7964번 - Fibonacci Sums 다국어

Fibonacci numbers are an integer sequence defined in the following way: Fib₀=1, Fib₁=1, Fib_i=Fib_i-2+Fib_i-1 (for i ≥ 2). The first few numbers in this sequence are: (1,1,2,3,5,8,…).

The great computer scientist Byteazar is constructing an unusual computer, in which numbers are represented in Fibonacci system i.e. a bit string (b₁,b₂,…,b_n) denotes the number b₁⋅Fib₁+b₂⋅Fib₂+…+b_n⋅Fib_n. (Note that we do not use Fib₀.) Unfortunately, such a representation is ambiguous i.e. the same number can have different representations. The number 42, for instance, can be written as: (0,0,0,0,1,0,0,1),(0,0,0,0,1,1,1,0) or (1,1,0,1,0,1,1). For this very reason, Byteazar has limited himself to only using representations satisfying the following conditions:

• if n > 1, then b_n=1, i.e. the representation of a number does not contain leading zeros.
• if b_i=1, then b_i+1=0 (for i=1,…,n-1), i.e. the representation of a number does not contain two (or more) consecutive ones.

The construction of the computer has proved more demanding than Byteazar supposed. He has difficulties implementing addition. Help him!

Write a programme which:

• reads from the standard input the representations of two positive integers,
• calculates and writes to the standard output the representation of their sum.

The input contains the Fibonacci representations (satisfying the aforementioned conditions) of two positive integers x and y - one in the first, the other in the second line. Each of these representations is in the form of a sequence of non-negative integers, separated by single spaces. The first number in the line denotes the length of the representation n, 1 ≤ n ≤ 1,000,000. It is followed by n zeros and/or ones.

In the first and only line of the output your programme should write the Fibonacci representation (satisfying the aforementioned conditions) of the sum x+y. The representation should be in the form of a sequence of non-negative integers, separated by single spaces, as it has been described in the Input section. The first number in the line denotes the length of the representation n, 1 ≤ n ≤ 1,000,000. It is followed by n zeros and/or ones.