
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
In rectangle $ABCD$, angle $D$ is bisected by $\overline{DG}$ and $\overline{DH}$, where $H$ is on $\overline{AB}$ and $G$ is on $\overline{BC}$, $HA=4$, and $GB=3$. Find the area of $ABCD$.

[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (3*sqrt(3) - 4), width = 3*sqrt(3);
draw(origin--(length,0)--(length,width)--(0,width)--cycle);
draw((0,width)--(length,4)^^(0,width)--(3,0));
dot("$A$",origin,SW); dot("$B$",(length,0),SE); dot("$C$",(length,width),NE); dot("$D$",(0,width),NW); dot("$G$",(3,0),S); dot("$H$",(length,4),E);
[/asy]

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