
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, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively of a right-angled triangle $\triangle ABC$ with right angle at $B$. Point $F$ is the midpoint of $\overline{BC}$. Calculate the area of quadrilateral $BDFE$.

[asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,4);
B=(0,0);
C=(6,0);
D=(0,2);
E=(3,2);
F=(3,0);
draw(A--B--C--A);
draw(B--D--F--E--B);
label("$A(0,4)$", A, NW);
label("$B(0,0)$", B, S);
label("$C(6,0)$", C, S);
label("$D$", D, W);
label("$E$", E, NE);
label("$F$", F, S);
[/asy]

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