
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 below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{4}{5}$ and $\sin C = \frac{2}{5}$, what is $DC$?

[asy]
pair A,B,C,D;
A = (0,0);
B = (10*sqrt(5),24);
D = (10*sqrt(5),0);
C = (10*sqrt(5) + 40*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]

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