Prove or find a counterexample.


Let $a\_1, a\_2, b\_1, b\_2 \in \mathbb{R}$ satisfy $a\_1+a\_2=1$ and $b\_1+b\_2=1$. Then,
$$
\frac{1}{a\_1^2+a\_2^2} + \frac{1}{b\_1^2 + b\_2^2} \geq (\frac{1}{(a\_1 b\_2)^2 + (a\_1 b\_1 + a\_2 b\_2)^2 + (b\_1 a\_2)^2})^2
$$



Inequality with four variables
$a\_1=1/2$, $a\_2=1/2$, $b\_1=1$, $b\_2=0$. 


The left hand side is $3$, the right hand side is $4$.


