
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
$ABCDEFGH$ is a rectangular prism where $AB=CD=EF=GH=1$, $AE=BF=CG=DH=2$, $AC=EG=2$ and $BF=DH=2\sqrt{2}$. Find $\sin\angle HAE$.

[asy]
import three;
triple A,B,C,D,E,F,G,H;
real a = 1;
real b = 2;
A = (0,0,0);
B = (a,0,0);
C = (a,a,0);
D = (0,a,0);
E = (0,0,b);
F = (a,0,b);
G = (a,a,b);
H = (0,a,b);
draw(B--C--G--F--cycle);
draw(D--C);
draw(E--A--B,dashed);
draw(E--F);
draw(E--H,dashed);
draw(H--G);
draw(H--D,dashed);
draw(A--D);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,SE);
label("$D$",D,E);
label("$E$",E,NW);
label("$F$",F,SW);
label("$G$",G,N);
label("$H$",H,NE);
[/asy]

Present the answer in LaTex format: \boxed{Your answer}
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