123. Decompose the sequence $\Phi_{11}^{0}$

$0,1,1,2,3,5,8,2,10,1,0,1,1,2,3,5,8,2,10,1, \ldots$
into the sum of two 11 -Fibonacci sequences, which are also geometric progressions ${ }^{1}$ ).
123. In the previous problem, all 11-arithmetic sequences that are simultaneously geometric progressions and Fibonacci series were found. It was established that all such sequences fall into two families. Now, let's take a sequence from the first family with the initial term $x$ and a sequence from the second family with the initial term $y$, and add them term by term. We will obtain the sequence

$$
\begin{array}{r}
x+y, 8 x+4 y, 9 x+5 y, 6 x+9 y, 4 x+3 y, 10 x+y \\
3 x+4 y, 2 x+5 y, 5 x+9 y, 7 x+3 y, x+y, \ldots,
\end{array}
$$

which, as can be easily seen, is again a Fibonacci series. Now, let's choose $x$ and $y$ such that the first term of this sequence is zero and the second term is one. Solving the system of equations

$$
\left\{\begin{array}{r}
x+y=0 \\
8 x+4 y=1
\end{array}\right.
$$

we find

$$
\begin{aligned}
& x=\frac{1}{4}=3 \\
& y=-3=8
\end{aligned}
$$

Thus, the desired decomposition of the Fibonacci series is given by the formulas

$$
\begin{aligned}
c_{0}=0 & =3+8, \quad c_{1}=1=3 \cdot 8+8 \cdot 4 \\
c_{3} & =1=3 \cdot 8^{2}+8 \cdot 4^{2}, \ldots, \quad c_{n}=3 \cdot 8^{n}+8 \cdot 4^{n}, \ldots
\end{aligned}
$$