Problem:
Score : 600 points

Problem Statement

Snuke is introducing a robot arm with the following properties to his factory:

The robot arm consists of m sections and m+1 joints. The sections are numbered 1, 2, ..., m, and the joints are numbered 0, 1, ..., m. Section i connects Joint i-1 and Joint i. The length of Section i is d_i.

For each section, its mode can be specified individually. There are four modes: L, R, D and U. The mode of a section decides the direction of that section. If we consider the factory as a coordinate plane, the position of Joint i will be determined as follows (we denote its coordinates as (x_i, y_i)):

(x_0, y_0) = (0, 0).

If the mode of Section i is L, (x_{i}, y_{i}) = (x_{i-1} - d_{i}, y_{i-1}).

If the mode of Section i is R, (x_{i}, y_{i}) = (x_{i-1} + d_{i}, y_{i-1}).

If the mode of Section i is D, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} - d_{i}).

If the mode of Section i is U, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} + d_{i}).

Snuke would like to introduce a robot arm so that the position of Joint m can be matched with all of the N points (X_1, Y_1), (X_2, Y_2), ..., (X_N, Y_N) by properly specifying the modes of the sections.
Is this possible?
If so, find such a robot arm and how to bring Joint m to each point (X_j, Y_j).

Constraints

All values in input are integers.

1 \leq N \leq 1000

-10^9 \leq X_i \leq 10^9

-10^9 \leq Y_i \leq 10^9

Partial Score

In the test cases worth 300 points, -10 \leq X_i \leq 10 and -10 \leq Y_i \leq 10 hold.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1
X_2 Y_2
:
X_N Y_N

Output

If the condition can be satisfied, follow the following format. If the condition cannot be satisfied, print -1.

m
d_1 d_2 ... d_m
w_1
w_2
:
w_N

m and d_i are the configurations of the robot arm. Refer to the problem statement for what each of them means.
Here, 1 \leq m \leq 40 and 1 \leq d_i \leq 10^{12} must hold. Also, m and d_i must all be integers.

w_j is a string of length m that represents the way to bring Joint m of the robot arm to point (X_j, Y_j).
The i-th character of w_j should be one of the letters L, R, D and U, representing the mode of Section i.

Sample Input 1

3
-1 0
0 3
2 -1

Sample Output 1

2
1 2
RL
UU
DR

In the given way to bring Joint m of the robot arm to each (X_j, Y_j), the positions of the joints will be as follows:

To (X_1, Y_1) = (-1, 0): First, the position of Joint 0 is (x_0, y_0) = (0, 0). As the mode of Section 1 is R, the position of Joint 1 is (x_1, y_1) = (1, 0). Then, as the mode of Section 2 is L, the position of Joint 2 is (x_2, y_2) = (-1, 0).

To (X_2, Y_2) = (0, 3): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, 1), (x_2, y_2) = (0, 3).

To (X_3, Y_3) = (2, -1): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, -1), (x_2, y_2) = (2, -1).

Sample Input 2

5
0 0
1 0
2 0
3 0
4 0

Sample Output 2

-1

Sample Input 3

2
1 1
1 1

Sample Output 3

2
1 1
RU
UR

There may be duplicated points among (X_j, Y_j).

Sample Input 4

3
-7 -3
7 3
-3 -7

Sample Output 4

5
3 1 4 1 5
LRDUL
RDULR
DULRD

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

import Data.List as L
import Control.Monad

bit::Int -> [Int]
bit n = [2^i|i<-[0..n-1]]

odd_even::[(Int,Int)] -> (Bool, Bool)
odd_even a = let (h:t) = (map (\(x,y)->mod(x+y) 2) a) in
  (and (map (h==) t),h==0)

ls_accum::(b->(b,c))->(b->Bool)->b->[c]
ls_accum f condition state = if condition state then [] else let (next,res) = f state  in res:(ls_accum f condition next)

bit_expand::Int->[Bool]
bit_expand n = ls_accum (\ i -> (div i 2 , (mod i 2)==1) ) (0==) n

sigma_bit::Int->Int->[Bool]
sigma_bit n m = let bits = bit_expand (div (m + (2^n - 1)) 2) in bits++(take (n - (length bits))) (repeat False)

solver::Int->(Int,Int)->[Char]
solver n (x,y) = (map to_char) (zip (sigma_bit n (x+y)) (sigma_bit n (x-y)))
                                  where to_char (b,c) = case (b,c) of
                                                        (True,True)  -> 'R'
                                                        (True,False) -> 'U'
                                                        (False,True) -> 'D'
                                                        (False,False)-> 'L'

adj_solver::Int->(Int,Int)->[Char]
adj_solver n (x,y) = if mod (x+y) 2 == 1
                          then solver n (x,y)
                          else 'R':(solver n (x-1,y))

main=do
  (n:_) <-L.map read . words<$>getLine::IO[Int]
  inputs<- forM [1..n] (\_->(return.(\(x:y:[])->(x,y)).(map read).words)=<<getLine)
  case odd_even inputs of
    (False,_) -> do
      putStr "-1\n"
    (True,b) -> do
      putStr $ (if b then show (d+1) else show d)
      putStr "\n"
      putStr $ concat (intersperse " "  (map (\x-> show x) ((if b then [1] else[])++(bit d))))
      putStr "\n"
      forM_ inputs (\(x,y)->do
        case b of
          True -> putStr $ 'R':solver d (x-1, y)
          False -> putStr $ solver d (x, y)
        putStr "\n"
        )
  where
    d=31