
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 $\angle ADB = 60^\circ$. If $\cos A = \frac{3}{5}$ and $\sin C = \frac{1}{2}$, then what is $DC$?

[asy]
pair A, B, C, D;
A = (0, 0);
B = (8*sqrt(5), 12 + 16*sqrt(3));
D = (8*sqrt(5), 0);
C = (8*sqrt(5) + 40, 0);
draw(D--B--A--C--B);
label("$A$", A, SW);
label("$B$", B, N);
label("$C$", C, SE);
label("$D$", D, S);
draw(anglemark(B, D, A, 100));
[/asy]

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