XOR-Powers (noch nicht übersetzt)

We use $x\oplus y$ to be the bitwise XOR of $x$ and $y$.

Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.

For example, $11 \otimes 11 = 69$, or in base $2$, $1011_2 \otimes 1011_2 = 1000101_2$:

$$ \begin{align*} \phantom{\otimes 1111} 1011_2 \\ \otimes \phantom{1111} 1011_2 \\ \hline \phantom{\otimes 1111} 1011_2 \\ \phantom{\otimes 111} 1011_2 \phantom{9} \\ \oplus \phantom{1} 1011_2 \phantom{999} \\ \hline \phantom{\otimes 11} 1000101_2 \\ \end{align*} $$ Further we define $P(n) = 11^{\otimes n} = \overbrace{11\otimes 11\otimes \ldots \otimes 11}^n$. For example $P(2)=69$.

Find $P(8^{12}\cdot 12^8)$. Give your answer modulo $10^9+7$.