive y direction;

if U_i is R, it flies in the positive x direction;

if U_i is D, it flies in the negative y direction;

if U_i is L, it flies in the negative x direction.

To help M-kun in his work, determine whether there is a pair of airplanes that will collide with each other if they keep flying as they are now.

If there is such a pair, find the number of seconds after which the first collision will happen.

We assume that the airplanes are negligibly small so that two airplanes only collide when they reach the same coordinates simultaneously.

Constraints

1 \leq N \leq 200000

0 \leq X_i, Y_i \leq 200000

U_i is U, R, D, or L.

The current positions of the N airplanes, (X_i, Y_i), are all distinct.

N, X_i, and Y_i are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1 U_1
X_2 Y_2 U_2
X_3 Y_3 U_3
:
X_N Y_N U_N

Output

If there is a pair of airplanes that will collide with each other if they keep flying as they are now, print an integer representing the number of seconds after which the first collision will happen.

If there is no such pair, print SAFE.

Sample Input 1

2
11 1 U
11 47 D

Sample Output 1

230

If the airplanes keep flying as they are now, two airplanes numbered 1 and 2 will reach the coordinates (11, 24) simultaneously and collide.

Sample Input 2

4
20 30 U
30 20 R
20 10 D
10 20 L

Sample Output 2

SAFE

No pair of airplanes will collide.

Sample Input 3

8
168 224 U
130 175 R
111 198 D
121 188 L
201 116 U
112 121 R
145 239 D
185 107 L

Sample Output 3

100

Solution in Lua (LuaJIT 2.1.0) (judge verdict: Time Limit Exceeded)

local read = setmetatable({}, {__index = function(t, k) local a = {} for i=1,#k do table.insert(a, '*'..string.sub(k, i, i)) end local r = io.read local u = table.unpack or unpack return function() return r(u(a)) end end})
read.N = function(N) local t={} for i=1,N do t[i]=read.n() end return t end
string.totable = function(s) local t={} local u=string.sub for i=1,#s do t[i] = u(s, i, i) end return t end
string.split = function(s) local t={} for w in string.gmatch(s, "[^%s]+") do table.insert(t, w) end return (table.unpack or unpack)(t) end
local function array(dimensi0n, default_val) assert(type(default_val) ~= 'table') local n=dimensi0n local m={}if default_val~=nil then m[1]={__index=function()return default_val end}end for i=2,n do m[i]={__index=function(p, k)local c=setmetatable({},m[i-1])rawset(p,k,c)return c end}end return setmetatable({},m[n])end
local function tostringxx(o, depth) depth = depth or 0 if depth > 10 then return "<too deep!>" end if o == _G then return "<_G>" end local indent0 = (" "):rep((depth) * 2) local indent1 = (" "):rep((depth+1) * 2) local indent2= (" "):rep((depth+2) * 2) if type(o) == 'table' then local keys = {} local types = {} for k in pairs(o) do types[type(k)] = true table.insert(keys, k) end local types_count = 0 local lasttype for k in pairs(types) do types_count = types_count + 1 lasttype = k end if types_count == 1 and (lasttype == 'string' or lasttype == 'number') then table.sort(keys) end local inside = {} for i=1,#keys do local k = keys[i] local v = o[k] if type(k) == 'string' then k = string.format('%q', k) end table.insert(inside, indent1 .. '['..tostring(k)..'] = ' .. tostringxx(v, depth + 1)) end return '{\n' .. table.concat(inside, ',\n') .. '\n' .. indent0 .. '}' else if type(o) == 'string' then o = string.format('%q', o) end return tostring(o) end end
local function richtraceback() local x = 2 while true do local info = debug.getinfo(x) if not info then break end local fname = '<' .. info.short_src .. ":" .. info.linedefined .. ">" if info.name then fname = info.name end print(info.short_src .. ":" .. info.currentline .. ": in " .. ("%q"):format(fname)) print(" LOCALS:") local p = 1 while true do local name, val = debug.getlocal(x,p) if not name then break end print(" " .. name .. ": " .. tostringxx(val, 3)) p = p + 1 end print(" UPVALUES:") for p=1,info.nups do local name, val = debug.getupvalue(info.func,p) if not name then break end print(" " .. name .. ": " .. tostringxx(val, 3)) end x = x + 1 end end
local function myassert(b) if not b then richtraceback() error("assertion failed") end end 
--
local DBG = false
local insert = table.insert
local min = math.min
local sort = table.sort
local function getkeys(t)
    local keys = {}
    for k,v in pairs(t) do
        insert(keys, k)
    end
    sort(keys)
    return keys
end

local function dbgpr(...)
    if DBG then
        io.write("[dbg]")
        print(...)
    end
end
--
local N = read.nl()
local X, Y, U = {},{},{}
local yy = {}
local xx = {}
local rr = {} -- right down
local ll = {} -- left down
local function push(tt, p, e)
    if not tt[p] then
        tt[p] = {}
    end
    insert(tt[p], e)
end
for i=1,N do
    X[i], Y[i], U[i] = read.nnl()
    U[i] = U[i]:sub(2,2)
    local e = i
    push(yy, Y[i], e)
    push(xx, X[i], e)
    local rrp = X[i] + Y[i]
    push(rr, rrp, e)
    local llp = X[i] - Y[i]
    push(ll, llp, e)
end
--richtraceback()
local INF = 10^18
local nearest_crash = INF
local function f1(name, tt, ttk, inv, fw, bk, sp)
    for _,w in ipairs(ttk) do
        local t = tt[w]
        dbgpr("on same " .. name , w)
        local inttk = {}
        for _,i in ipairs(t) do
            if U[i] == fw or U[i] == bk then
                insert(inttk, {inv[i], i, U[i]})
            end
        end
        sort(inttk, function (a1, a2)
            return a1[1] < a2[1]
        end)
        local state = 0
        local recentfwinv = -200000
        for _,ee in ipairs(inttk) do
            local i = ee[2]
            if U[i] == fw or U[i] == bk then
                dbgpr("plane #" .. i, "inv="..inv[i], "X="..X[i], "Y="..Y[i], U[i])
                if state == 0 then
                    if U[i] == fw then
                        state = 1
                        recentfwinv = inv[i]
                    end
                elseif state == 1 then
                    if U[i] == fw then
                        recentfwinv = inv[i]
                    elseif U[i] == bk then
                        local d = inv[i] - recentfwinv
                        nearest_crash = min(nearest_crash, d * sp)
                        dbgpr("crash!", i, nearest_crash)
                    end
                end
            end
        end
    end
    
end
f1('Y', yy, getkeys(yy), X, 'R', 'L', 5)
f1('X', xx, getkeys(xx), Y, 'U', 'D', 5)
local rrk = getkeys(rr)
f1('rr', rr, rrk, X, 'R', 'U', 10)
f1('rr', rr, rrk, X, 'D', 'L', 10)
local llk = getkeys(ll)
f1('ll', ll, llk, X, 'U', 'L', 10)
f1('ll', ll, llk, X, 'R', 'D', 10)

if nearest_crash == INF then
    print("SAFE")
else
    print(nearest_crash)
end