
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, point $D$ is the midpoint of $\overline{AB}$, and point $E$ is the midpoint of $\overline{BC}$. If points $A$, $B$, and $C$ have coordinates $(0,10)$, $(0,0)$, and $(12,0)$ respectively, calculate the area of $\triangle DBC$. 
Additionally, assume that there is another point $F$ which is the midpoint of $\overline{AC}$, forming a smaller triangle inside with $E$. 

[asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,10);
B=(0,0);
C=(12,0);
D=(0,5);
E=(6,0);
F=(6,5);
draw(A--B--C--cycle);
draw(D--E);
draw(F--E);
draw((0,0)--(14,0), EndArrow);
draw((0,0)--(0,12), EndArrow);
label("$A(0, 10)$", A, W);
label("$B(0, 0)$", B, SW);
label("$C(12, 0)$", C, S);
label("$D$", D, W);
label("$E$", E, S);
label("$F$", F, NE);
label("$x$", (14,0), dir(0));
label("$y$", (0,12), dir(90));
[/asy]

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