
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
[asy]
draw((0,0)--(0,3)--(2,3)--(2,0)--cycle,dot);
draw((2,3)--(0,0)--(0,1.5)--cycle,dot);
draw((0,3)--(1,0),dot);
MP("B",(0,0),SW);MP("A",(0,3),NW);MP("D",(2,3),NE);MP("C",(2,0),SE);
MP("E",(0,1.5),W);MP("F",(1,0),S);MP("H",(4/5,4/5),E);MP("I",(4/11,18/11),N);
dot((1,0));dot((0,1.5));dot((4/5,4/5));dot((4/11,18/11));
[/asy]
If $ABCD$ is a $3\times2$ rectangle, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then find the area of quadrilateral $BEIH$.

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