Matrix Cipher

Alice holds her message to Bob as a bitstring and encodes it into a matrix. She starts from the identity matrix

$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

and reads the bitstring one character at a time, starting from the leftmost bit. For a 0 bit she multiplies $A$ on the right by

$\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$

that is, $A \leftarrow A \cdot \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$. For a 1 bit she multiplies $A$ on the right by

$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

that is, $A \leftarrow A \cdot \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. After the last bit she transmits the resulting matrix $A$.

Bob deleted the program that decodes a message from Alice. Given the transmitted matrix, recover the original bitstring.

The input consists of two lines. Line $i$ has two integers $a_{i1}$ and $a_{i2}$ separated by a space ($0 \le a_{i1}, a_{i2} \le 2^{128} - 1$, $1 \le i \le 2$), where

$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$

is the matrix holding the encoded message.

The original bitstring has length 1 to 120, and the input is always the encoding of one such bitstring.

Print the decoded bitstring on one line. The bitstring that satisfies the conditions is unique.