7. Given $a, b, c \geqslant 0, t \geqslant 1$, satisfying
$$
\left\{\begin{array}{l}
a+b+c=\frac{1}{2}, \\
\sqrt{a+\frac{1}{2}(b-c)^{2}}+\sqrt{b}+\sqrt{c}=\frac{\sqrt{6 t}}{2} .
\end{array}\right.
$$

Then $a^{2 t}+b^{2 t}+c^{2 t}=$ $\qquad$ .