3. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. If
$$
a^{2}+2\left(b^{2}+c^{2}\right)=2 \sqrt{2} \text {, }
$$

then the maximum value of the area of $\triangle A B C$ is $\qquad$ .
3. $\frac{1}{4}$.

Let $M$ be the midpoint of $BC$.
By the median length formula, we have
$$
\begin{array}{l}
S_{\triangle A B C}=\frac{1}{2} a A M \sin \angle A M C \leqslant \frac{1}{2} a A M \\
=\frac{a}{4} \sqrt{2\left(b^{2}+c^{2}\right)-a^{2}}=\frac{a}{4} \sqrt{2 \sqrt{2}-2 a^{2}} \\
=\frac{1}{4} \sqrt{\frac{2 a^{2}\left(2 \sqrt{2}-2 a^{2}\right)}{2}} \leqslant \frac{1}{4},
\end{array}
$$

Equality holds if and only if $a^{2}=\frac{\sqrt{2}}{2}, b^{2}=c^{2}=\frac{3 \sqrt{2}}{8}$.