
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, triangle $ABD$ is a right triangle with $AB = 30$ and $\angle ADB = 90^\circ$. We are given that $\cos A = \frac{4}{5}$. Triangle $BCD$ is such that $\angle BDC = 90^\circ$ and $\sin C = \frac{4}{5}$. Find the length of $DC$.

[asy]
pair A,B,C,D;
A = (0,0);
B = (24*sqrt(5),30);
D = (24*sqrt(5),0);
C = (24*sqrt(5) + 75,0);
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}
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