
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, $\triangle ABC$ is right-angled at $A$, with $AB = 12$ and $AC = 16$. The point $D$ is on $BC$ such that $AD$ is perpendicular to $BC$. Determine the lengths of $DB$ and $DC$. [asy]
pair A, B, C, D;
B=(0,0);
A=(0,12);
C=(20,0);
D=(0,0);
draw(A--B--C--A--D);
label("B", B, SW);
label("A", A, N);
label("D", D, S);
label("C", C, SE);
label("12", (A+B)/2, NW);
label("16", (A+C)/2, NE);
[/asy]

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