7. Given $\left\{\begin{array}{l}25^{x}-13 \times 5^{x}+a<0, \\ 12 \sin ^{4} \pi x-\cos ^{4} \pi x=11\end{array}\right.$ for each value of $a$ there is at least one solution. Find all values of $a$.

For each value of $a$ there is at least one solution. Find all values of $a$.
7. Notice,
$$
\begin{array}{l}
12 \sin ^{4} \pi x-\cos 4 \pi x=11 \\
\Leftrightarrow 3(1-c)^{2}-\left(2 c^{2}-1\right)-11=0(c=\cos 2 \pi x) \\
\Leftrightarrow c^{2}-6 c-7=0 \\
\Leftrightarrow(c-7)(c+1)=0 \\
\Leftrightarrow \cos 2 \pi x=-1 \\
\Leftrightarrow x=0.5+n(n \in \mathbf{Z}) .
\end{array}
$$

To ensure that for at least one of the sought $x$ values, we have
$$
\begin{array}{l}
25^{x}-13 \times 5^{x}<-a \\
\Leftrightarrow t^{2}-13 t<-a\left(t=5^{x}=5^{n} \sqrt{5}\right) \\
\Leftrightarrow(t-6.5)^{2}<-a+6.5^{2},
\end{array}
$$

it must hold for such $t=5^{n} \sqrt{5}$. Let $t_{0}$ be the value that minimizes the left-hand side of equation (1) and is close to the point 6.5.
Since $\sqrt{5}<3<6.5<5 \times 2<5 \sqrt{5}$, we have
$$
t_{0}=\sqrt{5} \text {. }
$$

Therefore, the required $a$ satisfies
$$
\begin{array}{l}
t_{0}^{2}-13 t_{0}<-a \\
\Leftrightarrow(\sqrt{5})^{2}-13 \sqrt{5}<-a \\
\Leftrightarrow a<13 \sqrt{5}-5 .
\end{array}
$$