Death to Binary?

A group of eccentric calculation enthusiasts has found a wonderful new way to count. Instead of ordinary decimal numbers, they use the Fibonacci base. A number in this base is written as a string of $0$s and $1$s, just like a binary number, but the weight of each digit is not a power of two — it is an element of the Fibonacci progression. This progression starts with $F_0 = 1$ and $F_1 = 2$, and for $n \ge 2$ it satisfies $F_n = F_{n-1} + F_{n-2}$ (that is, $1, 2, 3, 5, 8, \dots$).

In a representation, the rightmost digit has weight $F_0$, and each digit further to the left carries the next Fibonacci weight. For example,

$1101001_{\mathrm{Fib}} = F_0 + F_3 + F_5 + F_6 = 1 + 5 + 13 + 21 = 40.$

Every integer can be expressed in this base, but not necessarily uniquely — for instance, $40$ can also be written as $10001001_{\mathrm{Fib}}$. However, for every integer there is a unique representation that contains no two adjacent $1$s; we call it the canonical representation. For example, the canonical representation of $40$ is $10001001_{\mathrm{Fib}}$.

To build a computer that calculates in the Fibonacci base, write a program that adds two numbers given in the Fibonacci base (not necessarily in canonical form).

The input consists of several test cases, each on a single line. Each line contains two numbers $X$ and $Y$ in the Fibonacci base, separated by a single space. Each number has at most $40$ digits. The end of the input is marked only by the end of file; there is no special terminator.

For each test case, print the following four lines:

1. Line 1: the canonical representation of $X$, left-padded with spaces.
2. Line 2: a plus sign + followed by the canonical representation of $Y$, left-padded with spaces.
3. Line 3: two spaces followed by a row of minus signs - whose length equals the length of the canonical representation of the sum.
4. Line 4: two spaces followed by the canonical representation of $X + Y$.

Left-pad $X$ and $Y$ with spaces so that the least significant (rightmost) digits of $X$, $Y$, and $X + Y$ line up in the same column. Put a single blank line between the outputs of consecutive test cases.