
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 the diagram, triangles $ABD$ and $ACD$ are right triangles attached along edge $AD$. Triangle $ABD$ is a 45-45-90 triangle, and triangle $ACD$ is a 30-60-90 triangle. If $AB = 6$, find the length of $CD$.

[asy]
import olympiad;
draw((0,0)--(6,0)--(0,6)--cycle);
draw((0,0)--(6,0)--(0,-6)--cycle);
label("6",(3,3),NW);
label("$x$",(3,-3),SW);
draw("$45^{\circ}$",(6.5,0),N);
draw("$60^{\circ}$",(-0.5,-3),NE);
draw(rightanglemark((0,6),(0,0),(6,0),4));
draw(rightanglemark((0,-6),(0,0),(6,0),4));
label("$A$",(0,0),W);
label("$B$",(6,0),E);
label("$D$",(0,6),N);
label("$C$",(0,-6),S);
[/asy]

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