In quantum computing, logic gates operate a bit differently. Quantum logic gates are reversible, and the number of input qubits is equal to the number of output qubits. Additionally, they can be represented by $$$2^N$$$ by $$$2^N$$$ matrices, where $$$N$$$ is the number of qubits.

A quantum circuit is a model for quantum computation where the computation is performed through a sequence of quantum logic gates and measurement devices. A sequence of logic gates can be represented by a matrix resulting from the multiplication of the matrices of the logic gates in the order of application, which is the reverse order of how they are graphically represented. For example, the circuit for adding two bits in its quantum form is:

In this circuit, we have two variations of a logic gate that we will call $$$\text{CNOT}(q_c, q_t)$$$ and $$$\text{CCNOT}(q_{c_1}, q_{c_2}, q_t)$$$. In the diagram, the qubit $$$q_t$$$ is marked with a $$$\oplus$$$. The logic gate $$$\text{CNOT}(q_c, q_t)$$$ can be seen as being equal to $$$\text{CCNOT}(q_c, q_c, q_t)$$$, that is, the application of the logic gate $$$\text{CCNOT}$$$ with $$$q_c = q_{c_1} = q_{c_2}$$$.

The logic gate $$$\text{CCNOT}(q_{c_1}, q_{c_2}, q_t)$$$ behaves by inverting the output qubit $$$q_t$$$ if both control qubits $$$q_{c_1}$$$ and $$$q_{c_2}$$$ are set. Mathematically, $$$q'_t = q_t \oplus (q_{c_1} \land q_{c_2})$$$. In its matrix form:

where $$$i$$$ is the row and $$$j$$$ is the column with $$$0 \leq i, j \lt 2^N$$$, and $$$i$$$ contains bit $$$k$$$ ($$$0 \leq k$$$) if $$$\left\lfloor \frac{x}{2^k} \right\rfloor \mod 2 = 1$$$. The operation $$$\oplus$$$ is the bitwise exclusive or operation, commonly represented as $$$\land$$$ in programming languages.

Thus, the matrix of the quantum circuit for adding two bits is given by

where the qubits $$$q_0, q_1, q_2, q_3$$$ are used with inputs $$$|A\rangle, |B\rangle, |C_{in}\rangle, |0\rangle$$$ respectively and result in $$$|A\rangle, |B\rangle, |S\rangle, |C_{out}\rangle$$$ respectively.

Your task is given the description of a circuit with logic gates $$$\text{CNOT}$$$ and $$$\text{CCNOT}$$$ in the order of application, to print the resulting matrix.