
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 \( \sin C = \frac{1}{4} \), what is the length of \( DC \)?

[asy]
pair A,B,C,D;
A = (0,0);
B = (6,8);
D = (6,0);
C = (6 + 24,0);
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