7. As shown in Figure 1, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$,

the points $E, F, G$ are on the edges $A A_{1}, A_{1} D_{1}, D_{1} C_{1}$ respectively, with $E$ being the midpoint of $A A_{1}$,
$$
\frac{D_{1} F}{D_{1} A_{1}}=\frac{D_{1} G}{D_{1} C_{1}}=\frac{1}{3}.
$$

Let the intersection line of plane $E F G$ and
plane $A_{1} B_{1} C D$ be
$m$. Then the tangent value of the angle formed by line $m$ and
plane $A B C D$ is $\qquad$
3. $\left\{-1, \frac{\sqrt{22}}{2}\right\}$.

It is known that, $A=(-2,3)$.
If $x \in A$, then $[x]=-2,-1,0,1,2$.
When $[x]=-2$, if $x \in B$, then
$$
2 x^{2}+6-5=0 \text {, }
$$
$x$ does not exist.
When $[x]=-1$, if $x \in B$, then
$$
2 x^{2}+3-5=0 \Rightarrow x= \pm 1 \text {. }
$$

Upon inspection, $x=1$ does not meet the requirements, $x=-1$ meets the requirements.
When $[x]=0$, if $x \in B$, then
$$
2 x^{2}-0-5=0 \Rightarrow x= \pm \frac{\sqrt{10}}{2} \text {, }
$$

neither meets the requirements.
When $[x]=1$, if $x \in B$, then
$$
2 x^{2}-3-5=0 \Rightarrow x= \pm 2 \text {, }
$$

neither meets the requirements.
When $[x]=2$, if $x \in B$, then
$$
2 x^{2}-6-5=0 \Rightarrow x= \pm \frac{\sqrt{22}}{2} \text {. }
$$

Upon inspection, $x=\frac{\sqrt{22}}{2}$ meets the requirements, $x=-\frac{\sqrt{22}}{2}$ does not meet the requirements.
Therefore, $A \cap B=\left\{-1, \frac{\sqrt{22}}{2}\right\}$.