
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)--(3,3)--(3,0)--cycle,dot);
draw((3,3)--(0,0)--(0,1.5)--cycle,dot);
draw((0,3)--(1.5,0),dot);
MP("B",(0,0),SW);MP("A",(0,3),NW);MP("D",(3,3),NE);MP("C",(3,0),SE);
MP("E",(0,1.5),W);MP("F",(1.5,0),S);MP("H",(3/4,3/4),E);MP("I",(3/7,9/7),N);
dot((1.5,0));dot((0,1.5));dot((3/4,3/4));dot((3/7,9/7));
[/asy]
If $ABCD$ is a $3\times3$ 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 $H$, find the area of quadrilateral $BEIH$.

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